• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 List of Tables
 List of symbols
 Abstract
 Introduction
 Instability mechanism
 Instability of stratified shear...
 Entrainment in stratified shear...
 Methodology
 Results and analysis
 Summary and conclusions
 Appendix A: Test materials
 Appendix B: A note on Richardson...
 Bibliography






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 89/013
Title: Response of fine sediment-water interface to shear flow
CITATION PDF VIEWER PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00076141/00001
 Material Information
Title: Response of fine sediment-water interface to shear flow
Series Title: UFLCOEL
Physical Description: xivi, 114 leaves : ill. ; 28 cm.
Language: English
Creator: Srinivas, Rajesh
University of Florida -- Coastal and Oceanographic Engineering Dept
Publisher: Coastal & Oceanographic Engineering Dept., University of Florida
Place of Publication: Gainesville Fla
Publication Date: 1989
 Subjects
Subject: Sedimentation and deposition   ( lcsh )
Sediment transport   ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M.S.)--University of Florida, 1989.
Bibliography: Bibliography: p. 110-113.
Statement of Responsibility: by Rajesh Srinivas.
General Note: Vita ...
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
 Record Information
Bibliographic ID: UF00076141
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: oclc - 20341167

Downloads

This item has the following downloads:

UF00076141 ( PDF )


Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Acknowledgement
        Acknowledgement
    Table of Contents
        Table of Contents 1
        Table of Contents 2
        Table of Contents 3
    List of Figures
        List of Figures 1
        List of Figures 2
    List of Tables
        List of Tables
    List of symbols
        Unnumbered ( 10 )
        Unnumbered ( 11 )
        Unnumbered ( 12 )
        Unnumbered ( 13 )
    Abstract
        Unnumbered ( 14 )
        Unnumbered ( 15 )
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
    Instability mechanism
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
    Instability of stratified shear flows
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
    Entrainment in stratified shear flows
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
    Methodology
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
    Results and analysis
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
    Summary and conclusions
        Page 101
        Page 102
        Page 103
        Page 104
    Appendix A: Test materials
        Page 105
        Page 106
    Appendix B: A note on Richardson Number
        Page 107
        Page 108
        Page 109
    Bibliography
        Page 110
        Page 111
        Page 112
        Page 113
Full Text




UFL/COEL-89/013


RESPONSE OF FINE SEDIMENT-WATER INTERFACE
TO SHEAR FLOW










By


Rajesh Srinivas





1989


Thesis























RESPONSE OF FINE SEDIMENT-WATER INTERFACE TO SHEAR FLOW


By

RAJESH SRINIVAS


















A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF MASTER OF SCIENCE


UNIVERSITY OF FLORIDA


1989
fOEstal Engineering Archives
University of florida















ACKNOWLEDGEMENTS


I would like to express my sincere gratitude to my advisor and chairman of

my graduate committee, Dr. Ashish J. Mehta, for his valuable and imaginative

guidance and ideas which have made this thesis possible. I am indebted to him

for going out of his way in acting like a mentor and guardian. My thanks also go

to Dr. R.G. Dean and Dr. D.M. Sheppard for serving on my committee. I am

also grateful to the personnel at the Coastal Engineering Laboratory, Roy Johnson,

Danny Brown and, especially, Vernon Sparkman for their help and suggestions in

building the flume and pump. Special thanks are also due to Shannon Smythe and

Barry Underwood for their excellent drafting work.

Finally, I would like to thank my parents for their unqualified support and faith

in me.

This study was supported by the U.S. Army Engineer Waterways Experiment

Station, Vicksburg, MS (contract DACW39-89-K-0012) with project manager, Allen

M. Teeter.















TABLE OF CONTENTS


ACKNOWLEDGEMENTS ........

LIST OF FIGURES ............

LIST OF TABLES ............

LIST OF SYMBOLS ...........

ABSTRACT ................

CHAPTERS

1 INTRODUCTION ...........

1.1 Need for Study of Fluid Muds .

1.2 Some Observations of Fluid Mud

1.3 Approach to the Problem ....

1.4 Objectives .............

1.5 Plan of Study ...........

2 INSTABILITY MECHANISM ....

2.1 Discussion .............

2.2 Kelvin-Helmholtz Instability ...

2.2.1 Case of a Vortex Sheet..

2.2.2 Generalized Form of Kelvil

3 INSTABILITY OF STRATIFIED SHE

3.1 Background ............

3.2 Literature Review ........

3.2.1 Browand and Wang (1971)

3.2.2 Smyth, Klaassen and Pelti


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



Entrainment . .














n-Helmholtz Instability

.AR FLOWS .....






er (1987)........









3.2.3 Lawrence, Lasheras and Browand (1987) . . ... 33

3.2.4 Narimousa and Fernando (1987) . . ... 34

3.3 Conclusions ........... ......... ......... 38

4 ENTRAINMENT IN STRATIFIED SHEAR FLOWS ........... 40

4.1 General Aspects ........... .... .......... 40

4.2 Moore and Long (1971) ......................... 40

4.2.1 Results of Two Layer Steady State Experiments ...... 41

4.2.2 Results of Entrainment Experiments . . .... 44

4.2.3 Summary ................... .......... 44

4.3 Long (1974) .. ... ... ...... . . .. .. 45

4.4 Narimousa, Long and Kitaigorodskii (1986) . . .... 47

4.4.1 Deduction of u, ............. ........ 48

4.4.2 Entrainment Rates Based on u . . . ... 49

4.5 Wolanski, Asaeda and Imberger (1989) . . . .... 50

4.6 Conclusions ..................... .......... 51

5 METHODOLOGY ................. .......... 53

5.1 Apparatus .................... .. .......... 53

5.2 Procedure .............................. 61

6 RESULTS AND ANALYSIS ......................... 67

6.1 Definition of Richardson Number . . . .. .. 67

6.2 Initial Conditions ............................. 68

6.3 Evolution of Characteristic Profiles . . . ... 70

6.4 Shear Layer ................... ............. 74

6.5 Observations on the Interface ...................... 80

6.6 Entrainment Rate ................... ......... 84

6.7 Discussion in Terms of Equilibrium Peclet Number . ... 95

6.8 Comparison with Soft Bed Erosion . . . ... 98










. . . . . . 101


7.1 Summary .............

7.2 Conclusions ............

7.3 Recommendation for Further Wor]

APPENDICES

A TEST MATERIALS ..........

A.1 Kaolinite .............

A.2 Bentonite .............

B A NOTE ON RICHARDSON NUMBER

B.1 Introductory Note ........

B.2 Small Disturbances ........

B.3 Energy Considerations ......

BIBLIOGRAPHY .............

BIOGRAPHICAL SKETCH .......


. . . . . 101

. . . . . 101

k .. ............ .. 104


. . . . . 105

. . . . . 105

. . . . . 105

R .................. 107

. . . . . 107

. . . . . 107

. . . . . 108

. . . . . 110

. . . . . 114


7 SUMMARY AND CONCLUSIONS














LIST OF FIGURES


1.1 Definition sketch for fluid mud (source: Ross et al. 1988).... 3

1.2 Evolution of Suspended Sediment Concentration (source: Kirby
1986) . . . . . . . . .. 5

1.3 Internal waves produced by the passage of sailing vessels in
the Rotterdam Waterway (source: van Leussen and van Velzen
1989) . . . . . . . . 6

1.4 Field evidence of gravity driven underflows (source: Wright et
al. 1988) . . . . . . .. 7

2.1 Definition sketch of the flow for the case of a vortex sheet 16

3.1 Offset Velocity and Density Profiles . . . ... 28

3.2 Physical description of the complete flow configuration, with
density and velocity profiles (adapted from Narimousa and Fer-
nando 1987. . . . . . . .. .....35

5.1 Recirculating flume of plexiglass used in the present investigation
(dimensions in centimeters) . . . . .. 55

5.2 Section A-A of the flume, from Figure 5.1 (dimensions in cen-
tim eters) ............................. 56

5.3 Section B-B of the flume, from Figure 5.1 (dimensions in cen-
tim eters) ............................. 57

5.4 Details of the disk pump system used in the present investigation
(dimensions in centimeters). . . . ..... 59

6.1 Sequence of concentration profiles of Run 9 with kaolinite de-
picting the evolution of concentration with time. IF denotes
interface. ............................... 71

6.2 Evolution of the velocity profile in the mixed-layer for Run 6
with kaolinite. IF denotes interface. . . . ... 73

6.3 Change in the mixed-layer depth with time for Run 10 with
bentonite................... ............. 75









6.4 Rate of change of mixed-layer depth in Run 10 with bentonite. 76

6.5 Non-dimensional shear layer thickness vs. Richardson number .78

6.6 Non-dimensional shear layer thickness vs. Richardson number
on a log-log scale ........................... 79

6.7 Turbulent entrainment at t ~ 0.5 minute. Sediment- kaolinite. 81

6.8 Interface at Ri, < 10. Sediment-kaolinite . . ... 81

6.9 Interface at Ri, < 10. Sediment-kaolinite. . . ... 82

6.10 Highly irregular interface at Ri, > 10. Sediment- kaolinite. ... 82

6.11 Scour of growing crest at Ri, > 10. Sediment- kaolinite. .... 83

6.12 Scour of grown crest at Ri, > 10. Sediment- kaolinite. . 83

6.13 Subsiding crest at Ri, > 10. Sediment-kaolinite. . ... 85

6.14 Smoke-like wisp being ejected from the tip of disturbances. Sediment-
kaolinite. .... ............... .......... 85

6.15 Appearance of the interface at high Richardson numbers, Ri, >
25. Sediment-kaolinite... ............... ...... 86

6.16 Non-dimensional buoyancy flux vs. Richardson number for all
the experiments. ........................... 93

6.17 Comparison of erosion rates of soft beds with the rates predicted
by equation (6.9). .......................... 100














LIST OF TABLES


6.1 Initial conditions of all Runs ................. .. 69

6.2 Relevant measured parameters for runs with kaolinite . 87

6.3 Relevant measured parameters for runs with bentonite . 88

6.4 Richardson numbers and entrainment rates for runs with kaoli-
nite . . . . . . . . 89

6.5 Richardson numbers and entrainment rates for runs with ben-
tonite . . . . . . . .. 90

6.6 Peclet numbers for equilibrium conditions . . .... 98

A.1 Chemical composition of kaolinite . . . .... 106

A.2 Chemical composition of bentonite . . .... 106














LIST OF SYMBOLS


b = buoyancy.

boo = buoyancy of unperturbed layer.

bi = rms buoyancy fluctuation.

C = concentration of the suspension.

C1 = mean concentration of the mixed-layer.

C2 = concentration of fluid mud at the level of the interface.

C2 = mean concentration of fluid mud.

c = disturbance wave speed.

c' = turbulent speed.

d = distance between the centers of the shear layer and the density interface.

dm = change in mass with time.

dt = time of the interval.

E = entrainment coefficient.

E = erosion rate.

Ef = floc erosion rate.

F = Froude number.

F, = vertical flux.

H = depth of the fluid mud layer.

H = total depth of the two-layered system.

i = V-.

h = depth of the mixed-layer.

J = local Richardson number.









k = horizontal (x-direction) wave number of the perturbation.

K, = eddy diffusion coefficient.

k = resultant horizontal wave number of the perturbation.

L, = mixing length.

1 = horizontal (y-direction) wave number of the perturbation.

11 = length scale.

M2 = mass per unit area of the fluid mud.

N = buoyancy frequency.

n = Manning's resistance coefficient.

P = probability that a particle reaching the bed will deposit.

Pe = Peclet number.

p = pressure in the fluid.

p' = perturbation in the pressure due to the disturbance.

Q = non-dimensional bouyancy flux.

q = bouyancy flux.

Ri = Richardson number.

Ricr = critical Richardson number.

Ri = minimum Richardson number.

Rio = overall Richardson number.

RiU = Richardson number based on the mean velocity of the mixed layer.

Ri. = Richardson number based on the friction velocity.

s = complex angular frequency of the disturbance.

T = surface tension.

Ta = advective time scale.

Td = diffusion time scale.

U = velocity of fluid.

u = representative velocity.


I









u' = perturbation in the horizontal (x-direction) velocity due to the disturbance,

or horizontal (x-direction) turbulent velocity.

= mean velocity of the mixed-layer.

S = rms turbulent horizontal velocity.

ue = entrainment velocity.

u, = friction velocity.

V = potential energy.

V1 = potential energy per unit mass.

v' = perturbation in the horizontal (x-direction) velocity due to the disturbance.

W = width of the side-walls.

w' = perturbation in the vertical (z-direction) velocity due to the disturbance,

or vertical turbulent velocity.

w, = particle settling velocity.

wl = turbulent fluctuation of the vertical velocity.

w, = friction velocity of the side-walls.

x = horizontal co-ordinate.

y = horizontal co-ordinate.
z = vertical co-ordinate.

a = horizontal (x-direction) of the perturbation.

a = a rate coefficient.

Pf = horizontal (y-direction) of the perturbation.

Ab = interfacial buoyancy jump.

Ap = interfacial density jump.

6 = thickness of the density interface.

6T = kinetic energy per unit volume of the flow.

6W = work done to overcome gravity.

6, = thickness of the shear layer.









6, = amplitude of the interfacial wave.

e = dissipation function.

r7 = displacement of the interface.

A = wavelength of the disturbance.

v = kinematic viscosity.

p = density of the fluid.

p = perturbation in the fluid density due to the disturbance.

pl = mean density of the mixed-layer.
= = shear stress.

= velocity potential.

= perturbation in the velocity potential due to the disturbance.














Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

RESPONSE OF FINE SEDIMENT-WATER INTERFACE TO SHEAR FLOW

By

RAJESH SRINIVAS

August 1989
Chairman: Ashish J. Mehta
Major Department: Coastal and Oceanographic Engineering
An experiment was conceived and executed to simulate the effects of turbulent

shear flow on fine sediment, specifically fluid mud. The tests were conducted in

a "race-track" shaped recirculating flume with a disk pump. Experiments were

run with two types of fluid mud, consisting of kaolinite and bentonite in water.

Shear layer thickness and the nature of the interfacial instabilities were qualita-

tively examined. Entrainment rates of fluid muds were examined as a function of

increasing Richardson number and an empirical relation was obtained between the

non-dimensional buoyancy flux and the Richardson number. This relationship was

then compared with that obtained by previous experimenters for salt- stratified sys-

tems. This comparison made apparent the effect of sediment particles in causing

additional dissipation of turbulent kinetic energy at higher Richardson numbers as

the entrainment rate decreased substantially. Peclet number consideration showed

that the mixed-layer at these higher Richardson numbers appears to behave like a

suspension in equilibrium. The effect of varying the clay constituent of fluid mud on

the entrainment rate could not be fully investigated, although within the limits of

data no discernible trend differences could be clearly identified. A brief comparison

of the fluid mud entrainment rate, which is proportional to the cube of the flow


xiii









velocity, with soft bed erosion rate, which is proportional to the square of the flow

velocity, showed that fluid mud entrainment can dominate over bed erosion at low

current velocities.














CHAPTER 1
INTRODUCTION


1.1 Need for Study of Fluid Muds

A challenging aspect of many coastal and estuarine problems is the elucidation
of fine sediment transport behavior. The compelling factors for such investigations

are both economical and environmental. The last couple of decades have seen

extensive effort being applied to experimental and theoretical studies with a variety

of mathematical models developed for simulation of fine and cohesive sediment

transport. The common aspect in the modeling approach is a soil bed subject to

layer by layer or massive erosion. However, experimental observations verify the

existence of the sediment population in three distinct states: mobile, upper column

suspensions, high concentration near-bed suspensions, and settled muds (e.g., see

Kirby and Parker 1983). In mobile suspensions, the particles are dispersed and

stay in suspension by turbulent momentum exchange. Near-bed high concentration

suspensions or fluid muds, are partially supported by the fluid and partially by their

particle network while in settled muds the particles rest at the bottom supported

by their infrastructure (soil matrix).

The relatively high concentrations of fluid muds play a substantial role in hori-
zontal transport to sedimentation-prone areas. Indeed, in spite of low near bed ve-

locities, the horizontal sediment mass flux can be considerable and can lead to "fluff"

accumulation in navigational channels. The movement of fluid muds has been cited

as the most likely cause of rapid sedimentation in ports located in muddy estuaries.

Obviously, ignoring fluid muds can lead to gross underestimation of sedimentation

rates. Almost totally neglected has been the issue of their upward turbulent en-







2
trainment and mixing due to vorticity generation by shear flows (of current) above

them. Their loose structure permits fluid muds to entrain into the water column

easily and contribute substantially to degradation of water quality. This facet of

transport is evidently not simulated in solely considering erosion of cohesive beds

(which have a measurable shear strength).

Thus, one can assert that the consideration of the entrainment behavior of

this state of fluid muds is necessary to comprehensively simulate the mechanics

of fine sediment transport effectively. This implies that the prediction of fluid
mud behavior to hydrodynamic forcing by shear flows is necessary. Entrainment

rates need to be established and possible physical mechanisms causing this kind of

response need to be formulated, neither of which are presently widely available in

detail. These aspects are briefly examined in this experimental study.

1.2 Some Observations of Fluid Mud Entrainment

Typical variations in concentration and velocity with depth for muds, and the
related definition terminology are presented in Figure 1.1. Fluid muds are confined

to the region between the lutocline, i.e., the zone with a steep concentration gradi-

ent, and the partially or fully consolidated bottom. The upper zone of fluid mud
may have both horizontal and vertical motion, while the lower zone may have some

vertical motion only. Using concentration as a measure, it is generally accepted

that these fluid muds fall in the range 20 to 320 g/1 (Ross et al. 1987). These
concentrations correspond to the bulk density range of 1.01 to 1.20 g/cm3, given a

sediment granular density of 2.65 g/cm3.

Fluid mud behavior is largely time dependent, varying with the physico-chemical

properties of both sediment and water. The theological properties of fluid muds are

strongly affected by factors such as pH, salinity, mineralogical composition and

particle size. Their mechanical behavior is generally pseudoplastic while at very

high concentrations they resemble Bingham plastics (Bryant et al. 1980), as at














CONCENTRATION
1 102 103 104


(mgl -')
105


VELOCITY (msec -1)


Figure 1.1: Definition sketch for fluid mud (source: Ross et al. 1988).


10
or


106







4
high concentrations strong inter-particle bonds provide an initial resistance to shear

deformation (when the applied stress is less than the yield stress, elastic deformation

is possible without any breakdown of structure leading to fluidization).

The dynamic behavior of fluid muds during a tidal cycle is well recognized by

presenting the sequence of concentration profiles recorded by Kirby (1986) (see Fig-
ure 1.2). These are given for accelerating flow, while the reverse sequence prevails

for decelerating flow. Zone 1 is a very low concentration suspension, Zone 2 is the

lutocline layer, i.e., the zone with steep concentration gradients, while, Zone 3 is

high concentration suspension (similar to fluid mud). At slack water, the destabi-

lizing shear forces are small compared to buoyancy stabilization and there is no

entrainment. The physical situation corresponds to a two-phased system with fluid

mud separated from the overlying water by a distinct interface. As the velocity

picks up, the resulting turbulent kinetic energy becomes sufficient to overcome the
stable stratification of the fluid mud and there is subsequent entrainment.
High concentration (- 300 g/l) fluid mud layers of thicknesses more than a

meter have been observed in the Rotterdam Waterway (van Leussen and van Velzen

1989). The passage of sailing vessels over these layers produces internal waves (see

Figure 1.3) at their surface, in spite of the fact that the bottom stresses are quite
low.

Wright et al. (1988) made field measurements of dispersion of concentrated

sediment suspensions over the active delta front of the Yellow River in China. They

provided evidence of the existence of both hypopycnal (buoyant) plumes as well as

gravity driven hyperpycnal (near bottom) dispersal modes. Downslope advection
within the hyperpycnal plume of mixed, lower salinity water from the river mouth

caused vertical instability as regards the excess bulk density (including sediment

concentration, salinity and temperature). Once deposition began, tidal currents

contributing to vertical momentum exchange resulted in instability induced en-
























1







Zone 1


--Conc.
Velocity --


L=Lutocline


--Conc.
--


--Conc.


--Conc.
0--,


4
Zone 1
Zone 2
L





Zone 3







--Conc.


Sequence 1-5:Accelerating Phase


Figure 1.2: Evolution of Suspended Sediment Concentration (source: Kirby 1986)


Zone 3


--Conc.






































I. J
16'




A sr;;-


Figure 1.3: Internal waves produced by the passage of sailing vessels in the Rotter-
dam Waterway (source: van Leussen and van Velzen 1989).





























E

I1

01


SE


5
Km


Figure 1.4: Field evidence of gravity driven underflows (source: Wright et al. 1988).








8
hanced mixing. They observed large amplitude high frequency internal waves at

close to the Brunt-Vaisili frequency.

1.3 Approach to the Problem

In a most general sense, it can be asserted that shear flow in a stratified fluid is

a natural occurence and a crucial mechanism for turbulence production in the at-

mosphere and oceans. A number of practical engineering problems, often associated

with a desire to thoroughly mix effluents entering the surroundings, also requires a

knowledge of the behavior of stratified shear flows.

There are numerous situations in nature where an understanding of the behavior

of velocity-sheared density interfaces is important:

Wind generated waves in the ocean can be a manifestation of Kelvin-Helmholtz

type instabilities at the air-water interface.

The tangential stress which occurs when the wind blows over the ocean gener-

ates a drift current in the upper layers of the ocean, which causes entrainment

of the stratified layers below. This has been cited as the mechanism responsi-

ble for bringing deep- sea nutrients into more accessible regions (Phillips 1977).

Substantial bearing on the world climate is attributed to drift currents in the

upper atmosphere causing growth of this mixed layer against previously stable

inversions.

The rising and subsequent spreading of methane gas in coal mines has an

important bearing on safety (Ellison and Turner 1959).

Gravity currents under a stratified layer over sloping bottoms are very com-

mon in oceans.

In estuaries, the oceanic salt-water wedge penetrates upstream and under

lighter river water.








9
Finally, as mentioned before, shear flows can cause entrainment of underlying

fluid mud, which is the focal point of interest of this study.

Again, it can be stated in general terms that vorticty generation by shear flows

causes instabilities to appear at the density interface and these seem to be the

prime cause for mixing across this interface. A gamut of literature exists for the

same general kind of problem, with density stratification caused by salinity, or ther-

mal effects, or both. These are analogous because of comparable density ranges

and statically stable arrangements. Salinity experiments have been conducted to

simulate oceanic situations which have velocity shear values similar to estuarine

environments, with resulting comparable values of the ratio of buoyancy to shear

forces. Interfacial instabilities and entrainment rates have been examined, theoret-

ically as well as experimentally. However, a peculiar feature of these studies is the

fact that most investigators seem to arrive at quite different results, which they

then generally proceed to explain satisfactorally. So, relative newcomers are sad-

dled with numerous and quite different relationships and explanations for observed

phenomena, without any explicit kind of unification. This is a potent indicator of

the fact that this process of production and dissipation of turbulent kinetic energy

which governs the buoyancy flux and generation, growth and collapse of instabilities

is a very complex process and far from being well understood.

Experiments considered here have additional complications due to non-Newtonian

rheology. Fluid muds are not autosuspensions. Settling is characteristic, and the

downward buoyancy flux due to particle fall velocity causes additional dissipation

of turbulence, which is obviously not the case for salinity and temperature stratified

experiments.

Defining, h as the the depth of the turbulent mixed layer, u, as a relevant

entrainment velocity = dh/dt (rate of propagation of the mixed layer), ul as the

turbulent velocity scale for the mixed layer, Ab as the buoyancy step across the







10
density interface = (gAp)/po, Ap as the interfacial density step, and po as a refer-

ence density, the Buckingham-7r theorem for dimensional analysis can be used for

determining the relevant non-dimensional parameters governing the dynamics of

this situation. Intuitively, one can see that density and acceleration due to gravity

should be coupled as buoyancy. We can in fact identify the pertinent variables to
be Ab, ul, u and h; the fundamental dimensions being that of length, L, and time,

T (as mass becomes implicit in buoyancy). Choosing ul and Ab as our repeating

variables we can form the combinations Ab"ugh and Ab'u'ue. Now, we demand the

exponents of L and T to be zero in each combination. So, we obtain a = 1, f = -2,
7 = 0, and 6 = -1, giving us the non-dimensional parameters A and -, the

first of which is the Richardson number (Ri), whereas the second is an entrainment
coefficient (E). The dimensional analysis is completed by the statement f(Ri,E) =

0, or, further,

E = 7(Ri) (1.1)

The fact that such a functional relationship exists is borne out by the experi-
mental results of many previous investigators, albeit in different forms.

This relationship between E and Ri represents interaction between mechanical
mixing energy and the potential energy stored in stratification that it is working

against. As entrainment is considered a turbulent process, effects of molecular
diffusion are largely ignored, although, some investigators have pointed out that

at high Ri, when turbulence is relatively weak, molecular diffusion does become

important for salinity and thermal types of experiments.

Experimenters have arrived at different power laws (of the form E oc Ri-")
for subranges of Ri (for example, see Christodoulou 1986 and Narimousa et al.
1986). More complicated relationships have also been derived by evaluation of the

turbulent kinetic energy budget (Zemen and Tennekes 1977; Sherman et al. 1978;

Deardorff 1983; Atkinson 1988).







11
1.4 Objectives

With the proceeding discussion in mind, and after an in-depth review of perti-

nent literature regarding the mechanism of instabilities and the consequent entrain-

ment, it was decided to run experiments to simulate entrainment of fluid muds by

turbulent velocity-shear flows in a specially-designed flume. A 'race-track' shaped

recirculating flume was constructed for this purpose in which a two-layered system

of fluid mud and water could be established. The flume was built of plexiglass, as

one of the prime objectives of the present investigation was to observe the nature of

interfacial instabilities. Shear flow was generated by using a specially designed disk
pump which is basically a system of interlocking plates on two parallel externally-

driven shafts rotating in opposite directions. The horizontal velocity of the driven

fluid was constant over the depth of the disk-pump. This disk-pump was instrumen-

tal in imparting horizontal homogeneity to the flow The velocity profile diverged

from the vertical at a distance from the level of the bottom disk of this pump, thus
producing flow with mean-shear.

The ultimate objective of this investigation was to run a series of experiments

to simulate the effects of shear flow on the fluid mud-water interface and the re-
sulting entrainment of relatively low to medium concentration fluid muds, and to

make phenomenological observations to obtain qualitative descriptions of interfacial

instabilities and quantitative expressions) for rates of entrainment by measuring

mass flux in relation to the destabilizing velocity-shear. Another objective was to

determine the effect of varying the degree of cohesion of sediment on rates of en-

trainment. This was done by using kaolinite and bentonite (see Appendix A), which

vary greatly in their degree of cohesion, since kaolinite is only weakly cohesive while

bentonite is cohesive and thixotropic.







12
1.5 Plan of Study

The following chapters document the investigation of the issue of entrainment

of fluid mud by shear flow to find a quantifiable relationship for this process, which,

as mentioned before, has hitherto remained largely unaddressed. Starting with
the justifiable surmise that fluid mud entrainment is a manifestation of interfacial

instability due to current shear, theoretical background for the production and

propagation of instabilities is first discussed, and thus the investigation begins in

Chapter 2 with a theoretical background of Kelvin-Helmholtz type of hydrodynamic

instability. The classic case of stability of a vortex sheet is discussed first in this
chapter, and this is followed by the more generalized version of Kelvin-Helmholtz

instability.

In Chapter 3, some of the more pertinent work of previous investigators on the

subject of instability of shear flows is reviewed. Considerable work has been done

in the area of numerical simulations of instabilities, but adequate support in the
form of accurately documented experimental evidence seems to be lacking. It must

be mentioned, however, that the recent work of Narimousa and Fernando (1987) is

both comprehensive as well as enlightening.

The question of entrainment rates due to shear flows of stably stratified fluids

is examined in Chapter 4. Again, the volume of work which has been done is

considerable, and only directly pertinent literature is considered for review.

Chapter 5 is devoted to the experimental methodology of the present investiga-

tion. The details of the flume and the disk pump constructed for the present study,
the procedure of experimentation and methods of measurement are documented.

In Chapter 6, the results of the investigation are presented and analysed, while

Chapter 7 gives the main conclusions of the study.

In Appendix A a description of the constituent materials of fluid mud, namely

kaolinite and bentonite, prepared in the laboratory is included, while Appendix B








13
traces the history of the definition of the critical Richardson number for stability of

a stratified shear flow.














CHAPTER 2
INSTABILITY MECHANISM


2.1 Discussion

In general, instability occurs when there is an upset in the equilibrium of the
external, inertia and viscous forces in a fluid. Examples of external forces are

buoyancy in a fluid of variable density, surface tension, magneto-hydrodynamic,

Coriolis and centrifugal forces. Surface tension and magnetic forces usually tend to

stabilize, while an interesting point to be noted regarding viscosity is that it can

both inhibit or amplify disturbances. An obvious effect is of dissipation of energy,

whence any flow is stable if viscosity is large enough. However, it's effect of diffusing

momentum may render flows unstable, as in parallel shear flows, which are stable
for the inviscid case.

The analysis is restricted to primarily steady flows, although tidal action in

estuaries is obviously unsteady. However, tidal flows may be considered to be steady

for the purpose at hand, since one is dealing with widely different time scales.
Analysis of unsteady flows is very complex in general. Boundaries of the flow are

an important factor, as well; the closer the boundary, the more efficient is the

constraining of disturbances, although boundary layer momentum diffusive effects

may serve to enhance instability.

Any flow is likely to be disturbed, at least slightly, by irregularities or vibrations
of the basic flow. This disturbance may die away, persist at the same magnitude,

or grow so much as to alter the very flow. Such flows are termed stable, neutrally

stable and unstable, respectively. Stability of parallel inviscid fluid flow has been

investigated since the latter half of the nineteenth century, when the instability







15
of homogeneous and non-homogeneous flows were considered. Subsequent analy-

ses have been with subtle modifications to this same basic problem, including for

compressible fluids, considerations for rotational systems, magneto-hydrodynamic

effects, etc. A wide range of literature has emerged, of interest to specialized sec-

tors in engineering. The consideration in this section will be for the most general
case, fluid dynamical, for studying this phenomenon of instability, rather than its

occurence or application.

2.2 Kelvin-Helmholtz Instability

2.2.1 Case of a Vortex Sheet

Formulation of the Problem

It has been understood since the nineteenth century that the dynamic insta-
bility of a weakly stratified parallel shear flow leads to the formation of vortex-like

structures called Kelvin- Helmholtz (KH) waves. Consider the basic flow of incom-

pressible, inviscid fluids in two infinite horizontal streams of different velocities and

densities, one above the other (see Figure 2.1), and given by


S= 2 U= 2 P= 2 P = P- P2gz (z > 0)

= 4 U =U1 P = P = P plgz (z < 0)

The interface has an elevation z = r7 (x,y,t), when the flow is disturbed.
The governing differential equation is


V2 4 = 0 (2.1)

i.e.,

V72 =O z >r

V201=0 z <


























--- U2


-- U1


Figure 2.1: Definition sketch of the flow for the case of a vortex sheet








Boundary Conditions

(a) The initial disturbance is constrained to a finite region


V a -'- U as z -- foo (2.2)

(b) A particle at the interface moves with it, i.e.,

D[z tl(x, y,t) 0 (2.3)
Dt

(c) Pressure is continuous across the interface

p2(C2 2 ( 2)' gz) =
84d 1
Pi (C1 (V1)2 gz) at z = 7 (2.4)
at 2

by Bernoulli's theorem.
Solution

The above equations pose the non-linear problem for instability of the basic
flow. For linear stability, we consider

02 = U2X + '2 (z > T) (2.5)

01 = Uzx + f'i (z < r7) (2.6)

Products of small increments '1, '2 and rj are neglected. There being no length
scale in the basic flow, it is difficult to justify linearization as regards r7. However, it
appears plausible assuming that the surface displacement and it's slopes are small,
and gl << U 2, U.
With these these assumptions, linearisation yields,

v22 = 0 Z>O (2.7)

v = 0 z<0 (2.8)

4, = 0 z -- +00 (2.9)







18
V1 = 0 z_ -oo (2.10)

a = a +U z = 0 (i = 1,2) (2.11)
az Wt ax
+Pt1 + g) p(U + + gr) z = 0 (2.12)
az at az at
We now use the method of normal modes, assuming that an arbitrary distur-
bance can be resolved into independent modes of the form,

(q,, 'i, '2) = (i, 1, $2) exp[i(kx + ly) + st] (2.13)

[s = a + iw, thus, if a > 0, the mode is unstable, if a = 0, the mode is neutrally
stable and stable asymptoticallyy) for a < 0 ]
Thus, equations (2.7) and (2.8) yield,

~, = Aie--A + Bje.' where k =- +12 (2.14)

From equations (2.9) and (2.10),

01 = Ae (2.15)

$2 = Ae-*z (2.16)
The coefficients can be evaluated from equation (2.11) as

A, = -(s + ikUi)/l (2.17)

As = -?i(s + ikU2)/k (2.18)
From equation (2.12), we can obtain,

pi(U1Alekzik +Alels + gl) =
P2(U2A2e-Iik + A2e-'s + gi) (2.19)

Thus, with the substitution of the coefficients,

pi{(s + ikUi) + kg}=

p2{-(s + ikU2)' + kg} (2.20)









which can be written as

82(Pi + P2) + 2iks(piUi + p2U2)+

[g(PI P2) k2(plU + p2U,2)] = 0 (2.21)

This yields

-ik(piUi + pU2) k2pip2(U U2)2 ) (2.22)
P1+P P2 (P +P2)2 PI +P2
Conclusions

Several conclusions are of interest here,
(1)If k = 0, then

S= gik(P P2) (2.23)
P1 + P2
i.e., perturbations transverse to the direction of streaming are unaffected by it's
presence.

(2)In every other direction, instability occurs for all wave numbers with

k > k( P-- ) (2.24)
plp2(U2l U2)
/
If the wave vector k is at an angle 0 to U, k = k cos 0, instability occurs for

> (2g(p-p) .25)
PIP,(U2 Ua) cos2 0
For a given relative velocity of the layers, instability occurs for the minimum wave
number when the wave vector is in the direction of streaming, i.e.,

g(p2 -P p)
kmi = 2 -- ) (2.26)
PxP2(U2 U2)
Instability occurs for k > k,,n.
This predicts the onset and development of instability, no matter how small
(U1 U2) may be. The presence of streaming overcomes the stability of the static
arrangement. This is the classic Kelvin-Helmholtz instability. Helmholtz (1868)
stated this as:







20
Every perfectly geometrically sharp edge by which a fluid flows must tear

it asunder and establish a surface of separation, however slowly the rest
of the fluid may move.

However, if the effects of surface tension are considered, stability is predicted if,

2g p2 2
(U1 U2)2 < 2 P(2.27)
kmin P1P2
where, kmin = minimum wave number for stability.

With this condition, we have stability for,


(U1 U2)2 < 2 Tg(p P) (2.28)
PIP2
where T is the surface tension.

2.2.2 Generalized Form of Kelvin-Helmholtz Instability

From the above discussion, for the case without surface tension, it can be in-
ferred that the onset of Kelvin-Helmholtz instability is by the crinkling of the in-
terface by shear, and this is independent of the magnitude of the relative velocity
of the two layers. A natural question to confront the reader is whether this result

is entirely fortuitous, due to the sudden discontinuity in the density and velocity

profiles, and not be true for continuous distributions. Thus, now, we take the case
of the stabilizing effect of gravity on a continuously stratified fluid and of the desta-
bilizing influence of shear in a generalized form of Kelvin- Helmholtz instability. We
start with a basic state in dynamic equilibrium,

u, = U,(z,) (2.29)

P. = P.(z.) (2.30)

P. = (Po). g p(z) dz (2.31)

for zl. < z, < z2., where, z. is the height and zl. and zz. are the horizontal

boundaries of the flow. The subscript indicates dimensional quantities. Taking L,







21
U and Po to be the characteristic length, velocity and density, respectively, of the
basic flow and further assuming the fluid to be inviscid and density to be convected
but not diffused, we non-dimensionalize the equations of motion, incompressibility
and continuity to get,

u ,
p(t +u.Vu) = -VP-F-2pk (2.32)

v.u = 0 (2.33)
ap
-+u.VP = 0 (2.34)
at
where F = V/I/gL is a Froude number.
Perturbations are introduced into the flow,

u(z,t) = U(z)-+u'(x,t) (2.35)

p(x,t) = P(z)+p'(x,t) (2.36)

p(, t) = Po F- (z)dz' + p'(x, t) (2.37)

The form of the equations obviously permits us to take normal modes of the
form,

{u' (, t), p (, t), p(, t)} =

{u(z), (z), A(z)} exp[i(ax + 0y act)] (2.38)

where, the real part is understood. The fact that the solutions must remain bounded
as x, y -+ oo implies that a, must be real; but, the wave speed c may, in general,
be complex, i.e., c = c, + ici thus representing waves traveling in the direction

(a, /,0) with phase speed ac,/V/a2 + 2 and grow/decay in time as exp(acit). Thus,
aci > 0 implies instability, acj < 0 stability, while acc = 0 implies neutral stability.
Introducing these into equations (2.32)- (2.34), and linearizing by neglecting
quadratic terms of the primed quantities and using equation (2.38) we obtain,


iap(U c)u + pU'ti = -iap


(2.39)







22
iap(U c)0 = -if# (2.40)

iaA(U c)b = -D F-2 (2.41)

iauc + if + Dtw = 0 (2.42)

ia(U c)A + p'w = 0 (2.43)

where differentiation with respect to z of a basic quantity is denoted by prime
whereas that of a perturbation by D.
Thus, from equations (2.39) and (2.40),
i A AU'^A
t = a (2.44)
ia,5(U c)
S= (2.45)
ap(U c)
Using these in conjunction with equation (2.42), we can obtain,

--i' p U'8 iP2,
+ Dtb = 0 (2.46)
p(U c) ap(U c)
Eliminating ^ and 5, we finally arrive at,
(2 +I
(U c) D (a2 + )} (a + + {(U c)Do U'}
a2F2 (U c)p p
(2.47)
Yih (1955) applied Squire's transformation to the system to show that for a
three-dimensional (3-D) wave with wave number (a, #), there is a 2-D wave with
the same complex velocity c, but wave number (Viai + 2, 0) and Froude number
aF/I1/xa +2, which thus has effectively reduced gravity but magnified growth rate

(a2 + P2)c, and thus is more unstable.
Equation (2.47) indicates that F-2 occurs as a product of -p'/p, so an overall
Richardson number is defined as

'R = gL2 dp.
pF2 V2 p, dz.
The Brunt-Viisill frequency (or buoyancy frequency) N, is defined as

N(z.) = -g I/. = RiN2(z)V/L2








Thus, we get,

RiN2/U' L2 g d-A dU
V2 F. dz. dz

-g /{ .(dUz)2}
dz dz.
as the local Richardson number, J, of the flow at each height z,, such that

2z dU,2
J-= N(z)/( )2 (2.48)
dz.
In many applications, Fp (z,) varies more slowly with height than U, (z,) such that
-p',/p < 1; whence Ri is of the order of magnitude unity as F < 1. Thus, as in the
Boussinesq approximation the last two terms of equation (2.47) are neglected; hence,
the effect of variation of density is neglected in inertia but retained in buoyancy.
With this approximation and considering only 2-D waves we get,

2d 2 2U 1 RiN2
dz' dzz U c (U c) 2

which can be written as

(U c)(D2 a2)0 U"q + RiN20/(U c) = 0 (2.50)

with the corresponding boundary conditions at z = zl and z2, which is the Taylor-
Goldstein equation, where

i = af/az (2.51)

w = -iao(z) (2.52)

u = ak'/laz (2.53)

w = -ao'/la (2.54)

0' = O(z)exp{ia(x- ct)} (2.55)

Here, a > 0 can be assumed without any loss of generality, and also that each
unstable mode has a conjugate stable one.








Assuming ci 7 0, define

H = //VU c (2.56)

Substituting into equation (2.50) yields,

U" U"
D{(U c)DH} {2(U c) + +( RiN2)/(U c)}H = 0 (2.57)
2 4
Multiplying by the complex conjugate, H* and integrating,

Li 1 U'/4 RiNC
f{(U- c){IDHI2 + a2 jH2} + UIH2 + U2/4 RiN2 H12} dz = 0 (2.58)
12 U-c
The imaginary part gives,

'\jDH\2 + a2JH2 + (RiN2 U/4) IH/1U cl2} dz = 0 (2.59)

Thus,

0 > I DHI2 dz
1
= f'{(RiN2 U'2/4) +a U c12}H2/IJU- c2 dz (2.60)

(assuming ci $ 0). Thus, the local Ri has to satisfy RiN2/U" < 1/4 somewhere in
the field of flow for instability.
The same can also be established, although somewhat heuristically, by analyzing
the energy budget; the essential mechanism of instability being the conversion of
the available kinetic energy of the layers into kinetic energy of the disturbance,
overcoming the potential energy needed to raise or lower the fluid when d{./dz* < 0
everywhere. Consider two neighboring fluid particles of equal volumes at heights z.
and z. + 6z. being interchanged.
Thus, 6W = work per unit volume needed to overcome gravity = -g6ps6z..
For horizontal momentum to be conserved, the particle at z, will have final
velocity (U, + k6U.)tand the particle at z. + 6z. have (U, + (1 k)6U,)ias it's final
velocity, where, k = some number between 0 and 1, and

SU. = ( )6z. (2.61)
dz,







25
Thus, the kinetic energy per unit volume released by the basic flow is,

1 1
6T = pU. + (.+ 6.)(U +6U.)2
2 2
AP,(U. + k6U,)2 (p + 6p,)(U, + (1 k)6U.)5 (2.62)
2 2
= k(1 k)p,(6U.)2 + U,6U,6p, (2.63)

< -(6U,)2f + U.6U.6p. (2.64)
4
A necessary condition for this interchange, and consequently, instability is 6W <
6T, and therefore, somewhere in the field of flow,

dp. 1 dU. dU_.dp._
g- < -p.( )2 + U. (2.65)
dz. -4 dz. dz, dz,
i.e.,
_h 1
d < (2.66)
neglecting the inertial effects of the variation of density.

Miles (1961) stated that the sufficient condition for an inviscid, continuously

stratified flow to be stable to small disturbances is that the local Richardson num-

ber should exceed 1 everywhere in the flow (a modified result is presented in Ap-
pendix B). This does not imply that the flow becomes unstable if this falls below

somewhere. Counter examples have been found, for example, with a jet-like velocity

profile uoc sech2 z and an exponential density profile, in which case the flow can

become unstable if Ri,, < 0.214. Hazel (1972) has demonstrated the stabilizing ef-

fect of rigid boundaries. One must consequently surmise that the entire profile (the
boundary conditions, viscosity, etc.) matters in determining the critical Richardson

number.

Thus, it is seen that the effect of velocity-shear on statically stable stratification
can be to cause disturbances to appear at density interfaces which grow with time.

Intuitively, one can sense that after a period of sustained growth, the wave should

break, with the natural ramification being upward mixing of the denser fluid, i.e.

entrainment.







26
With the preceding background of the theory of velocity-shear induced inter-

facial instability, we now proceed to Chapter 3 where pertinent work on the same

phenomenon is reviewed. Some examples of numerical and laboratory simulations

are covered to give a feel for the magnitude as well as different facets of the problem.














CHAPTER 3
INSTABILITY OF STRATIFIED SHEAR FLOWS


3.1 Background

As noted in Chapter 1, shear induced instabilities are a very important factor
in the generation of turbulence and mixing in stratified flows. When 6, 6 and
d = 0 (see Figure 3.1), at sufficiently low Ri (= ~A), the primary instability

is of the Kelvin-Helmholtz (KH) type; however, the process of growth by pairing

becomes limited by the stabilizing effects of buoyancy (Corcos and Sherman 1976)

and a sufficiently large density difference will stabilize the flow.

As it is relevant in geophysical situations, the case of 6, > 6, with d = 0 was

studied by Holmboe (1962), who predicted a second mode of instability, now called

the Holmboe mode, which has been further studied by a number of researchers, for
example Hazel (1972). Theoretically, this comprises of two trains of growing interfa-
cial waves traveling in opposite directions to the mean flow, eventually resulting in
a series of sharply cusped crests protruding alternately into each layer, with wisps

of fluid being ejected from these cusps (but, more often, experimental results indi-
cate cusping only into the high speed layer which may possibly be attributed to the
selective vorticity concentrations in the high speed layer). Thus, when 6,/6 > 1,
theoretically, there is always a range of wavenumbers which is unstable, however

large Ri may be, with this second mode having maximum amplification rates at
non-zero Ri.

For small Ri transition to turbulence is by the first mode (i.e., KH) regardless
of 6,/6 values, with collapse by overturning due to the concentration of the available

vorticity into discrete lumps along the interface (Thorpe 1973). This results in finer

















U2 P2
U(z) p(z)









-U. -

IU PI


Figure 3.1: Offset Velocity and Density Profiles







29
scales of turbulence, and in a homogeneous fluid these lumps continue to pair with

the growth of the mixed layer. However, with stratification, entrainment of fluid into

the mixing layer degrades this vorticity in these lumps and this mixed layer growth

eventually stops, and if the initial Ri is small, turbulence grows till length scales

become large enough for buoyancy to play an important role, followed by collapse.

If 6,/6 > 1, this collapse is followed by mode 2 waves (Browand and Winant

1973). These seem to be like internal waves within the mixing layer, with nearly
horizontal wave crests and small wavelengths (Delisi and Corcos 1973); and, finally,

there is decay of the turbulence structure. Fernando (1988) mentions that turbulent

patches in stratified media may be generated by the mechanism of instability (by

wave-breaking and double diffusion).

Thus, stratification has this ability to destroy turbulence which may be a pos-
sible explanation for it's intermittent character, as found in nature. McLean (1985)

observed longitudinal ripples on the bed while modeling deep ocean sediment trans-

port, which he postulated to occur during deposition after high energy erosional

events due to helical circulation owing to a non-uniform turbulence field. This kind
of turbulence field can result because of lateral homogeneity of turbulence damping

by the aforementioned density stratification. Physically, this turbulent mixing layer

is destroyed by the stabilizing effect of gravitation on the largest scales of Ri.

When the initial Ri is large enough, say > 0.1, then turbulence production
depends strongly on the d/6 ratio, with initial instability of the mode 2 waves.
These decay by breaking at sharply peaked crests (Browand and Winant 1973),

with fluid ejected into the higher speed layer as thin wisps from these crests.







30
3.2 Literature Review

3.2.1 Browand and Wang (1971)

Background

A velocity shear interface of thickness 6, is considered between two horizontal
streams of velocities U1 and U2 and densities P1 and P2, with the density interface

of thickness 6. They define Ri = Abb,/(AU)2.

The velocity profiles agreed remarkably well with the hyperbolic function, often
used in stability analysis. The difference between the stability of a sheared layer

which is homogeneous and that which has a stable density interface was demon-

strated.

Discussion

The effect of stratification on sheared layers is complex, with the mode unstable
in the absence of stratification, called Rayleigh waves, being stabilized while a new

one, the Holmboe mode is now unstable. The mode destabilized by gravity has

a non-zero wave speed when riding at the mean velocity (U1 + U2)/2. In these
co-ordinates, the disturbance is assumed to consist of one wave traveling upstream

and one traveling downstream, with the interface a standing wave of increasing
amplitude.

Disturbances in the case of a homogeneous shear layer can be thought of as

two almost independent distortions of the upper and lower boundaries of the con-
stant vorticity region. Short wave length disturbances are totally independent. The
amplitude of the disturbance oscillates as the two distortions alternately reinforce

and obstruct. However, long wavelength disturbances influence each other to such

an extent that "slippage" of the upper and lower distortions can be stopped. The
relative phase is fixed in the position most favorable for growth (PFMMG) of the
perturbation. In the stratified case, additional vorticity is generated by the distor-







31
tion of the central interface. This baroclinic vorticity is responsible for inhibiting

instability at low Richardson numbers (Rayleigh waves); however, at high Ri, strat-

ification alters the slippage of the distortions such that the wave lingers more at the

PFMMG than in unfavorable regions (Holmboe waves).

In the regions of instability of Rayleigh waves, both Holmboe and Rayleigh
waves are indistinguishable, both being phase locked, and non-linear growth is by

roll-up or overturning. Previously well distributed vorticity is now concentrated

into discrete lumps along the interface and breaking is violent.

In regions where Holmboe waves are unstable, no roll-up occurs. Interface
displacement simply grows in magnitude with each succeding oscillation, ultimately

breaking at the crests, which may be on both sides or not, according to as the

excitation is unforced or not, respectively.

3.2.2 Smyth, Klaassen and Peltier (1987)

These investigators performed numerical simulations of the evolution of Holm-
boe waves. A series of simulations using progressively lower levels of stratification
led to Kelvin Helmholtz (KH) waves. The effect of strong statification on KH
waves depends on the ratio of the vertical distances over which the density and flow
velocity, i.e., 6 and 6, change.

If 6 > 6,, increasing stratification stabilizes the flow.

If 6 < 6,/2, increasing stratification causes the KH wave be replaced by Holm-
boe type oscillatory waves.

From linear theory, the relationship between KH and Holmboe type instabilities

can be shown to be equivalent to a damped oscillator, governed by,

A"(t) + bA'(t) + cA(t) = 0

Stratification, represented by c, provides the restoring force. Shear, represented by

b, serves to transfer energy into or out of the oscillation.








32
Solutions are of the form A elt, where a = a, + ioi, subscripts denoting real

and imaginary parts respectively. If ai = 0, we have a monotonically growing distur-

bance, i.e., KH waves. However, if c/b2, which is analogous to the bulk Richardson

number, grows beyond a certain value, this train gives way to oscillatory Holmboe

waves.

A linear analysis of the governing hydrodynamic equations was performed to

determine, for a given level of stratification and Ri (with 6, being the length scale),

the value of a, the wave number, which has maximum growth rate, a,, to determine
the horizontal wave length to impose on the non-linear model.

The plot of a(a, Ri) showed that, for small values of Ri(< 0.3), the fastest

growing modes had ai = 0; while for higher Richarson numbers, ai had non-zero

values, i.e, Holmboe instability. Two points were taken from the Holmboe regime

and one from the KH regime for non-linear analysis. By analysing the evolution of
the non- dimensional perturbation kinetic energy for the three points they confirmed

the nature of the instabilities predicted by the linear analysis : slow exponential

growth coupled with fast oscillations characterising disturbances in the Holmboe
regime and monotonically growing waves in the KH regime.

Holmboe waves have two components, with equal growth rates and equal but

oppositely directed phase speeds. The position most favorable for growth (PMFFG)

is just before the "in-phase" configuration in accordance with Holmboe (1962). In

the "in- phase" configuration, the kinetic energy is maximized. The phase speed

is maximum just beyond this "in-phase" position. This implies that as the level of
stratification decreases, the maximum phase speed increases relative to the cycle

averaged speed, resulting in a greater time spent in the PMFFG and thus effecting

increasing growth rates. When this level of stratification is further decreased, the

phase speed at the PMFFG should vanish, with phase locking of the two compo-

nents. They should now rotate as a unit and grow into intertwined fingers of heavy









and light fluid as in KH waves.

With decreased stratification in the Holmboe regime, growth rates and oscil-

lation frequency reduced as predicted, and also, the phase speed increased after

leaving the "in-phase" position. With evolution, thin plumes of fluid were ejected

from the peaks of the waves, primarily after passing the "in-phase" configuration.
The KH regime simulation, too, was in accordance with linear predictions.

3.2.3 Lawrence, Lasheras and Browand (1987)

Two layers of different velocities and densities were separated by interfaces of
thicknesses 6, and 6, respectively. The centers of the two interfaces were separated

by a distance d.

Theoretical Analysis

An eigenvalue relation was derived from the Taylor- Goldstein equation and
stability diagrams are plotted of Ri vs. a, for different values of e, where, Ri =

Af5, a = k6, = instability wave number, e = 2d/6,, Ab = 12 AU = UIi Us2,
k = 27r/A, and A = wavelength.

With E = 0, there were two modes of instability : a non-dispersive Kelvin -

Helmholtz type for Ri < 0.07 and a dispersive one, the Holmboe type, for all
(positive) Ri. In the overlap region, 0 < Ri < 0.07, KH had higher amplification
rates. For e > 0, the KH mode as well is dispersive and has higher growth rates.

For e > 1, the Holmboe mode disappeared.

Experimental Observations

For e > 0, concentrated spanwise vorticity was observed above the interface, in
the high speed layer only (and none in the lower low speed layer), causing inter-

facial cusping into the upper layer. Initial instability was two dimensional. As Ri

decreased, the wavelength of the disturbances increased. At lower Ri, disturbances

developed considerable three dimensionality, with wave breaking, similar to KH bil-

lows. This billowing was only in small wisps, demonstrating the inhibiting effect of







34
buoyancy. With increasing Ri, at fixed e, this tendency decreased and thin wisps

were lifted almost vertically into the upper layer. Instabilities were observed to pair

in the same manner as KH instabilities in unstratified fluid, with wisps ejected, just

after this pairing.

3.2.4 Narimousa and Fernando (1987)

The investigators discuss the effects of velocity induced shear at the density
interface of a two-fluid system. One of their most important conclusions has been

regarding the entrainment- Richardson number relationship : Eu oc (Ri;"), where,

Eu is an entrainment coefficient = Ue/u, u, = entrainment velocity, u = scaling
velocity, Ri = Richardson number = Abh/u2, Ab = interfacial buoyancy jump,

h = mixed layer depth, and n = a coefficient.

The investigators used a recirculating flume, which was free of the rotating

screen of the more popular annular flume experiments. Their two-fluid system
consisted of initially fresh and salt water layers. The mixed layer (of initially fresh
water) was selectively driven over the heavier quiescent fluid by using a disk pump,

developed by Odell and Kovasznay (1971). The velocity of the mixed layer was

varied between 5 15 cm/s using variable pump rotation rates. Shear layer velocity
profile appeared linear while that in the viscous diffusive momentum layer resembled

Couette flow profiles.

For moderately high Richardson numbers, Ri, > 5, the density interface was

found to be topped by a thin layer of thickness 6,, with a weak density gradient

which had not yet got well mixed. This partially mixed fluid results owing to

the fact that energy of the eddies is not strong enough to entrain the fluid from

the stable interfacial layer, and mixing can only occur by wave breaking resulting

from the mixed layer turbulence at higher Richardson numbers, i.e., eddies assist

entrainment in two stages, from the interface to the intermediate layer and from

there into the mixed layer.




















Mixed
Layer








Non-Turbulent
Layer


Figure 3.2: Physical description of the complete flow configuration, with density
and velocity profiles (adapted from Narimousa and Fernando 1987.


P(z)


- SI


`_17







36
Fluid above this layer was homogeneous. At low Ri,, with high rates of en-
trainment, the intermediate layer was absent. The entrainment interface consisted

of regularly spaced billows with high spatial density gradients within, with their

centers having small scale irregularities which could be the effect of local instability

regions due to the entrainment of heavy and light fluid into the core. However,

the final stage of mixing within these billows was fairly slow, with breakdowns into

regions containing small scale structures which may be due to the interaction of

two adjacent vortices. With increasing Ri, the frequency of billows progressively

decreased and entrainment was dominated by a wave breaking process, with wisps

of fluid being ejected into the upper layer. This kind of behavior was seen over

a whole range of Ri,(5 < Ri, < 20), with decreasing frequency as Ri, increased.
Also, large amplitude non-breaking solitary waves were seen over Riu = 10 20.

The shear layer is very important as it is responsible for the turbulent kinetic

energy of entrainment and thereby controls the size of the energy containing eddies

at the interface. The investigators found that 6,/h was independent of Ri' (and

about 0.2) indicating that the size of the eddies should be scaled by h.

The average measured value of 6/h was also independent of Ri,, and around
0.04-0.08. This ratio was also confirmed by another interpretation of data as follows:

Observing that the buoyancy in the mixed layer and the gradient in the interface
are constant,


b(z) = bo + Ab(z h- 6)/(6) for (h < z < h + 6) (3.1)

where, b(z) = mean buoyancy at elevation z, bo = buoyancy of lower unperturbed

layer, and z is positive down from the free surface.

Assuming horizontal homogeneity, Long (1978) integrated the buoyancy conser-

vation equation,
Ob Oq
-t = (3.2)
at 8z









37
where, q(z) = -bw = buoyancy flux; b and w being the values of buoyancy and

vertical velocity fluctuations, respectively.

This yielded,


q(z) = q2z/h (0 < z < h) (3.3)

q2 = -hd( (3.4)
dt
Sr2 d(Ab) Abr2 d6 Ab dh
q(z) = q2+( -T) (h < z < + ) (3.5)
26 dt 62 dt 6 dt

where, q2 = buoyancy flux at the entrainment interface, and r = z h.

As q(h + 6) = 0, it is possible to obtain

d{Ab(h + 6/2)} 0 (3.6)
dt

By defining a characteristic velocity scale based on the initial buoyancy jump

and the depth of the initially homogenous layer, i.e.,


Vo2 = hoAbo

and defining 6 = ah one finally arrives at


h(1 + a/2) = Vo0/Ab (3.7)

Plotting this equation showed 6 ~ 0.06h.

Energy Budget Analysis

Analysis of the energy budget yielded the result that buoyancy flux, turbulent

energy production and dissipation terms were of the same order and that E ~ Ri1'.

Wave amplitudes at the interface, 6,, scaled by h were of the order of Ri1/2.
This may possibly be due to the energy containing eddies impinging on the interface.

The vertical kinetic energy of the eddies = w2 (where wi is the rms fluctuation of

the vertical velocity). Then the generated potential energy of the waves ~ N26.

Thus, 6, wi/N, where N = (Ab/6)1/2 = boundary frequency of the interfacial


I









layer.

6 ~ h (3.8)

wl ~ Au (3.9)

Hence,

S~ Ri-1/2 (3.10)

Summary

(1)During entrainment, two layers, the density interfacial layer and the shear

layer, having direct bearing on the entrainment process developed and increased

linearly, independent of Ri,.

(2)Billows, formation and breakdown of large ordered vortices cause mixing at

low Ri,, while breaking waves cause it at higher Ri,.

(3)Wave amplitudes scaled well with the size of the energy containing eddies of

the size of the mixed layer.

(4)The rates of work done against buoyancy forces, kinetic energy dissipation
and shear production of turbulent kinetic energy were of the same order.

3.3 Conclusions

The preceding discussion documents some of the modes of interfacial instability
which are possible. The mode of instability is dependent on the stratification and
the ratio of the thicknesses of the shear layer and the density interface. When

6,/6 > 1, increasing stratification causes monotonically growing Kelvin-Helmholtz

waves to be replaced by the oscillatory Holmboe mode. The physical nature of the

modes differs as well, in that Kelvin-Helmholtz waves are associated with billowing

and lumping (and pairing) of vorticity near the interface, while Holmboe waves are

characterized by a series of non-linearly crested waves cusping generally into the high

speed layer only. Billowing as well as cusping into the high speed layer were observed

in laboratory experiments by Narimousa and Fernando (1987) with the transition







39
in the mode of instability occurring with increasing Richardson number. Moore and

Long's (1971) experiments to determine entrainment rates in velocity-sheared salt-

stratified systems (see Chapter 4) also describe some of these phenomena in detail.

In effect, it can be concluded that velocity-shear has a destabilizing effect on stable

stratification and can cause upward mixing of the heavier fluid. This effect of the

growth and breakdown of instabilities is examined in the next chapter.














CHAPTER 4
ENTRAINMENT IN STRATIFIED SHEAR FLOWS


4.1 General Aspects

The effect of interfacial instabilities in causing entrainment across the (stat-

ically) stable density interface is considered in this chapter. As a considerable

amount of worthwhile and interesting work has been done on both shear flows and

flows without mean shear, a complete review is beyond the current scope. Thus,

only directly pertinent studies as regards shear flows are reviewed. Moore and Long

(1971) discuss their results with respect to those obtained by previous investiga-

tors and Long (1974) theoretically examines many of these results, thereby making

this literature especially riveting. A recent experimental study by Narimousa, Long

and Kitaigorodskii (1986) is also reviewed. Not much published work is available

specifically regarding vertical entrainment of fluid muds, and thus the study using

kaolinite by Wolanski, Asaeda and Imberger (1989) is reviewed in spite of it being

for a mean-shear free environment.

4.2 Moore and Long (1971)

The experiments were run in a racetrack shaped flume with a system of holes

and slits in the floor and in the ceiling, allowing fluid injection and withdrawal

to produce required steady state horizontally homogeneous shearing flows. Their

steady state was defined as keeping the level of the density inflexion point constant.

In the steady state two-layer experiments, the density and velocity profiles were

kept constant by adjusting the flow rates and replenishing salt to the lower saline

layer. This amount of salt per unit time, on dividing by the horizontal cross section







41
area of the flow tank, gave the salt mass flux.

In the entrainment experiments, the tank was filled with fluid with a linear
density gradient and then circulation of either fresh or salt water was started and

the density profile observed as a function of time.

4.2.1 Results of Two Layer Steady State Experiments

The investigators' overall Richardson number was defined as, Rio = ItAb/(2Au)
where, H = total depth, Ab = buoyancy difference between the top and bottom

layers of fluid, and 2AU = velocity difference between the top and bottom layers of

fluid. Also, q = buoyancy flux, and, Q = q/Ab(2AU) = non- dimensional buoyancy
flux.

A layer of thickness 6,, with a velocity gradient, separating two homogeneous

layers of depths h each, developed. At low Rio, 6, was very large and decreased

with increasing Rio, until it ultimately became quite small.

For values of Rio greater than about three, turbulence in each homogeneous layer
caused erosion to a considerable extent of the layer over which the density gradient

initially manifested. The interface was clearly visible. The surface of the interface

was irregular in shape (with amplitudes ~ 0.5 cm, wavelength ~ 3-4 cm and width

~ 1 cm) with wisps of fluid being detached from the crests of disturbances, this

phenomenon being more observable for disturbances cusping into the lower density

layer. The speed of these waves was less than of the homogeneous layer above.

These grew in amplitude and then simply disappeared with a wisp of fluid ejected

from the tip, indicating that the original disturbance may well have been caused by

eddies scouring the interface, with it's "roller action" drawing dense fluid up into a

crest before it sharpened and was sheared off.

For values 1.5 < Rio < 3.0, the interface was less sharp and more diffuse

(with 6, increasing). The thickness of the region with the density gradient, 6,

also increased, as did the salt mass flux. Mixing now seemed to be more due to







42
internal wave breaking. For Rio < 1.0, very large eddies extended through the

diffused interface. For low values of Ri, 6, ~ 6 while for higher values of Rio,

6, > 6. Richardson number, Ri,,, defined using the average density gradient and

average velocity gradient over 6, had a value close to one.

Plotting the non-dimensional buoyancy flux with Rio yielded the functional

relationship

Q = C1/Rio (4.1)

with C1, which may be weakly dependent on kinematic viscosity and diffusivity,

having a value ~ 8 x 10-.

Other researchers have obtained relationships between E and Ri, where


E = u,/u (4.2)

with the entrainment velocity u, defined as the normal velocity of the interface, or for

steady flow experiments, the volume flow rate of the fluid being entrained divided

by the cross sectional area over which this is occurring, u = some representative

velocity and Ri = Richardson number computed for that particular experiment,

with Ap always representing the density jump between the turbulent homogeneous

layer and the fluid being entrained.

Rouse and Dodu (1955) used a two layer fluid system with turbulence being

generated by a mechanical agitator and pointed out that if the entrainment rate is

proportional to Ri-1, the implication is that the rate of change of potential energy

due to entrainment is proportional to the rate of production of turbulent energy by

the agitator.

Ellison and Turner (1959) discussed entrainment rates of a layer of salt water

of thickness D flowing with velocity i under a layer of fresh water. Defining Ri =

AbD/U2, they obtained E ~ Ri-1 for Ri < 1.

Lofquist (1960) got a similar relationship for Ri < 1, but his data were scattered







43
for Ri > 1, with a faster decrease in entrainment rates than is indicated by E ~

Ri-1.

Turner (1968) studied mixing rates across a density interface with turbulence

being generated on either or both sides by a mechanical agitator and obtained

E ~ Ri-1 for Ri < 1, but E ~ Ri-3/2 for Ri > 1.

Kato and Phillips (1969) applied a constant shear stress r = pu2 at the upper

surface of a linearly stratified fluid and obtained E ~ Ri-1, with values of Ri.

equivalent to Rio < 1. These investigators also demonstrated that the entrainment

coefficient E represented a time rate of change of potential energy per unit mass V1,

in non- dimensional terms, i.e.,

2po dVI u,
2po dV1 E = KRi-1 (4.3)
gApu, dt u,
with, Ri, = g--h/u2 and K is some constant.

Moore and Long (1971) used this basis to compare their functional relationship

with other researchers and showed that the non-dimensional flux is essentially the

same as an entrainment coefficient. Another way of showing this relationship is as

follows :

If the injection-withdrawal system at the top is turned off and the interface

allowed to rise a distance dh = u, dt, then [mass(t + dt) mass(t)] = mass added

at the bottom = dm.

Letting lower density = pi + Api/2 and upper density = pi Api/2,
1 1
(Pi + 2Ap)(HT/2 + dh)A + (p -Api)(H/2 dh)A
2 2
1 1
-(Pl + 2API)(t/2)A (pi Apl)(A/2)A

= dm (4.4)

Thus,

Ue = dh/dt

= (1/ApiA)dm/dt (4.5)









Therefore,

UeAb = q (4.6)

If u, is defined thus for the steady state experiment, too, we get,


Q =E (4.7)

Thus, E ~ Q ~ Ri-1 should be valid over 0 < Rio < 30, as evidenced by the

Moore and Long experiments. Lofquist's results maybe attributed to the horizontal

inhomogeniety of his experiments, while Turner's maybe due to the absense of a

mean velocity to his flow, his method of definition of the Richardson number, or

the absence of what he calls fine structure in his experiments.

These relationships were considered in terms of energy changes and it was shown

that the rate of change of potential energy of the system or the buoyancy flux and

the rate of dissipation of kinetic energy per unit volume were of the same order.

4.2.2 Results of Entrainment Experiments

The initially linearly stratified fluid was eroded and replaced by a homogeneous

layer of depth h(t), when the injection- withdrawal system was applied to only one

side of the channel. The results showed that h3 oc t, similar to Kato and Phillips

(1969).

4.2.3 Summary

Over the range of Richardson numbers studied, results showed that the existence

of turbulent layers on either side of a region with a density gradient caused erosion

of this region to occur, with the formation of two homogeneous layers separated by

a layer with strong density and velocity gradients. The gradient Richardson number

of this transition layer tended to have a value of order one. The non-dimensional

buoyancy flux Q was functionally related to the overall Richardson number, Rio,

by Q ~ Ri;o for 0 < Rio < 30. Entrainment experiments of an initially linearly








45
stratified fluid with the application of shear on one side resulted in the formation of a

homogeneous layer separated by an interface from the stratified layer,with h3(t) cc t.

4.3 Long (1974)

Long critically analyzed mixing processes across density interfaces including

cases without and with shear, which have been shown by previous investigators to

have different relationships with an overall Richardson number, Ri., based on the

buoyancy jump across the interface, the depth of the homogeneous layer and the

intensity of turbulence at the source.

At large Reynolds (Re) and Peclet (Pe) numbers, the fluxes of heat or salt and

the entrainment velocity appear to be proportional to minus one and minus three

halves powers of Ri. for flows with and without mean shear respectively, where the

higher entrainment rate for shear flows is attributed to the decrease of rms velocities

near the interface for increasing Ri. for cases of zero shear. Conforming to our area

of interest, this discussion will be restricted to the cases with mean shear.

Kato and Phillips (1969) applied a constant shear stress r = pu2 at the surface

of initially linearly stratified fluid in an annular flume using a rotating screen. This

resulted in the development of an upper homogeneous layer and lower stratified fluid

with an interfacial buoyancy jump Ab. Defining the rate of downward propagation

of this interface as u,, the investigators arrived at


Ue/u* = KiRi,1 (4.8)

with, Ri, = hAb/u*, h = depth of the homogeneous layer and K1 is some constant.

They also found that U/u, increased with time, where U is the speed of the
screen, with u, held constant. A simple analysis also reveals this quantity to be

independent of the Richardson number.

In Moore and Long's (1971) experiments in a race track shaped flume with salt







46
and fresh water, buoyancy flux q was measured at steady state, yielding

q = K2(Au)3/h (4.9)

with Au being the mean velocity difference of the two layers and K2 is some con-

stant. Defining the entrainment velocity by, u,Ab = q, this yields equation (4.8) on

making the plausible assumption that Au/u. is independent of Ri., where pu2 is

the constant momentum flux in the tank.

The theory (Turner 1973) that erosion of the interface should depend on the

properties of turbulence near the interface (and not at the source), especially on the

rms velocity scale ul and the integral length scale 11 near the interface proposes a

relationship of the form

U,/u, = f(Ri) (4.10)

with, Ri = liAb/u2 assuming no dependence on any other quantities, and large Re

and Pe.

Now, we have,
ar a ((4.11)
8z at
with i as the mean horizontal velocity at depth z.

For the Moore and Long steady state experiments,

ar
-=0
oz

As r = -pu'w' and the correlation coefficient is of order one in the homogeneous

layers, thus, u. = r/ is proportional to ul and li ~ h. Thus,

u,/ul = KsRi-' (4.12)

where Ks is a constant. The energy equation for these experiments is,

a /2) (4.13)
w(C /2) = -[w (c r/2 + p'/po) + rn + q E(4.13)

where, c' = turbulent speed, p' = turbulent pressure, and e = dissipation function.







47
Now, the velocity difference is proporional to V/F and the two energy source

terms as well as the dissipation function are of order ul/h or u/l1 near the interface.

Assuming that q ~ uAb is of the same order, one again arrives at

Ue/ul ~ Ri-1

as in equation (4.12).

The shearing experiments indicate that

q ~ ulh u./h (4.14)

In the homogeneous layer near the interface, q ulbl where bl is the rms

buoyancy fluctuation. With the assumption that this correlation is of order one,

we obtain that u2/(bih) 1, thus, the kinetic energy and the available potential

energy, b1hl, are of the same order.

Long thus interprets the experiments to indicate that turbulence causes poten-

tial energy to increase at a rate proportional to the rate at which kinetic energy is

supplied to the region of the interface and not necessarily to generation at the source.

A plausible unifying argument leads to the conclusion that entrainment rates in

cases with or without shear are proportional to Ri-' defined on the buoyancy jump

and velocities and lengths characteristic of turbulence near the interface.

4.4 Narimousa, Long and Kitaigorodskii (1986)

The flume and pump section in this study were the same as used by Narimousa

and Fernando (1987). Experiments were run with two kinds of systems : a linearly

stratified system and a two layered fluid system. We will confine our discussion to

the latter which comprises of fresh water over salt water.

During entrainment, interfacial Kelvin Helmholtz instabilities and wave break-

ing were easily observed. At low Richardson numbers (Ri), turbulence caused the

disturbances to be highly irregular, however, this irregularity decreased as Ri grew

and finally internal waves developed and occupied the entire interfacial layer. These







48
instabilities (disturbances) were larger for higher pump speeds and smaller density

jumps.

These investigators attempted to find a relationship between E,, where the

entrainment velocity was scaled by the friction velocity u,, and Ri,, based on u, as

well, i.e.,

E. = ue/u, (4.15)

Ri, = hAb/u* (4.16)

Measurements were made of the mean mixed layer velocity, U, the mixed layer
depth, h, and the entrainment velocity u,, while the friction velocity u, was deduced

from the mean momentum balance equation for homogeneous turbulent shear flow.

Plots of h vs.t revealed that ue = dh/dt was constant.

4.4.1 Deduction of u,

The streamwise momentum equation for the mixed layer, the interface and a
thin layer below it where the velocity drops to zero, is


d(Uh)/dt = u w2hlW (4.17)

where u, is identified with the pressure gradient force and Reynolds stress force
accelerating the flow due to pump action and is the friction velocity of the pump;

while w, is the friction velocity of the side walls, W being the width of the side

walls. The second term accounts for the retarding action of the side walls.

The Blasius resistance formula for turbulent channel flow is,


U/w, = 8.74(Ww,/2v)1'/ (4.18)

i.e., w, = 0.15U7/8(2v/W)1/8 (4.19)
d(Uh) (Uh) ah
also, dt A at
SUdt (4.20)h
= Ucu, (4.20)









Thus,

u. = U,u, + {0.15U7/8( )1/8}2h/W (4.21)

As U,(= a ( h) was obtained from graphs, u, was easily calculated, and found to
increase very slowly with h.
The measured S = Uh, on being plotted against h had two distinct regions,
initially increasing linearly and then remaining constant, leading to the interesting

observation that U decreases with h, first slowly and then faster as the pump term

is balanced by the wall friction term. Experimentally, it was determined that U -
10.65u*.
4.4.2 Entrainment Rates Based on u,

The investigators start out with the assumption that hAb is constant, in slight
contrast to Narimousa and Fernando (1987) who had

hoAbo = hAb(1.03) (4.22)

which may be due to the fact that the region of the thin density interface was not
considered while employing the buoyancy conservation equation. This enabled them

to develop plots of E, vs. Ri,.
No simple unifying relationship was found over the entire range of Ri,, but for
subranges they found,

E, 0.65RiC1/2 15 < Ri, < 150 (4.23)

E, 7Ri-1 150 < Ri, < 800 (4.24)

E, 5Ri;3/2 Ri, > 150 (4.25)

This can be attributed to the difference in the very nature of the entrainment
process over the three ranges. Initially, the mixed layer as well as the base of

the mixed layer are fully turbulent; the high turbulent shear and weak density
jumps result in the eddies of the mixed layer directly producing entrainment; in








50
the next range, the interface becomes less chaotic and Kelvin- Helmholtz type of

instabilities occur which are less efficient in causing entrainment. Finally, due to

even lesser shear, entrainment decreases further, and this is similar to previous shear

free experiments (oscillating grid type), where Long (1978) obtained E ~ Ri7/4,

which is close to the one obtained here.

4.5 Wolanski, Asaeda and Imberger (1989)

Turbulence was generated in a plexiglass cylinder using oscillating grids along

it's walls. The cylinder was filled with a fluid mud mixture of kaolinite and tap water,

with initial concentrations always greater than 40 g/1. The grids were stopped after

fully mixing the fluid mud. A lutocline formed, separating the clear, upper layer

from a turbid bottom layer, and moved down with a constant velocity wfo, which

depends on the suspended sediment concentration. As the oscillation was started

again, the fall velocity wf reduced to less than wpo. There was no mixing for stroke

frequencies w < we, the critical frequency at which billowing and wisp formations

occurred. In this frequency range, wf/wf, decreased for increasing values of w, which

can be attributed to the break-up by turbulence of the clay flocs. At w = we, there

was active mixing across the interface, which eroded by moving downwards at a
velocity greater than wf/, thereby implying the presence of a higher intensity of

turbulence in the upper layer. However, very soon, the fall velocity decreased to

below wfo again as a balance prevailed the upward turbulent entrainment and the

downward gravity settling at the lutocline, with no more erosion of the lutocline.

For w > we, the lutocline was convoluted with large internal waves cusping into
the upper layer where the intensity of turbulence was lower due to sediment induced

dissipation. The height to which the fluid was entrained increased with increasing

stroke frequency. Also, the onset of turbulence occurred at almost the same value of

the Richardon number for all the experiments.

Thus, similar to heat and salt stratified experiments, buoyancy effects are dom-







51
inant in inhibiting mixing across the lutocline. However, an additional feature

affecting the process is the extraction of turbulent kinetic energy by the sediment

to counteract the buoyancy flux due to sediment fall velocity, which, here, causes

a collapse of turbulence in the bottom layer with resultant erosion of the lutocline

only from the top. It must be mentioned that Wolanski and Brush (1975) found, in

oscillating grid type of experiments, that the entrainment rate decreased much faster

with increasing Richardson numbers than was the case with salt or heat stratified

experiments.

The dependence of the fall velocity on the suspended sediment concentration

served to limit the height of entrainment into the top layer and also stopped lutocline

erosion after an initial period of active mixing, which is what E and Hopfinger (1987)

as well had observed.

4.6 Conclusions

The preceding review of flows with mean shear show that many investigators

have found relations of the form E oc Ri-' for salt-stratified systems, although the

range of validity of this relation varies according to the method of defining the

Richardson number (see, for instance, Appendix B). Moore and Long (1971) re-

lated the non-dimensional buoyancy flux, Q, to the Richardson number according

to a similar (Q oc Ri-') relation and showed that the entrainment coefficient, E

is equivalent to Q. Narimousa et al. (1986) arrived at different entrainment re-

lations for (three) sub-ranges of the Richardson number defined on the basis of

the friction velocity u,. The exponent in the E, oc Ri;" increased with increasing

Richardson number, thus resulting in decreasing rates of entrainment with increas-

ing Richardson number, which they attributed to the difference in the very nature of

the entrainment processes over the sub-ranges. Wolanski et al. (1989) demonstrated

that the behavior of sediment particles or aggregates is different from salt-stratified

systems because of greater dissipation of turbulent kinetic energy to counteract the







52
sediment fall velocity thereby implying a lesser rate of entrainment for sediment-

stratified systems, which is the focal point of the present investigation.














CHAPTER 5
METHODOLOGY


5.1 Apparatus

Turbulence generated by grid stirring has been the most popular mode of labo-

ratory studies of dynamics of stratified systems. In a two-layered system separated

by a density interface, the grid is placed in either layer or there might be a system of

grids placed in vertical succession and extending into both layers. It can be shown

that the grid may be replaced by a virtual source of energy at a horizontal plane,

the "action" of the source being determined by a single "action parameter" (Long

1978) having the dimensions of viscosity and proportional to the constant eddy

viscosity in the turbulent fluid above the source. However, the issue of entrain-

ment of fluid muds in estuarine situations is obviously the result of current-shear

induced turbulence. Thus, it appears more prudent and realistic to simulate this

phenomenon with a laboratory apparatus which can produce the required turbulent

kinetic energy for mixing by velocity shear.

In this respect, most previous experimenters have used flumes with annular ge-

ometries, with a rotating screen applying shear stress at the surface of the stratified

fluid within it (Kato and Phillips 1969; Kantha, Phillips and Azad 1977; Deardorff

and Willis 1982). This annular flume has the advantage of being free of end walls

and thus avoids undesirable recirculating flow (as in some previous surface shear

free experiments with salt and fresh water of Ellison and Turner 1959, and Chu

and Baddour 1984). But, this kind of arrangement seems affected by secondary cir-

culations in the radial direction (Scranton and Lindberg 1983) causing substantial

interfacial tilting. Deardorff and Yoon (1984), after an in-depth study concluded







54
that the cause for this tilting lies in the uneven angular momentum distribution

across the annulus due to the solid body rotation of the screen. The fluid possses

a mean velocity towards the outside resulting in higher entrainment rates at the

outer wall relative to the inner wall.

The experiments were carried out in a specially designed recirculating flume (see

Figures 5.1, 5.2 and 5.3) which basically consists of two sections, the pump and

the observation sections, joined together by two semi-circular annuli. This kind of

flume has been used by previous experimenters (for example, Moore and Long 1971;

Narimousa, Long and Kitaigorodskii 1986) and is free of the effects of end walls and

that of a rotating screen. Some secondary circulation is introduced in the process

of bending the flow; however, this is possibly to be minimized by the large radius

of curvature and the relatively long straight section used for observations. Here, it

might be noted that in some experiments of this general nature, secondary circula-

tions are not undesirable (even though the geometric dimensions of the apparatus

will dictate their transverse length scales), since turbulent geophysical flows also

contain them (e.g., longitudinal rolls or Langmuir cells), albeit with independent

preferred wave numbers. Although the effect of streamline curvature is not too well

understood, the effects of variation in transverse length scales does not appear to

cause substantial variation in entrainment (Scranton and Lindberg 1983).

The flume was entirely made of plexiglass to enable visualization of the flow and

other desired parameters. Except for the walls of the two semi-circular sections, the

plexiglass used was 1.25 cm thick everywhere, including the bed and the floor of the

flume. The walls of the semi-circular section were 0.32 cm thick, this merely being

expedient to afford ease of bending to the design radius of curvature. The flume

was 61 cm over the floor throughout. The entire unit was placed on a specially built

table.

A 'bed' was constructed first and placed on the table. The floor of the flume
































1.27cm Plexiglass


0.32cm /
Plexiglass


0.32cm
Plexiglass


1.27cm Plexiglass


PLAN


Figure 5.1: Recirculating flume of plexiglass used in the present investigation (di-
mensions in centimeters).

















-Spur Gears (R=8.5)


Splitter
Plate
Flume
Wall

Drive
Mechanism
Support


Figure 5.2: Section A-A of the flume, from Figure 5.1 (dimensions in centimeters).












Propeller Drive Motor

Flexible Shaft




Flume
Walls




Propeller

6 T' Splitter Plate








30.8





Floor

Bed


SECTION B-B


Figure 5.3: Section B-B of the flume, from Figure 5.1 (dimensions in centimeters).







58
was bolted and glued onto this. Next, the walls were cut, and bolted and sealed to

this floor. The walls were supported with 20 cm high and 5 cm wide sections at

periodic intervals and also connected with brackets at the top. Joints were sealed

with gussets and rubber to prevent any leakage.

The required turbulent shear flow was obtained by using a disk pump (see

Figure 5.4), first introduced by Odell and Kovasznay (1971), to selectively drive the

upper fresh water layer over the quiescent fluid mud.

The width of the flume was 10 cm everywhere, except at the pump section.

The disks of the pump were between walls 47.5 cm apart. To maintain a constant

cross-section of flow as much as possible, flow seperator sections (triangular in plan,

of dimensions 73.7 x 73.7 x 31.8 cm) were placed both up and downstream of the

pump, which created two channels of 5 cm width each on either side of the pump.
These channels guide and blend the flow into the semi- circular section in front of

the pump, while upstream of the pump, these split and guide the flow onto the

pump. The flow seperators were of the same height as the flume, i.e., 61 cm. Also,

the two flow seperator sections were connected by a weighted wooden box of height

30.8 cm which served to maintain the required 5 cm width. The hole of the intake

valve for fluid mud was directly under a transverse slot (of dimensions 38 x 13 x 3

cm) at the bottom of this box. The entire pump section had vertical supports at

every 40 cm.

The radius of curvature of the curved section was 51 cm. Here, supports were

put at closely spaced intervals of 25 cm, to take into account the additional stresses

due to bending. The observation section was completely straight and 200 cm long.

Small holes were drilled into the outer wall of the observation section, near the

entrance (in the flow direction) to the observation section and thin, flexible pipes

were inserted into them, taking care that they did not intrude into the interior of

the flume. The other ends of these pipes were closed with metal clips. These holes

















IV-26-- I=


0.325
0.s.






















Nut Nut



All dimensions in centimeters
Vertical dimensions are greatly
exaggerated.


Figure 5.4: Details of the disk pump system used in the present investigation (di-
mensions in centimeters).







60
were at closely spaced vertical intervals-1-4 cm apart. This was done to extract

samples of fluid mud for the estimation of concentration profiles (as a function of

time). At the center of the outer wall of the observation section, a 38 x 56 cm size

grid of 2 cm mesh on a transparency was pasted to record the rate of progress of

injected dye-lines. The entire flume had brackets at the top, every 10 cm, to tie the

walls together.

A Sony Betamax video recording system was set up about 1 m from the obser-

vation section of the flume such that the line of vision of the camera was normal to

the sidewall of this section. The camera was focused onto the grid at the center of

the wall. Two powerful (1000 W) lamps were set behind the camera to provide the

requisite illumination for recording.

The reason for a disk pump was that it could produce quite homogeneous hor-

izontal streaming of the flow. The pump imparted only a horizontal component of

velocity to the fluid. The disks of the disk pump were of two different diameters,

8 and 26 cm, which were alternately stacked on each of the shafts. The larger

disks were 0.325 cm thick, while the smaller ones were 0.65 cm thick. These shafts

were so positioned that a large disk of one shaft meshed with the smaller of the

other and so on. Thus, the two stacks meshed, leaving almost no space in between

them, but creating gaps at the outer edges, between the larger disks. When the

two shafts were driven in opposite directions by a 1/8th h.p. Dayton Permament

Magnet Gearmotor (F/L rpm 50, F/L torque 130 inch- pounds) via a.chain and

sprockets arrangement, fluid was pulled around the outer channels by the viscous

drag of the larger disks and ejected as horizontal jets from within these gaps. The

disks were sand-blasted to improve surface roughness to increase the efficiency of

the pump by increasing drag. With the disk pump in place, the bottom-most disk

of the pump was just above the elevation of the splitter plate (described next).

Preliminary calibration tests were performed with a homogeneous fluid to test








61
the range of velocities obtainable with this pump. As the maximum mean velocity

obtained was only about 9 cm/s, it was decided to augment the velocity with the

assistance of a screw propeller. This propeller was placed (the axis of the propeller

was ~ 2 cm above the level of the splitter plate which is described below) in the

curved section downstream of the pump and before the entrance to the observation

section (Section B-B, Figure 5.3), and driven by a motor placed outside via a flexible

shaft. The result was quite satisfactory with the maximum obtainable velocity with

the two-layered system of fluid mud and water in place being 14 cm/s. Also,

the horizontal homogeneity of the flow was not disturbed. To impart additional

horizontal homogeniety to the streamlines, a sidewall-to-sidewall thin metal splitter

plate was constructed in a horizontal plane 30.8 cms from the floor and this splitter

plate covered the entire pump section (which is the only region with a width greater

than 10 cm and having the flow seperators and the disk pump) and the entire curved

section downstream of the pump. This also served to prevent any suction effects

either due to the disk pump or the propeller (which together are referred as the

pump system hereafter) from affecting into the fluid mud below.

5.2 Procedure

Each experimental Run was divided into intervals of ~ 8-20 (generally ~ 10)

minutes each. At the end of each interval, the required measurements were "in-

stantaneously" made, and these were considered to be the representative conditions

for that interval. It must be mentioned is that in the case of experiments with

salt-stratified systems, the sole cause for the deepening of the mixed-layer is tur-

bulent entrainment across the density interface. In a velocity-sheared two-layered

system of fluid mud and water, the rate of propagation of the visual density inter-

face (i.e., the rate of change of depth of the mixed-layer) is not the result of vertical

entrainment of fluid mud alone, but is also due to the settling characteristic of fluid

mud below the level of the interface. Thus, in this case, one cannot easily quantify







62
turbulent entrainment in terms of changes of the mixed- layer depth. Therefore, a

more direct approach was adopted. The most basic effect of the turbulent kinetic

energy of the system is mass/buoyancy transfer across the interface. It therefore

appears to be the most logical quantity to measure and relate to a suitably defined

Richardson number.

From an estimate of the initial depth of the mixed layer, the flume was first filled

with the requisite pre-determined height of tap water. Two types of test sediment

were considered-kaolinite and bentonite (see Appendix A), with the objective of de-

termining the effect of varying degrees of cohesion on entrainment rates. Bentonite,

a montmorillonitic clay, is highly cohesive (and thixotropic) whereas kaolinite is not

as cohesive and properties of bentonite aggregates are not as uniform as that of

kaolinite. Sediment was well-mixed with tap water (for composition of tap water,

see Dixit 1982) in a vertical, steel cylinder of 77 cm diameter with the aid of a

Ingersoll-Rand two-stage 10 h.p. air compressor with a maximum discharge rate of

14.0 kg/cm2. The compressed air was introduced into this vertical mixing tank at a

high flow rate through tiny holes in a T-shaped PVC pipe section placed at the bot-

tom of the cylinder. This agitation was continued long enough until the fluid mud

was well mixed and quite homogeneous. In the case of kaolinite, the sediment-water

mixture was thoroughly agitated for at least an hour which provided quite "homo-

geneous" mixing, while bentonite was not as tractable in this respect. Bentonite,

which is highly thixotropic, formed lumps with a wide range of sizes (of upto 20

cm diameter) even when the sediment was introduced at a slow rate into agitated

tap-water. These lumps were dry inside although covered by a wet "skin". The

mixture was allowed to equilibrate for ~ 5 days with periodic agitation (upto ~ 6

hours every day) before a fairly uniform, workable mixture resulted.

The well-mixed fluid mud was instantaneously pumped into a horizontal cylin-

drical feeder tank above the elevation of the flume. This fluid mud was then in-







63
produced into the flume through the intake valve at the flume bottom. The mud
entered the flume with a vertical (upward) velocity at the position of the slot at

the bottom of the wooden box and on encountering the wooden obstruction turned

at right angles and flowed horizontally into the bottom of the flume, displacing the

lighter tap water upwards. With all stops open, the filling rate was about 2.5 cms

per minute per unit area of the flume. The time required to fill the flume with the

requisite volume of fluid mud was ~ 15 minutes. In the first three experimental

runs, the filling rate was slightly slower, while in the remaining runs the resulting

fluid mud layer underneath water was essentially homogeneous initially ( except

very near the bottom). The interface was always positioned so that it was just

under the level of the splitter plate such that the internal boundary to the diffusion

of momentum (Narimousa and Fernando 1987) formed by the interface, and the

physical boundary of the horizontal splitter plate would be almost continuous. In

all the runs, the method of filling fluid mud under water always resulted in the

formation of a diffuse intermediate layer (of thickness ~ 5-7 cms) just above the

interface. The density of this layer was found to be minimal (~ 10-5 g/cm3) and

this layer completely eroded within 1-1.5 minutes after starting the run.

As soon as the two-layered system was in place, the depth of the mixed layer, h,

was noted, and samples (N 10 cm3 each) of fluid mud, at discrete vertical intervals
(~ 5 cms), were withdrawn via the flexible tubes in the outer wall of the observation

section of the flume to obtain the initial concentration profile (across depth). These

samples were directly withdrawn into small (capacity ~ 60 cmS each), clean glass

bottles which were then tightly capped. The elevation of the position (from the

bottom of the flume) from which the sample had been taken was marked on the
corresponding bottle. The time was also noted. As the instantaneous concentra-

tion profile was required, it was not considered expedient to spend more than 1-1.5

minutes for sampling, by which the number of samples was limited to a maximum







64
of six each time. The video recording system was turned on (to record the entire

experiment) and the experiment was begun by starting the pump system (the disk

pump and the propeller were started simultaneously) to rotate at a predetermined

rotation rate (which, in conjunction with varying buoyancy jumps across the inter-

face, provided a wide range of Richardson numbers N 4 32). The rotation rate of

the disk pump was always at the maximum, while the rotation rate of the propeller

was adjusted such that the pump system could produce the desired predecided mean

initial velocity in the mixed-layer.

After the experimental run was in progress, with velocity- shear causing fluid

mud entrainment across the density interface through massive undulations convo-

luting the interface, sets of samples, for gravimetric analysis, were systematically

withdrawn at discrete time intervals (~ 10 minutes). Consecutively, dye lines were

also injected to get the corresponding velocity profile for that interval. The depth

of the mixed layer was noted.

Dye-lines (of diluted rhodomine such that it would be almost neutrally buoy-
ant in water) were injected into the flow and their movement across the grid was

recorded by the video camera. A syringe with a long needle (~ 35 cm) was used

for this purpose. The needle was introduced vertically into the observation section

through a slot in a bracket tying the sidewalls together at the top. The needle was
aligned with the upstream vertical edge of the grid and it's end was well within the

lower layer of fluid mud. The plunger was depressed and the spewing needle was

"instantaneously" pulled out leaving a clearly visible dye-line (which became diffuse

with downstream progress). The velocity profile could be easily determined by mea-

suring the rate of downstream progress of this injected dye-line. For each interval,

dye-lines were injected at least twice (and frequently three times) and averaged to

get a more accurate velocity profile. A problem which could not be circumvented

was that, below the level of the visual density interface, the turbidity of fluid mud







65
prevented visualization of the dye-line. However, Narimousa and Fernando (1987),

using a similar flume and pump system, found that the velocity rapidly decreased

to a very insignificant value at the interior of a density interface of finite thickness,

6 ~ 0.06x (depth of mixed layer). Thus, the contribution of this portion to the

mean overall velocity was assumed to be negligible. In the present investigation

as well, visual observation seemed to be in conformance with this argument. The

resulting velocity profile was integrated, and knowing the depth of the mixed layer,

the representative mean velocity for the interval could be obtained. Temerature

recordings of the mixed-layer were also made throughout the course of some of the

runs which showed that the increase of temperature of the mixed-layer by the end

of a run was not more than 2 OC (mean temperature was ~ 170C.

The recording of the experiment (on the video recording system) was played

back to obtain the rate of progress of injected dye-lines. The representative velocity

distribution was thus obtained. This was drawn on a graph-paper to measure the

area which further gave the mean representative velocity for each interval. The

point of inflection of the velocity profiles (see Figure 3.2) were also noted as the

velocity-gradient is responsible for the shear production causing entrainment. The

vertical distance of this point from the interface was designated 6,.

For the purpose of gravimetric analysis, Millipore Filtering System was used in

conjunction with a small, vacuum pump (which could produce a vacuum of upto

65 cms of Hg). Millispore filters (Filter Type HA, Pore Size 0.45 pm) were first

dried in an oven at a temperature of 50 oC for at least 3 hours. These were then

removed from the oven and allowed to equilibrate in a room (whose temperature

and relative humidity were monitored with an air-conditioning unit) for a minimum

of 8 hours. These filters were then weighed in the same room on a Mettler balance

(Type H80) which was accurate upto 1 mg. These pre-weighed filters were then used
to dewater known volumes of sediment samples. In the case of bentonite, the sample







66
volume that could be used for this process of dewatering was only 0.5 cm3 as the

filters got clogged with the sediment particles for greater volumes of fluid mud. To

improve accuracy in obtaining concentration profiles for bentonite, this procedure of

dewatering was done for at least three sub-samples for each base sample withdrawn

from any elevation of the flume at any time, and these were averaged.

The filtrate was allowed to remain on the paper which was then heated in the

oven again (at 50 OC for at least 6 hours) to remove the last vestiges of water.

The dried filter paper with dry sediment on it was again equilibrated in the same

monitored room and then weighed on the Mettler balance from which the mass of

sediment in a known volume of sample was easily obtained. This procedure was

carried out for all the samples, and, thus, the concentration profile of fluid mud was

known for each interval. Knowing the depth of the fluid mud, the mass flux (and

hence, the buoyancy flux) across the interface could be calculated.














CHAPTER 6
RESULTS AND ANALYSIS


6.1 Definition of Richardson Number

Vertical mixing across a density interface is dependent on the local Richardson

number (Turner 1986), e.g., the gradient Richardson number across the interface

in terms of the velocity and density differences across the interface. However, mea-

surement of the local Richardson number is generally difficult (the thickness of the

interface needs to be determined) and a common procedure is to define an over-

all Richardson number. The most suitable definition in the present case as well is

such an overall Richardson number in terms of the depth of the mixed layer and

the buoyancy jump across the interface. The depth of the mixed layer controls the
length-scale of the energy-containing eddies, with the interface acting as an internal
boundary. Regarding the velocity- scale, most of the previous researchers tend to

identify with the friction velocity, u,. In flume experiments without rotating screens,

it can be seen that most of the turbulence is produced at the density interface and
the side-walls (Narimousa and Fernando 1987). However, in wall bounded flows,

most of the sidewall induced turbulence dissipates near the walls itself and only
a small portion diffuses outwards (Hinze 1975, p. 648). This is also confirmed by

Jones and Mulhearn (1983). Thus, most of the energy required for turbulent mixing

is a direct result of shear production at the interface and the most important scaling
velocity should be the velocity difference between the two layers, AU (e.g., Ellison

and Turner 1959, Lofquist 1960, Moore and Long 1971). In the present case, the

velocity of fluid mud at and below the level of the density interface was considered

negligible (although it could not be expressly measured, visual observations seemed







68
to confirm the fact) as in Narimousa and Fernado (1987). Thus, the mean velocity

of the mixed-layer was taken as the most representative velocity scale. With this,

the Richardson number is defined as

hAb
Ri, = g9hA (6.1)

with the interfacial buoyancy jump being

P2 P1
Ab = gP2 l (6.2)
PI
where #1 is the mean mixed-layer density and p2 is the density of fluid mud at the

level of the density interface.

6.2 Initial Conditions

The initial conditions for all the experimental Runs are listed in Table 6.1 (for

a physical description of the flow configuration, refer to Figure 3.2). The associated

terminology is as follows:

Ms = mass per unit area of fluid mud
h = depth of mixed layer
H = depth of fluid mud
C2 = mean concentration of fluid mud
C2 = concentration of fluid mud at the level of the interface
ii = mean velocity of the mixed layer
Ab = buoyancy step across the interface

The subscript 0 denotes initial conditions. It must be noted that in Runs 4-10,

the pump system was kept at some fixed (by not altering the speed controls) rotation

rate (the rate was tuned such that a pre- determined mean velocity could be achieved

in the mixed-layer) throughout the course of the each run, the velocity profiles

being allowed to evolve with time, while the speed settings of the pump system

(specifically, only the propeller) was varied during the course of the experiment

for the remaining runs. The initial values of the mean velocity were in the range

7.4-13.1 cm/s. However, in the first three runs, mean velocity values even exceeded

these initial values as the rotation rate of the pump system was increased. In Runs





























Table 6.1: Initial conditions of all Runs
RUN (M2)o ho Ho (C2)0 (C2)0 (U)o (Ab)o
NUMBER g/cm2 cm cm g/l g/l cm/s cm/s2
1 1.1685 32.2 19.8 59.0 29.0 7.4 17.7
2 3.0625 28.8 26.5 115.6 94.5 7.5 57.7
3 3.2375 26.0 25.0 129.5 110.0 11.8 67.2
4 2.5410 26.5 29.5 86.1 76.0 13.1 46.4
5 2.3450 25.3 27.7 84.7 76.0 11.9 46.4
6 1.6125 22.2 28.5 56.6 35.0 9.5 21.4
7 1.2450 28.0 27.0 46.1 30.0 11.0 18.3
8 1.9918 25.0 31.2 63.8 50.0 13.0 30.5
9 2.2350 23.6 28.2 79.3 62.0 9.6 37.9
10 1.0154 24.5 29.5 34.4 30.5 9.2 18.6
11 1.0800 23.6 28.0 38.6 28.0 9.9 17.1







70
1-9, kaolinite was the constituent sediment of fluid mud while the fluid mud was

of bentonite for Runs 10 and 11. The initial mean concentration of fluid mud was

in the range 45-130 g/1 which corresponded to bulk density range of ~ 1.03-1.08

g/cm3. The upper limit of this range was imposed by the performance capabilities

of the pump system so as to obtain reasonable (for which entrainment was possible)

values of the Richardson number In the case of bentonite, higher values of mean

initial concentration could not be used because of difficulty in obtaining a fairly

uniform, well-mixed suspension.

6.3 Evolution of Characteristic Profiles

A typical time-evolution of the concentration profile below the level of the den-

sity interface is shown in Figure 6.1. The data are for Run 9 with kaolinite as the

constituent sediment of fluid mud. Initially, i.e. at t = 0, the fluid mud was essen-

tially quite well-mixed (with generally mild lutoclines) with obviously the steepest

gradient at the interface. Although the interface is shown to have an infinite gradi-

ent, it is a well-known fact that in similar and geophysical situations, the interface

is a region of thin but finite thickness (of the order of 1/20th the thickness of the

mixed layer) with a steep density gradient, see Narimousa and Fernando (1987).

The settling characteristic of the suspension caused a lutocline to develop for about

5 cm directly below this interface. The bottom 8 cm show a slightly steep lutocline

as well, which might be due to settling. With the passage of time, the interface

sharpens in the sense that the lutocline below it disappears. The concentration

of fluid mud at the level of the interface generally increases with time (except for

the profile at t = 21 minutes, which could be due to the local settling rate being

more than the rate of scour of the interface due to entrainment, temporarily). The

bulk concentration of the fluid mud always increased with time as the mud settled.

The major lutocline progressively steepened. It must also be noted that the mean

concentration of the mixed layer is simultaneously increasingly as well. However, as






















50.0
Symbol t IF U
(minutes) (cm) (g/I)
0 0 28.2 79
o 9 23 95
S21 19.7 101
40.0 31 16.8 114
SA 44 14.8 121
E U 59 13 136
0
O.
0
1 30.0
OU

m

20.0




10.0




0.0 1 1 t li t I
40 60 80 100 120 140 160

CONCENTRATION (g/l)








Figure 6.1: Sequence of concentration profiles of Run 9 with kaolinite depicting the
evolution of concentration with time. IF denotes interface.







72
will be seen later, the buoyancy jump across the interface generally increased with
time (except for the second interval t = 9 to t = 21 minutes, when it appeared to

decrease). Entrainment progressed until the buoyancy jump became strong enough

to overcome the excess (after dissipation) turbulent kinetic-energy which tended to

increase the potential energy of the system by causing entranment, at which point

the entrainment apparently decreased a lot.

Figure 6.2 shows the typical evolution of velocity profiles in the mixed layer for

Run 6 (with kaolinite). Initially, the profile was homogeneous without any gradient

at all in the mixed-layer, i.e. an apparent step velocity profile resulting in the case

of a vortex sheet discussed in Section 2.2.1. There was much entrainment at these
earliest times with massive convolutions covering the entire extent of the interface.

Closer examination revealed that the (thin, but finite) interface might itself be tur-

bulent at these times. Initially, the mean velocity of the mixed-layer increased very

rapidly with time as the inertia of the system was being overcome. This generally
took between 3 to 4 minutes, by which time the mean velocity peaked. Next, the
mean velocity of the mixed-layer slowly decreased with time which might be due to

three reasons : (1) with the passage of time during the course of a run, with en-

trainment (and settling), the elevation of the interface decreased, and progressively

more and more volume of fluid was being driven (considering the interface to act
as an internal boundary to the diffusion of momentum) by the pump system which

had a constant energy input; however, calculations to check conservation of mass

(ht) and momentum (h]2) for each run revealed discrepencies indicating that more

accurate measurements of velocity profiles need to be made if these quantities (mass

and momentum for each run) need to be accurately estimated, (2) sidewall friction

may not always be negligible, and (3) as the mixed-layer concentration increased

with time (due to mass flux into it), there was consequently increasing dissipation of

turbulent kinetic energy in the mixed-layer to counteract the downward buoyancy





















60.0
Symbol t IF U
(minutes) (cm) (cm/s)
A 2 26 6.6
0 8 24 9.9 .
50.0 18 21.3 8.6
0 28 19.2 7.8
0 48 16.7 7.8


40.0
0
0
O-
UL
> 30.0
0



.2 20.0



10.0-



0.0 I
0.0 4.0 8.0 12.0
VELOCITY, (cm/s)







Figure 6.2: Evolution of the velocity profile in the mixed-layer for Run 6 with
kaolinite. IF denotes interface.








74
flux due to the sediment particle's fall velocity (see also Wolanski et al. 1989). This

can be considered in terms of the energy equation (see Abraham 1988)

dK --, 9 g ,
dK= u7- -wp E (6.3)
dt 8z p

with the primes denoting turbulent fluctuations, K the turbulent kinetic energy and

E the dissipation function. Assuming the turbulence to be in local balance, diffusive

transport is neglected and dK = 0. Thus,

uw' = p +e (6.4)
dz p
whence the production term of kinetic energy is balanced by the buoyancy term

(which is the conversion of input energy into the potential energy of the system) and

the dissipation function. Hence, at constant input of kinetic energy, as the buoyancy

term decreases, the dissipation of energy increases. Thus, with the passage of time,

at fixed input of energy due to the pump system, the available energy to effect

entrainment decreased. Visually, this resulted in decreased amplitudes of the waves

at the interface. Figure 6.3 shows that the mixed-layer depth (for Run 10 with

bentonite as the constituent sediment of fluid mud) increased very rapidly with

time initially, but slowed down after ~ 20 minutes. Figure 6.4 is more illustraive

as it plots the rate of change of the mixed-layer depth against time (obtained by

differentiating the curve fitted in Figure 6.3). As expected, the curve asymptotes

towards zero after about -25 minutes.

6.4 Shear Layer

The shear layer is obviously very important as it is directly responsible for

overcoming the static stability of the two-layer system (of fluid mud and water) and

causing mixing. The vertical distance, between the point where the velocity profile

deviates from the vertical in the mixed layer to the the level of the interface, was

taken to be the thickness of the shear layer, 6, (refer to Figure 3.2) (although the

velocity may decrease to zero inside the thin interface). At the start of each run

















































0 10 20 30 40 50
TIME (minutes)


60 70 80 90


Figure 6.3: Change in the mixed-layer depth with time for Run 10 with bentonite.


W
0




.
w


W
w

















3.5 -


3.0


2.5


S2.0
E

S 1.5


1.0


0.5


0 I I I
0 20 40 60 80 100

TIME (minutes)


Figure 6.4: Rate of change of mixed-layer depth in Run 10 with bentonite.







77
when, initially, the upper mixed layer was equilibrating to the energy input of the

pump system, the interface appeared to be in turbulent motion, but the velocity

below the level of the visual interface could not be determined. Above it, the mixed

layer was fully turbulent and dye injection showed an apparent step velocity profile

with the vortex sheet at the interface. Actually, as mentioned before, it might be

more realistic to assume the thin density interface (Narimousa and Fernando 1987)

to have a steep density gradient. This appears to be most plausible as the interface,

at those times, was convulsed by massive undulations of heights of the order

of ~ 6-8 cms causing much mixing. The effect of the splitter plate in causing

additional entrainment was also visible. Thus, the first interval of each run was not

considered while plotting data in Figure 6.5 where the data are from all the runs

(i.e., for both kaolinite and bentonite) have been included. Unfortunately, the data

are quite scattered for any definitive conclusions to be made. The values of 6, range

from 0.18 0.34 h. To avoid illusionary appearances on account of disparate

scales of the axes, the same was also plotted on a log- log scale (see Figure 6.6)

which indicates that the 6, may be ~ 0.23h for 4 < Ri, < 20 with a slight increase

beyond 20. Long (1973) reported that Moore and Long's (1971) data indicated

6,/h ~ Ri'0-5 while Narimousa and Fernando (1987) found the non-dimensional

shear layer thickness to be independent of Richardson number and about 0.2. In

the present case, the scatter of data may not actually be very surprising, as (1) the

diffusion of momentum into and maybe even below the level of the interface may

not always have been totally negligible, and (2) the conditions for two different runs

were not exactly duplicated for the same Richardson number, e.g., a value of Riu

= 10 may have been obtained in the second time interval of a run while it may

have occurred in the later intervals of the other. There will be greater dissipation

of kinetic energy in the second case in trying to counteract the settling tendency

of more sediment particles, as the concentration of the mixed layer increases with



















0.35
O




6 0.30




0
-
* 0.25 0 0

0o o0
So00 o o0
S O 0 0
< O GD O O
S0.20 IO 0 0
00 O
C 0 0

Lu
-I

2 0.15

Z-
0


0.10 i i I I 'I t I I I -

0.0 10.0 20.0 30.0 40.0

RICHARDSON NUMBER, Riu


Figure 6.5: Non-dimensional shear layer thickness vs. Richardson number





















10
u6
U)





r)
UJi


w






0
uJ










-1
cc


RICHARDSON NUMBER, Riu







Figure 6.6: Non-dimensional shear layer thickness vs. Richardson number on a
log-log scale


10 102








80
time. Thus, it appears that the thickness of the shear layer increased with increasing

Richardson number, unlike in Moore and Long (1971). More accurate methods of

measurement are required for velocity profiles before any definitive conclusions can

be reached.

6.5 Observations on the Interface

When the pump section was turned on to start the experiment, the entrain-

ment process started out with turbulent entrainment of the diffuse intermediate

layer which formed when the fluid mud was introduced under the water layer while

setting up the two-layer system. Figure 6.7 was taken within half a minute of start-

ing the pump system. The grid squares are 2 times 2 cm. This intermediate layer

eroded completely within 1-1.5 minutes. Although the contribution to the density

of the mixed layer was minimal, there was a significant contribution to the turbidity

of the mixed layer, thereby rendering it opaque and obstructing visibility. Small

amounts of dye were injected and this dye stained the mixed-layer as it moved

around the flume and gave a color contrast with respect to the fluid mud layer. Ini-

tially, the effect of the splitter plate in producing additional vorticity at the entrance

to the observation section was quite pronounced and was visible as deepening of the

interface there (this tilt was generally perceptible for about the first 8-10 minutes).

Mixing was caused by massive internal waves (upto 8 cm wave height in the up-

stream portion of the observation section) breaking, as the steep velocity gradient

in the thin interface caused significant scour (of the interface). Figures 6.8 and 6.9

were taken back-to-back in the same run at Ri, ~ 7. Wave heights were about

4 cm. The interface was highly irregular and action of eddies causing entrainment

is visible towards the left of the photographs. Entrainment due to internal wave

breaking seemed to cause most of the mass flux at Ri, < 10. Another mechanism of

entrainment, evident for Riu > 10, was as seen in the sequence of Figures 6.10, 6.11,

6.12 and 6.13. Figure 6.10 shows the highly irregular interface with eddies scour-
































t o
t t


Vt

I- t- -
t -


-4
UPt-)-


Figure 6.7: Turbulent entrainment at t ~ 0.5 minute. Sediment- kaolinite.


ItILM XLF


Figure 6.8: Interface at Ri, < 10. Sediment-kaolinite.































4 1'


Figure 6.9: Interface at Ri, < 10. Sediment-kaolinite.


Figure 6.10: Highly irregular interface at Riu > 10. Sediment- kaolinite.




































Figure 6.11: Scour of growing crest at Ri, > 10. Sediment- kaolinite.


Figure 6.12: Scour of grown crest at Ri, > 10. Sediment- kaolinite.









84
ing the interface. Some of the undulations formed pronounced crests which grew
in amplitude and sharpened with entrainment due to eddies mostly scouring their

backs and tips (Figures 6.11 and 6.12). The remaining portion of the crest then

subsided back towards the interface (Figure 6.13). When the same mechanism was

active at slightly lower Richardson numbers (- 10-15), after the wave sharpened at

the crest, instead of breaking to form an eddy, this crest suddenly disappeared with

a thin of fluid being 'ejected' from the tip (see Figure 6.14). It appears possible

that the original undulations were caused by eddies from the mixed layer scouring

the interface, with it's 'roller action' causing crest growth and entrainment across

(the crest). When the eddy was strong enough, it could shear off the crest. These

phenomena of cusping into the upper (mixed) layer and appearance of 'smoke-like

wisps' from these crests cusping into the faster layer seemed to indicate the ex-

istence of Holmboe (mode 2) type of instabilities which was discussed earlier in

Section 3.1 and 3.2. Referring again to this review, this appears to be feasible as

6, > 6 and the levels of stratification attained were always quite high (as compared

to the lower, Ri < 3, similarly defined Richardson numbers obtained in experiments

with salt-stratified systems). When the Richardson number still increased (beyond

~ 25), the interface was convoluted with smaller (less than 1 cm) disturbances (see

Figure 6.15) which appeared to be slightly more regular.

6.6 Entrainment Rate

Table 6.2 documents the parameters that were measured during the course of

each run for all the runs with kaolinite as the constituent of fluid mud (i.e., Runs

1-9). Table 6.3 does the same for runs with bentonite (i.e., Runs 10 and 11).

The listed parameters are the representative ones for each time interval of each

run. Table 6.4 contains the calculated parameters which further lead to the non-

dimensional buoyancy flux and Richardson number for each interval of each run

with fluid mud of kaolinite. Reynolds number (Re) calculated according to Re =


































Figure 6.13: Subsiding crest at Ri, > 10. Sediment-kaolinite.


Figure 6.14: Smoke-like wisp being ejected from the tip of disturbances. Sedi-
ment-kaolinite.




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs