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11. TITLE (Include Security Classification)
PRELIMINARY STUDIES ON MIXING AND ENTRAINMENT OF FLUID MUD
12. PERSONAL AUTHOR(
Mehta. A.J.; Scarlatos, P.D.; Srinivas, R.
13a. TYPE OF REPORT i13b. TIME COVERED 14. DATE OF REPORT (Year.Month, Day) 15. PAGE COUNT
Final FROM 2/28/89 TO7/31./90 February, 1991 24
16. SUPPLEMENTARY NOTATION
17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
FIELD GROUP SUBGROUP Fluid mud
Sediment entrainment
Sediment stratification
19. ABSTRACT (Continue on reverse if necessary and identify by block number)
PART I. A matter of particular interest to estuarine sedimentation problems is the dynamic
behavior of fluid muds. It is suggested that shear flowinduced interfacial instabilities
leading to the erosion of fluid muds is a significant resuspension mechanism. Thus, the
erosion process is controlled by the scale of nearbottom turbulence and the vertical
velocity and density profile structures. Under strongly entraining conditions in which
settling is small, the density profile microstructure of the entraining material is governed
by buoyancy stabilized turbulent diffusion. A stepwise profile of the suspended sediment
is theoretically developed under simplified physical conditions. Field and experimental data
support the theoretical assumptions, while experience from other stratified flow phenomena
provides valuable insight relative to studies required to quantify fluid mud erosion response
to hydrodynamic forcing.
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19. ABSTRACT (Continued)
PART II. An experiment is described to simulate fluid mud entrainment by current shear. A
recirculating flume with a disk pump system to produce shear was used for this purpose.
Significant interfacial instabilities were observed with resultant fluid mud entrainment.
An empirical relationship is obtained between the nondimensional buoyancy flux and the
Richardson number, which is then compared with that obtained for saltstratified flows. This
comparison makes apparent the additional dissipation of turbulent kinetic energy in sediment
water systems relative to saltstratified systems, particular at higher Richardson numbers
as the entrainment rate decreases substantially.
UFL/COEL91/005
PRELIMINARY STUDIES ON MIXING AND
ENTRAINMENT OF FLUID MUD
by
A. J. Mehta
P. D. Scarlatos
and
R. Srinivas
February, 1991
Sponsor:
U.S. Army Engineer
Waterways Experiment Station
3909 Halls Ferry Road
Vicksburg, MS 391806199
UFL/COEL91/005
PRELIMINARY STUDIES
ON MIXING AND ENTRAINMENT OF FLUID MUD
Submitted by:
A. J. Mehta
P. D. Scarlatos
R. Srinivas
Coastal and Oceanographic Engineering Department
University of Florida
336 Well Hall
Gainesville, FL 32611
Submitted to:
U.S. Army Engineer
Waterways Experiment Station
3909 Halls Ferry Road
Vicksburg, MS 391806199
February, 1991
PREFACE
Material presented herein constitutes the final report for the contract
DACW3989K0012: Separation (Stratification) and Mixing of FineGrained Dredge
Sediment Slurries A "Bench" Study, through the U.S. Army Engineer Waterways
Experiment Station, Vicksburg, MS. The report is divided into two parts as
follows. Part I: Some Observations on Erosion and Entrainment of Estuarine Fluid
Muds, by. P.D. Scarlatos and A.J. Mehta (1990), has been published in Residual
Currents and Longterm Transport, R.T. Chen ed., Coastal and Estuarine Studies
Vol. 38, SpringerVerlag, New York: 312332. Part II: Observations on Estuarine
Fluid Mud Entrainment, by R. Srinivas and A.J. Mehta (1989) has been published
in International Journal of Sediment Research, 5(1): 1522.
TABLE OF CONTENTS
PREFACE ii
LIST OF FIGURES iv
PART I. Some Observations on Erosion and Entrainment of Estuarine Muds 1
Abstract 1
Introduction 1
Physical Properties of Fluid Muds 2
Erosion and Entrainment 3
Interfacial response 3
Entrainment 5
Experimental observation 6
Microstructure and Density Profile 6
Concluding Remarks 9
References 10
PART II. Observations of Estuarine Fluid Mud Entrainment 13
Abstract 13
Introduction 13
Background 14
Interfacial response 14
Entrainment 14
Experimental Methodology 15
Experimental Observations 17
Conclusions 19
Acknowledgement 20
References 20
LIST OF FIGURES
FIGURE PAGE
PART I.
1. Schematic definition of fluid mud relative to mud density
profile 2
2. Characteristics of density and velocity profiles in the
interfacial region 4
3. High concentration mud underflow over Yellow River delta
(adapted from Wright et al., 1988). 4
4. Interfacial disturbance at RiU = 5 (dp/p1 = 0.007). Wave
height ranged from 1.3 to 2.5 cm, period from 3 to 5 s.
Grid squares are 2.54 cm2. 6
5. Microstructure of three sequential sediment vertical
profiles in the Severn estuary, England, taken over a
16 min interval (adapted from Kirby, 1986). 7
6. Simulated development of suspension density microstructure. 9
PART II.
1. Definition sketch for density and velocity profiles in mud
stratified twolayer system. 15
2. Experimental flume. 16
3. Variation of mixed layer velocity profile with time. 16
4. Massive interfacial convolution (Ri, ~ 7). 18
5. Nondimensional buoyancy flux. Q, Vs Richardson number, Ri 19
PART I.
Some Observations on Erosion and Entrainment of Escuarine Fluid Muds
Panagiotis D. Scarlatos1 and Ashish J. Mehta2
ISouth Florida Water Management District
P.O. Box 24680
West Palm Beach, FL 334164680.
2Coastal and Oceanographic Engineering
University of Florida
Gainesville, FL 32611
ABSTRACT
A matter of particular interest to estuarine sedimentation problems
is the dynamic behavior of fluid muds. It is suggested that shear
flowinduced interfacial instabilities leading to the erosion of
fluid muds is a significant resuspension mechanism. Thus, the
erosion process is controlled by the scale of nearbottom
turbulence and the vertical velocity and density profile structures.
Under strongly entraining conditions in which settling is small, the
density profile microstructure of the entraining material is
governed by buoyancy stabilized turbulent diffusion. A stepwise
profile of the suspended sediment is theoretically developed under
simplified physical conditions. Field and experimental data support
the theoretical assumptions, while experience from other stratified
flow phenomena provides valuable insight relative to studies
required to quantify fluid mud erosion response to hydrodynamic
forcing.
I. Introduction
Simulation of estuarine residual cohesive sediment transport leading to
sedimentation is an important but also a complicated task. This complexity is
largely defined by our level of understanding of a wide variety of nearbed flow
sediment interactions which characterize the overall transport problem. Due to the
small particle size, cohesive sediment particle movement is controlled not only by
inertial, gravitational, buoyancy and drag forces, but also by electrochemical
ones. Since however a microscopic description of the processes involved in cohesive
sediment transport is not presently feasible, for simulation purposes a macroscopic
approach using lumped parameters is usually applied (Scarlatos and Partheniades,
1986).
During a tidal cycle or an episodic event, spatial and temporal redistribution
of the suspended sediment concentration occurs by virtue of the effects of the
different phases of the sedimentation cycle. From field and laboratory observations
it has been noticed that, generally, as shown in Figure 1 the vertical concentration
(density) profile of fine sediments can be categorized into three zones: an upper
zone of mobile, relatively low concentration suspension, a middle zone of higher
concentration suspension or fluid mud, and the lower zone of settled bed (Kirby and
Parker, 1983). The fluid mud zone is of particular practical importance because,
due to its high concentration and weak internal structure, it can be easily
entrained and can substantially contribute to turbidity even under relatively low
energy inputs (Ross, 1988). Therefore, assuming an initially sharp interface
between the overlying water and fluid mud during quiescent periods as occurs for
example during slack water, it is important to understand the erosion behavior of
the mudwater interface, and the evolution of the interface as the fluid mud
entrains into the water column due to shear flow.
Coastal and Estuane Studies. Vol. 38
1 R. Cheng (Ed.)
Resdual Currents and Longten ansport
0 SpingerVerlag New'~rk, Inc., 1990
Many past studies on cohesive sediment transport seem to have neglected the
fluid mud zone, emphasizing instead the erosion of the deposited bed (Hehta, 1986).
Such has been the case in spite of the fact that it has been long recognized that
fluidized muds do not erode in the same manner as a settled bed (Parker and Kirby,
1982). Yet, because of inadequate knowledge of the underlying mechanism for fluid
mud erosion, numerical models have had to rely on conjecture and empiricism
(Ariathurai, 1974; Hayter, 1983). The purpose of this paper is to briefly describe
the physical properties of fluid muds, to review erosion and entrainment processes
of initially stationary high concentration suspensions, to present a simple model
for simulation of vertical suspension density distribution which results from
entrainment, and, thereby, to highlight some areas for further research.
II. Physical Properties of Fluid Muds
The suspended sediment concentration typically increases rather dramatically
downwards from the surface to the bottom of the water column (Figure 1). The
concentration profile may be smooth or stepwise depending on, among other factors,
the concentration of total sediment mass in suspension (Kirby, 1986). The
transition from a mobile suspension of low concentration (< 300500 mg/l), to a
stationary suspension of high concentration (> 10 g/1), usually occurs within a very
narrow vertical distance, creating thus a steep density gradient. The zone of the
steepest gradient within the water column is called the lutocline layer (Figure 1).
The thickness of the various zones and the degree of vertical stratification depend
on the magnitude of energy input by the tides, wave action, wind stress, geophysical
motions, bioturbation or sometimes even human activity.
i Mobile Suspension
I
..* .'.*". ":*. *. Lutocline Layer
S .*' ..... .
.. Fluid Mud, '.> ,

j" /1 l 7777
//7J "7 "7" *v "*/"y7 7 '7'7"/< ~ 
Settled Bed
DENSITY
FIGURE 1. Schematic definition of fluid mud relative to mud
density profile.
Fluid muds are usually contained between the lutocline and the settled bed. The
thickness of the fluid mud layer may be from a few centimeters to several meters.
Typical concentrations for fluid muds are usually within the range of 10 to 320 g/l,
which" correspond to bulk densities approximately between 1.03 to 1.2 g/cm3,
respectively (Bradley et al., 1988).. However, as Been and Sills (1981) have noted
from their laboratory work on quiescent settling, a weak solid structure is usually
present when the concentration exceeds about 220 g/l.
Fluid muds exhibit a strong timedependent, nonNewtonian fluid response (Bryant
et al., 1980); therefore any realistic description of their dynamic behavior must
account for this essential property. Various constitutive models have been used for
simulation of their mechanics including kinematics and energy dissipation. These
include: elastic, poroplastic, viscous fluid, Bingham plastic, and viscoelastic
models (Maa, 1986). Laboratory and field tests indicate that the theological
behavior of fluid muds, i.e., the relationship between stress and rate of strain,
may change from pseudoplastic at relatively low concentrations, to Bingham plastic
at high concentrations (Sills and Elder, 1986; Maa, 1986). The Bingham yield stress
documented in high concentrations provides an indication of the work required to
disrupt the floc structure in the cohesive .suspension. Under very high shear
stresses fluid muds approach Newtonian behavior, i.e., a linear relation between
stress and rate of strain. This is due to the fact that increasing shear rates
break down the flocs and reduce the particle's partial support by the cohesive
structure.
Besides concentration, the theological properties of fluid muds are greatly
influenced by the physicochemical characteristics of the supporting fluid and the
composition of the sediment, which essentially indicates the importance of cohesion
in defining mud behavior. Experimentally it has been observed that, for example,
very weakly cohesive kaolinitequartz mixtures tend to behave like a pseudoplastic
material, while highly cohesive estuarine muds may apparently behave as Bingham
plastics (Krone, 1963). Very few muds however seem to be truly Bingham plastics;
most exhibit pseudoplastic character at very low rates of shear, and consequently
creep under even mild forcing due to shear flow or gravity (Bryant et al., 1980;
Kirby, 1986).
III. Erosion and Entrainment
III.l Interfacial response
Under low energy conditions, such as during slack water, when the suspended
sediment settles rapidly, fluid mud is formed near the bed, provided the rate of
deposition is high enough to prevent dewatering of the underlying material. The
thickness of the fluid mudwater interface depends on the concentration, the
particle size distribution, and the time from the initiation of deposition. Based
on these physical characteristics, Davis and Hassen (1988) estimated the spreading
of the interface during deposition by considering hindered settling effect, particle
interaction, and a slight polydispersity of the suspension. Hindered settling
effect, caused by increasing sediment concentrations in the lower layers and very
important in influencing sediment dynamics near the bed, was shown to lead to a
decrease of the interface thickness.
Given a twolayer flow with a distinct interface, erosion of the fluid mud layer
is believed to be controlled by the mechanisms of shear flow interfacial
instability. This is unlike, for example, the erosion of a settled bed which erodes
by dislodgement and entrainment of surface flocs due to applied stress. Liu (1957)
was first to suggest the notion of interface instability in sediment transport
studies, while attempting to explain ripple formation in moveable beds.
Unfortunately his application may not have been entirely appropriate, because the
motion of granular beds is more discrete than continuous, as identified by rolling
and saltation of individual grains. On the other hand, fluid mud and the overlying
water column can be treated in many respects as a system of two miscible continue
with different densities and different theological properties separated by an
interface of a certain thickness.
In general, under shear flow conditions, the thickness of the density interface
(d) affects the mode of instability that will erode the interface. The mode of
instability is equally affected by the thickness (D) of the region of high velocity
gradient (Figure 2). In case the midaxis of the velocity and density gradients
coincide, then when d is approximately equal to D, the prime mode of instability is
that of the KelvinHelmholtz type; when d is smaller than D, the instability is of
the Holmboe type. The main characteristic of the KelvinHelmholtz instability is
the rollup and pairing of the interfacial vortices (Delisi and Corcos, 1973). The
Holmboe mode of instability is recognized by the sharp crests along the interface
which finger alternatively into both fluids (Browand and Wang, 1972).
Velocity
2Z Density
0
D d
_ii
dU d d
VELOCITY OR DENSITY
FIGURE 2. Characteristics of density and velocity profiles in the
interfacial region.
Large scale internal wave instabilities over high concentration mud underflows
have recently been recorded in the Bohai Gulf offshore near the mouth of the Yellow
River, China (Wright et al., 1988). These features (Figure 3), thought to be
ubiquitous, were of rather high frequency (2.55.0 x 10 3 Hz) close to the local
BruntVaisala frequency (3.3 0.8 x 10 3 Hz). They were generated by shear arising
from the relative motion between the gravity underflow and the overlying water, and
were believed to contribute to underflow deceleration. Undoubtedly, such large
disturbances are likely to play a significant role in mixing the mud flow with the
upper fluid in which another, more diffused, low concentration plume was observed.
Bohal Sea
DISTANCE FROM MOUTH (km)
FIGURE 3. High concentration mud underflow over Yellow River
delta (adapted from Wright et al., 1988).
Instability can be studied by superimposing small perturbations on the mean
flow, and representing the continuity and NavierStokes momentum equations in terms
of the perturbated quantities. By using the theory of normal modes, the linearized
governing equation for stability is given by the wellknown OrrSommerfeId equation
(Betchov and Criminale, 1967). The flow is considered as stable if the
perturbations decay in time. Analytical solutions of the OrrSommerfeld equation
are feasible only when the twophased fluid system under consideration is either
inviscidinviscid or inviscidviscid, i.e., when there is a velocity discontinuity
at the interface. The case of interfacial stability between viscousviscous fluids
requires numerical treatment of the governing stability equation. The case of
instability, where the velocity and density gradient axes do not coincide, have been
studied by Lawrence et al. (1987).
The analytical results obtained from the OrrSommerfeld equation provide an
understanding of the conditions that lead to instability and breaking of the
interface. These results are given in terms of the physical flow characteristics
such as flow velocity, viscosity, surface tension, and density variations. in the
system. For inviscidviscid flow, which can be considered in a very simplified
treatment of the waterfluid mud type problem, the linear stability criterion is
given as
(2T3/2/[v2p2(gAp)1/21)P/2 > 1 (1)
where T is the surface tension, v is the kinematic viscosity of the upper fluid, g
is the acceleration of gravity, Ap P2pi, and Pl, P2 are the densities of the
upper and lower fluid layers, respectively. Eq.(l) was generated considering the
fact that the time scale of the perturbation must be much shorter than vorticity
diffusion in the mean flow. It is illustrative of the number of characteristic
parameters which are involved even in this simplified approach. The nature of the
assumption made concerning the physical properties of the two fluids critically
influences stability. Thus, for example, the stability criterion for viscid
inviscid fluid systems is more restrictive than the criterion for inviscidinviscid
fluids (Hogan and Ayyaswamy, 1985).
A realistic assessment of the instability of the waterfluid mud interface,
presently unavailable, requires a more rigorous approach which considers not only
the viscous effects, but also the theological properties of the fluid mud. The non
Newtonian nature of fluid mud complicates the instability problem by introducing
variable viscosity, nonzero normal stress differences, and out of phase
relationships between stress and rate of strain (Maa, 1986). Depending on the
magnitude of the characteristic flow velocity, characteristic length and fluid
relaxation time, the fluid mud system can be simplified and treated either as a non
Newtonian fluid with nonzero normal stresses and inphase stress versus strain
rate, or with zero normal stresses and outofphase stress versus strain
relationship (Harris, 1977). Attention must also be given to the fact that Squire's
theorem, which states that twodimensional flows provide more strict instability
conditions than three dimensional flows, may also not be valid for nonNewtonian
fluid and, consequently, mud flow. These issues are the subject of ongoing research
at the University of Florida, and will be examined in a subsequent publication.
111.2 Entrainment
Once the interface breaks, the sediment particles entrain into the overlying
water column, and there is mixing between the fluid mud and the overlying water.
Entrainment processes have been extensively studied for the cases of thermal and
salinity stratification, but apparently not for fluid muds. However, the role of
this mechanism in modeling fine sediment resuspension has been recognized by, among
others, McLean (1985). Entrainment is usually quantified as a function of the bulk
Richardson number, Ri, which expresses the balance between buoyancy and inertia
forces. There are different laws pertaining to the rate of entrainment based on the
degree of stratification. The entrainment velocity, ue, can for example be
normalized either by the mean flow velocity, U, the shear velocity, u,, or the
velocity jump, dU. Accordingly, depending on which of these is used as the
characteristic velocity, the Richardson number is defined as RiU, Ri*, or RidU'
Narimousa et al. (1986) suggested the shear velocity as the scaling velocity, and
proposed the expression:
e ib (2)
where a and b are numerical coefficients defined accordingly as: a 0.65, b 1/2
for 15< Ri <150, a 7.00, b 1 for 150< Ri, <800, and a 5.00, b 3/2 for
800< Ri,. Tiese exponential laws can be explained on physical grounds as follows:
for low Ri*, shear is high and the density gradient is small, so that mixing is
produced by the larger, energy carrying eddies. For intermediate values of Ri, the
processes are less chaotic, so that entrainment is more ordered and caused by
KelvinHelmholtz instability. Finally, for large Ri, shear is very weak and
entrainment is due primarily to the kinetic energy of the turbulent layer. If the
density gradient is extremely large, then entrainment is supressed and the only
mixing is caused by molecular processes. In another publication, Narimousa and
5
Fernando (1987) indicated that' the best scaling velocity is the velocity jump at the
interface. The explanation of their suggestion was based on the fact that the
energy for turbulent mixing of the interface is produced at this high shear zone.
III.3 Experimental observation
Five very preliminary experimental runs were performed at the University of
Florida in order to monitor the interfacial instability of a twolayered flow of
clear water over kaolinitewater suspensions. The tests were conducted in a 18.3 m
long and 0.6 m wide recirculating flume. A description of this flume is given by
Dixit (1982). Initially the kaolinitevater suspension was thoroughly mixed through
high speed velocity, and then by reducing the velocity substantially, the sediment
was allowed to settle. After 1020 minutes when a distinct interface was formed,
the velocity was increased to a predetermined magnitude and the interfacial activity
was filmed for a duration of approximately 30 seconds. The initial density jump,
dp/p1 (P2Pl)/Pi, was varied from 0.005 to 0.009.
Results from these experiments showed that for RiT, 19.0 there was a distinct
wave at the interface but no breaking (no turbulent entrainment). For RiU 5.0
(Figure 4) there was breaking of the wavy interface, while for Ri, 1.7 the wave
breaking caused smokelike wisps that were entraining into te upper layer, and
resembled the Holmboe type of instability. Furthermore, the interfacial thickness,
d, was found to be on the order of 5% of the depth of the mixed upper layer. The
observed value of RiU 19.0 for stability (nonentrainment) of a watermud system
coincides with the experimental data presented by Narimousa t al. (1986), for fresh
watersalt water system. However, more data are necessary in order to establish
confidence in the range of applicability of the existing stateoftheart
understanding of instability and entrainment to the case of fluid mud systems, as
suggested by Odd and Cooper (1989).
Qualitative differences (e.g., in the meaning of coefficients a and b in Eq.(2))
are likely to arise in replacing salt water by fluid mud due to differences in the
rate of dissipation of turbulent kinetic energy in these two fluids, as suggested by
the recent laboratory results of Srinivas (1989) and Wolanski et al. (1989), in
which sedimentinduced collapse of turbulence was observed at the waterfluid mud
interface.
Sv. Wave Breaking 
FIGURE 4. Interfacial disturbance at Ri 5 (dp/pl 0.007).
Wave height ranged from 1.3 to 2.5 cm, period from 3 to 5 s. Grid
squares are 2.54 cm2.
IV. Microstructure and Density Profile
Continuous monitoring of the vertical density profile in stratified estuaries
has indicated the occurrence of a stepwise profile structure. This microstructure
has been identified in cases of salinity, temperature, and sediment stratification.
Typical profiles of suspended cohesive sediment, recorded a few minutes apart in the
Severn Estuary in England, are shown in Figure 5 (Kirby, 1986). The time scale of
the microstructure development, in the differentiated (stratified) layer between the
mixed upper layer and the main or primary lutocline near the bed, was on the order
of 10 to 15 minutes. The general location of the primary lutocline is largely
controlled by hindering effects during settling (Ross, 1988), while turbulent
diffusion strongly influences mixed layer thickness. Posmentier (1977) presented
data for salinity microstructure in the Hudson River with a longer representative
time scale on the order of 90 minutes. Another difference between Kirby's and
Posmentier's data is that the former documented a dynamic system where the number
and position of secondary lutocline steps changed in time, while the latter
presented a profile where the microstructure apparently stabilized after a certain
time. Since the microstruccure is indicative of the vertical stratification
process, its understanding is essential for quantifying sediment mass exchange
between stationary and mobile sediment suspensions.
00 0
0
Mixed
SLayer
1. 10 Secondary
CL. Lutocline
LU
2 Differentiated
M Layer
S20
Primary
Lutocllne
30
S I I II 1 111
0 5 9
CONCENTRATION (gl 1)
FIGURE 5. Microstructure of three sequential sediment vertical
profiles in the Severn estuary, England, taken over a 16 min
interval (adapted from Kirby, 1986).
A full understanding of the mechanisms of density microstructure have not as yet
been established. There are a number of theories which attribute the stepwise
density structure to different causes such as KelvinHelmholtz instability, breaking
of internal waves; viscous overturning, or salt fingering. A common agreement among
these theories is that the microstructure is due to turbulent energy patches
resulting in vigorous and uniform mixing which creates sharp gradients with the rest
of the density layers. Since the representative length scale of the phenomenon is
the profile step, the turbulent eddies affecting the system are expected to be of
the same magnitude (Monim and Ozmidov, 1986).
Theoretical analysis of the vertical suspension density distribution under
general conditions is quite complicated. Assuming steady state flow conditions,
homogeneity along the longitudinal axis, and considering only the entraining phase
pf flow when gravitational settling effects may be ignored for simplicity, the
vertical concentration profile can be studied very approximately by means of the
onedimensional diffusion equation written as
C .. .. . .. . (3)
where C is the sediment concentration, F is the sediment flux, z is the vertical
axis, and t is time. A comma in front.of a subscript denotes partial differentia
tion with respect to that subscript.. Applying Fick's law for turbulence induced
fluctuations the sediment flux reads
F KC ()
,z
where K is the coefficient of vertical diffusivity for mass. In sediment stratified
systems turbulence is strongly affected by the stratification characteristics
(McLean, 1985). Therefore, the buoyancy stabilized coefficient of diffusivity can
be expressed by a semiempirical form as
K A (1 + sRi) r (5)
where A is the coefficient of vertical eddy diffusivity under neutral stability, and
s, r are empirical coefficients (Okubo, 1970). The coefficient A is related to the
velocity and density fluctuations in the vertical direction. Using the mixing
length theory, A can be approximated as
A = (1.)2 ju (6)
where 1. is the mixing length in a neutrally stratified flow, and u is the depth
average horizontal velocity. For fluid muds the Richardson number can be expressed
as (Scarlatos and Hehta, 1988)
Riu (g C, [(s P)I/Pw/uz (7)
where p p are respectively the density of water and sediment. Assuming for
simplicity a logarithmic profile for the velocity, the mixing length is given as
1. (kz [1 (z/H)])1'2 (8)
where k is the von Karman's constant, and H is the depth of the boundary layer.
Setting X C, Eqs.(34) yield
Ct  Fx Cz (9)
,t ,x ,zz
Due to the existence of the gradient F Eq.(9) is a nonlinear partial differential
equation. Posmentier (1977) suggested'Ehat when the factor F,x is negative then the
system becomes unstable. As a result of the instability, there is thorough mixing
which creates steep density gradients and a more stable system. When stability
increases the factor F,x becomes positive. Combining Eqs.(48) yields
F R [1 + (1 + r)BX] (1 + BX)rl (10)
,x
where R and B are defined as
R kz [1 (z/H)] ju J (11)
B s g[(ps p)/p]/2 (12)
Since R is always positive, the sign of the quantity F,x depends on the sign of the
remaining right hand side of Eq.(l0). From field observations it has been estimated
that the values for the coefficients s and r are approximately between 3.33 < s < 10
and 2.0 < r < 0.5, (Ross, 1988).
It is not known whether all the values between the upper and lower inequality
limits of s and r are physically realistic. For r 1, F, is always positive, and
therefore Eq.(9) is wellposed. In this case, it is not possible to generate
instability and consequent microstructure. For r 2 the stability requirement is
given as
(BX)2 (sRiu)2 < 1 (13)
Assuming s 4, Eq.(4) yields Ri < 1/4, which is the wellknown stability criterion
for shear flows (Okubo, 1970). X similar expression can be derived for r 0.5.
Whenever there is violation of the stability criterion, internal waves break and a
*patch of weak turbulence is generated. This turbulent patch spreads horizontally
and creates a layer of homogeneous water which causes the generation of
microstructure.
Simulation of the microstructure development process can be achieved by solving
Eqs.(912). The numerical scheme used for the solution was the Newton linearization
iterative technique which applies to the governing equation in a form of:
C, G(z; t; C; C ; C ) (14)
zz ,' ,t
Then, Eq.(14) is expanded as
(n+1) (n) (n+l) (n) (n+) (n) C
zz + I C z z ,x
+ (C(n+) (n) (15)
where n is the number of iterations (Ames, 1977). The term C is discretized by
using a central difference scheme. For this numerical scheme'tne solution matrix is
tridiagonal while the iteration converges quadratically. Boundary conditions for
the system must be specified at the free water surface and at the consolidated bed.
At the free surface the concentration is assumed as constant while at the bed a mass
flux is defined to essentially account for net erosion of sediment (Mehta, 1986).
Simulated result for a hypothetical case in a 20 m deep unit width channel is
presented in Figure 6. In this simulation a logarithmic velocity profile with a
surface current of 0.2 m/s was used, and the values for the coefficients s and r
were s 4.0 and r 2.0. These values for s and r are typical values estimated
from field data (Ross, 1988). The time step used was 5 s while the flow depth was
discretized into 80 segments for one run and into 50 segments for another. The
microstructure generation was however not measurably affected by the number of
segments over the depth. It can be clearly seen that starting from a smooth
profile, a stepwise profile developed after 90 s of simulation time. The lutocline
steps defining the differentiated layer are not however as dramatic as those
recorded in nature (Figure 5). Neglect of horizontal advection may be one reason
for this discrepancy. 'For any velocity profile other than logarithmic, e.g. for a
bottom reverse current, the vertical mixing coefficient must be redefined since
Eqs.(48) will not be valid. In general, the mechanics of generation of the
differentiated layer is the same as that in a halocline, and is physically explained
by the nonlinear dependence of diffusion flux on the density gradient (Postmentier,
1977; Ross, 1988).
F0
z/H t =0 t = 90s
Differentiated
Layer
C (Relative)
FIGURE 6. Simulated development of suspension density
microstructure.
V. Concluding Remarks
It appears that fluid mud erosion and entrainment mechanisms can be simulated by
approaches applied to salinity or thermal density flows. Erosion of fluidized mud
water interface depends strongly on the velocity and density gradients, and
resembles interfacial instability phenomena typical to stratified flows
characterized by haloclines and thermoclines. Laboratory experiments with water and
kaolinitewater suspension seem to show a qualitative agreement with other existing
instability data. However, due to differences in the rate of turbulence dissipation
near the interface, entrainment rates of fluid muds, while undoubtedly Richardson
number dependent, are likely to show qualitative differences from those for salt and
heat.
The vertical profile of the mobile suspension at the upper water levels exhibits
a steplike density microstructure. This microstructure can be simulated very
approximately by using a diffusion type equation with an appropriate value for the
vertical mixing coefficient. The simulated density "steps" are not however as
dramatic as those observed in nature. The discrepancy may result from the neglect
of horizontal advection. In addition, the use of the Fickian closure approach is
inherently limited by empiricism, and a higher order closure scheme may be essential
for quantitative simulation.
Due to the nonNewtonian nature of fluid muds, additional theoretical and
experimental research is needed on the instability and entrainment mechanisms in
shear flows arising from hydrodynamic forcing under generalized conditions of out
ofphase stress versus strain rates, variable viscosity, and nonzero normal stress
differences.
Support from the U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS
through contract DACW3989K0012 is acknowledged.
VI. References
Ames, W. F., 1977: Numerical Methods for Partial'Differential Eauations. Academic
Press.
Ariathurai, R., 1974: A finite element model for sediment transport in estuaries.
Ph.D. Thesis, University of California, Davis.
Been, K., and G. C. Sills, 1981: Selfweight consolidation of soft soils: an
experimental and theoretical study. Geotechnicue, 31(4), 519535.
Betchov, R., and W. 0. Criminal, Jr. 1967: Stability of Parallel Flows. Academic
Press.
Bradley, J. B., R. C. MacArthur and B. J. Brown, 1988: Analytical and numerical
modeling at high sediment concentrations. Proc. Hydraul. Eng, Meet., S.R.
Abt and J. Gessler Eds., ASCE, Colorado Springs, 230235.
Browand, F. K., and Y. H. Wang, 1972: An experiment on the growth of small
disturbances at the interface between two streams of different densities
and velocities. Proc. Int. Svmp. Stratified Flows, Novosibirsk, USSR, 491
498.
Bryant, R., A. E. James and D. J. A. Williams, 1980: Rheology of cohesive
suspensions. Industrialized Embavments and Their Environmental Problems,
M.B. Collins, et al. Eds., Pergamon Press, 279287.
Davis, R. H., and M. A. Hassen, 1988: Spreading of the interface at the top of a
slightly polydispersed sedimentation suspension. J. Fluid Mech., 2, 107
134.
Delisi, D., and G. M. Corcos, 1973: A study of internal waves in a wind tunnel.
Boundary Layer Meteorol., 5, 121137.
Dixit, J. G.,* 1982: Resuspension potential of deposited kaolinite beds. M.S.
Thesis, University of Florida, Gainesville.
Harris, J., 1977: Rheology and NonNewtonian Flow, Longman.
Hayter, E. J., 1983: Prediction of cohesive sediment movement in estuarial waters.
Ph.D. Thesis, University of Florida, Gainesville.
Hogan, J. M., and P. S. Ayyasvamy, 1985: Linear stability of viscousinviscid
interface. Phys. of Fluids, 28(9), 27092715.
Kirby, R., 1986: Suspended fine cohesive sediment in the Severn estuary and
Inner Bristol channel. Rep. ETSUSTP4042, U.K. Atomic Energy Authority,
Harwell, United Kingdom.
Kirby, R., and W. R. Parker, 1983: The distribution and behaviour of fine sediment
in the Severn Estuary and Bristol Channel. Can. J. Fish. and Aauat. Sci.,
40(1), 8395.
Krone, R. B., 1963: A study of theologic properties of estuarial sediments. Rep.
SERL 638, Hydraulic Engineering Laboratory, University of California,
Berkeley.
Lawrence, G. A., J. C. Lasheras and F. K. Browand, 1987: Shear instabilities in
stratified flow. Proc. 3rd Int. Svmr. Stratified Flows, Session A.3,
California Institute of Technology, Pasadena, 113.
Liu, H. K., 1957: Mechanics of sedimentripple formation. J. Hydraul. Div.,
ASCE, paper 1197., 83, 123.
Maa, P. Y., 1986: Erosion of soft muds by waves. Ph.D. Thesis, University of
Florida, Gainesville.
McLean, S. R., 1985: Theoretical modelling of deep ocean sediment transport. Mar.
Geol., 66, 243265.
Mehta, A. J., 1986: Characterization of cohesive sediment properties and transport
processes in estuaries. Estuarine Cohesive Sediment Dynamics, A. J. Mehta
Ed., SpringerVerlag, 290325.
Monin, A. S., and R. V. Ozmidov, 1986: Turbulence in Ocean. D. Reidel.
Narimousa, S., R. R. Long and S. A. Kitaigorodskii, 1986: Entrainment due to
turbulent shear flow at the interface of a stably stratified fluid. Tellus,
38A, 7687.
Narimousa, S., and H. J. S. Fernando, 1987: On the sheared interface of an
entraining stratified fluid. J. Fluid Mech., 174, 122.
Odd, N. M. V., and A. J. Cooper, 1989: A twodimensional model of the movement of
fluid mud in a high energy turbid estuary. J. Coast. Res., Special Issue
5, in press.
Okubo, A., 1970: Oceanic Mixingt Management Oceanic Services, Detroit, Michigan.
Parker, W. R., and R. Kirby, 1982: Timedependent properties of cohesive sediment
relevant to sedimentation management. Estuarine Comparisons, V. S. Kennedy
Ed., Academic Press, 573589.
Posmentier, E. S., 1977: The generation of salinity finestructure by vertical
diffusion. Phvs. Oceanor., 7, 298300.
Ross, M. A., 1988: Vertical structure of astuarine fine sediment suspensions.
Ph.D. Thesis, University of Florida, Gainesville.
Scarlatos, P. D., and A. J. Mehta, 1988: Microstructure of cohesive sediment
suspensions. Proc. Hydraul. Eng. Meet., ASCE, Colorado Springs, Colorado,
218223.
Scarlatos, P. D., and E. Partheniades, 1986: Numerical simulation of fine sediment
motion in estuaries. Proc. Int. Conf. Hydraul. Eng. Software, Southampton
University, United Kingdom, 111123.
Sills, G. C., and McG. Elder, 1986: The transition from sediment suspension to
settled bed. Estuarine Cohesive Sediment Dynamics., A. J. Hehta Ed.,
SpringerVerlag, 192205.
Srinivas, R., 1989:. Response of fine sedimentwater interface to shear flow.
HM.S. Thesis, University of Florida, Gainesville.
Wolanski, E., T. Asaeda and J. Imberger, 1989: Mixing across a lutocline. Limnol.
and Oceanogr., (in press).
Wright, L. D., W. J. Wiseman, B. D. Bornhold, E
Keller, Z. S. Yang and Y. G. Fan, 1988: Marine
yellow river silts by gravitydriven underflows.
12
1. B. Prior, J. N. Suhayda, G. H.
dispersal and deposition of
Nature, 332(6164), 629632.
international Journal of Sediment Research
Volume 5, No. 1. January 1990
PART II.
OBSERVATIONS ON ESTUARINE FLUID MUD ENTRAINMENT
Srinivas. R.1 and Mehta. A. J2.
ABSTRACT
An experiment is described to simulate fluid mud entrainment by current shear. A recirculating flume
with a disk pump system to produce shear was used for this purpose. Significant interfacial instabilities were
observed with resultant fluid mud entrainment. An empirical relationship is obtained between the nondimen
sional buoyancy flux and the Richardson number, which is then compared with that obtained for saltstratified
flows. This comparison makes apparent the additional dissipation of turbulent kinetic energy in sedimentwater
systems relative to saltstratified systems, particularly at higher Richardson numbers as the entrainment rate
decreases substantially.
Key Words: Estuary, Mud entrainment, Fluidized mud, Instability, Interface, Richardson number,
Stratified flow, Disk pump, Buoyancy flux, Shear flow.
I. INTRODUCTION
Prediction of mud bed erosion by forcing due to tidal currents usually requires a numerical solu
tion of the advectiondispersion equation for sediment mass transport. Key role is of course played in
this by the bottom boundary conditions defining erosion and deposition fluxes. The issue of erosion is
briefly considered here, .noting that it is customary to calculate the rate of erosion as a function of the
bed shear stress in excess of the erosion shear strength of the bed (Mehta et al., 1982).
While it is found from practice that such erosion rate expressions work reasonably well for nu
merical model application in low to moderate concentration environments (Hayter and Mehta, 1986),
questions regarding their applicability persist in cases involving high concentration fluid muds (Odd and
Cooper, 1989). Fluidized mud has effectively no shear strength, and the mudwater interface is easi
ly destabilized under shear flow. It is therefore instructive to focus attention on what might be the en
trainment behavior of the fluidized mud layer underlying initially sedimentfree water which starts
1. Graduate Research Assistant, Coastal and Oceanographic Engineering Department, University of Florida,
Gainesville, FL 32611. U.S.A.
2. Professor. Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FL
32611 U.S.A.
16 Srinivas, R. and Mehta A. J.
from rest, thus shearing the mudwater interface.
I. BACKGROUND
Interfacial Response
Under low energy conditions, such as during slack water, when suspended sediment rapidly set
tles, fluid mud is formed near the bed, provided the rate of deposition is high enough to prevent rapid
dewatering of the freshly settled material. Erosion of this mud layer and its entrainment into the over
lying water column following slack water is essentially controlled by the mechanisms of shear flowin
duced interfacial instability. In general the thickness of the density interface, d, affects the mode of
instability that will erode the interface. The mode of instability is equally affected by the thickness, D,
of the region of high velocity gradient (Fig. 1). In case the midaxis of the velocity and density gradi
ents coincide then, When d is approximately equal to D, the prime mode of instability is that of the
KelvinHelmholtz type; when d is smaller than D, the instability is of the Holmboe type. The main
characteristic of the KelvinHelmholtz instability is the rollup and pairing of the interfacial vortices
(Delisi and Corcos, 1973). The Holmboe mode is recognized by the sharp crests along the interface
which finger alternatively into both fluids (Browand and Wang, 1972).
ll. ENTRAINMENT
Once the interface breaks, the sediment aggregates entrain in the water column, and there is mix
ing between the fluid mud and the overlying water. Entrainment processes under unidirectional shear
flows have been extensively studied for the cases of thermal and salinity stratification, but apparently
not for fluid muds. However, the role of this mechanism in modeling fine sediment resuspension has
been recognized by, among others, Wolanski and Brush (1975), McLean (1985) an Odd and Cooper
(1989). Entrainment is usually quantified as a function of the Richardson number, Ri, which ex
presses the balance between buoyancy and inertia forces. There are different laws pertaining to the rate
of entrainment based on the degree of stratification, since the entrainment velocity can be normalized
either by the mean flow velocity, the shear velocity, or the velocity jump.
However, the velocity jump across the interface, AU, appears to be the most significant velocity
scale for entrainment across sheared density interfaces, as shear production in the entrainment zone is
mainly responsible for turbulent mixing. Turbulence is also produced at the sidewalls, but most of this
is dissipated at the walls themselves, with the portion diffusing outwards being only a small fraction of
the total amount (Hinze 1975), as confirmed by Jones and Mulhearn (1983). Visual observation of
injected dyelines seemed to indicate only minimal diffusion of momentum below the level of the densi
ty interface. This was aswell the case with Narimousa and Fernando (1987). It was thus decided to
use the mean velocity of the turbulent mixed layer, for scaling purposes. Thus, the Richardson
number for the problem at hand was defined as Ri, = hAb/u2, where h is the depth of the mixed layer,
Ab is the interfacial buoyancy step; buoyancy being defined as b = g (Pi Po) /Po, with pi and po being
Observations on estuarine fluid mud entrainment 17
the density of the fluid and a reference density, respectively.
Density Velocity
Profile,p(z) Profile, u(z)
Mixed
Layer
"Visual"
SDensity
Interface
T T
d D
Fluidized
Mud
Fig. 1 Definition sketch for density and velocity profiles in mud stratified twolayer system
IV. EXPERIMENTAL METHODOLOGY
Laboratory experiments were carried out in a plexiglass "race track" flume (Fig. 2), in order to
investigate the influence of Ri. on entrainment of fluid mud (Srinivas, 1989). Similar flumes have
been used successfully previously by other investigators to study salt stratified flows (Narimousa and
Fernando, 1987). In this apparatus the initially clear water layer can be made to flow relative to the
higher density fluid below by means of disk pump system, which is designed to minimize any intrusive
effect of pumping on the interfacial dynamics.
The annular geometry at the flume ends is meant to gradually turn the flow without undue inter
ference. As the width of the flume is necessarily large at the pump section, triangular plexiglass flow
separators were placed upstream and downstream of the pumpsection so as to create two equal chan
nels of half the width elsewhere. The 200 cm long straight test section was used for observations.
The disk pump consists of two vertical shafts driven by a motor and rotate in opposite directions.
Each shaft is stacked with a number of thin disks of sand blasted plexiglass. These are of two diame
ters, 4 and 13 cm, stacked alternately on each shaft, and so arranged that a small disk on one shaft
meshes with the larger of the other.. thus almost sealing the center of the pump, while the fluid is
thrown in a series of horizontal jets around the outside of the sfnaller disks and between the larger
disks. In order to prevent the disks from "sucking up" the density interface, horizontal partitioning of
the flume became essential in the region in proximity of the pumps. This "splitter plate" was extended
18 Srinivas. R. and Mehta A. J.
1.27cm Plexiglass
0.32cm PlexIglass
1.27cm Plexiglass
PLAN
Fig. 2 Experimental flume
5 10
VELOCITY IN MIXED LAYER, u (cms 1)
Fig. 3 Variation of mixed layer velocity profile with time
into the downstream curved flume segment in order to minimize the effect of helical secondary flows
on the processes occurring within the linear test section. An impeller had to be added just upstream of
Observations on estuarine fluid mud entrainment 19
the test section and above the splitter plate to enhance diskgenerated velocities.
The investigation herein reported was restricted to kaolinite and bentonite as sediments. The ini
tial bulk density of fluid mud was varied over 1. 03 1. 08 gem3. The lower limit was selected such
that the mud would be in the hindered gravitational settling range, which is critical for mud' s exis
tence in the fluidized state while the upper limit was dictated by the performance constraints of the ex
perimental setup.
The flume was first filled to the requisite preselected height of tap water. Premixed fluid mud
was then introduced through the intake at the flume bottom. The resulting fluid mud layer underneath
water was nearly homogeneous initially. The initial height of the fluid mud layer was generally always
kept at just under the elevation of the splitter plate.
As the pump was started to the required rotation rate, it took 2. 53. 0 min for the inertia of
the system to be overcome as the upper layer mean velocity increased to its peak value. With time, as
the fluid mud was entrained, the density interface moved downwards. The cumulative effect of this
increase in mixed layer depth and increasing dissipation of turbulent kinetic energy on the entrained
sediment aggregates was a slow decrease in the mixed layer mean velocity. Fig. 3 is an example of the
variation in the mixed layer velocity profile with time.
V. EXPERIMENTAL OBSERVATIONS
The experiments were carried out over a range of Ri. from about 4 to 30. Higher velocity values
required for lower Ri. values could not be achieved. When the pump was turned on, the energy of the
generated turbulent shear flow resulted in the development of substantial instabilities at the density in
terface. At low Ri. ( 5), entrainment appeared to be turbulence dominated, and the base of the
mixed layer appeared to be turbulent as well. This resulted in a rather diffuse and highly irregular in
terface. As Ri ,increased to above 5, the interface became better defined and convoluted with mas
sive, irregular undulation; see Fig. 4. Entrainment appeared to be dominated by wave breaking in
which wisps of fluid were ejected into the turbulent mixed layer. Interspersed with these breaking
waves were also some largeamplitude solitarytype waves which decayed without breaking (see also
Narimousa and Fernando, 1987). The frequency and amplitude of these disturbances decreased with
increasing Ri., and the interface became more regular. When the Ri. was above 10, the distur
bances could be seen to grow slightly in amplitude, sharpen into nonlinear crests and disappear sud
denly as the "rolleraction" of an eddy sheared off the crest (similar to Moore and Long, 1971). Be
yond Ri, of 25, the intensity of entrainment was highly diminished and appeared to taper off.
Thus, the effect of turbulence in the upper layer was entrainmentof the stably stratified fluid
mud across the density interface, with resulting buoyancy transfer. It can be shown (Moore and
Long, 1971) that the buoyancy flux is essentially the same as the rate of change of potential energy
per unit mass (of the entire system). Using a result form Kato and Phillips (1969), it can further be
shown that the nondimensional buoyancy flux is equivalent to the nondimensional rate of deepening
of the mixed layer due to entrainment alone, which is well understood to be a function of the relevant
20 Srinivas. R. and Mehta A. J.
I 4t1
Fig. 4 Massive interfacial convolution ( Ri, ~7)
Richardson number. The buoyancy flux is q = dm/dt g/Po, where dm/dt is the mass flux into the tur
bulent mixed layer, and po is the mean mixed layer density. Thus, we define the nondimensional
buoyancy flux to be Q = q/ (uAb) : see Srinivas (1989).
A plot of Q as a function of Ri, is presented in Fig. 5. Notwithstanding obvious data scatter, a
trend line can be drawn through the data according to the relationship
Q = A Ri..9/[B2 Ri.2]" (1)
where A = 0. 27, B = 20 and m = 0. 66. It is believed that the main effect of the splitter plate on this
relation might have been to cause some discrepency in the values of the constants (A, B and m) of the
above relation, and data scatter particularly at low Ri., but that the basic relationship between Q and
Ri, seems to be adequately quantified. The dashed line relation indicates a trend
Q cc Ri, (2)
with n = 0. 9, which is similar to the relation obtained by experimenters with saltstratified systems
with n =1 (e. g., Moore and Long, 1971: Narimousa and Fernando, 1987).
The n = 1 relationship can be considered in terms of energy changes. Kato and Phillips (1969)
showed that the rate of increase of potential energy per unit mass is related to the rate of dissipation of
turbulent energy per unit mass. It is possible to obtain from their equations the relationship
dvi/dt = u3. /h (3)
where V is the potential energy per unit mass and u. is the friction velocity. Batchelor (1953) found
that the rate of change of turbulent kinetic energy per unit vloume in isotropic turbulence, e. is given
by
e = d () /dt (u) /1 (4)
where 11 is the length scale of the energy containing eddies, and u the rms turbulent velocity.
Assuming that u. =u5=u (Towmsend, 1956: Moore and Long, 1971), and that the homoge
Observations on estuarine fluid mud entrainment
x RI.9 + 0R )n 0 Kaolinite
0Q ARIfa/(5+R R Bentonite
103 
0
00.9
Oa a Rfs
RICHARDSON NUMBER, RIu
Fig. 5 Nondimensional buoyancy flux. Q, Vs Richardson number, Ri.
neous layer depth h is of the the size of the energy containing eddies,
E () 3/h (5)
Thus,
105 l
/dt1 10 102 10e (6)
RICHARDSON NUMBER, Riu
Fig. 5 Nondimensional buoyancy flux. Q, Vs Richardson number, Ri.
neous layer depth h is of the the size of the energy containing eddies,
Thus,
From our experiments, Q = q/ (uAb) = K/Ri. = K (u) /hAb, where K is a proportionality con
stant. Because d V, /dt= q. therefore, d V, /dt= K (u) /h and thus d VI /dt (u)3/h, as in the
form of eq. 6. This evidence seems to indicate that in geophysical situations and similar experiments,
q (= dV,/dt) e, with the "constant" of proportionality probably depending on the coefficients of
viscosity and diffusivity.
In the present experiments, at low mixed layer concentrations, there apparently was not much
additional dissipation of kinetic energy to counteract the downward buoyancy flux due to sediment fall
velocity, as indicated by the trend of eq. 2 with n = 0. 9 (as compared with the n = 1 trend for salt
stratified systems). However, at Richardson numbers greater than 25, the buoyancy flux is ob
served to fall off much more drastically when, it is surmised that, a greater fraction of the input ener
gy is used up merely in maintaining the sediment particles in suspension in the mixed layer.
VI. CONCLUSIONS
The hitherto neglected phenomenon of fluid mud entrainment due to unidirectional current shear
in an estuarial environment prompted this study. It appears that the nondimensional buoyancy flux,
___
22 Srinivas. R. and Mehta A. J.
Q, is related to the relevant Richardson number, Ri., by the relation described by eq. 1 for the range
of Richardson numbers considered. This relationship indicates that at low Richardson numbers, the e
quation is similar to that obtained for many saltstratified systems, which can be further shown to indi
cate that the rate of increase of potential energy of the system is of the same order as dissipation. How
ever, for Richardson numbers >25, the trend line indicates a much steeper fall off in the dimen
sionless entrainment rate than for saltstratified experiments, indicating additional dissipation of kinetic
energy as the mixedlayer concentration increases.
ACKNOWLEDGEMENT
This study was supported by the U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS (contract
DACW39 89 K 0012).
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