Hydraulic Modft,T,Ing of Circulation in Reservoirs

HYDRAULIC MODFT,T,ING OF CIRCULATION IN RESERVOIRS

An experimental study of jet-induced circulation in water-supply reservoirs.

Stephen John Robinson B. Sc. (Eng.) London.

Thesis submitted to the University of London for the Degree of Master of Philosophy in the Faculty of Engineering, and to Imperial College of Science and Technology for the Diploma of Membership.

Submitted March, 1979 ABSTRACT

HYDRAULIC MODELLING OF CIRCULATION IN RESERVOIRS BY S. J. ROBINSON

At certain times of the year large volume water supply reservoirs can become thermally stratified due to an increase in solar energy and a decrease in the natural mixing and circulation induced by wind action at the water surface. Since thermal stratification can cause serious water quality problems artificial circulation and mixing is often induced by introducing the inflow through nozzles which form momentum jets. This thesis describes an experimental study of jet-forced circulation in reservoirs.

Since it is virtually impossible to study prototype reservoir circulation, hydraulic models are often used to investigate this phenomenon. A major problem encountered when using hydraulic reservoir models is the measurement of the extremely low water velocities which occur. This difficulty has been overcome in this research by the development of a simple method in which photogrammetry is used to determine the deflection (and hence the velocity) of a:number of tethered buoys.

The velocity measuring technique has been used to study the circulation in several model reservoirs by the determination of the velocity fields and circulation parameters for various geometric and dynamic parameters. The aspect ratio has been shown to be very important for reservoir circulation suggesting that the use of vertically exaggerated reservoir models could produce misleading results. TABLE OF CONTENTS

PAGE

ABSTRACT i

TABLE OF CONTENTS ii

LIST OF FIGURES vi

LIST OF TABLES xiv

LIST OF SYMBOLS xv

ACKNOWLEDGEMENTS xx

CHAPTER 1 STATEMENT OF PROBLEM 1 1.1 INTRODUCTION 2

1.2 WATER QUALITY IN RESERVOIRS - TEE PROBLEMS 5

1.2.1 Algal Growth 5

1.2.2 Thermal Stratification 6

1.2.3 Stagnation 9

1.2.4 Short-Circuiting 10 1.3 WATER QUALITY CONTROL - REMEDIAL METHODS 13

1.3.1 Selective Withdrawal 13

1.3.2 Artificial Destratification by Vertical

Mixing 14

1.4 WATER QUALITY CONTROL - PREVENTIVE METHODS 17

1.4.1 Multi-Inlet Systems 17

1.4.2 Momentum Jet Inlets 17

1.5 BACKGROUND TO THE USE OF JET INLETS 21

1.5.1 Early Research 21

1.5.2 Current Use 21

1.5.3 Economic Considerations 24

1.5.4 Further Investigation 24

CHAPTER 2 RESERVOIR AND ASSOCIATED FLOW-LifERATURE

SURVEY 25

INTRODUCTION 26

2.1 RESERVOIR FLOW SYSTEMS 27

2.1.1 Laboratory and Mathematical Studies 27 2.1.1.1 The Generation of Circulation by

Throughflow 27 2.1.1.2 The Generation of Circulation by Wind 33 2.1.1.3 Other Studies 34 2.1.2 Field Studies 36 2.2 ANALAGOUS FLOW SYSTEMS 47 2.2.1 Lake Flows 47 2.2.1.1 The Generation of Circulation by Wind 47 2.2.1.2 Laboratory and Mathematical Studies 52 2.2.1.3 Field Studies of lakes 5.9 2.2.2 Other Studies 60 2.3 THE NECESSITY FOR RESEARCH 63

CHAPTER 3 THE HYDRAULIC TURNTABLE MODEL 65 INTRODUCTION 66 3.1 MODEL ROTATION 69 3.1.1 Ring Beam and Rollers 69 3.1.2 Levelling of Turntable 70 3.1.3 Model Bed and Basin 73 3.1.4 Turntable Drive Unit 75 3.2 PUMPED SUPPLY 80 3.2.1 Water Circuit 80 3.2.2 Flow Measurement and Control 80 3.2.3 Inlet and Outlet 84 3.3 AIR-PRESSURED SUPPLY 87 3.4 PROVISIONS FOR INSTRUMENTATION 88 3.4.1 Photography 88

iii

3.4.2 Grid 90

3.4.3 Photogrammetric Control and Survey 91 CHAPTER 4 THE VELOCITY MEASURING TECHNIQUE 92 INTRODUCTION 93

4.1 PREVIOUS METHODS USED 94 4.2 THE TETHERED SPHERE METHOD OF S'1'tAN AND • SCHIEBE 96

4.3 AN ALTERNATIVE BUOY 99 4.4 POLYETHYLENE BUOY - DESIGN AND CALIBRATION 102 4.5 POLYETHYLENE BUOY - THEORETICAL ANALYSIS 110 4.5.1 General Analysis 110 4.5.2 Deflection-Velocity Relationship for the

particular Buoy geometry 116 4.6 MEASUREMENT OF THE BUOY DEFLECTION IN THE HYDRAULIC MODEL 121 4.6.1 The Photography 121 4.6.2 Photogrammetric Control 125 4.6.3 Interpretation of negative observations 131 4.6.4 Photogrammetric Errors 134 4.7 TEE COMPLETE PROCESS 148 CHAPTER 5 THE JET-INDUCED CIRCULATION IN SEVERAL

MODEL RESERVOIRS 150 INTRODUCTION 151 5.1 DIMENSIONAL ANALYSIS OF RESERVOIR CIRCULATION 152 5.2 A DYNAMIC RESERVOIR MODEL 158 5.2.1 Dynamic Similarity 158 5.2.2 Model Considerations 162 5.2.3 A Circular Reservoir Model 166

iv 5.3 INVESTIGATIONS OF A CIRCULAR RESERVOIR MODEL 169 5.3.1 Tangential Jet and Central Outlet Model 171 5.3.2 Radial Jet and Diametric Outlet Model 190 5.3.3 Asymmetric Jet and Outlet Model 205 5.3.4 Comparison of the three types of circulation 209 5.3.5 Unsteady Flows 212

5.3.5.1 Short Term Stability 212 5.3.5.2 Long Term Stability 215 5.3.5.3 Decay of a Velocity Field 216 5.3.6 The Influence of the Earth's Coriolis Acceleration 220 5.3.7 Influence of Bed Topography 225 5.4 INVESTIGATION OF A MODEL OF TURRIFF RESERVOIR 228

5.5 INVESTIGATION OF A MODEL OF RESERVOIR 4 (ALI, HEDSRS AND WHITTINGTON, 1978a) 235 CHAPTER 6 CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH 251 6.1 CONCLUSIONS 252 6.2 SUGGESTIONS FOR FURTHER RESEARCH 262 APPENDICES A VELOCITY FIELDS'FOR THE RESERVOIR MODELS 265 B THE SPIN-UP TIME OF A SHALLOW CIRCULAR BASIN 330

C PHOTOGRAMMETRIC CONTROL DATA 338 D COMPUTER PROGRAM USED TO DETERMINE THE

VELOCITY FIELDS 342

REFERENCES 352

V LIST OF FIGURES

FIGURE PAGE 1-1 (a)- Section of a lake or reservoir during summer stratification. 8 (b)Section of a lake or reservoir with full

circulation and mixing during autumn.

(c)Typical summer stratification in a lake (Thompson, 1954).

1-2 Schematic diagram of momentum jet action 19 1-3 The inlet arrangement at Queen Mother Reservoir,

Thames Water Authority. 23

2-1 Summer Energetics of King George VI and Queen

Elizabeth II Reservoirs during March-August, 1965,

(Steel, 1972). 41 2-2 Wind-driven circulation in a long, deep, unstra-

tified lake. 50

2-3 Wind-driven circulation in a shallow circular lake. 51

3-1 General view of experimental turntable (1973). 67 3-2 General view of experimental turntable (1978). 68 3-3 Roller elevations before and after levelling. 72 3-4 Model bed levels in contour form. 74

3-5 Calibration graphs for rotation controls. 78 3-6 Turntable rotation control unit with avometer and oscilloscope. 79

3-7 Delivery side of pumped supply. 82 3-8 Rotameter calibration graphs. 83 3-9 Outlet arrangement. 86 3-10 Scaffolding bridge above turntable. 89 4-1 The Tethered Sphere of Stefan and Schiebe (1968) 97

vi -FIGURE PAGE

4-2 The wax sphere/glass rod buoy. 100 4-3 The low-density polyethylene buoy. 104

4-4 Buoy in flume with cathetometer above. 105 4-5 Graphs used for selection of gradient change point.

(a)Correlation coefficient v gradient change point.

(b)Degree of agreement v gradient change point. 107 416 Data sample, regression lines, 5% confidence limits for calibration experiments of the poly-

ethylene buoy. 108 4-7 Forces acting on the polyethylene buoy. 111

4-8 Surface tension forces acting on the polyethylene buoy.

(a)with no flow.

(b)with a flow. 115

4-9 Theoretical and experimental calibrations for

the polyethylene buoy. 120

4-10 Pye Universal Measuring Microscope. 122

4-11 Zeiss Jena Stecometer. 123

4-12 Schematic diagram of the photogrammetric control. 127

4-13 Threaded post for theodolite and surveying target. 128

4-14 General view of turntable showing theodolite and

subtense bar. 129

4-15 Subtense bar and surveying target. 130

4-16 Definition sketch for negative centre co-ordinate

determination. 132

vii FIGURE PAGE 4-17 Enlarged photograph of test field used for

camera calibration. 135 4-18 Plot of lens distortion vectors. 136

5-1 Definition sketch for dimensional analysis. 153 5-2 Components of the Coriolis force due to the

Earth's rotation. 161

5-3 Definition sketch for azimuthal velocity profiles

of Sobey (1973a). 176 5-4 Definition sketch for wall jet flowing over a

surface in still surroundings (Newman, 1969). 177 5-5 Sample velocity profile at a representative section, tangential jet inlet Ko=3.66 x 106m4s-2. 178 5-6 Sample velocity profile at a representative section,

tangential jet inlet Ko=28.25 x 10 6m4s 2. 179 5-7 Graphs of Qc and Vm against Kol, L/h = 62.5 and Qc against Kol., L/h = 18.75 for tangential

inlets. 184

5-8 (a) Graphs of Qc/Ko1L and Vm/(ghA against Fr., L/h = 62.5 and Qc/Ko2L against Frj, L/h = 18.75 for tangential inlets.

(b) As (a) but against Rej. 185 5-9 Graphs of Qc and m against Kol, L/h = 62.5,

radial jet with diametric and asymmetric outlet. 195 5-10 (a) Graphs of Qc/Ko L and Vm/(gh)1 against Frj,

L/h = 62.5, radial jet with diametric and

asymmetric outlet.

(b) As (a) but against Re.. 196

viii

FIGURE PAGE 5-11 Definition sketch for the equations of motion

of an axisymmetric turbulent jet. 198 5-12 Graphs of IIc against (x-xo)-1 for experimental

and theoretical results. 204 •5-13 . Graphs of Qc and Vm against o , L/h = 62.5,

asymmetric jet and outlet. 206 5-14 (a) Graphs of Qc o1L and Vm/(gh) against Fri, L/h = 62.5, asymmetric'jet and outlet.

(b) As (a) but against Re j. 207

5-15 Graphs of Qc against ō'l for tangential, radial

and asymmetric jet directions, L/h = 62.5. 210

5-16 Graphs of Vm against Kot for tangential, radial

and asymmetric jet directions, L/h = 62.5. 211 5-17 Variation of Qc and Vm with time for test on short term stability of a velocity field. 214 5-18 Graphs of Qc and Vm against time to test the decay of a velocity field. 219 5-19 Field velocity measurements for Turriff Reservoir, (Courtesy of the Water Research Centre). 231 5-20 Flow patterns measured by Ali et. al. (1978a) (reproduced by permission of the Institution of

Civil Engineers). 238 5-21 Graphs of Qc and Vm against Kol, L/h = 58 and

Qc against K04., L/h = 27, Reservoir 4. 240

5-22 Graphs of Qc/Ko L and Vm/(gh)2 against Fr., 1 L/h = 58 and Qc/K02L against K04", L/h = 27, Reservoir 4. 241

ix FIGURE PAGE 1 5-23 Graphs of Qc/KolL and m/(gh)1 against Rej, L/h = 58 and- Qc/KolL against K0-1, L/h = 27, Reservoir 4. 245 5-24 Graphs of Qc/Ko againstit Ko1L/Qo for L/h = 58 and 27. 246 A-1 Tangential jet and Central Outlet, -6 Ko = 0.92 x 10 2 . 266 A-2 Tangential jet and Central Outlet,

Ko = 3.66 x 10-6m4s2 . 267 A-3 Tangential jet and Central Outlet, Ko = 7.06 x 10-6m4s2 . 268 A-4 Tangential jet and Central Outlet,

Ko = 13.84 •x 10-6m4s2 . 269 A-5 Tangential jet and Central Outlet, 270 K0 = 21.88 x 10-6m4s2 . A-6 Tangential jet and Central Outlet, Ko = 28.25 x 10-6m4s 2. 271 A-7 Tangential jet and Central Outlet, Ko = 0.92 x 10 6m4s-2. 272 A-8 Tangential jet and Central Outlet,

-6m4s2 . 273 Kb = 3.66 x 10 A-9 Tangential jet and Central Outlet,

Ko = 7.06 x 10-6m4s2 . 274 A-10 Tangential jet and Central Outlet, Ko = 13.84 x 10-6m4s2 . 275

A-11 Radial Jet and Diametric Outlet, -2. Ko = 3.66 x 10-6m4s 276

x FIGURE PAGE A-12 Radial Jet and Diametric Outlet,

Ko = 7.06 x 106m4s2. 277 A-13 Radial Jet and Diametric Outlet, Ko = 13.84 x 106m4a2 . 278 A-14 Radial Jet and Diametric Outlet,

ō = 28.25 x 106m4s2 . 279 A-15 Radial Jet and Asymmetric Outlet,

Ko = 3.66 x 10-6m4a2 . 280 A-16 Radial Jet and Asymmetric Outlet, Ko = 7.06 x 106m4a2 . 281 A-17 Radial Jet and Asymmetric Outlet, = 13.84 x 10 282 K0 -6m4a 2. A-18 Radial Jet and Asymmetric Outlet, Ko = 28.25 x 10-6m4a2 . 283 A-19 20° Jet Asymmetry and Asymmetric Outlet, 4s ō=3.66x106m 2. 284 A-20 20° Jet Asymmetry and Asymmetric Outlet, K° = 7.06 x 10- 6ma2 . 285 A-21 20° Jet Asymmetry and Asymmetric Outlet,

K° = 13.84 x 10-6m4a2 : 286 A-22 20° Jet Asymmetry and Asymmetric Outlet, Ko = 28.25 x 10-6m4s2 . 287 A-23 Short Term Stability Test, Ko = 13.84 x 10-6-2 ms , 5 minutes after initial jet entry. 288

A-24 10 minutes after initial jet entry. 289 A-25 15 minutes after initial jet entry. 290 A-26 20 minutes after initial jet entry. 291 A-27 25 minutes after initial jet entry. 292

xi FIGURE PAGE

A-28 30 minutes after initial jet entry. 293 A-29 35 minutes after initial jet entry. 294

A-30 40 minutes after initial jet entry. 295 A-31 45 minutes after initial jet entry. 296

A-32 50 minutes after initial jet entry. 297 A-33 55 minutes after initial jet entry. 298

A-34 60 minutes after initial jet entry._ 299 A-35 Short Term Stability Test, o = 13.84 x

106m4s2 , final velocity field. 300

A-36 Long Term Stability Test, Ko = 7.06 x

106m4a2 , 1 hour after initial jet entry. 301

A-37 2 hours after initial jet entry. 302

A-38 3 hours after initial jet entry. 303 A-39 4 hours after initial jet entry. 304

A-40 5 hours after initial jet entry. 305 A-41 Decay of Velocity Field - Initial Condition. 306

A-42 Decay of Velocity Field - Ave.

6.5 minutes after jet off. 307

A-43 16.5 minutes after jet off. 308 A-44 26.5 minutes after jet off. 309 A-45 36.5 minutes after jet off. 310

A-46 46.5 minutes after jet off. 311

A-47 Radial Jet and Diametric Outlet, Ko = 13.84 x 10 6m4a 2, with rotation of the model. 312 A-48 Radial Jet and Diametric Outlet, Ko = 28.25 x 106m4s 2 , with rotation of the model. 313 A-49 Radial Jet and Diametric Outlet, Ko = 13.84 x

10 6m4s2 , uniform slope in one half of the model, 314

xii FIGUxr; PAGE A-50 As figure A-49 with model rotation. 315 A-51 - A-58 Turriff Reservoir Model. 316-323

A-59 Reservoir 4 Model, Ko = 5.6 x 10-6m4s2 . 324 A-60 Reservoir 4 Model, Ko = 9.85 x 10-6m4a2 . 325 A-61 Reservoir 4 Model, Ko = 17.53 x 106 m4s2 . 326 A-62 As Figure A-61 with model rotation. 327 A-63 Reservoir 4 Model, Ko = 48.37 x 106 m4s2 . 328 A-64 Reservoir 4 Model, o = 65.87 x 10-6m4s2 . 329 B-1 Spin-up times for the 5m. diameter turntable. 337 C-1 Photogrammetric control. 341 LIST OF TABLES

TABLE PACE

3-1 Variation of Inflow Temperature with Time. 84

4-1 Summary of the Results of Camera Calibration. 137

4-2 Effect of Amendment of Interior Orientation. 139-140

4-3 Comparison of Standard Errors of Estimate. 141

4-4 Effect of Accidental Observation Errors. 144-145

5-1 Typical Figures for 3 Thames Valley Reservoirs. 156-157

5-2 Experimental Results, Tangential Jet and Central

Outlet, Large Model. 182

5-3 Experimental Restults, Tangential Jet and Central

Outlet, Small Model. 188

5-4 Experimental Results, Radial Jet and Diametric

Outlet, Large Model., 193

5-5 Experimental Results, Radial Jet and Asymmetric Outlet, Large Model. 194

5-6 Experimental and Theoretical Centreline Velocities

for a radial jet. 202

5-7 Experimental Results, Asymmetric Jet and Outlet,

Large Model. 208

5-8 Experimental Results - Long Term Stability. 216

5-9 Experimental Results - Decay of Velocity Field. 218

5-10 Effect of Model Rotation. 223

5-11 Experimental Results Reservoir 4, 1:116 Model 242

5-12 Experimental Results Reservoir 4, 1:250 Model

of Ali et. al. (1978a). 243

C-1 Control Point Space Co-ordinates. 340

xiv LIST OF SYMBOLS

SYMBOL DEFINITION

Ao Area of jet inlet

a Radius of cylinder

Bo Reference length

CD Drag coefficient

C Loge 2 D Deflection of buoy from no flow position

Do Diameter of jet orifice

DR Diameter of tank

d Cylinder diameter c dR Mean diameter of polyethylene rod

F Densimetric Froude number

FB Buoyancy force on polyethylene rod

FD Drag force on polyethylene rod

FG Weight of glass rod

FM Weight of the plastic target

FR Drag force on glass rod

Fr Froude number

Frj Ko/Qo(gh)2, a jet Froude number

f Coriolis parameter fc Absolute magnitude of the Coriolis force

g Local acceleration due to gravity

g k g

H Velocity head

h Typical reservoir water depth

Ia Integral, f2 (2 ) dr) J O i, j and k Unit vectors in the x, y and z directions

respectively SYMBOL DEFINITION 2 J Richardson number, g ktj Ii/j0 U K 0 Kinematic jet momentum flux per unit density k Average bed roughness height s L Typical reservoir length

LA Length of glass rod

Lc Length scale

LO Length scale

LR Length of polyethylene rod

LS Submerged length of glass rod

1 Length of a section

MD Moment of FD about the hinge m Mass of plastic target

n Mean diameter of glass rod

p Pressure Forced circulating flow or discharge Qc Jet discharge or volume flux Qo q A velocity vector, i u + jv + kw R Dimensionless co-ordinate, r/a

Rc Cylinder Reynolds number

Re Reynolds number Re. 1)KjIA , a jet Reynolds number Ro Rossby number

Ro. Ko/QofL, a jet Rossby number

R Rate of rotation r Radial distance R1 r Radial distance S Length of jet trajectory

SG Specific gravity of glass rod

xvi SYMBOL DEFINITION

SR Specific gravity of polyethylene rod

SS Select speed setting of the rotation control unit

Tv Tachogenerator voltage

T1 Spin-up time t Time

U Axial jet velocity

Uc A velocity scale

U Maximum velocity m Uo Jet efflux velocity

Mean velocity over a section

u, v, w Velocities in the x, y and z directions

Velocity (Reynolds stress)

✓ Current velocity Vm Mean velocity magnitude

Vo Jet efflux velocity or reference velocity

VR Velocity over length of glass rod

✓ Standard deviation of Vm V Mean flow velocity A ✓ Maximum velocity Velocity (Reynolds stress)

X Horizontal distance-

DC Co-ordinate direction x e Horizontal distance of line of action of FB from hinge point.

Horizontal distance of line of action of FM

from hinge point

X co-ordinate of buoy 'zero' position or virtual

origin of jet

xvii SYMBOL DEFINīTION

Horizontal distance of line of action of F XS s from hinge point

y Co-ordinate direction

Height of line of action of F above bed YD D y co-ordinate of buoy 'deflected' position Yd ym y co-ordinate when velocity is m

y co-ordinate of buoy 'zero' position Y0o yR Height of line of action of FR above bed

yi y co-ordinate when velocity is Um/2 Z Depth co-ordinate p Relative density difference

Angle (to the vertical) of inclination of the e polyethylene buoy

eJ Angle (to the horizontal) of jet inclination

Earth's rotation rate

~1 Basin rotation rate

p(a Jet growth rate

Spin-up parameter

Kinematic eddy viscosity

Dimensionless lateral jet co-ordinate / Dynamic viscosity of water

F`k■ Kinematic viscosity of water Koh/fL2

2 Ko1L/Qo 7v3 L/n 74 / oi/p 75 Ko/gL ]t6 ks/h

xviii

SYMBOL DEFINITION -1C, q/ j) P Density of water Latitude of the reservoir

Angular velocity (73 -k it sin 0 ll Vector operator, I Q ax

V 2 Scalar operator, a2 2 axz abz +

Notes Subscript min denotes minimum. Subscript max denotes maximum.

Subscript m denotes model quantity.

Subscript p denotes prototype quantity. Subscript r denotes the ratio of a model to

a prototype quantity. Prime denotes variation from the mean.

xix ACKNOWLEDGEMENTS

This research project was conducted between May 1975 and October

1978 whilst the author was seconded from the Metropolitan Water

Division of the Thames Water Authority to the Department of Civil

Engineering of Imperial College of Science and Technology and

was jointly financed by the Thames Water Authority and the Science

Research Council to whom thanks are due.

Professor J. R. D. Francis and Mr. P. Minton of Imperial College

supervised the research and the author was grateful for their

assistance throughout the three years and especially during the

preparation of this manuscript. Many other members of the college

staff also assisted and those notable of mention are Mr. A. Scott-

Moncrieff for his continued work on the turntable drive unit,

Mr. B. Chiat for untiring efforts with the photogrammetry, Mr. G.

Thomas for technical assistance and sharing his room with me,

Ms. J. Gurr and staff for co-operation with the photography and

the technicians of the hydraulics laboratory for unlimited tech-

nical skill.

The author is also indebted to Mr. P. Cooley and Mr. D. B. H.

Pawsey of the Thames Water Authority for their interest and

encouragement and Mr. J. Hornsby, also of T. W. A., who prepared

the plates.

Thanks to additional financial support from the Thames Water

Authority it was possible to make use of the analysing facilities

of the Terrestrial Photogrammetric Unit of City University and the technical assistance of Mr. N. Lindsey and Mr. A. Kenny and the manpower of Mr. J. Davies and Mr. J. Hooker was appreciated.

Acknowledgement is also due to Mr. R. Collingwood and Mr. J. Davis of the Water Research Centre, Medmenham for their field data which they were kind enough to give to the author and Dr. R. A.

Falconer (now of Birmingham University) for many valuable dis- cussions whilst researching at Imperial College.

Finally I would like to express my sincere thanks to my wife

Jane who devoted many hours of her spare time to the typing of this manuscript and remained patient throughout despite the added strain of my appalling writing.

xxi CHAPTER 1

STATEMENT OF PROBLEM

1 1.1 INTRODUCTION

The principal types of reservoir considered for this study are

those where the original primary function of the reservoir is the

storage of raw (untreated) water to average out fluctuations in

demand and to guard against restrictions on abstraction from

rivers at times of 1-ow flow. However, recently consideration has

also been given to the short-retention time bankside storage

'reservoirs and these will be discussed later. An additional benefit

of the large storage reservoirs, with a fairly high retention time,

is that they have the effect of acting as a primary water treatment

process since the decay of..bacteria with storage time is quite

dramatic. The biological processes involved in this improvement

are maximised if the reservoir is well-mixed vertically and the

individual fluid elements have an equal retention time within the

reservoir. Unfortunately large. water masses are subject to complex

meteorological and biological effects and the mixing processes are

often either weak or unpredictable. A lack of vertical mixing

can allow stratification while extreme differences of residence

time can cause stagnation within the reservoir and a resultant

deliterious effect on local or overall water quality.

Stratifications are most often caused by seasonal temperature

variations or, occasionally, by dissolved solids and can impose a

layered structure on much of the stored water volume. The inter-

faces are quite stable and effectively prevent the exchange of

water between layers, this results in a considerable difference in

water quality from layer to layer.

2 Stagnation is merely the non-movement of water in particular areas, which can be extensive. These areas of stagnant water

often occur due to the particular geometry of the reservoir or the poor siting of inlets or a combination of both. This lack of

horizontal movement invariably means there is also a lack of vertical

mixing as well as horizontal mixing and a subsequent decline in

water quality occurs. In some cases the stagnant area can become

so isolated that it effectively reduces the useable volume of

stored water.

Slow-moving large scale current systems, both horizontal and vertical,

can be induced naturally, by the wind, or artificially, by the

throughflow. The resulting flow patterns in the reservoir are

largely determined by the reservoir geometry (plan shape, bottom

topography etc.), the inlet characteristics (momentum, position in

the reservoir, direction and inclination of discharge) and, if

there is a wind, the direction and intensity of the wind stresses

at the water surface, stratification and, depending on the surface

area and depth of the reservoir, the Earth's rotation.

Stratification tends to confine the slow moving current systems to

particular layers although some shear can be transmitted through

an interface to induce motion in an adjoining layer. The Earth's

rotation induces an additional (Coriolis) acceleration acting to

the right (in the Northern hemisphere) of the usual temporal and

convective acceleration. This acceleration component may influence

but would not be expected to dominate the current system as it does

oceanic circulation. In a reservoir which is poorly mixed in the horizontal plane the current system may be significant but could allow water to flow directly from inlet to outlet whilst other

water remains stagnant.

Much of the previous work concerned with improving the water quality in reservoirs has concentrated on remedial rather than preventive techniques. There is a considerable amount of literature on selective withdrawal, artificial destratification etc. but

these are measures intended to relieve a deterioration in water

quality and as such they are often expensive and unsatisfactory.

Preventive techniques such as the introduction of the inflow as

subsurface directional momentum jets, siting of inlets and design

of suitable reservoir shapes have received limited attention

although their potential would seem considerable.

In order to effectively apply preventive techniques such as momentum

jet inlets to reservoirs their action and effect must be predicted.

The present study has aimed at a better understanding and prediction

of the effect of momentum jet inlets by the use of hydraulic models

in order to aid the present operation and future design of reservoirs. 1.2 WATER QUALITY IN RESERVOIRS - THE PROBLEMS

The necessity for techniques to control and maintain reservoir water quality has become more important over a number of years with increasing consumer demand dictating a trend towards deeper

(and hence larger volume) reservoirs. The major water quality problems encountered in these reservoirs are outlined below.

1. a 1 Algal Growth

One of the disadvantages of impounding large volumes of water is that the subsequent slow water movements and the exposure of large areas of the water surface to radiant heat often encourages the growth of algae in the upper layers. This growth becomes particularly prevalent when the reservoir water is eutrophic

(nutrient rich); this is becoming more often the case as the storage of eutrophic water is becoming more common as underground water supply sources become depleted.

Large algal crops in reservoirs may cause difficulties at the treatment works during filtration either by dead algae physically clogging the filter or by proliferation on the water surface.

They may also interfere with chlorination and other processes and can lead to taste and odour problems in the water supply.

Wind blown algae may transfer inlet bacteria to the outlet and, in addition, algae on a reservoir surface may be unsightly, smell and generally interfere with any recreational activities on the reservoir. Thompson (1954) has given examples of the problems arising from algae and Ridley(1971) has given an account of the annual cycle of a stratified reservoir with particular reference

5 to the physical, chemical and biological stages and notes the effect on algal growth.

Attempts to reduce the incidence of algal problems have included the use of large volumes of algicide, eg. copper sulphate, but this has proved an expensive measure; selective withdrawal has no effect on the algal growth but allows algae-free water to be used for supply and the removal of nutrients from the river to inhibit growth has had limited success. Artificially induced mixing and circulation by momentum jets has proved the most successful

(section 1.4.2).

1.2.2 Thermal Stratification

Thermal stratification is the term applied to the segregation of the water of a lake or reservoir into horizontal layers exhibiting differences of temperature, and consequently density and viscosity.

For most of the year reservoirs are isothermal and even moderate winds can produce currents which extend to the bottom of the reservoir and cause efficient mixing in the vertical plane. However, in the spring and early summer, when the inflow and upper layers are warm from increased solar radiation, the wind becomes less and less able to mix the water over the depth. The surface water then heats up even more rapidly and, if the weather remains calm, thermal stratification occurs.

When fully established thermal stratification produces three distinct layers. The upper layer, which may extend 7-10m. below the surface, is known as the epilimnion and has a high and fairly uniform temperature and is kept in frequent horizontal circulation by the wind stresses on the water surface. Beneath the epilimnion is a zone in which the temperature decreases rapidly with depth (sometimes

1°C per metre) which is known as the thermocline. The lowest layer extends from the bottom of the thermocline to the bed of the reservoir and is called the hypolimnion and has a fairly uniform low temperature. The hypolimnion. is completely isolated from surface influences and little or no horizontal or vertical mixing occurs within it. A typical vertical temperature profile for a stratified reservoir or lake is shown in figure 1-1.

Mild winds can often worsen a stratification by equalising the temperature within the epilimnion thereby accentuating the thermocline 'step' between it and the hypolimnion; such a thermocline can prove particularly persistent and difficult to disperse. The establishment of a stable stratification means that the hypolimnion is effectively isolated from the atmosphere. Algal growth takes place at the surface with the strong illumination and, as they die, the algae drop to the reservoir bed where they and other organic debris decay using up the residual dissolved oxygen in the hypolimnion. Once the hypolimnion has become completely deoxygenated anaerobic decomposition takes place and reduction products (particularly manganese and iron), and hydrogen sulphide may be formed. The result is a drastic decline in the hypolimnion water quality which gives rise to taste and odour problems if the water is used for supply. Often the quality of the water in the hypolimnion deteriorates to such a level that it cannot be used and the volume of potable water in the reservoir is considerably reduced. An additional constraint imposed on the use of water during a period of thermal stratification is as a result of the 17c wind --- 5c wind

.1.■■. E - - -

16c - - ± 7 ) r__ //////////////7 H 7°c 5°c summer au umn E—epilimnion T—thermocline H — hypolimnion (a) (b)

Depth m.

5 10 15 Temperature °C

(c) Figure 1-1 Section of a lake or reservoir during summer stratification. Section of a lake or reservoir with full circulation and mixing during autumn. Typical summer stratification in a lake (Thompson, 195+). rapid growth of algae in the epilimnion as described in section 1.2.1.

Towards the end of August the strength and duration of solar radiation

declines and the average air temperature tends to decrease. The

epilimnion water cools and, as the surface water becomes cooler

than that below, it sinks. Slight wind also causes circulation in

the epilimnion which continues to cool throughout and begins to extend

into the thermocline, eventually eliminating it. Only a modest

wind is then required to cause complete vertical mixing of the

whole body of water ('overturn'). Hypolimnion water is then dis-

tributed throughout the reservoir together with algae from the

epilimnion causing further supply problems. A similar 'overturn'

may also be caused by the strong spring winds which cause large

quantities of nutrient-rich water to be distributed throughout the

reservoir. Under the correct light conditions algae can feed on

these nutrients and form blooms.

Steel (1975) has stated that stable thermoclines can form in reservoirs

which have surface areas C 1Km2 and/or maximum depths > 10m. These figures would include many of the large water supply reservoirs

in the United Kingdom and in particular the pumped storage reservoirs

of the Thames Valley (see table 5-1). A good account of the process

of thermal stratification has been given by Thompson (1954) and

other literature on the subject includes Hutchinson (1957), Ridley

(1964) and Symons (1969).

1.2.3 Stagnation

Stagnation occurs at times when there is little or no horizontal

circulation in a reservoir or when the circulation is ineffective. Large areas of the reservoir may become isolated, possibly due to the particular geometry of the reservoir or the poor siting of inlets and outlets or a combination of all three. The isolated water being stagnant is liable to deteriorate in quality in a similar manner to that in a stratified reservoir and thus reduces the effective volume of the reservoir. Under normal circumstances the wind can be relied upon to prevent the formation of stagnant regions for most of the year but periods of calm often occur during the summer months when biological and bacterial activity is high.

1.2.4 Short-Circuiting

Although the primary purpose of a large storage reservoir is to act as a reserve against restrictions in river abstraction during periods of drought or pollution its importance as an initial stage of water treatment has long been recognised. Using E. Coli as an indicator of the degree of bacterial pollution there is a greater than 90% reduction in the numbers of this pathogen when water is stored for ten days, the reasons for such a bacterial decline have been discussed by Young, Wallingford and Smith (1972). In view of the benefits of such an improvement in water quality at no extra cost it is obviously a desirable objective for the throughflow to utilise the entire reservoir volume ie. for all water particles to have a retention time in the reservoir close to the theoretical mean

(Reservoir Volume Throughflow Rate).

Unfortunately in windy weather the upper currents are strongly influenced by the wind and if the incoming river water is warmer than the reservoir water it may rise quickly to the water surface and be carried directly to the outlet by the wind (White, 197$).

10 During calm weather the currents are determined by the direction and momentum of the inflow and are more easily controlled and predicted. However, under either condition stagnant areas may be left isolated from the main currents and may rely on a fortuitous wind to disperse them. The streaming of water directly from inlet to outlet is known as short-circuiting.

It is convenient in this section to mention the short retention time bankside storage reservoirs (see section 5.4). The Depart- ment of the Environment Steering Committee on Water Quality (1971) have recommended that bankside storage reservoirs should provide a minimum of 7 days storage between abstraction and arrival at the treatment works, this period is intended to minimise the possibility of any river-borne toxic pollution passing into the supply system. It is of interest to record a controversy that has occurred concerning the operation and type of mixing employed in these reservoirs.

Given that the reservoir provides a minimum of 7 days protection to the water supply system there is considerable debate as to whether the reservoir should be operated as a 'completely mixed' system or as a 'plug flow' system. The two systems are completely different in concept and operation; with the 'completely mixed' system a pollutant entering a reservoir is instantaneously mixed with the existing stored water and a small amount of pollutant arrives at the outlet immediately. The reservoir water and the pollutant continue to be mixed and although the concentration of pollutant increases, the level should never become toxic before the pollutant is detected and the inflow ceased.

11 With a 'plug flow' system no attempt is made to mix the pollutant and the pollutant gradually passes to the outlet taking 7 days or more (depending on the nominal retention time) to arrive. The reservoir can then be isolated, inflow and outflow ceased and the water and pollutant removed from the reservoir. The relative merits of the two systems hinges on the view as to whether it is better to have a low level of pollution arriving immediately at the treatment works or a potentially dangerous level arriving at some time in the future but with the possibility that no pollutant will pass into supply. The choice seems to be largely controlled by the possibility of detecting the pollutant. If the pollutant can be detected then 'plug flow' would seem preferable, however, if there is the possibility of not noticing the pollutant then 'completely mixed' is better as it minimises the chances of toxic pollution entering the supply.

12 1.3 WATER QUALITY CONTROL-REMEDIAL METHODS

The advantage of controlling water quality within a reservoir rather than treating to improve the quality after storage is principally economic. Other benefits that are derived are improved reservoir recreational use and larger volumes of potable water available in the reservoir. The approach to controlling reservoir water problems can be divided into two sections, remedial methods and preventive methods. In this section examples of the former will be given, that is methods which are applied after water quality problems have occurred.

1.3.1 Selective Withdrawal

Selective withdrawal is aften used when reservoirs have become stratified with contrasting water qualities between layers

(section 1.2.2.). The method consists of drawing the reservoir outflow from any of several levels throughout the reservoir depth and allows any selected stratum to be withdrawn independently thereby avoiding layers of poor quality, eg. deoxygenated hypolimnion water or algal-infested epilimnion water. Much research has been done on the flow currents around such an outlet and in particular the effect of its shape and size in relation to the position and depth of the layer being withdrawn. Brooks and Koh (1969) have given a good account of the subject and other work includes Bryant and Wood (1976) and Darden, Imberger and Fischer (1975).

Unfortunately selective withdrawal only provides a means of making the best use of a reservoir in a poor condition with it only being

possible to abstract water from limited volume layers which may

13 become exhausted during periods of high demand. The method does nothing to improve the overall quality of the water and the reservoir, at such times, is proving a waste of capital expenditure being unable to fulfil its primary purpose as a reserve storage.

An additional problem that may occur is that an internal seiche

(Smith, 1975) may cause poor quality water from another layer to leave the reservoir via the selected outlet. Also selective withdrawal has no effect on a sudden pollution or short-circuiting and may in fact worsen the situation in these cases by limiting water movement to one layer.

In summary selective withdrawal is a method of abstracting water from a stratified reservoir but is only likely to be viable in the short term, however, it has some merits and may prove a useful reserve measure in reservoirs using a preventive method (section 1.4).

1.3.2 Artificial Destratification by Vertical Mixing

The practise of transferring water from one level to another with the intention of destroying or modifying thermal stratification has been practised for some years. Ridley (1964) has summarised the development of devices to achieve this vertical mixing.

Such devices have taken the form of mechanical pumps, diffused air injectors, air curtains or bubble guns. Generally the application of these devices has the effect of transferring hypolimnion water to the epilimrion thereby lowering the thermocline and provided sufficient energy is applied the thermocline can eventually be eliminated and the reservoir become isothermal.

Davis (197$) has summarised many of the techniques and has reported considerable success with an unconfined air injection

14 technique applied to small bankside storage reservoirs (eg. volume

2 x 105m3) which had water quality problems. The extension of such a method to large pumped storage reservoirs (typically two orders of magnitude larger in volume), however, is debatable.

In general the capital and running costs of these remedial techniques are modest compared with the benefit derived. Symons (1969) has published a compilation of research papers including costs, designs and details of the beneficial effects of vertical mixing. Other literature on the subject of vertical mixing techniques has been contributed by Tolland (1977), Symons (1971), Symons, Irwin and

Robeck (1967) and Ridley, Cooley and Steel (1966).

Although vertical mixing does successfully overcome thermal stratification the mixing it induces in the horizontal plane will be small. Hence, stagnant areas and short-circuiting are still possible; with this in mind vertical mixing can be said not to solve all the problems mentioned in section 1.2 but it may eli- minate the worst.

1.3.3 Aeration

A method of improving reservoir water quality immediately which is__occasionally used is by the injection of oxygen directly into the hypolimnion. This injection increases the amount of dissolved oxygen in this layer and thereby prevents the highly undesirable anaerobic decomposition from occurring, however, it has no effect on the thermocline,which persists. As a result the water must be selectively withdrawn since the stratification remains stable but it is possible, when the dissolved oxygen level has recovered

15 sufficiently, to abstract hypolimnetic water. The method is not very satisfactory being very inefficient, however, in old reservoirs where the provisions for vertical mixing are limited it may offer a temporary solution.

16 1.4 WATER QUALITY CONTROL - PREVENTIVE METHODS

The methods described in this section are preventive, ie. they are applied before a deterioration in water quality has a chance to occur.

1.4.1 Multi-Inlet Systems

Stagnant areas in a reservoir can be avoided to a large extent by the provision of multiple inlets which cause the flow to become a steady stream across the full width of the reservoir. The system is particularly applicable to reservoirs which have a large length: breadth ratio but does not prevent the possibility of thermal stratification or provide good vertical mixing. Another disadvantage is that the additional cost of providing several inlets may not be justified considering the small increase in improved performance.

1.4.2 Momentum Jet Inlets

In large pumped storage reservoirs, where the inlet water has to be regularly pumped from a river or where it is fed under a high head by gravity, the introduction of the inflow into the reservoir via directional sub-surface jetted inlet offers a solution to, and more importantly prevents, the water quality problems described in section 1.2.

Vertical mixing is achieved and stratifications destroyed or

prevented by directing the jet upwards from the bottom of the reservoir at an acute angle to the bed. Hypolimnetic water is

17 entrained by the jet and the mixture carried up into the epilimnion, depressing an existing thermo"cline or,more normally, ensuring that the reservoir remains isothermal. The overall water quality is therefore maintained fairly constant over the depth. Mixing in the horizontal plane is achieved by directing the jet in such a way that it entrains large volumes of water and'carries the mixture, with the jet stream, away from the local mixing zone preventing local recirculation. In order to maximise the horizontal and vertical mixing it is necessary to produce the longest possible entraining length in the deepest water masses, the trajectory of the jet being such that the mixture of inflow and hypolimnetic water eventually rises to the surface and also such that the horizontal component of momentum is as large as possible and generates a significant horizontal circulation in the reservoir which prevents stagnant zones and maximises the mean detention time. A schematic diagram of jet action is shown in figure 1-2. The jet momentum is the critical factor in driving the horizontal circulation being transferred horizontally and vertically to the reservoir water and forming large scale swirling currents.

For a pumped storage scheme both the initial costs of installation and the operating costs of the jets are small in comparison with the total expenditure and when considering the benefits derived

(discussed in section 1.5.3), however, there are certain drawbacks and these should be mentioned.

A major disadvantage of relying on jets is that during periods of restricted abstraction (eg. low river flow or a polluted source) there is little or no inflow and hence water quality problems can

18 input

Elevation

Plan

Figure 1-2

Schematic diagram of momentum jet action.

19 occur during such periods. To guard against such a situation

(eg. summer 1976) the reservoirs are provided with a remedial reserve measure, eg. auxiliary vertical mixing pumps or provisions for selective withdrawal. This provision, of course, is an additional source of expenditure and must be taken into account when considering costs.

Whilst the jet induced circulation and mixing of a reservoir with regard to overall water quality is of benefit there is some evidence

(Steel, 1975) that natural biological processes within the reservoir are disturbed and certainly the natural processes of sedimentation of algal particles and organic debris may be hindered at certain times of the year. This may result in a decline in water quality as more solids-associated bacteria reaches the outlet, Ridley et.al.

(1966) have quoted an example of this for a particular reservoir.

To overcome this problem many reservoirs are provided with 'low- velocity' inlets which have a very large discharge area and hence the inflow has low momentum. These inlets are principally used during the winter when the river water has a high suspended solids content and the high momentum jet inlets are not required for mixing.

20 1.5 BACKGROUND TO Thr; USE OF JET INLETS

1.5.1 Early Research

The Metropolitan Water Board was responsible for commissioning the original research work that resulted in the use of jetted inlets. In 1946 the problems of thermal stratification and the necessity to increase the stored volume by constructing deeper reservoirs came to a head and a research team led by Professor

C. M. White began examining ways of overcoming this problem.

They proposed that thermal layers in a reservoir could be dispersed by the incoming water if it was discharged as an angled jet entraining the surrounding water. Their experiments are described in Chapter 2 so it is sufficient to say here that they were successful. Cooley and Harris (1954) published an account of the Imperial

College work and its application to the reservoirs of the Metro- politan Water Board (now part of Thames Water Authority) and this is a good summary of their final report to the Board with-White

(White et. al., 1955).

1.5.2 Current Use

As a result of the work of White et. al. (1955) all the large storage reservoirs in the Thames Valley built since and considered liable to stratify have had jetted inlets incorporated in their design and several accounts (Ridley et. al., 1966; Windle Taylor, 1966 and Steel, 1975) of their successful operation have been published.

The latest reservoirs to include jetted inlets are the two largest of the Thames Valley group, these being at Wraysbury and the Queen

Mother Reservoir at Datchet (34 and 38 x 106m3 respectively in

21 volume). The design of the inlets and outlets of these reservoirs

is one which allows a considerable flexibility of operation. For

example, at Datchet there are 7 high velocity jets ranging in

orifice diameter from 0.6m. to 1.5m. and having inclinations to

the horizontal of 22i° or 45°, the steeper jets being a reserve

measure for use in destroying an established thermocline. They issue in essentially two directions (figure 1-3), a combination of several jets being used at any one time. The data for this reservoir,

Wraysbury reservoir and the Queen Elizabeth II reservoir is summarised in table 5-1. The Queen Mother reservoir also has a low velo-

city inlet and recirculating pumps and the outlet tower is fitted

with pairs of penstock valves at five levels giving a selective

withdrawal facility in the event of stratification. The measures installed are very conservatively designed and no problems have, as yet, been encountered.

Despite the success of jetted inlets in the Thames Valley reservoirs they have not been adopted elsewhere to any great extent. In the

U.S.A. experience seems almost entirely with vertical mixing devices; the Silvola reservoir in Finland (constructed in 1962) uses a jetted system (described by Priha, 1969) and a proposed reservoir in Hong

Kong (Symons, 1969) was also to use jets. In the United Kingdom several other reservoirs use jets eg. Bough Beech, Chew Valley,

Foremark (under construction), Farmoor (Hammond and Evans, 1964) and a pair of reservoirs near Liverpool investigated by Ali, Hedges and Whittington (1978a) being particularly known to the author.

22 t111 ,lout tt►W1{ 11111►1t, t T i l itt 1 71,40 ‘ nnrt ,, Jv ~11 ttt111n:° ~iiijrttn ~Uj~~ i~. i

Figure 1-3

The inlet arrangement at Queen Mother Reservoir, Thames Water Authority 23 1.5.3 Economic Considerations

As mentioned in section 1.4.2 the costs of installing and operating jet inlets in a pumped storage reservoir are quite small in comparison with the overall construction costs and the cost of pumping water from the river into the reservoir respectively. Installation of the jets during construction of the reservoir leads to little extra cost and will certainly be a small fraction of the total, typical values being less than 2% for Wraysbury Reservoir (Cooley,

1970) and 5% for Silvola (Priha, 1969). The extra pumping head required for the operation of the jets is small (less than 0.5m. according to White et. al., 1955; approximately 19KW.) and the operation and maintenance of jets is a small percentage of total costs (from the same sources 2-3% at Wraysbury and 6% at Silvola).

Jets are therefore economically viable and may actually show a financial benefit if the value of the increased storage is included in the calculations.

1.5.4 Further Investigation

Jetted reservoir inlets offer a powerful means of aleviating and preventing reservoir water quality problems although their action and hence their design is not fully understood. The study described in this thesis was initiated by the Thames Water Authority in conjunction with the Science Research Council in response to the necessity for a better understanding in this area of reservoir hydraulics. The first stage of the study was concerned with identi- fying the particular areas that needed to be considered and with ways of solving the problems. Chapter 2 reviews the literature relevant to jet-induced reservoir circulation which has aided the decisions made and the solutions found.

24 CHAPTER 2

RESERVOIR AND ASSOCIATED FLOW - LITERATURE SURVEY

25 INTRODUCTION

Before proceeding to the description of the work undertaken in this study it is both necessary and worthwhile to consider previous studies connected with or relevant to reservoir circulation and, since the amount of literature in this field is scant, also studies of analagous flow systems.

26 2.1 RESERVOIR FLOW SYSTEMS

2.1.1 Laboratory and Mathematical Studies

2.1.1.1 The Generation of Circulation by Throughflow

Although the title of this section mentions throughflow it is more relevant here to concentrate on momentum jets. The early work on jet-induced reservoir flows was concerned with the possible use of jets to break up an established thermal stratification, their ability to generate substantial currents in a reservoir at other times was not immediately recognised. White et. al. (1955) investigated the ability of a buoyant jet to break up the thermal layers of a stratification by the entraining properties of the jet. They computed the thermal history of a reservoir by considering the stratification as a series of superimposed horizontal layers. Each layer travelling downwards and becoming thinner as part of it is entrained by the jet or lost through the reservoir outlet. As the buoyant jet rises to the surface it entrains cold water from the lower layers and replaces the surface layers left by the moving strata. By making use of the mathematical analysis of Tollmien (1958) of the behaviour of a jet of fluid discharging into the same fluid they were able to derive an equation for the trajectory of a buoyant jet in terms of the density difference between the inflow and the ambient density of the reservoir water, the head applied to the jet and the diameter of the jet nozzle. Hence, knowing the density difference, and the depth of the nozzle below the free water surface solutions to this equation gave values of the head and nozzle diameter for any desired entrainment. The theory was tested and found to be correct by using salt water to represent the ambient density with a freshwater jet

27 and tracing its path by a tracer such as potassium permanganate or rhodamine dye.

To check the validity of their models as used for mixing studies a 1:450 model of the Queen Mary reservoir of the Metropolitan Water

Board (now Thames Water Authority) was compared with records of the thermal changes in the full scale reservoir for three months in 1936.

Unfortunately these experiments were not entirely successful as models without inclusion of the effects of surface wind stresses and solar radiation cannot be expected to reproduce the complex thermal processes occurring in a prototype reservoir and furthermore the solar radiation heating of the upper layers of the reservoir will quite probably influence the entrainment capability of the jet and its dispersion. However, in general their experiments, with some compensation made for the solar effects, showed that the horizontal strata could be 'turned over' and mixed with the incoming water in two days (prototype time). The authors also recognised that a jet which mixed too vigorously would become ineffective in the upper part of its trajectory whereas too dilute a mixture would lack the buoyancy required for it to spread rapidly over the surface and hence could accumulate in one position, this drew the authors into a consideration of the induction of horizontal reservoir circulation to sweep away such stagnant areas and to keep the reservoir water continually flowing. On the basis of their study they proposed that an ideal reservoir would be nearly circular in plan shape with inlet situated on the floor of the reservoir close to the outer boundary and orientated to give an anti-clockwise circulation. The outlet would be positioned at the centre of the reservoir and would have its main outflow ports at the bottom,

28 supplemented with ports at higher levels for occasional use. As a result of their successful work the inlets at the Queen Elizabeth

II Reservoir at Walton were designed as momentum jets.

Since the construction of this reservoir in 1962 two other, larger and deeper reservoirs have been constructed in the Thames Valley,

Wraysbury and the Queen Mother. With the knowledge acquired from the study by White et. al. (1955) the inlets and outlets in both reservoirs have been designed on the basis of the results of tests on scale models (Metropolitan Water Board, 1965 and 1967). These tests were performed to investigate the ability of jets to break- up stratification and bring•about near isothermal conditions and with the following additionāl specifications:

1)the avoidance of a fast passage of water from inlet

to outlet so that as much bacteria and suspended matter

may be lost as possible,

2)the sweeping'of the greater part of the area but

particularly the periphery to discourage the establish-

ment of algal reserves,

3)the generation of currents that effectively disperse

mixtures forming at the inlet.

The 'worst case', a well-established thermocline occurring in early

June, was assumed. Laboratory tests performed in a small tank stratified by a salt solution and freshwater showed that for a horizontal jet only one sixtieth of the entraining potential of the jet would be acting on the upper layers and hence a situation could be reached where the lower part of a reservoir was continually mixed yet the upper part was effectively stratified. 22i-° jets were found to be far more successful and this was selected as the

29 principle jet inclination; 45° jets were reserved for destroying a well-established thermocline. The position of the jet inlets in plan was determined by experiments on scale models, a 1:1000 model was rotated so as to include the Coriolis effects whereas a larger, static 1:250 model was used to verify the flow patterns in the small model. The position and direction of the jet discharge were varied so that conditions 1) to.3) were satisfied.

Priha (1969), in his description of the hydraulic functioning of the Silvola Reservoir of the Helsinki Water Company, mentions a model investigation as part of the preliminary work on the design.

An angle and inclination of the jets was sought such that, at the expected flow rates, they would prevent stratification whilst maintaining an essentially horizontal flow in the reservoir. Salt and freshwater were used to similate the critical condition when the inflowing river water and the reservoir water differed in temperature by 12°C, coloured dye was used to aid visual interpretation.

Sobey and Savage (1973b and 1974) have reported some of the earlier work done at Imperial College by Sobey (1973a) on jet-induced reservoir circulation. The physical processes responsible for large-scale current motions in water supply reservoirs were examined using a simplified mathematical model of a single reservoir gyre.

The forced circulating flow (QC) was adopted as an indicative mixing parameter with the slightly questionable inference

(Alonso, 1975) that an increase in this mixing parameter relative to geometric and dynamic reservoir parameters would improve the overall reservoir water quality. The forced circulation (Q6.) was shown to be significantly influenced by the reservoir geometry,

30 as described by the aspect ratio (L/h) and the boundary resistance, as described by the relative resistance (ks/h). Some of the other reservoir characteristics eg. stratification were-not modelled but the influence of the jet geometry (Ko-1L/Qo), the jet Reynolds 1 number (K 1/0 ) and the jet 'Froude number (K0/Q0 (gh)l2 ) was shown to be small.

Other aspects of Sobey's work are well-documented in papers by

Sobey (1973b), Sobey and Savage (1973a), and Savage and Sobey

(1974 and 1975). In the former paper a theoretical study of the slow-moving interior flow within a reservoir gyre is recorded; this three-dimensional flow is likened to an axisymmetric laminar boundary layer flow as previously studied by Badewadt (1940). The influence of the Coriolis acceleration on this type of flow was shown to generate a significant swirling boundary layer flow.

This mechanism for large scale mass convection or interchange of fluid was suggested to have important consequences for reservoir mixing particularly as it was concerned with areas where there is very little flow. The joint literature is concerned with a study of a turbulent jet issuing horizontally from a circular orifice into a rotating basin. A series of flow visualisation experiments are described and from these the path of the jet was found to depend significantly on a similarity parameter h/Lc (where h is the water depth and Lc a length scale dependent on S.1 , the basin rotation rate and Ko the kinematic jet momentum flux). The theory is also presented which confirms these experimental observations and pre- dicts the effects of the Earths rotation on the jet path in a typical full scale reservoir. These reservoirs are normally shallow enough for the flow to be essentially two-dimensional and for this

31 effect to be small and this was confirmed, however, Sobey and

Savage have shown that because these reservoirs are not inordin-

ately shallow tests on vertically exaggerated hydraulic models

where the model is rotated could lead to incorrect conclusions

being drawn from misleading results.

Most recently Ali, Hedges and Whittington (1978a) and All and Hedges

(1975) have described an extensive experimental investigation into

the existing water circulation in two reservoirs near Liverpool and

ways of improving it. Their results confirmed the conclusions of

Sobey and Savage (1974) with respect to the effect of reservoir

geometry on the circulating flow and they once more emphasised the

dangers of conducting tests on vertically exaggerated hydraulic

reservoir models.

Falconer (1976) has mathematically modelled the unsteady two-

dimensional circulation in a circular reservoir by computing the

depth-averaged velocity field for a confined jet. The time

dependent non-linear equations of motion were formulated to include

the effects of bottom roughness, wind action, the Earth's rotation

and the turbulent diffusion of momentum and were integrated ovdr

the depth. A finite difference technique was used to solve these

depth-averaged equations of motion. The reproduction of the velocity

fields for jet-forced circulation was acceptable although limitations

on computer time and storage restricted the number of time steps

possible and the grid spacing meant that the fields were not always realistic and in fact the comparitively large outlet area had a

strong effect on the flow pattern when it has consistently been

found (Minton and Robinson, 1978; Ali, Hedges and Whittington, 1978b)

32 that the outlet has'little or no effect on the circulation'.

Falconer showed that the convective acceleration terms of the equationsof motion were important in increasing the vorticity and hence the circulation. This inferred that the jet momentum for a given throughflow should be maximised by minimising the area of the inlet. Unfortunately the mathematical model was not rigorously verified and would seem to require further work to include three- dimensional effects of stratification and secondary currents as well as a more realistic representation of the inlet and outlet conditions to be able to accurately predict a prototype situation.

2.1.1.2 The. Generation of Circulation by Wind

To the author's knowledge there have been no laboratory studies of reservoir flows where the effects of a wind acting on the water surface have been taken into account. The reasons for this are quite apparent; as was discussed in Chapter 1 for most of the year the wind alone promotes sufficient circulation and mixing in a water supply reservoir to ensure that the reservoir remains isothermal, any water quality problems only occurring when the weather is calm for some time and there is intense solar radiation warming the surface layers. In the past the study of hydraulic models has been concerned with improving the existing or expected water quality and as such the 'worst' conditions have been assumed.

Consequently models are tested windless; there is also the additional extreme difficulty of physically modelling the wind stress transfer to the water surface of hydraulic models which has discouraged studies of,the complex interaction between wind and throughflow- induced circulations.

33 Falconer (1976) included the surface wind stress as one of the

variables in his mathematical study of jet-forced circulation in

a circular model reservoir and found that, whatever its direction,

the surface wind stresses improved the generation of gyres and

increased the vorticity and in doing so could be considered bene-

ficial. to the overall flow and hence the reservoir mixing. His

- findings may be a confirmation of the hypothesis of Steel (1972

and 1975) that the mixing is improved by the interaction of the

wind with the currents generated by throughflow (discussed in

section 2.1.2).

2.1.1.3 Other Studies

Mayer (1975) has described a model study concerned with the location

of the spillway and power house for four reversible pump turbine

units of a pumped storage reservoir project in the U.S.A. Two

models were constructed, initially a 1:400 undistorted model which

was used to gather information for the second, a vertically exag-

gerated and considerably more comprehensive model with a horizontal

scale of 1:400 and a vertical scale of 1:60. The jet penetration

and overall mixing were studied by dye injection for both the

pumping and generating cycles by the use of time-lapse colour

photography and cine film.

Density-stratified man-made lakes (reservoirs) have been studied by

Wunderlich and Elder (1971) for the Tennessee Valley Authority who

manage some 30 major reservoirs with a total water surface area

of 2400 Km2, a total water volume of 27 x 1012m3 and depths of up

to 137 metres. The essential function of most of these reservoirs

is the regulation of large temporary run-off for flood control

31+ and other purposes with the releases from these reservoirs being steady or intermittent and of varying volume. The reservoirs being so deep are prone to stratify and this, together with the irregular and unspecified inflow and outflow, makes the flow mechanics quite complex. The study was concerned with a laboratory and field research programme to improve the understanding of these complex flows and thereby to aid and enhance the management of the reservoirs. Theoretical as well as field and laboratory data was analysed, the field measurement of the water currents being made by using a deepwater isotropic current analyser with a measur- -1 ing range from 0.001 to 0.3 ms . Their results were inconclusive and many discrepancies encountered remained unsolved particularly pertaining to selective withdrawal and the water movement of the inflows.

In a somewhat similar study Grace and Brown (1975) coupled physical and mathematical models of density-stratified reservoir flows in an attempt to improve the understanding of these flow mechanisms for the management of existing reservoirs and design of proposed reservoirs. Dynamic and unsteady state meteorological, hydrological and operational characteristics were modelled together with the selective withdrawal properties and potential of various regulating structures. The prime purpose of the research was to simulate various physical, chemical and biological regimes of specific reservoirs and then to extrapolate the findings so as to predict the performance of an existing reservoir under various operating conditions or similarly for a hypothetical reservoir.

An experimental investigation where the stratification below the

35 thermocline was modelled has been reported by Darden, Imberger and

Fischer (1975). A line jet was discharged at mid-depth below the thermocline in a density-stratified prismatic rectangular two- dimensional laboratory tank. They discovered that the jet reached an equilibrium level and then moved horizontally towards the rear wall of the tank pushing fluid ahead of it and forming a system of reversing internal jets throughout the model. The pattern of induced flow showed no change when the model length (ie. the Lfh ratio) was varied and very little when the jet discharge was varied. These results are interesting in the light of the findings of Sobey and

Savage (1974) and Ali et. al. (1978a) who, for homogenous reservoir models, found the aspect ratio (ī,,/'h) to be very important for the flow patterns. The discrepancy here may possibly be due to the thermocline having an effect such that it suppresses any changes in the flow patterns.

There are many other studies and descriptions of thermal stratification and its effects (eg. Harleman, 1961; Dake and Harleman,

1969) however the relevance to this study is purely academic, the study being concerned with homogenous reservoirs, but is mentioned here for completeness.

2.1.2 Field Studies

The field studies dealt with in this section fall into two cate- gories, the first type can be termed specific where the reservoir was chosen and visited with the express purpose of making particular measurements as part of a special study. The second type are not really studies but reports on the operation of a reservoir based

36 on routine measurements taken during the day-to-day management of the reservoir. The experiences gained over a number of years of operating reservoirs with momentum jet inlets can provide much information useful to the design of future reservoirs. As such they are, in some ways, more significant than the particular studies as they can reveal whether a reservoir is behaving as expected and predicted during design or from model studies.

Priha (1969) has described field measurements taken in the Silvola

Reservoir. This reservoir is similar. to many British pumped storage reservoirs being approximately 1Km. long and 500m. wide and having a fairly uniform water depth of about 15m. It is kept in continuous artificial circulation by three momentum jet inlets directed tangentially at a depth of approximately 6-8m. from three inlet towers.

In order to determine the existing velocity magnitudes and their

directions measurements were taken at various depths and at various

points throughout the reservoir by using a current meter. The results were compared with theoretical flow velocities within the reservoir on the basis of forced vortex flow in a circular basin

of uniform depth (and neglecting frictional losses). The agreement

was quite good, considering the assumptions made and the circulation

patterns were in good agreement with those determined in the original

model study. Unfortunately no mention is made of the weather con-

ditions, eg. windy or calm, or stratifications during the study.

A noticeable feature of Priha's results is that the variation of

horizontal velocity with depth is quite small emphasising the

essentially two-dimensional nature of the flow in such shallow

basins.

37 Seager (1972) also attempted to make field measurements but by tracking floats in an existing reservoir at Wraysbury, a reservoir typical of the large pumped storage schemes which use momentum jet inlets to create an artificial circulation (see table 5-1). His

His work was hampered by the not uncommon practical difficulties encountered when using floats in the field as well as by a limited time constraint. However, his work is of interest and some very useful observations were made, concerning different aspects of the currents in the reservoir. He measured currents in the range

27-144 mms1 in the northern circulation and noted their unsteadiness and variation with depth, he also noted, by the use of surface floats, that the flow in the uppermost 2m. of the reservoir was strongly influenced by the prevailing wind with short-circuiting

(see section 1.2.3) sometimes occurring. The theoretical circulation velocities calculated in the same way (forced vortex theory) as by Priha (1969) were of the same order of magnitude as the observed velocities.

Ali et. al. (1978a) have reported some fieldwork undertaken in conjunction with their laboratory study of two reservoirs near

Liverpool. The total area of the reservoirs was covered by a rectangular grid of 30m. x 30m. mesh which was determined by theodolite surveying from five theodolite stations, later used for the location of surface floats and temperature stations. The vertical temperature profile was determined during the summer of

1975 and the greatest temperature difference between surface and bed found to be 1.7°C, thus the entire reservoir water may be supposed to be in the epilimnion; this is not surprising since according to Steel (1975) thermal stratification can only be ex-

38 pected when the reservoir depth is >10m. surface currents in the two reservoirs were determined by tracking surface floats from

two theodolite stations, the wind speeds were measured at heights

of 5-10m. above the water surface using a cup anemometer. The

surface currents measured were in broad agreement with the findings

of Harleman, Bunker and Hall (1964) who stated that 'the current

velocities at the surface are about 2% of the wind speed measured

30ft. above the surface'. In the field these wind-induced surface

currents were 4-16 cms 1 corresponding to model velocities of _ t 34-136cmrun. for their model shown in fig. 5-20(a). It is clear

from examination of fig. 5-20(a) that these wind-induced currents

would dominate the surface flow patterns, however, experiments

undertaken by the author and described in Chapter 5,.where depth-

average velocities were measured, indicate that the jet-induced

velocity field is substantial and consequently the effect of the

wind overall may not be so profound. In any case Ali et. al. were

concerned with the 'worst' case which would occur when there was

no wind, the currents induced by a wind were considered to be only

beneficial.

Some field studies concerned with.reservoir mixing but in particular

the use of unconfined air injection to destratify and improve

reservoir water quality have been described by Tolland (19?7) and

Tolland, Davis, Johnson and Collingwood (1977) for Sutton Bingham

Reservoir, (2.6 x 106m3) where an established thermal stratification

was destroyed in two days and the dissolved oxygen of the water

improved from 20% to 80% saturation, and for various reservoirs by

White (1978) and Davis (1978). They are mentioned here for reference

although the work described was concerned with a remedial rather than

39 preventive measure.

Steel (1972 and 1975) has described the management of the Thames Valley reservoirs of the Thames Water Authority, and in particular the Queen Elizabeth II reservoir at Walton, and has made some very interesting observations as a result of the experience gained after operating these reservoirs over a number of years. The reservoirs are all artificially circulated at certain times of the year and are thus very often subject to mixing forces due to the throughput and also due to the wind work at the reservoir surface.

He has noticed that greater vertical mixing work is done by a wind when artificial mixing is also attempted and suggests that there may be some interaction between the two which allows more of the wind work to be used in mixing; he has demonstrated this by the use of real reservoir data obtained during operation.

Steel defined the total amount of work required to maintain isothermal conditions in a reservoir during any part of the critical heating period (March-August) to be equal to the sum of the work required to produce the observed distributions of the summer heat income (assuming that all the heat enters the reservoir as a result of solar energy warming the water surface) and the work required to produce isothermal conditions in a stratified water mass.

The surface wind work was evaluated from the surface stress which was, in turn, calculated from the wind speed. Fig. 2-1 illustrates the total'surface work'of the wind, the total incident radiation, the'mixing wind work' and the stratification stability from two reservoirs, King George VI and Queen Elizabeth II, from March to mid-August in a particular year. The reservoirs are of similar

4o Total 70 70 Surface Total Windworl~0 Radiation Zww 60 3 M Io (x 10 50 50 (x103 gm-cm. cal.cm-2 ) cm 40 40

mar: apc may June july aug.

a Total mixing work requirement a 'Mixing wind work c Stability

Figure 2-1

Summer energetics of King George VI and Queen Elizabeth II Reservoirs during March-August, 1965, (Steel, 1972)

41 size, 16m. and 17.2m. deep and 1.3 and 1.4Km2 in surface area

respectively. The only difference being that during this critical

period the Queen Elizabeth II reservoir was subject to the effects

of a steady throughflow (0.8% of the total volume per day) through

jet inlets. From the graphs in fig. 2-1 it is quite apparent that

greater mixing occurred in this reservoir during this period there-

by giving weight to Steel's theories.

Steel (1975) has also used buoyant jet theory to analyse the re-

lative performances of the three principle jet inclinations used

in the reservoirs of the Thames Water Authority. Although in

model studies a neutral jet is used it is generally found that

the inflow from the river is warmer than the ambient temperature

of the reservoir water and hence the jet is buoyant. The trajectory

of such a jet is fundamentally dictated by a densimetric Froude

number:

F = U 0 (2-1)

"Y gD06, where Do = diameter of jet orifice

IIo = jet efflux velocity

and A p = relative density difference

Assuming the decision has been made to use jets in a reservoir, the choice lies between the inclination of the jets, the discharge per jet and the value of the densimetric Froude number, the latter two are interrelated by the discharge velocity at the jet orifice.

In controlling thermal effects in the reservoir the jets are required to perform two fundamental tasks:

42 1)they must entrain deep reservoir water and carry it

to, or near to the water surface,

2)they must cause the mixture formed to flow away from

the mixing zone and thereby prevent a local re-circulation.

In order that these objectives can be met it is necessary for the longest possible entraining length to be produced in the deepest waters, with the jet trajectory being such that the mixture even- tnally rises to the surface but also such that the jet has as large a horizontal component of momentum as possible. Abraham

(1965) has suggested that momentum jets become buoyant plumes if

S/D° ? 3F, where S is the trajectory length,hence providing this inequality is satisfied theory pertaining to momentum jets can be used to describe the flow. For the Queen Elizabeth II reservoir

(0.92m. diameter jets)

F = 0.33U o (2-2)

)77 -

For given jet dimensions the discharge per jet determines U0 and hence broadly the cost of jetting by generating a velocity head

(frictional losses being ignored). The value of Uo chosen must be large enough to imply a momentum jet but also low enough to ensure that an excessive velocity head is not generated. Relatively arbritary constraints may also have to be imposed so that the jet may be defined, eg. a minimum axial velocity may determine the jet end and then this value may not be allowed to occur until the jet reaches the surface. Abraham has given:

43 U = U0 6.2 (2-3) S/o

2gH max U 0.33 S JD0 g D 0

for a > (0.5 U min)2 Do g

and 2 g H max . U 0.16 S U min

N( D 0

for dQ ~, (0.5 U min)2 D g 0o

Since S, the possibly curved relative path length of the jet, is

D 0 itself a function of F it is necessary to assess the likely trajectory in order to determine its magnitude. Bosanquet, Horn and Thring

(1961) have given, for an inclined jet where ej is the angle to the horizontal

Z X sec ej sine j + 0.048 X sec e j 2 Do Do F2 Do

Steel (1975) has observed, during calm weather at the Queen Elizabeth

44 II, that the 'boil' where the jet is seen to have reached the surface agrees quite well with the predicted position derived from equation (2-4). He has also used the relationships (2-1) to (2-4) to suggest that horizontal momentum jets can be successfuly and economically used for 30k; F 4:i:100-200 and generating 0.25 - 0.5m. velocity head, this represents approximately 2-3% extra on the cost of raising the river water against the reservoir's head. If

D 5 x 10-5 (0.5° temperature excess) during the critical heating period horizontal jets will not prove as successful as inclined jets, however, an inclined jet could be used above this threshold value of Ai. In general horizontal jets are preferable as for 22.5° jets the horizontal momentum is only 92% of the total and more importantly the entrainment potential of 0° jets is vastly superior to acutely inclined jets particularly for deep strata.

A horizontal jet may start entraining immediately whereas a 22.5° jet will not start entraining until it has reached 3 diameters above the discharge level and will thus have difficulty in drawing bottom water into the jet stream especially as the bottom roughness will induce a stress which tends to inhibit lateral flow. As a result of this Steel has recommended that if inclined jets are to be used it would be best to sink them into the reservoir floor so that they discharge at bed level.

During'the April - June quarters of 1973 and 1974 the greatest temperature excess of river water over Queen Elizabeth II water was 5°C. but in general was less. In 1974 when Addp ‘ 1 x 10-3 the reservoir was operated with inflow via three 0° jets from

April - September and was kept isothermal, the dissolved oxygen distribution suggested very good vertical mixing. Steel (1975)

45 has found that as long as the reservoirs are isothermal by mid-

June there is no possiblity of a long term well-established

stratification. At Wraysbury reservoir experience has shown that if 22.5° jets are used when there is a NNW wind and the value

of F is such that the jet travels directly to the surface the

mixture can drift very quickly to the outlet. This is a case

where the inclusion of 0° jets, ensuring a maximum dilution before

the mixture reaches the surface, would have been worthwhile.

Clearly it is necessary to have a combination of jets of different

sizes and inclinations and even directions to ensure that all

inflow and weather conditions can be met. Despite the added

cost of construction the provision of a multiple choice could

prove invaluable.

46 2.2 ANALAGOUS FLOW SYSTEMS

Since there is little literature available which deals directly with reservoir circulation and mixing problems it has been necessary to survey literature on different flow systems but with similar properties. For example much literature exists on oceanic circulation and is referred to in texts such as Neumann and Pierson

(1966). The overall general circulation is induced by wind action over the ocean surface with the well-known 'western intensification', a large feature of the circulation, being caused by the curvature of the Earth (known as thefleffect). These two effects, however, would not be expected to significantly effect reservoir circulations.

Lake circulations are somewhat similar in some respects to reservoir flows, lakes being wide and relatively shallow and although the major currents are induced by the wind the throughflow, although usually being small, does contribute to the overall circulation in some cases. The principle differences between reservoir and lake circulations occur firstly, due to the larger lake surface area making the effects of the surface wind stress on lakes overall more important than on reservoirs and secondly because the natural lake throughflow has a much lower momentum than an artificially pumped reservoir inflow.

2.2.1 Lake Flows

2.2.1.1 The Generation of Circulation by Wind

The precise manner by which momentum is transferred from the atmospheric boundary layer by the wind to a water surface of

47 large area is not completely understood. The effect, however, is the- generation of two basic types of motion, waves and currents, which baize differing time scales and are not normally interactive.

Wave motions are of very limited interest as far as large scale reservoir circulations are concerned and will be neglected in this thesis.

The currents generated by winds have been shown to be directly related to the surface wind stress by Ekman (1905). He assumed the Coriolis parameter to be constant and that stratifications, horizontal momentum transfer and convective accelerations could be neglected. Ekman was primarily interested in wind-forced oceanic circulation where the water is quite deep. His analysis led him to propose the now classic Ekman spiral description of momentum transfer from a surface wind stress. However, his assumptions are now generally accepted as being invalid for deep

water but Welander (1957) has argued that such Ekman-type dynamics,

when generalised, can be appropriately applied to small shallow

seas (eg. lakes). This generalised Ekman-Welander theory has

been applied to lakes by several authors (eg. Liggett and

Hadjitheodorou, 1969- and Liggett, 1969).

The form of lake currents is controlled by the same forces as

the wind ie. friction, Coriolis and any externally applied forces

(which for lakes is normally the action of the wind on the water

surface). The mechanics of lake currents are further complicated

by the proximity of shores, with the resulting deflection of currents,

by the tendency of lakes to stratify and by the fact that any

change in wind speed or direction can induce water surface

48 oscillations (seiches). Internal seiches due to the oscillation of the hypolimnion can also occur. Smith (1975) and Hutchinson

(1957) have given good accounts of the flows occurring in lakes.

The classic Ekman spiral theory assumes that spiral motions extend right down to the lake bed, so that, at a certain depth, the flow is in an opposite direction to that at the surface. Alternatively with a strong wind blowing over shallow water the Reynolds stresses may be at least as large as the Coriolis force so that the current velocity is more or less constant in direction although the variation of velocity with depth may be complex. In long deep unstratified lakes where there is return flow and where the shape restriction inhibits the Coriolis deflection the general current form should conform approximately to that derived by Hellstrōm (1941) shown in figure 2-2.

At the other extreme is a shallow circular lake where Csanady

(1968) postulates the general form of circulation to be as shown in figure 2-3. Neither the extent of the Coriolis deflection nor the form of the velocity profiles is known for this case.

The nature of the current patterns in irregularly-shaped (in plan) reservoirs where there may be return currents in the deeper parts of the reservoir as well as the shallow is clearly more complicated than the two preceding simple cases. In large basins where the

Coriolis force due to the Earth's rotation is significant complex surface and internal oscillations can occur which do not remain stationary but rotate around the basin. A theoretical analysis of the response of an idealised circular lake has been carried out by Birchfield (1969).

wind

Figure 2-2

Wind-driven circulation in a long,deep, unstratified lake

50 Figure 2-3

Wind-driven circulation in a shallow circular lake

51 2.2.1.2 Laboratory and Mathematical Studies

Much of the present day information on lake circulation has been acquired over the last fifteen years mainly as a result of the need for water quality management of lakes and in particular the

North American Great Lakes. The decline in the water quality of the lakes was largely due to their use as a medium for waste water disposal but increasing recreational demands required an improvement and research was undertaken to this end.

It had been suggested that the Great Lake currents were caused by the hydraulic gradient from west to east and that there was no clear evidence of large scale geostrophic effects. This last statement was shown to be incorrect by Verber (1966) who demonstrated the presence of oscillations at near inertial frequency. The

Coriolis force due to the 1arth's rotation has been shown to far outweigh the inertial forces in steady or slowly-changing currents by Csanady (1968) and also the hydraulic gradient at the centre of

Lake Michigan has been shown to be infinitesimal. The water movements in the Great Lakes can therefore be expected to depend on the following essential dynamic factors:

1)the wind stress exerted on the lake surface,

2)the continuity constraint imposed on water movements

by the boundaries, eg. the shores of a closed lake,

3)the Coriolis force due to the rotation of the Earth.

The throughflow is not considered to contribute significantly to the currents occuring in the Great Lakes as, for Lake Michigan,

computations indicate that the annual major outflow at the Straits of Mackinac is only 1% of the total volume of the lake (Harleman,

Holley, Hoopes and Rumer, 1962). This can be contrasted with a

52 typical large pumped storage reservoir such as the Queen Mother

Reservoir at Datchet where, under normal operating conditions,

1% of the total volume passes through the reservoir in less than one day (table 5-1). The surface areas vary considerably between lakes and reservoirs, for Lake Michigan the overall length is approximately 500Km. (mean depth 80m.) and for the reservoir the maximum length is approximately 1.5Km. (mean depth 20m.). The mechanism by which wind energy is transferred to a reservoir must be basically similar to the mechanism for a lake such as Lake

Michigan, however, the induced currents will be additionally influenced by the interaction with the throughflow, the increased constraining action of the boundaries and to a lesser extent by the Coriolis force.

Narleman et. al. (1962) reported a feasability study on the possible use of a dynamic model of Lake Michigan and concluded that such a model was feasible but that its success would largely depend on the availability of the field measurements needed to verify the model. This verification is particularly necessary in physical hydraulic models where wind effects are to be'included, the normal way- in which the wind is correctly scaled is to use the relationship between prototype lake currents and the wind speed to determine the model wind speed that induces the correctly scaled model currents. Harleman et. al. also pointed out that previous experience with rotating laboratory models, a necessary provision if all the factors affecting lake currents are to be included, was limited. This was a major influence on their decision to construct

a small scale pilot model before proceeding with a larger model.

They considered hydrological circulation but concluded that the

53 throughflow of 1% per annum would make it insignificant compared with meteorological circulation. The results obtained from the pilot model study (Harleman et. al., 1964) showed that the major circulation patterns could be reproduced and that the modelling of gravitational and Coriolis forces was possible, in addition the rotation of the model produced major changes in the surface current pattern. Their intention was tb then build a larger model and to perfect it by using prototype data, the model could then be used to determine critical points for investigation in the prototype. Unfortunately the results from the larger model are not at present available, to the author's knowledge.

There are numerous studies of wind-forced lake circulation in the literature, however, many of these are theoretical studies where mathematical models have been used with only a few physical studies being reported. Rumer and Robson (1968) and Rumer (1970) have reported a physical model study of the wind-driven circulation in

Lake Erie where the results obtained were compared with field observations revealing 'many similarities but some notable differences'. They also noted that the throughflow generated significant patterns.

Whilst not being physical model studies of particular lakes Baines and Knapp (1965) and Mortimer (1951) have done work on the transfer mechanism of the wind stress to the water surface whereas Moretti and McLaughlin (1977) have described the results obtained from a hydraulic model of mixing in a particular stratified lake. The size of their lake and the rate of inflow are more typical of reservoirs than lakes and consequently the study is of particular

54 interest. Their investigation was intended to develop the modelling of stratified lakes to the point where reliable predictions of prototype lake mixing phenomenons could be made confidently and with few similar studies documented their initial work was concerned with deciding upon the model criteria which best described the particular prototype situation. Their hydraulic model was vertically exaggerated to preserve turbulent flows with, since the model was -a to be stratified, the overall Richardson number (J = g J QL/ p 2) and the Reynolds number (Re = IIL/u ) chosen as the two important non-dimensional parameters. However, the choice of the characteristic length posed certain problems; the lake depth would normally be chosen but Fischer and Holley (1971) have pointed out that, for steady flow in distorted hydraulic models, the dispersive effects of the vertical velocity gradients are magnified and the transverse gradients diminished. Experiments were undertaken to resolve this point and Moretti and McLaughlin showed that for this type of problem complete similarity cannot be achieved in vertically exaggerated models but that selective analysis of the data could lead to similarity in either the vertical or lateral dispersion alone. A model of a particular lake was constructed and a prototype destratification process which used an inverted propeller, was modelled. Previous field experiments had shown that vertical

mixing induced by the propeller was the most important physical

process. Comparison of the respective density profiles showed

that the modelling was an accurate representation of the process

with the non-dimensionalised destratification time predicted by

the model being within 15% of the prototype value.

55 On the question of the use of distorted hydraulic lake (and for that matter reservoir) models one of the unscaled forces, the frictional force at the model boundary, is very important in the prototypes and with their also being a sparsity of good field data the usual methods of model verification can prove difficult.

Shiau and Rumer (19?3 and 1974) have contributed two papers to the body of literature in which they discuss the possibility of using the decay of mass oscillations (seiches), as given by the non- dimensional Proudman number (() 2/gh5 k s2), to check the validity of a hydraulic model; complete similarity would be present when the Froude number and Proudman number were equal in model and prototype. The method needs further analysis and verification but may offer a solution to a persistent difficulty.

King and Rhone (1975) have published details of an unusual situation where excessive throughflow induced circulation was undesirable.

They were concerned with two interconnected high mountain lakes in the U.S.A. which both tended to stratify during the summer and were dimitic with natural mixing in spring and autumn. The lower lake produced large populations of freshwater shrimp providing a food source for a valuable lake fishery and its bed was covered with a fine rock material known as glacial flower. The study was initiated in response to the imminent construction of a pumped storage power plant and the possible environmental effects of this construction were investigated. Several potential adverse environmental effects were studied notably destratification, re-suspension of the glacial flower causing high turbidity and entrainment of fish and their subsequent passage through the pump turbines.

Three hydraulic models were constructed, the first, with scales

56 of 1:6000 horizontal and 1:84 vertical, was used to demonstrate destratification effects, the second, 1:600 and 1:100 respectively, was used to study both destratification and circulation patterns and the third, an undistorted model, was used to study the near field characteristics of the jet and to develop the design of the tailrace channel. Model current velocity measurements were made by an electromagnetic current meter and circulation patterns and jet movement were detected by single-frame photography of dye clouds. The experiments confirmed that de-stratification would occur and that there would be a noticeable movement of the bed sediment, as a consequence further investigations are under way.

Amongst the numerous mathematical studies of wind-forced circulation in lakes Gedney and Lick (1971) have described a mathematical model of Lake Erie and compared their results with field measurements.

The three-dimensional steady state wind-driven water velocities were calculated numerically as a function of depth and horizontal position using a shallow lake model (that is using an 'Ekman-

Welander' type of analysis with an eddy viscosity model for the vertical turbulence structure) with prevailing SW winds. The velocities were found to vary considerably from position to position and to be strongly dependent on the bottom topography and the boundary geometry. The predicted velocities and average eddy viscosity measurements were in reasonable agreement with field measurements taken at mid-depth.

Bonham-Carter and Thomas (1973) had similar success with Lake

Ontario and the Rochester Embayment and Cheng and Tung (1970) used a finite element technique to solve the equations of motion

57 in their mathematical model of lake circulation. The use of the finite element technique in solving the equations of motion (as opposed to the alternative finite difference technique) has the advantage of being more suitable for lakes with irregularly shaped boundaries and in this case allowed special attention to be paid to geometric, effects on the circulation patterns;' unfortunately the validity of the model was not tested.

An alternative to the Ekman-Welander 'eddy viscosity turbulence model' has been used by Murty and Rao (1970). A steady state linear model was devised with the effects of bed topography and

Coriolis force taken into account but the model was homogenous

(ie. stratifications were not included). The lake coastlines were approximated by a grid and a finite difference method used to solve the equations of motion.. A depth configuration was considered by vertically integrating the equations of motion and the equation df continuity after making the hydrostatic approximation for the pressure field. By doing this it is possible to specify, quite simply, the bottom friction stress as opposed to other models where it is necessary to prescribe the nature of the vertical variation of the eddy viscosity.

Hamblin (1969) presented a mathematical model of a 'hydraulic'

circulation forced by a throughflow. He followed the depth-

averaged analysis of Birchfield (1969) of wind-forced lake

circulation but included the horizontal momentum transfer (eddy

viscosity) terms whilst setting the surface wind stress to zero.

The model was linear, the convective accelerations being ignored,

58 and the additional equation for bed shear being taken as EIIQnan's deep ocean solution. Sobey (1973a) has since shown that Hamblin's stream function equation reduces, to a good approximation, to

Laplace's equation and indeed his constant depth stream pattern looks very similar to a potential flow. Unfortunately Hamblin does not seem to have recognised this.

Uzzell and Ozi9ik (1977) have described a three-dimensional mathematical model for the prediction of the circulation in the far field region of a lake. The lake was studied in connection with a project for the supply of cooling water for electric power plants.

The results were presented in the form of temperature and velocity fields after analytically solving Laplace's equations derived for a constant depth lake with a uniform surface wind and a constant horizontal temperature'gradient, by reducing the equations of motion.

Csanady (1968) has given a good review of analytical and numerical . studies of model Great Lakes and other studies include Liggett

(1970 and 1975), Gallagher, Liggett and Stevens (1973), Lee (1972),

Lee and Liggett (1970), Csanady (1967, 1970 and 19?5) and Bye (1965).

2.2.1.3 Field Studies of Lakes

The evaluation of both physical and mathematical models depends to a large extent on their relative success in predicting real lake circulations. The collection and analysis of the necessary field data is a daunting task for reservoirs let alone for lakes with surface areas of several hundreds of square miles. The problem is further complicated by the unsteady nature of the prevailing wind fields.

59 Amongst the few successful studies Harleman et. al. (1964) mentioned a field study where measurements of currents were taken in Lake

Michigan. The fundamental relationship of wind speed to induced current was shown to agree fairly well with the 45° shift predicted by Ekman (1905). Current velocities at the surface were normally about 2% of the wind speed measured 9m. above the surface.

This calculation was based upon an average wind speed of 5ms1 and, an average current of 25cros1.

Progress in gaining field measurements has also been made by

Yeske, Green, Scarpace and Terrell (1972) who have published a preliminary report of current measurements in Lake Superior using aerial photography to track surface drift cards. The paper records the first analysis of 379 current vectors (less than 3% of the total) and these compare well with sample results obtained by using drogues.

2.2.2 Other Studies

There are a few other studies which are of some relevance to the present investigation: Fossett and Prosser (1949) investigated the'time for complete jet-induced mixing of fluids in large circular petrol storage tanks. Their investigation showed that a simple jet directed across a diameter of the tank caused complete mixing in a time approximately equal to 8DR2/ Qo U0 , they also discovered the necessity of having a sufficiently large vertical component of the jet velocity to avoid stratiication. Unfortun- ately, although their study is of interest, the extrapolation of their results to water supply reservoirs, which are relatively very much wider, would be rather questionable.

6o' Iamandi and Rouse (1969) were concerned with the design of submerged jets and bubble screens and decided to hydraulically model the manifold releasing the bubbles. They soon encountered measurement problems, finding the water velocities too small to be measured by using conventional equipment such as pitot tubes. As a result they changed from water to air and used a hot-wire anemometer to measure the air currents. Their principle conclusion was that submerged jets provide an alternative, more adaptable, means of maintaining a pattern of flow for which bubble screens are often used.

An experimental study of the influence of a free surface on a horizontally-orientated axisymmetric turbulent jet has been reported by Maxwell and Pazwash (1967). The locus of maximum velocity was found to be attracted towards the free surface and the lower part of the velocity distribution closely resembled that for a deeply submerged jet; the upper part was affected by wave action and was seen to spread at a faster rate. They subsequently extended this work (Maxwell and Pazwash, 1970) to include the effect of horizontal bed, a similar attraction and increased growth rate was observed at the solid boundary. Both the free surface and the bed tended to attract the jet and both caused an increased spreading of the flow but one boundary generally proved to be dominant, depending on the geometry.

On the subject of particular studies of jets there is much literature dealing with many aspects of jet theory. Sobey (1973a) has given a good review of many of the analyses and other studies include Abraham (1963 and1974) and Abraham and Eysink (1969).

61 For the purposes of this study the well-established jet theory has not been extensively reproduced but used when necessary in the general text (eg. chapter 5).

62 2.3 THE NECESSllf FOR RESEARCH

As can be seen from the preceding survey the literature available pertaining to jet-induced reservoir circulation is limited with little work having been done, on the effectiveness of jet inlets in preventing the formation of stagnant areas and short-circuiting.

In the past the design of reservoir inlet systems has been based on scale hydraulic model studies and the use of largely unproven theory. The hydraulic modelling has been simple by any standards involving the observation of the flow patterns shown by dye for different jet configurations (Metropolitan Water Board, 1965 and

1967) with no attempt being made to measure velocities in the models.

Sobey (1973a) has made a major contribution to the literature on reservoir circulation and mixing but his study was primarily theoretical and used a simple mathematical model to predict reservoir flows. His attempts at experimental measurements in a scaled model were hampered by his inability to measure the very low .velocities occuring, -to overcome this he .distorted the velocity scale in order to obtain measurable velocities, his experimental results must therefore be questionable. Falconer (1976) also mathematically modelled reservoir flows but his model was more sophisticated and probably more representative of real reservoir flows. His results are interesting but unfortunately were not satisfactorily verified by field or hydraulic model data.

63 The question of the scale of hydraulic reservoir models has consistently been raised (Moretti and McLaughlin, 1977; Fischer and Holley, 1971; Sobey and Savage, 1974; and most recently Ali et. al., 1978a) particularly where, in order to preserve turbulent

flows, a vertically exaggerated model has been used. In addition the relationship between model and prototype flows in reservoirs is largely unproven: _

With these points in mind and obvious gaps in the knowledge on reservoir flows apparent it was clear that there was plenty of

scope for further research particularly in the form of a detailed

hydraulic model study. Such a model study had the aims of over-

coming some of the previous problems of measurement of the_low

water velocities, scale effects etc. and to clarify the signifi-

cance of possible geometric and dynamic parameters of the reservoir

and its inlets and outlets. It should be borne in mind that the

study of All et. al.. (1978a) did not become known until March 1978

and unfortunately some of the work was duplicated.

It is surely not sufficient to be content that the operation of a

reservoir is improved by the introduction of the inflow as momen-

tum jets, the benefits should be maximised and the complex processes

more fully understood as the benefits of improved knowledge could

be operationally, economically and ecologically significant.

64 CHAPTER 3

Thii HYDRAULIC TURNTABLE MODEL

65 INTRODUCTION

The 5m. diameter rotating turntable in the Civil Engineering

Hydraulics Laboratory at Imperial College was originally designed as a permanent feature of the laboratory when the new buildings were constructed in 1963. At the initiation of this study in

May 1975 the arrangement of this piece of apparatus was much as it was left in 1973 by Sobey (1973a) and is shown in figure 3-1.

Various changes, which will be described in this chapter were made between May 1975 and October 1978 with the turntable finally being as shown in figure 3-2..

The turntable is based upon a large roller race on which a rigid framework, supporting an experimental basin, is mounted. Rotation of the basin is via an electronic-hydraulic friction drive and the model has a self-contained pumped water supply.

66 Figure 3-1 General view of experimental turntable (1973)

67 Figure 3-2 General view of experimental turntable (1978)

68 3.1 MODEL ROTATION

The novel feature of the experimental facility is the ability to rotate the hydraulic model during experimentation. The rotation

of a hydraulic reservoir model, where large water areas in the

prototype are concerned, is desirable in order that the Coriolis effects, due to the Earth's rotation, may be included, particularly as the scale relationships applicable to a reservoir on a rotating

earth dictate an increased rate of rotation of the model. Sobey

(19?3a) has described this feature in some detail , however, in

view of various modifications and improvements made during this

study it is worthwhile to re-capitulate and to record the updated

aspects.

3.1.1 Ring Beam and Rollers

The original mechanical engineering design and construction of the

turntable was based on a 4.88m. diameter ring beam that rotates

in a horizontal plane on twelve rollers equispaced in a circle and

with their mountings firmly fixed to the laboratory floor. The

ring beam is constructed from eight individual sectors and rotates

about a conical bearing at its hub with structural rigidity

maintained by sixteen evenly-spaced 32mm. diameter steel spokes.

The rollers are designed as a section of a right circular cone

which can rotate on an inclined axis such that the upper surface

(in contact with the ring beam) is always horizontal and such

that one turn of the bearing screw thread results in a 0.075mm.

vertical adjustment of the upper surface of the roller. The

maximum vertical adjustment of a roller is 0.11mm. beyond which

69 the whole roller assembly must be shimmed. To aid the levelling of the turntable Sobey marked the twelve rollers R1 to R12 and the sixteen spokes I to XVI.

3.1.2 Levelling of Turntable

The design of the turntable incorporates individual vertical adjustment of each of the twelve rollers over which the ring beam rotates, this is achieved by turning the screw thread such that the cone moves in or out. To attain steady rotation of the model it is necessary for all the rollers to always bear simultaneously and for the upper surface of each roller, where it is in contact with the ring beam, to be in the same horizontal plane'. In practise this is an impossible situation to achieve although it is possible to approach it quite closely by determining the elevations of each of the rollers to precise tolerances and then by adjusting each roller accordingly.

Sobey (1973a) devised a successful technique for determining the elevations of the rollers, which was again used for this study, by using a water manometer- with both water surfaces open to the atmosphere and with one side being formed by a very large glass aspirator bottle placed at the approximate centre of the turntable and with its large water surface area being assumed to have an effectively constant elevation. The other side of the manometer, where the measurements were made, is formed by a small water surface area glass-panelled container with a micrometer-gauged fine water surface pointer attached. The two ends are connected by a length of plastic tubing and the whole system filled with water. In re-levelling the turntable rollers it was convenient

70 to follow Sobey and use the spoke marked IX as the standard reference position above which the measuring side of the manometer was placed. The turntable was then rotated a little at a time with a micrometer reading of the water elevation being taken when spoke IX was above each roller in turn. At each position it was necessary to check that'the-ring beam was actually bearing on the roller and not merely being supported above this roller by adjacent rollers. The relative water elevation above each roller is a measure of the horizontal plane in which the roller is rttating.

When Sobey initially levelled the rollers in October 1969 he found a difference in elevation from one side of the turntable to the other in excess of 2mm. which had apparently been caused by differential settlement of the laboratory foundations. Considerable shimming of many of the rollers was necessary to bring them to a position where the required level was within the range of their screw adjustment, eventually, after much averaging of the height discrepancies,the maximum variation in level was reduced to

0.005mm.:

The levelling of the turntable rollers in May 1977, prior to any experiments including model rotation and before the calibration of the upgraded rotation drive, revealed that settlement had once more occurred but, fortunately, not to such a large degree; the maximum variation in level being 0.7mm. Since an improvement in the turntable drive unit made the elevation of the rollers less critical it was decided that the work required to shim the rollers would not be worthwhile and therefore the largest variation in level was reduced to 0.5mm. by the use of the maximum adjustment

71 Relative Elevation

mm. , ,

0.4 IMP

before levelling 0.2

after levelling

-02

-0.4 l , ; R1 2 3 4 5 6 7 8 9 10 11 12 Roller

Figure 3-3

Roller elevations before and after levelling

72 of 0.11mm. on each roller. The roller elevation before and after adjustment are shown in figure 3-3.

3.1.3 Model Bed and Basin

The model bed is supported on an octagonal frame of double RSJ's bearing on the ring beam and spanning to the central bearing of the turntable by means of eight single RSJ's within the outer double frame. The joists all have latticed webs which reduces the dead weight and minimises the vertical deflection.

The waterproof model bed is formed from an aluminium-bonded plywood top plate placed over the frame and the water-sealed model basin completed by building a 38cm. high sheet metal side wall to an octagonal plan. The largest possible inscribed circle has a diameter of 5.08m.

With there being a dormant period of several years during which the basin was only occasionally filled with water it was thought that the model bed, being basically of timber, might have warped during this period and so the relative elevation at each intersection point of a 40cm. grid, projected onto the model bed, was found. The levels were determined by the use of a standard surveyor's optical level and staff and are shown in contour form in figure 3-4. The maximum variation in level was 2mm. but in general the variation was less than 1mm., this was thought to be acceptable and to be unlikely to seriously affect the flow patterns in the model during the experiments.

73 Spoke XVI Spoke I

Levels in mm, Datum 0.0 mm, Grid Centreline

Figure 3-4

Model bed levels in contour form

74 3.1.4 Turntable Drive Unit

The drive unit existing at the beginning of this research project, designed by Hawker Siddley Dynamics Ltd., consisted of a hydraulic motor coupled to the turntable by means of a rubber-tyre wheel pressed against the turntable periphery (outer web of the ring beam) by a hydraulic ram. The ram force is balanced by the forces from two further rams which press similar rubber-tyred wheels against the ring beam at the two load roller units. Rotation of the hydraulic piston motor is slaved to that of a stepping motor.

Originally the stepping motor received its controlling signals from an electronic oscillator (described by Sumray, 1968) but this proved unsatisfactory both for the long and short term stability of the rotation and as a result several changes were made in this part of the unit during Sobey's research programme.

The short term stability problems were associated with minor track irregularities.and flexibility in the transmission and were generally considered to be mechanical engineering problems requiring extensive and costly work in relation to the small benefits to be derived. The long term stability problems were more serious and stemmed from the long term performance of the two oscillators controlling the speed of rotation.

Improvements made in 1972 were aimed at eliminating the principle points of flexibility in the transmission, monitoring the instantaneous turntable speed by means of a tachogenerator driven by the ring beam and deriving the long term stability from a constant voltage source. As a result of the improvements there was a general advance in the performance of the turntable drive unit but

75 the transient build-up time to a steady-state condition was around

5 hours and obviously further work was required to achieve satisfactory rotation.

With a new research programme underway investigations of the drive unit were renewed during 1976. It soon became apparent that the performance of the drive unit was directly related to the temperature of the oil delivered from a reservoir to the hydraulic motor, as the temperature of the oil increased the drive unit performance became more erratic and on a closer examination the oil temperature was found to be exceeding the design specification. To counter this effect a cooling system was installed which intercepted the oil before it was delivered to the hydraulic motor and maintained it at a constant temperature compatible with the design specification.

The cooling system consisted of a standard Serck oil -cooler with cold water as the cooling medium. The flow of water through the cooler was controlled by a Danfoss thermostatic water valve which responded to the rise and fall of the oil temperature. The thermostatic control was not only intended to prevent the oil from overheating but was also to ensure that the oil was not overcooled, which might have led to condensation problems. In view of the previous overheating of the oil the complete hydraulic system was drained and the oil renewed. With the installation of this cooling system the operation of the turntable drive was noticeably smoother and allowed further investigation of the electronic aspects of the drive unit to proceed.

76 The electronics of the drive unit were overhauled by Churchill

Controls Ltd. and involved moving the tachogenerator to the rim of the turntable, as opposed to part way through the drive. The stepping motor was also modified so as to include 16 steps per

90° as opposed to 2 steps per 90° and the feedback from the tachogenerator tuned through variable resistance and capacitance controls. These changes to the control unit produced a considerable improvement in the rotation performance of the turntable.

With the improvements to the rotation of the turntable complete it was possible to calibrate the select speed setting dial and the tachogenerator voltage (from an avometer) against the actual rotational speed in order that the turntable could be set to rotate at a required rate. The calibration was achieved quite simply by setting the speed setting to a series of suitable values in turn,-and allowing the rotation to stabilise (this only took several minutes). The tachogenerator voltage from the avometer was then recorded and the rotation rate determined by taking the average of several recorded times to complete one revolution. The calibration graphs are shown in figure 3-5 together with the relationships of select speed setting (Ss) and tachogenerator voltage (Tv) to the rate of rotation (R) as

determined by a linear regression analysis of the data. These relationships were used to ensure the correct rate of rotation

for the experiments where the model was required to be rotating;

either for the spin-up tests or modēl tests where the Coriolis

effects were to be included. The control unit, avometer and an

oscilloscope, used to monitor the short-term stability of the

rotation, are shown in figure 3-6.

77 Tachogenerator Speed Voltage (Tv) Setting (S,) 12 50

10 40

30 6

4 20

2 10

4 6 8 _10 12 14 16 .18 .20 22 _ 24 Rate of Rotation (Revhr ') (Rr) Figure 3-5

Calibration graphs for rotation controls Figure 3-6 Turntable rotation control unit with avometer and oscilloscope

79 3.2 PUMPED SUPPLY

3.2.1 Water Circuit

The throughflow to the model is provided by a 38mm. (12in.)

Safran ACE/E5 2900 r.p.m ill* centrifugal pump mounted underneath the model bed and such that it rotates with the model. The pump draws water from a circular reservoir underneath the turntable by means of a vertical delivery pipe fitted with a foot-valve to ensure that the line is always primed, the pipe and foot-valve also rotate with the model. The pump was originally used by

Sobey (1973a) who had it delivering to the model through a length of 38mm. galvanised steel water pipe via a venturi meter (for flow measurement) and being controlled by two valves in parallel. It soon became clear that this arrangement was intended for much higher flow rates than the author was expecting to use and was too insensitive at low flows to be modified. As a result the delivery side of the pump had to be re-designed.

3.2.2 Flow Measurement and Control

The original length of 38mm. (1i" tube) leaving the pump delivery side was retained but the valve arrangement and the final length of delivery tube were removed and by means of several interconnected reducing sections the delivery tube from the pump was reduced to a 6.35mm. (i".) nipple. To this was attached a length of 6.35mm. (i".) nylon-reinforced plastic tubing through a 9.52mm.

(fig".) Enots screw-down regulating valve, used for controlling and regulating the flow, fitted with 9.52mm. to 6.35mm. reducing nipples and connected via a further length of the plastic tubing to the bottom of a Ca1lenkamp Gapmeter Lab-kit rotameter which

8o was used for flow measurement. A final length of plastic tubing supplied the water flow to the model from the top of the rotameter to a nozzle fixed to the model bed. This inlet arrangement can be seen in figure 3-7.

The Gallenkamp rotameter is capable of measuring water flows from

7.5 to 5000 cm3min 1 by the choice of four graduated Pyrex glass measuring tubes of different bores and three different indicating floats. Although the instrument was supplied with a set of manufacturers calibration graphs it was thought necessary to perform an individual calibration for the single tube and two floats used by the author for measuring the inflow during this study, particularly as the ambient temperature of the laboratory (18°C approximately) was slightly less than the 20°C calibration temperature. The rotameter was calibrated quite simply by setting the float at each of the graduations in turn by adjusting the flow regulating valve and then measuring the volume of water collected during a certain length of time. The average of several measurements was taken for each setting. The resulting calibration graphs are shown in figure 3-8 and were used to fix the inflow rate during this study.

In view of the behaviour-of a buoyant jet it was also thought necessary to check that prolonged use of the centrifugal pump over a period of 5 or 6 hours did not significantly affect the temperature of the model inflow. The temperature of the inflow was taken at intervals over a period of nearly seven hours and the results are shown in table 3-1.

81 Figure 3-7 Delivery side of pumped supply

82 Rat am eter Scale Reading

hollow ss. float

solid s.s. float

1000 2000 3000 4000 5000 Rate of Flow ccrnin-i (18°C)

Figure 3-8

Rotameter calibration graphs

83 Table 3-1, Variation of Inflow Temperature with Time

Time Temperature (hours) of water (°c)

0 20.5 0.5 18.5 2.25 18.5 4.5 18.5 5.75 18.5 6.75 18.5

The results show that after 30 minutes the temperature settled down to the ambient laboratory temperature and that there was no variation in temperature over the following period, in excess of six hours.

3.2.3 Inlet and Outlet

The inlet used by Sobey was mounted on an adjustable frame with a length of 38mm. diameter plastic tubing providing flexibility in height and radial direction of the inlet. For this study the

plastic tubing from the top of the rotameter to the nozzle was

of a smaller diameter (6.35mm.) and was consequently far more

flexible, as a consequence it was possible to remove the frame

from the model. At the end of the tubing a nozzle of the required

size could be fitted and held in the correct position by a small

pipe clip screwed to the model bed.

The outlet consisted of varying lengths of 50.8mm. (2".) Durapipe

plastic tubing fitted together so as to allow complete flexibiltiy

of outlet type and plan location in the model. This suction

piping must pass over the basin wall and a suction tapping point

has been provided at a suitable high point to allow priming of

84 the line. The line must be primed whilst the turntable is stationary as the suction pressure is taken from a nearby venturi pump.

The depth of water in the model is controlled by the siphoned outlet which is connected to a small outflow tank fitted with an. adjustable overflow weir. The outlet arrangement is shown in figure 3-9.

85 Figure 3-9 Outlet arrangement

86 3.3 AIR-PRESSURED SUPPLY

Experimental difficulties created by the rotation of the turntable,

particularly due to the long spin-up time and experiment time,

dictate the provision of a permanent arrangement for dye injection

into the pumped supply especially where flow visualisation is

required. Sobey (1973a) has described in some detail the provi-

sions originally made for flow visualisation by using a compressed-

air reservoir to force dye from a tank, both the reservoir and the

tank being mounted on the turntable and rotating with it. During

this study flow visualisation using dye has only occasionally

been used to check or test all-.observation or a prediction from

the measured velocity fields. The same system has been used in

this study with the only modification to Sobey's original design

being the ommission of the rotameter which measured the rate of

flow of the dye into the model.

87 3.4 PROVISIONS FOR INSTRUMENTATION

Sobey (1973a) has pointed out many of the instrumentation problems created by a large scale rotating laboratory model. Firstly he mentioned the purely physical problems created by the large (by laboratory standards) width of the model and by its rotation. In addition by their nature model reservoir flows are typically very slow and susceptible to small disturbances. The measurement of the low velocities has been overcome in this study by the use of a particular technique which is described in detail in chapter 4 but it is convenient to describe here the alterations to the experimental rig necessary to make use of this technique.

3.4.1 Photography

The low velocity measuring technique makes use of photogrammetry and most of the modifications were involved with the photography of the model and provision of the ground control required for the photogrammetry. The original photographic tripod designed and used by Sobey was not suitable for use on this project, this was mainly due to the constraints imposed by the resolution and accuracy requirements of the photogrammetry. The Hasselblad ELM camera with its 80mm. planar lens was used and required an object distance of approximately 4.5m. to cover the model area with four photographs each taking a quadrant of the model (the reasons behind this are discussed in chapter 4). The original tripod was dis- mantled and a bridge, spanning the laboratory between concrete beams and with access at one end, was constructed from scaffolding poles supported on three tubular steel trusses and with a timber walkway. The bridge shown in figure 3-10 was designed to support

88 a substantial load including several experimenters. Four brackets on which the Hasselblad camera could be mounted are carried, two on each side of the bridge walkway, by transverse scaffolding poles projecting outwards and are clamped in such a position that a quadrant of the model is encompassed when a photograph is taken with the Hasselblad camera held in an approximately vertical position. The camera can be quite easily transferred from camera station to station and is easily accessible to an operator on the bridge being activated by him from the bridge or remotely from the model. The structure of the bridge is sufficiently stiff to ensure that the load due to the operator does not influence the camera.

Unfortunately this arrangement does not rotate with the turntable as Sobey's original tripod did, however, since the targets being photographed were fixed relative to the model, as opposed to being in motion when streak photography of floats was used by Sobey, this was not a problem. Poor definition and blurring of the photographs could be avoided by using a faster camera shutter speed. There were obviously limitations as far as the positions at which

photographs could be taken whilst the model was rotating but these

were not restrictive.

3.4.2 Grid

Sobey's original surveyed string grid across the top of the model

was used to fix a grid on the model bed for use with the tethered

buoys (chapter 4). This was achieved by making use of the string

grid and projecting these grid lines onto the model bed by using

a carpenters spirit level, the grid lines were recorded by a hard

pencil. The bed was painted white several times during the course

90 of this study but care was always taken to make sure that the grid could be re-drawn in the same manner. This grid of 40cm. mesh was used throughout the study as the basis for the positioning of the low-velocity measuring buoys.

3.4.3 Photogrammetric Control and Survey

As was mentioned earlier it was necessary to provide photogrammetric control in the model for the velocity measuring technique. For convenience the details of how this control was achieved and the survey are given with the velocity measuring technique in chapter 4.

91 CHAPTER 4

THE VELOCITY MEASURING TECHNIQUE

92 INTRODUCTION

A major difficulty encountered by workers modelling lakes and reservoirs has been the measurement of the low water velocities occuring in the model. In prototype reservoirs the water is, in general, very slow moving and currents of the order of cms 1 have been measured (Priha, 1969; Seager, 1972 and Ali et. al.,

1978a). When an undistorted vertical scale hydraulic model is constructed the model velocities are proportional tb corresponding prototype velocities through the square root of the vertical scale (see Dimensional Analysis, section 5.1). This results in water velocities in the model of-the order of mms1 . Sobey

(1973a) was unable to measure such low velocities and avoided the problem by modifying his experiments. This he achieved by artificially increasing the velocity scale, thus giving an increased, jet momentum, and confining his measurements of velocity, using a propeller current meter, to the strongest flowing regions of the model. With a tangential inlet and central outlet (a wall jet arrangement) he was able to take velocity measurements close to the boundary of a circular model. He was, however, aware of the limitations of the experimental measurements, in what could be termed unrealistic conditions, and included the measurement of the very:low water velocities in his thesis as a major recommendation for futher study.

A large percentage of the time spent working on this research

project has been devoted towards the development and use of the

simple but relatively effective velocity measuring technique,

described in some detail and evaluated in this chapter.

93 4.1 PREVIOUS METHODS USED

Attempts have been made in the past to measure the low water velo-

cities occuring in large hydraulic models of lakes and reservoirs.

The problem is compounded by the requirement for measurements over

an area of several square metres- and previous attempts have, in

general, had only limited success.

One of the most common methods has been to use integrating or

surface floats which move with the mean or surface flow and have

their progress in motion recorded by time-lapse photography. A

recent study by Ali. et. al., (1978a) used such a technique with

considerable success. Their floats basically consisted of candles

moving over the surface of the model measuring the surface flow, in

a darkened laboratory. A film was exposed for a series of egiialJy

spaced intervals of several seconds and the movement of the floats

appeared as dotted lines on the photograph. Results obtained by

this method are open to several criticisms. The surface currents

measured are not likely to represent accurately the overall motion

over the whole depth occuring in the model. Also, the collection

.of data must by its nature take several hours and hence cannot

give an instantaneous representation of the flow patterns in the

model. A further disadvantage of this type of method, found in

the present study when using depth-integrating floats, is their

tendency to congregate around the model boundaries and at the

outlet points and to follow the strongest currents in the model,

which are not always the regions of particular interest. This

last difficulty can be overcome by introducing the floats at care-

fully selected points but in a large model there may be certain

94 problems in gaining access to remote areas of the model.

The use of normal laboratory velocity measuring techniques such

as pitot tubes and current meters is generally precluded by their

inability to measure the very low velocities occuring and where

they can measure them their lack of sensitivity. The more

- sophisticated devices such as hot-film and laser-doppler anemo-

meters are extremely expensive, particularly as a large number

of pieces of equipment would be required to provide multiple

measurements simultaneously at a number of different points.

The support of a number of pieces of measuring equipment over

a large model would also present a considerable problem. A

further difficulty encountered in the use of all these techniques

for velocity measurement would occur due to it being necessary

to know the direction of flow;, clearly in a large model the flow

can be extremely complex and the prediction of its direction

practically impossible.

Since there were a number of reasons for not pursuing any of

the conventional methods further it was necessary to devise and

develop an alternative technique. A suitable method would have

to measure depth-averaged velocities of the order of mms 1

simultaneously at a large number of points over an area of several

square metres.

95 4.2 l'tiii TETHERED SPHERE METHOD OF STEFAN AND SCEIEHE

Stefan and Schiebe (1968) have published details of a method of measuring low water velocities using a small buoyant wax sphere tethered to the stream bed by a fine rayon line (see figure 4-1).

The displacement, relative to the point of a rigid support, of the buoyant sphere due to the drag force caused by flow was measured by the use of a travelling telescope mounted vertically, with two- degrees of freedom, in a horizontal plane above the sphere. A theoretical analysis of the forces acting on the sphere and tethering line, together with a knowledge of the degree of movement, allowed the velocity to be calculated. The unique relationship between the deflection of the sphere and the flow velocity was checked by towing the arrangement inverted through stagnant water at a constant rate with the deflection of the sphere being measured from photographs.

Good agreement between the calculated and measured deflections was found. Changes in the geometry of the sphere and line allowed different velocity ranges to be covered with velocities as low as

3mms1 measured.

This method satisfied part of the specification for a useful velocity measuring device but had certain limitations. The submergence of the sphere meant that the drag force on it was due only to velocities over a limited part of the flow depth with the composite nature of the probe (sphere and line) complicating the area of sampling.

It also meant that the deflection of the sphere had to be measured by a travelling telescope, a photogrammetric method being imprac- tical due to the likelihood of complex refraction problems affecting the measurements. The use of a number of travelling telescopes

96 reference j paint r ~--..t ~ D2--~ - - buoy'ant /s(tJ.ere ______rigid flow • ~ support I

Figure 4-1

The Tethered Sphere of Stefan and Schiebe (1968)

97 mounted above a 5m. diameter model was not considered feasible.

With these limitations in mind attention was paid to developing an alternative buoy which would overcome difficulties associated with the submerged sphere.

98 4.3 AN ALTERNATIVE BUOY

The first design of a new velocity measuring buoy consisted of a

paraffin wax sphere (12.5mm. in diameter) through which a pyrex

glass rod (approximately 0.5mm. in diameter) passed symmetrically.

Analysis of the forces acting on this buoy suggested that this

arrangement (shown in figure 4-2) would provide the necessary

small overall buoyant force needed to maintain stability, as well

as a deflection of approximately 10mm., at a water velocity of

10mms-1. The glass rod was tethered to the stream bed by a very

short length of cotton, acting as a hinge, attached to the end

of the rod and the stream bed. The wax sphere was centred about

the portion of the glass rod immersed in the water with a short length of glass rod protruding above the water level carrying, at its tip, a plastic target (usually 4mm. square).

The buoys were manufactured by using a aluminium mould for the

spheres and were cast with the glass rod in position. Control

tests based on accurate weighing of the components of the buoy

and the completed buoys showed that the method of manufacture

could produce identical buoys. A batch of these buoys were

calibrated individually in a laboratory flume (see section 4.4)

and found to be capable of measuring velocities in the range -1 5-40mms with a good agreement between different buoys. Although

this design of buoy could have been used in the hydraulic model

the range of velocities which, could be measured was a little

higher than desired and the composite nature and shape of the

buoy meant that an effective depth-averaging of the flow was

unlikely.

99 plastic target (4m m., sq.)- T TT 20 mm.

WIMP •

40mm.

80mm. paraffin wax pyrex rod sphere (12.5 mm, (0.5mm.dia) diam.) cotton tether / / / / / / / / / / / / ! / / / / / / / / / I /

Figure 4-2

The wax sphere/glass rod buoy

100 Unfortunately the drag on the glass rod is a significant part of the total drag. Hence it is not possible to place the spheres in different positions on the rods to measure the velocities at a number of depths and so construct a velocity profile.

101 4.4 POLI.i'HYLENE BUOY - DESIGN AND CALIBRATION

As a result of the previous experience it was decided to design a 'full depth' cylindrical buoy which, by virtue of its constant section, would be affected by the water velocities at all levels.

However, even with this type of buoy the effect of different velocities at different depths is not the same since the forces on the rod exert a moment about the bed which depends on their height above the bed. This limitation was considered to be acceptable when considering the virtues of the cylindrical buoy.

In order to measure lower velocities it was necessary to use an alternative material which would minimise the ratio of buoyant

force to drag force. Plastic rod was thought to be suitable for the sub-surface portion of the buoy and an investigation suggested that low-density polyethylene would be appropriate. The rod had to be sufficiently thin to ensure minimal flow disturbance whilst

maintaining its stiffness and also to have a density just less

than that of water to maintain the small buoyant force necessary

for stability. Low-density polyethylene has a specific gravity

of between 0.92 and 0.94 and is sufficiently stiff.at a diameter

of approximately 4mm.

The original intention was for the buoy to be totally composed of

polyethylene. However, calculations showed that if the above-

surface portion was a continuation of the plastic rod (at a diameter

of 4mm.) its weight would be larger than the buoyant force, and

so make the buoy unstable. Such a large diameter piercing the

water surface might also produce large surface disturbance effects

102 (see section 4.5.1). The buoy was subsequently developed with the above-surface portion a short length of pyrex glass rod

(approximately 0.5mm. diameter) carrying a plastic disc mounted at its tip as with the spherical buoy. A short length of cotton attached to the base provided the tether and the arrangement is shown in figure 4-3.

The plastic buoys were calibrated in the same manner as the spherical buoys but far more extensively since it was soon apparent that they would be suitable for the model tests. The calibration was performed in fully-developed turbulent flow near the downstream end of a 12m. long by 0.3m. wide laboratory flume with the water

80mm. deep (the proposed model water depth). The flow was assumed one-dimensional and the mean velocity (V) found by dividing the flow rate (measured by weighing) by the flow cross-sectional area.

The deflection (D) of the buoys was measured by a cathetometer used as a horizontal travelling telescope. It was mounted along the centre line of the flume and axisymmetrically and horizontally above the flume (see figure 4-4). Nine buoys were chosen randomly from over one hundred to be used in the 5m. diameter hydraulic model and calibrated in the flume. The deflection produced for a given velocity was noticeably larger than for the spherical buoy thereby reducing the effect of measurement errors and allowing lower velocities (1-25mms1 ) to be measured. The plastic buoy was also found to be easier to manufacture uniformly than the other buoy (wax sphere-glass rod) and, being less fragile, was easier to handle.

The results obtained from the nine calibrations combined showed that the velocity-deflection relationship was in the form of a

103 black plastic target 4mm. square pyrex D mm. odglass (O.51mm. diam ._

AMMO

polyethylene rod // / m. mean flow mm velocity(V) mms1 // 80 mm: //

1/ .

cotton thread tether //!//l t/i7'1 / / / / / /

Figure 4-3

The low-density polyethylene buoy

104 Figure 4 -4 Buoy in flume with cathetometer above

105

power law with non-constant index. When plotted logarithmically

the results appeared to form two intersecting straight lines of different gradients (figure 4-6). Since it was not possible to determine graphically where the gradient change point occurred

and hence to derive the equation of each straight line directly

it was necessary to perform a least-squares linear regression

analysis on the data expressed in logarithmic form. Examination

of the scatter of the.points indicated that the gradient change point occurred when the deflection of the buoy was between 8 and 14mm. A series of gradient change points were chosen in this

range. The regression analysis for each portion gave the gradient

and intercept and the correlation coefficients at,the particular

gradient change point chosen. Two graphs were plotted, figure

4-5(a) shows the variation in correlation coefficient with gradient

change point for the upper and lower portions of the curve and

figure 4-50) shows the degree of agreement (as expressed by their

difference) between the velocity predicted by the equations of the

upper and lower portion for each chosen gradient change point.

Ideally the correlation coefficient should be 1.0 and the agree-

ment absolute at the same gradient change point, however, a com-

promise. had to be reached and from an examination of the graphs

a gradient change point at a deflection of 13mm. (velocity

approximatelypp y 9mms 1)was chosen. This resulted in the adoption

of the following calibration relationships:

D < 13mm. V = 1.71 D0.65 mm s-1 (4-1) D0.83 D > 13mm. = 1.08 mm s-1 (4-2)

106 torrt lation coe- i ci ent

1.0

0.9

0.8

6 8 10 12 14 point of intersection (a) (D mm.) absolute velocity difference (mm s-') 0.6-

0.4 +

0.2

6 8 10 12 14 point of intersection (D mm.) -0.2 (b) Figure 4-5 Graphs used for selection of gradient change point (a)Correlation coefficient v gradient change point (b)Degree of agreement v gradient change point

107 8

0 0ō 0 ° / - Q83 1.0 V =1,08D S /

0 08 o/

0,6 o 0~

.65 0,4 V=1.71D0

0.2 04 0.6 0,8.. .1.0 12 14 1.6 1.8 Figure 4-6 I o g 1 0(D ) Data sample, regression lines, 5% confidence limits for calibration experiments ōf the polyethylene buoy where D (mm.) is the deflection of the buoy from its position with no flow to a stable position when the mean flow velocity is

V (mm:s-1).

With both the dependent variable (V) and the regressor variable

(D).being subject to measurement errors during the calibration it is difficult to assess quantitatively the error in V for a measured deflection D using the relationships of equations 4-1 and 4-2.

It is a fair assumption, however, that the use of a good data sample, with no discrimination, in the linear regression analysis will tend to compensate for measurement errors. A possible way of demonstrating the quality of the data sample_ is by determining the standard error of estimate for the linear regression analysis and hence the 5% confidence limits. These are shown, together with the data sample and regression lines (equations 4-1 and 4-2), in figure 4-6. As only two calibration points fall outside the 5% confidence limits, these being at the very lowest velocities where a more significant error might be expected and where fewer measurements were taken, it was considered unnecessary to calibrate

further buoys from the batch.

109 4.5 POLYETHYL BUOY - THEORETICAL ANALYSIS

4.5.1 General Analysis

With experimentally derived relationships linking the velocity and

deflection obtained the buoy can be theoretically analysed and a

comparison of.the experimental and theoretical relationships made.

In order to obtain a theoretical relationship between the deflection

and the velocity several basic assumptions must be made:

1)the horizontal velocity profile is assumed to be

approximated by a 1/7th power relationship,

2)vertical velocity components which might produce

some lift forces are negligible,

3)any surface tension forces are small and can there-

fore be neglected ( a discussion of this assumption

will be given-later).

The forces acting on the polyethylene buoy and their lever arms

are shown in figure 47.

The polyethylene rod experiences an unbalanced buoyancy force of:

FB =( 7 LR (1-SR)g (4-3) ` 4 / where dR = mean diameter of polyethylene rod,

LR ī length of polyethylene rod,

p = water density,

JSR = specific gravity of polyethylene rod, and g acceleration due to gravity; and a horizontal drag force of:

FD = 0.5O V2 LR dR CD (4-4)

110 1 D mm.

OMNI T

mean flow velocity (V mms -1 ) 80 mm,

i/ / // // / / / // % / ///7

1 xn

Figure 4-7

Forces acting on the polyethylene buoy 111 where CD is the drag coefficient.

The drag coefficient, CD, varies with the polyethylene rod diameter

through the cylinder Reynolds number Rc where:

Idc V (4-5)

where do is the cylinder diameter (in this casedo = dR) andp is th dynamic viscosity of water.

The following relationships for the variation of CD with Rc were derived from Hoerner (1965):

10 < Rc < 100 CD = 5.01R -0.3 (1+-7) and the assumption made that due to the high aspect ratio of the cylinder (17.7 : 1) and the proximity of the model bed and the

free water surface to the ends of the cylinder the drag on the cylinder ends will be negligible.

Substitution of equations 4-6 and 4-7 into 4-4 and 4-5 then gives:

1 Rc < 10 FD 6.28(fdR)0.3/A V1.3 LR (4-8) • 0.3 V1•7 1045 R15,5 100 FD 2.515p dR)0.7 LR (4-9)

If the velocity distribution were uniform the moment of the drag

force about the hinge (assumed to occur where the cotton meets

the stream bed) would be:

(4-10) __D = FD YD

112 where yD is the height of the line of action of FD above the model bed.

However, assuming a 1/7th power law for the horizontal velocity profile and taking equation 4-9 for FD:

= 1.12 (4-11) FD yD

The glass rod experiences a drag force of:

FR (4-12) = 0.519s ~D where n = diameter of the glass rod,

VR = velocity acting on the rod,

Ls = submerged length of glass rod.

Since the glass rod is close to the water surface it can be assumed that:

A = V (4-13)

A where V is the maximum velocity, at the surface, and since for a 1/7th power law relationship:

A V = 8 V (4-14)

then

8- VR V (4-15) = 7

113 and equation 4-12 then becomes:

FR = 0.37Q n V2 Ls CD (4-16)

Since for the glass rod Rc<10 generally, substitution of equation

4-6 into equation 4-16 and then equation 4-5 gives:

-3 )& 7 V1.3 Ls FR = 7.16(p n)0 (4-17)

The glass rod also exerts a force due to its weight:

2 FG K(1 )LA SGy g (4-18) where LA is the total length of glass rod and SG is the specific gravity of the glass, and acting on the submerged portion of the glass rod there is a buoyancy force of:

F Z(( )L s g (4-19)

The other force considered is the weight of the plastic target:

Fm = mg (4-20)

Another force which may possibly influence the deflection of the buoy is the surface tension. When the buoy is vertical in the

water, with no flow, the surface tension forces have no horizontal resultant, the vertical component being balanced by the

weight of the meniscus as shown in figure 4-8(a). However, when

114

(a) (b) with no flow '- with a flow

Figure 4-8

Surface tension forces acting on the polyethylene buoy

115 the buoy is deflected by the flow the meniscus becomes asymmetric as shown in figure 4-8(b). Once again the vertical component of the overall surface tension force is balanced by the weight of the meniscus but the horizontal components of the forces do have a resultant. If one assumes that the meniscus is formed in a similar way to as shown in figure 4-8(b) it is clear that the horizontal component-of P2 is going to be larger than the horizontal component of P. This will result in a moment about the bed which will tend to increase the deflection of the buoy. Unfortunately it is not possible to quantify the surface tension forces acting on the meniscus when it is not- axisymmetric (ie. when the angle of contact varies around the circumference) and the fluid is flowing so it is not possible to include the surface tension forces in this analysis.

Assuming the buoy to be in equilibrium with a deflection of D for a mean velocity V and taking moments about the hinge :

1.12 FD, yD + FR yR = FB xB + Fs xs - FG xG - Fm xm (4-21)

B, xs, xG and xm are lever arms as specifically defined yD' yR' x in figure 4-7.

A manipulation of equation (4-21) when the values of the forces and their lever arms are substituted in gives relationships linking V and D for each of the two cylinder Reynolds number ranges.

4.5.2 Deflection-Velocity Relationship for the particular Buoy geometry

The particular buoy designed for and used in the 5m. diameter

116 hydraulic model with an 80 mm. depth of water had the following physical characteristics: Polyethylene Cylinder Length, LR 75 mm. Mean Diameter, dR = 4.24 mm. Specific Gravity, SR = 0.926

Pyrex Glass Rod Length, LA = 20 mm. Mean Diameter, n = 0.51 mm. Specific Gravity, SG = 2.6

Plastic Target Mass, m = 0.005 gm.

Physical Constants assumed Density of water,/ = 103K-3 Dynamic Viscosity of water ,)1( = 10- 3m2s1 Acceleration due to gravity, g = 9.81 ms-2

Using these values to determine the magnitudes of the individual forces from their equations:

FB. = 769 x 10-6 N. (Eq. 4-3) 0 L - 6 N. Fs = 2. s x 10 (Eq. 4-19) -6 FG = 104 x 10 N. (Eq. 4-18) -6 Fm = 49 x 10 N. (Eq. 4-20) -6 Rc'10 FD = 5771 V1'3 cosA x 10 N. (Eq. 4-8) -6 10 RX100 FD = 65146 V1'7 cose x 10 N. (Eq. 4-9)

117 where 8 is the vertical angle of tilt of the buoy as defined in figure 4-7.

-6 FR = 41 V1'3 Ls x lo N. (Eq. 4-17)

and the lever arms and unknown lengths:

xB = 39.5 sine mm. L (8o -771 mm. $ - cse lJ

xs = (77 + sin8 = 38.5 + 40 \ sine mm. Ls J cos8 xG = 87 sine x m = 97 sine yD = 39.5 cose yR = ( 77 + Ls cos9 = 38.5 + 40 cos8 2 ( cos8

Now sine = D 97

(•1) 2 2 Hence cos28 = 1 - and cos2e = 1 +C' + . ) 97)

To solve equation 4-21 it is necessary to assume 2

Cir7) 0

Then solving equation 4-21 for the values of the forces and lever arms etc. given above:

118 1 R 10 V = 3. 53 D0'77 mm s-1 (4-22)

10.1R'-.5.100 D = 16776 11.7 + 55 V1-3 (4-23)

(V in ms-1)

Equation (4-23) cannot be expressed as V in terms of D as it stands however the second term is small in comparison with the first and can be neglected hence:

10:5.1145.100 V 3.27 D0'59 (4-24)

Figure 4-9 shows the theoretical and experimental relationships

(equations 4-1, 4-2, 4-22 and 4-24) together with the calibration relationship of another independent study which formed part of an undergraduate project (MacGilchrist and Morris, 1978).

The discrepancy between the predicted and measured deflections for a given velocity is coniderable, the measured deflection being larger than predicted.. It is considered that this difference is chiefly due to the unknown asymmetric surface tension forces acting on the glass rod when the buoy is deflected. A consideration of figure 4-8 earlier suggested a reason for this effect and this contention has been supported by observations during tests which have shown that when detergent is introduced onto the water surface close to the buoy the deflection is initially quite significantly reduced but that the buoy eventually returns to its previous position when the detergent has dispersed. Since the theoretical.analysis could not give exact relationships the ex-

perimentally determined deflection-velocity relationships were adopted.

119 25 I Mean Velocity theoretical (eqns. 4-22 & 424) V mm. s-1 -IA • ~--experimental (undergrad pro j.) 20 —o— —o— experimental (eqns. 4-1 &42

15 _ /g INJ 0

5 10 15 20 25 30 35 40 Deflection D mm. Figure 4-9 Theoretical and experimental calibrations for the polyethylene buoy 4.6 MEASUREMENT OF THE BUOY DEFLECTION IN THE HYDRAULIC MODEL

With a suitable buoy designed and calibrated it was necessary to

develop a method of determining the deflection of a number of

buoys in the hydraulic model. The method needed to attain a

suitable level of accuracy and be capable of providing data,

recorded simultaneously for a number of buoys, which could be

converted into deflections. A photogrammetric method was an

obvious solution.

4.6.1 The Photography

At an early stage of buoy development preliminary investigations

were made as to the feasability of using a photogrammetric method

to determine the amount and direction of movement of a buoy.

These early experiments were principally aimed at evaluating the

cameras:and.lenses readily available in order to decide which

was; the most suitable in the light of the image resolution required at a scale compatible with the measuring capabilities of the

photograph analysers available.

Two analysers were available, the first. a Pye Universal Measuring

Microscope was used on a light table and is shown in figure 4-10.

It is capable of determining a single photograph co-ordinate

(ie. x or y) to within ± 0.010 mm. with an experienced operator.

The second, a Zeiss Jena Stecometer (used as a mono-comparator)

shown in figure 4-11, has a manufacturers estimated accuracy of

0.0025 mm. in single co-ordinate measurement.

121 Figure 4-10 Pyc' Universal measuring microscope

122 Figure 4-11 Zeiss Jena Stecometer An early- development in the photography of the buoys was the discovery that a black target against a white background (the model bed) produced a far better definition than the more obvious white target against a black background and also had the advantage that the natural laboratory lighting provided sufficient illumination. It should, of course, be borne in mind that photograph analysis was to be on the negative where black on white in the model appears as white on black in the negative.

It was obviously desirable to encompass the whole of the 5m. diameter circular model in one photograph if possible and early tests assumed this condition. Of the cameras available the most suitable was a non-metric Easselblad 500 ELM camera fitted with its standard 80mm.. Zeiss Planar lens. A metric camera, that is a camera specifically designed for photogrammetric work, would have provided' the best solution, however, such a.camera was not available. The object distance, using the Hasselblad camera and lens together with a .70mm. filmback (55mm. square negative format), required to ensure-complete coverage of a 5m. field is 7.35m.

The definition of a 4mm. square target photographed from this distance in a test field was.found to be poor; this was due to the.faot--that the image size of 0.04mm. on the photograph at the scale .1:91- was close to the silver grain size of the film.

A-second attempt to cover the model with one photograph, whilst satisfying the resolution requirements, was unsuccessful. The camera, an MPP Technica with a shorter focal length (65mm.) lens and a larger format (90mm. square) was able to cover the model

from the much shorter object distance of 3.4m., however, the

124 definition,- particularly at the edges and in the corners, was

poor and undoubtedly of insufficient quality to enable accurate measurements of the order of 0.020mm. to be made.

The only alternative open at this stage was to abandon the single

photograph ideal in favour of a less staisfactory multiple photograph method of coverage. The most viable solution was to cover the model area by taking four photographs each covering a quadrant of the,model and with a small overlap on two edges of each photograph and one edge-of each of the adjoining photographs. The

field size chosen to ensure an adequate overlap between photographs

was 3m. which when using.,the Hasselblad 500 ELM camera fitted with the 80mm. lens and having a 55mm.-square format, required an object

distance of approximately 4.5m. The definition of the buoy targets

attained with this arrangement was much improved and of'sufficient

quality for measurements to be taken. The scala of the photographs

was approximately 1:55 which, since it was desirable to measure

minimum deflections of the order of 1-1.5mm. (0.019-0.O28mm. on the

negative) was within the measuring capabilities of the analysers

mentioned earlier.

4.6.2 Photogrammetric Control

Since very *small measurements were required from 'the negatives it

was necessary - to have good photogrammetric ground control over the

whole model area. This ground control was needed to facilitate

the determination of the exterior orientation of the camera and

hence the ground co-Ordinates of the buoy targets at their positions

in the model.

125 Photogrammetric control was provided by points on the edges of

nominally, horizontal white-faced steel surveying tape suspended

across the model basin at two levels under tension, and by the

centre of a cross on each of four plates projecting from the model

wall into the interior of the circular model. The arrangement is

shown schematically in plan in figure 4-12. A control point on a tape was defined as the intersection of a selected graduation line with an edge of the tape. The space co-ordinates of 34 control points (including the four crosses on the plates) were

determined from a triangulation survey using a Wild one-second theodolite, with readings observed from the four stations A, B,

C and D shown in figure 4-12.. The stations consist of threaded posts (figure 4-13), on which the theodolite (figure 4-14) could be screwed, welded to the tops of vertical steel columns which formed part of the structural support for the outer model wall.

Standard surveying targets mounted on a base, with tribrach; screws for levelling, were positioned at the three redundant stations when theodolite observations were being made from the fourth, one of these targets is shown in figure 4-15. The distance between the stations was determined by subtense bar (also to be seen in figure 4-15) observations of the diagonals of the quadrilateral ABCD. A suitable origin for an arbitary 3-dimensional co-ordinate system was chosen to ensure that all the control points had positive space co-ordinates and that the x and y directions coincided with the 0.4 grid described in section 3. These co-ordinates have no particular significance but are included in

Appendix C for reference.

126 Steel tape

Control • 100 mm. above bed

points o 400 mm. above bed A Theodolite stations

Figure 4-12

Schematic diagram of the photogrammetric control 127 Figure 4-13 Threaded post for theodolite and surveying target

11 ? P Figure 4-14 General view of turntable showing theodolite and subtense bar ------

Figure 4-15 Subtense bar and surveying target 4.6.3 Interpretation of negative observations

With a photogrammetric ground control system established it was

then possible to proceed with a full scale test in the hydraulic

model to ensure that the complete system could produce viable

results. A series of photographs were taken of the model with

a radial jet ,direction and a diametric outlet (section 5.3.2) and

a representative jet momentum to facilitate the test, which was

principally concerned with photogrammetric rather than hydraulic

aspects.

Photograph co-ordinates were measured on the negatives (Ilford

FP3 polyester-based aerial film) by using the Pye Universal

Measuring Microscope; this being the least accurate of the two

analysers mentioned in section 4.6.1. On each negative 7 control

points (6 on the tapes and 1 plate cross) as shown in figure 4-12

were visible. Between 36 and 40 buoy targets and 2 points on each

of the 4 edges of the negative, as near the corners of the frame

as. possible, were also observed.

The 8 edges points shown in figure 4-16 were used to derive a

position'for the negative centre. This was done by first. deter-

mining the straight line equation for each edge by using the two

observed edge -points (1 and 2, 3 and .4, 5 and 6, 7 and:8) on the

appropriate edge. Once the straight line equation for each edge

_ was known the four intersection points of these four straight

lines XY, YZ, ZW and WX could be found. X, Y, Z and W are the

four corners of the negative and their co-ordinates could then be

used to calculate the straight line equations for the two diagonals.

Finally, these two equations could be solved to determine their

131 y (xs,y5 ) 5 z

"x analyser axes

Figure 4-16

Definition sketch for negative .centre co-ordinate determination

132 intersection point 0 which was assumed to be the perspective centre of the negative. A program was written to perform this simple but tedious calculation on. the Hewlett Packard 9100B programmable calculator.

With the negative• centre co-ordinates determined- relative to the measuring instrument axes the exterior orientation of the camera and hence the planimetric"rectangular co-ordinates of the buoy targets could be determined by using the formulae quoted by

Hirvonen (1971) for the absolute orientation of one photograph.

The author was fortunate in being able to use an established computer program, written by Mr. B. Chiat, to determine the six unknown parameters of exterior orientation (ie. the tilt about the x and y axis, the swing, the camera or 'flying' height and the absolute perspective centre co-ordinates). To facilitate the determination it was assumed that the principal distance was

81.43mm., the image distance consistent with the range setting

(4.5m.) and the focal length of the objective as stated (80mm.), and that the principal point was coincident with the photograph centre (see section 4.6.4)..

With the orientation of the negative determined the six defining parameters were used to convert the analyser measurements of the buoy target positions to absolute ground co-ordinates.. Comparison of the relative positions of each buoy on two appropriate negatives, one taken with no flow and the other with the flow to be determined, gave the deflection of each buoy target. This early analysis of negatives demonstrated that the system worked and could detect the small deflections•occuring.

133 4.6.4 Photogrammetric Errors

Since a non-metric camera was used to make very small measurements there was a possibility of particular errors. During the course of this study many of these possible sources of error were analysed and it is convenient to record them and comment on their significance at this stage of the description.

The first necessary check was as to the distortion of the lens.

A timber frame was_ constructed which was approximately 2m. high and stood on the dry bed of the model. Strips of white-faced steel tape similar to those already described were affixed to batons crossing the top of the frame and to the model bed. A magnified photograph of the test field is shown in figure 4-17; it should be noted that the blurring of some of the tapes, notably those on the top of frame, is due to the focussing of the.camera to the standard 4.5m. object distance. 55 points were chosen, approximately 5 on each tape, at regular intervals along-the tapes and their space co-ordinates determined as before using the theodolite stations A, B, Gand D. These points were along the edges of the strips of tape on the batons and the: model floor as well as the two diametric control tapes shown in figure 4-12.

Two negatives taken by the Hasselblad ELM camera were measured on both analysers and the residuals of one set of measurements are shown in figure •.4-18 where the well-known 'pin-cushion' distortion pattern can be seen.

The results of the camera-calibration tests=are given in table

4-1, note that omitting the four control points in the corners of

the negative (which had the highest standard errors) changed the

134 U)

Figure 4-17 Enlarged photograph of test field used for camera calibration

135

•

r • „

0 10mm 0 .010mm I I p Scale of Scale of Photograph Distortion Vectors

Figure 4-1$

Plot of lens distortion vectors

136 principal distance and the co-ordinates of the principal point relative to the. centre of the photograph as origin.

Table 4-1 Summary of the Results of Camera Calibration

Number of Exposure Principal Co-ordinates of Control Setting Distance Principal Point Points Used (mm.) Relative to Centre of Photograph as origin (mm.)

x y

55 1 sec. @ f22 81.03 0.46 0.051 54 ►, 81.06 0.274 0.029

49 II 81.04 0.43 0.048

45 1 sec. @ f22 81.37 0.375 0.091 54 0.5. sec. @ f22 81.07 .0.44 0.072

54 ►I 81.06 0.?+51 0.065

50 0.5 sec. @ f22 • 81.37 0.398 0.098

* Notes: Readings for the four control points in the corners of

the frame omitted.

A series of photographs of the test field were also taken with a non-metric camera, the Hasselblad MK70, these showed a lower standard error of photograph co-ordinate estimation, however, because the lens was nominally of 60mm. focal length the test frame did not completely fill the negative format. As a result the distortion properties of this lens were not fully tested. In any event it was intended that the 500 ELM would be used and, as will

137 be seen, this camera more than satisfied the accuracy requirements

and justified its use.

To test the effect of amending the interior orientation parameters

one set of target deflections were re-calculated with the principal

distance set equal to 81.06 and the principal point co-ordinates

taken as (0.45, 0.07mm.) as found from a representative calibration

photograph (table 4-1). The values of the (x, y) ground co-ordinates

from the 'zero' and 'deflected' negatives are given together with

the values. of the absolute deflections in the x and y directions

for one quadrant and for each orientation in table 4-2. It can be

seen that the particular values of the ground co-ordinates for

corresponding points are different but the absolute deflections

agree very well; out of 62 rectangular components of deflection

calculated for this quadrant 18 are unchanged, 42 are changed by

0.1mm. and just 2 by 0.2mm. It was therefore accepted that small

errors in the interior orientation parameters were acceptable.

In view of the pattern of lens distortion it was not surprising

to find that the residuals at control points resulting from ex-

terior orientation of photographs of the model based on the central

perspective assumption were relatively large. The standard errors

for 37 arbitarily selected photographs, 17 analysed on the Pye

Microscope and 20 on the Zeiss Stecometer, are given in table 4-3.

-They varied between 0.018mm. and 0.044mm. averaging 0.028mm.

Surprisingly enough there was no apparent advantage in using the

Zeiss Stecometer. For the random sample of 20 negatives analysed

on this instrument the mean standard error was 0.029mm. whereas for

the sample of 17 negatives analysed by the Pye instrument the mean

138 Table 4-2 Effect of Amendment of Interior Orientation

PRINCIPAL DISTANCE = 81.43mm PRINCIPAL DISTANCE = 81.06mm CO-ORDINATES CO-ORDINATES PRINCIPAL POINT = (0,0)mm PRINCIPAL POINT = (0.451, 0.065)mm

'ZERO' 'DEFLECTED' ABSOLUTE 'ZERO' 'DEFLECTED' ABSOLUTE POSITION POSITION DEFLECTION POSITION. POSITION DEFLECTION

x y xo y - y x y xd Y -Y x xd d d o mm.o mm. mm.d mm.d mm. mm.d o mm.o mm. mm. mm. mm. mm.

4153.3 1020.0 4151.2 1018.3 -2.1 -1.7 4176.3 1028.3 4174.2 1026.7 -2.1 -1.6 3747.7 1007.9 3744.4 1006.6 -3.3 -1.3 3772.5 1016.2 3769.3 1015.0 -3.2 -1.3 3354.9 977.9 3349.5 977.7 -5.4 -0.3 3381.4 986.4 3376.o 986.1 -5.4 ,-0.2 4550.4 1430.9 4547.7 1428.3 -2.6 -2.6 4571.5 1437.5 4569.0 1435.0 -2.5 -2.5 4145.5 1414.6 4142.7 1413.2 -2.7 -1.4 4168.4 1421.2. 4165.7 1419.9 -2.7 -1.3 3740.5 1398.o 3739.1 1397.5 -1.3 -0.4 3765.2 1404.6 3763.9 1404.2 -1.3 -0.4 3336.4 1380.7 3334.8 1381.1 -1.7 0.3 3362.9 1387.4 3361.3 1387.7 -1.6 o.4 4964.2 1855.2 4961.7 1850.1 -2.5 -5.2 4983.6 1860.0 4981.1 1855.0 -2.4 -.5.1 4547.5 1837.7 4545.6 1834.6 -2.0 -3.1 4568.7 1842.5 4566.8 1839.4 -1.9 -3.o 4144.8 1816.1 4143.1 1814.9 -1.7 -1.3 4167.7 1820.9 4166.0 1819.8 -1.6 -1.2 3743.0 1809.0 3741.8 1809.8 -1.2 0.8 3767.7 1813.8 3766.5 1814.6 -1.2 0.8 3333.7 1791.5 3332.8 1791.8 -0.9 0.3 3360.1 1796.4 3359.3 1796.7 -o.8 0.4 2942.9 1774.7 2948.2 1822.9 5.3 48.2 2971.1 1779.5 2976.4 1827.5 5.3 48.o 4951.1 2249.3 4947.3 2240.1 -3.8 -9.1 4970.4 2252.4 4966.7 2243.3 -3.7 -9.0 4549.2 2235.9 4547.7 2232.8 -1.5 -3.1 4570.3 2239.o 4568.8 2236.0 -1.4 -3.o 4148.2 2221.5 4147.9 2221.3 -0.4 -0.2 4171.0 2224.6 4170.7 2224.5 -0.3 -0.1 3745.3 2211.2 3745.3 2212.4 0 1.2 3769.9 2214.2 3769.9 2215.5 0.1 1.3 3330.5 2185.9 3329.5 2187.7 -1.0 1.8 3356.9 2189.0 3355.9 2190.9 -0.9 1.9 Table 4-2 Contd.

'ZERO' 'DEFELCTED' ABSOLUTE . 'ZERO' 'DEFLECTED'D' ABSOLUTE POSITION POSITION • DEFLECTION ,POSITION POSITION DEFELECIION

xo yo xd yd xd xo ydyo x y xd yd xd xo yd yo mm. mm. mm. mm. mm.. mm. mm. mm. mm, mm. mm. mm.

5353.4 2672.7 5353.2 2668.9 -0.2 -3.7 5371.0 2674.0 5370.9 2670.4 -0.1 -3.6. 4956.0 2654.4 4955.4 2651.1 -0.6 -3.3 4975.3 2655.8' 4974.8 2652.6 -0.5 -3.2 4546.7 2642.3 4546.3 2641.0 -0.4 -1.3 4567.7 2643.6 4567.4 2642.5 -0.4 -1.2 4138.9 2628.5 4138.9 2628.2 0 -0.2 4161.7 2629.8 4161.7 2629.7 0.1 -0.2 3735.9 2611.4 3735.3.' 2612.3 -0.7 0.8 3760.5 2612.8 3759.8 2613.7 -0.6 0.9 3344.9 2595.1 3343.5 2598.9 -1.5 3.8 3371.1 2596.4 3369.7 2600.3 -1.4 3.9 5350.6 3075.6 5350.0 3065.2 -0.6 -10.4 5368.1 3075.3 5367.7 3065.0 -0.5 -10.2 4938.0 3061.4 4938.5 3056,5 0.5 -4.9 4957.3. 3061.1 4957.9 3056.3 0.6 -4',.8 4541.3 3037.8 4541.3 3036.4 0 -1.4 4562.3 3037.5 4562.4 3036.2 0.1 -1.3 4142.2 3022.2 4142.5 3022.8 0.2 0.6 4164.9 3021.9 4165.2 3022.6 0.3 0.6 3745.4 3014.3 3745.7 3014.7 0.3 0.4 3769.8 3013.9 3770.2 3014.5 0.3 0.5 3339.0 2984.9 3340.1. 2990.2 1.2 '5.3 3365.1 2984.6. 3366.3 2990.0 1.2 5.3 2944.6 2971.6 2954.9 2998.1 10.3. 26.5 2972.4 2971..3 2982.8 2997.8 10.3 26.5 standard error was 0.026mm. This discrepancy in mean standard

errors may-have been due to operator differences, however, in

general the,Stecometer was more suitable having many advantages.

It is more comfortable to operate and less tiring but most impor-

tant of all it has an automatic recording system via a teleprinter

and computer paper tape puncher which overcomes the possibility

of the type of accidental errors which have occasionally occurred

when using the microscope.

TabJ.e-4-3 Comparison of Standard Errors of Estimate

Pye Universal Zeiss Jena Microscope Stecometer

.033 mm. .036 mm. .021 .025 .021' .018 .034 .044 .028. .035 .020 .024 .026 .023 .021 .029 .029 .032 .023 .029 .036 .033 .028 .023 .020 .032 .021 .032 .031 .023 .029 .027 .028 .028 .027 .030 .024

In order to- assess the contribution of accidental observation

error towards the standard error figures discussed above, a

special set of photographs were taken of targets (identical to

those of the buoys) mounted on the tops of 16 vertical timber posts

98mm. long which were evenly distributed over the floor of the

141 model in one quadrant. This also pre-

sented an opportunity to partially check the state of film flatness.

One of the major criticisms of the use of non-metric cameras in photogrammetric work is aimed at the possible lack of flatness of the film (K8lb1, 1976; Davies, 1977). Many metric cameras such as the UMK use plates as opposed to film thereby overcoming the flatness problem, alternatively, if they use a film-back it is vacuum-formed over a flat plate. The Hasselblad MK70, the metric equivalent of the 500 ELM, uses a reseau plate, which is

4 glass plate with a very accurate grid of fiducial marks etched on, to hold the film flat. The fiducial marks (crosses) appear on the film and since their separation is known accurately the film flatness can be checked during analysis. Since the Hassel- blad ELM camera is not metric and therefore does not have a reseau plate to maintain the film flat it is not possible to determine the degree of film flatness absolutely. The only possible way to check this aspect is to take and analyse photographs in a way which maximises the likelihood of the film not being flat. If then, the errors are small, it is probable that the film is being held flat enough.

The photographs of the quadrant with the targets on the timber posts were taken over a maximum length of film with the magazine being removed from the camera body and replaced randomly. Although the six photographs analysed were taken from the same camera station the camera was detached from its bracket between exposures and various other changes introduced, such as those mentioned

142 above, to make the six randomly chosen photographs distinct. For

this experiment the exterior orientation determination was based

on the measurements from five of the usual control points, this

represented the least number of control points ever used for a

exterior orientation determination.

The set-of ground co-ordinates determined for each of the six

photographs are given in table 4-4 as the deviations from the mean

values. An analysis of the deviations of the residuals from their

mean values produced a standard error of observation of 0.003mm.

This value was checked by comparing residuals at 7 control points on one pair of photographs for each of the four quadrants taken at different times; this gave the standard error to be 0.004mm.

Comparisons between the ground co-ordinates of the 16 targets on the timber posts as found from the 6 photographs (table 4-4) produced a value for the standard error of estimate of a ground co-ordinate of 0.4mm. Therefore the standard error of a displacement of a buoy target given by the subtraction of two sets of co-ordinates is o-f the order of 0.6mm. (0.412--) for each component. Once more an independent check on this figure was possible. by using the displacement of targets which appear on overlaps between photographs (ie. those targets on the intersecting centre-line•diameters of the 0.4m. grid) and are determined twice for completely distinct negatives. From the -differences between 21 such pairs of displacements the standard error of a ground co-ordinate is 1.1mm. which is reduced to 0.6mm. when 4 points close to the frame corners are omitted. Since the 21 images concerned here are close to one edge of their respective negative these higher figures are

143 Table 4-4 Effect of Accidental Observation Errors

Deviations (mm)

Target Mean Neg. Ground Co-ordinates 59A 50 44A 37 28 22A

1 2970.E 'o 0 0.3 0.4 -0.5 0.3 2972.2 0.5 -0.2 0.8 -0.2 0.2 0.1

3740.5 0.5 0 -0.1 0.6 . -0.7 0.5 2 3003.2 0.2 0 0:3 0.3 -0.8. 0.5

4547.0 0.6 0.1 0.1 0.4 -0.3 0. 3 3032.3 -0.2 0.1 -0.3 0.1 -0.4 0.2

5353.9 -0.1 0.4 0.2 0.5 fax= -0.2 3073.4 0.4 0 -0.6 o.4 o.4 0.2

4145.3 O 0 -0.2 0.4 -0.3 0.3 5 3421.7 0.4 0.4 0 -0.2 0 0.3

3334.7'. o.. -0.3 -ō.1 0.1 -o.4 0.5 3389.7 0.4 0.1 -0.1 0 -o _4 ' : 0.8

2935.8 0.2 -0.2 -0.1 0.7 -0.4 0.1 3776.0 0.2 0.6 -0.4 o -o.3.. 0.7

3739.3 0.4 -0.1 0. 0.5 -0.2 ,-0.1 3810.6 0 0.4 -0.4 0.7 -o.8 0.1

4540.1 0.1 -0.2 -0.3 •0.3 -0.1. -0.1 3845.7 0.7 0.4 -0.3 0.3 -0.2 0.6

1 5280.0 0.1 0.4 0.1 -0:1 -0.5 0.2 0 3869.5 -0.4 0.3 -0.5 0 -0.7, 0.6

11' 4146.6 o.4 o o 0.4 0 -0.1 4232.o 0.2 -0.1 -0.2 0.4 -0.8 0.6

2934.8 O -o.4 -0.1 0.8 -0.7 o.4 12 4584.5 0.3 0.1 0.2 0.3 0.4 -0.5

144 Table 4-4 contd.

Target Mean Neg. Ground Co-ordinates 59A 50 44A 37 28 22A

3331.2 0.4 -0.3 -0.2 0..7 -0.5 0.5 4604.2 -0.4 0.2 0.3 -0.3 0.5 -0.4

4546.6 0.3 o.1 0.3 o.4 -0.4 0 4643.7 -0.1 -0.2 0.2 0.4 -0.2 -0.3

15 2939.4 0.4 0.1 -0.1 0.6 -0.2 -0.1 5385.2 -0.4 0 0.4 -0.4 -0.2 0

16 3730.5 0.3 " 0 0 0.2 0 -0.1 5413.8 -0.7 0.6 0.1 -0.7 -0.8 0.1

145 readily explained.

In determining the ground co-ordinates of the buoy targets from the exterior orientation parameters it is necessary to specify the height of the buoy targets relative to the arbitary datum

plane of the control point co-ordinate system. Throughout it has been assumed, in determining the ground co-ordinates of the buoy, that the buoy targets have a constant height (98.0mm.) above the datum plane. Since this is technically an invalid assumption, the height of the buoy target varying as the buoy tilts, itq, was necessary to check that in making this assumption significant errors were not generated in the ground co-ordinates derived. The formula to calculate such an error is given below:

Error in Ground Co-ordinate = Error in height x Radial dis- of a point tance of Camera height the point

(4-25)

where the camera height is the height of the camera perspective centre above the datum plane and. the radial distance of the point is the real distance from the centre of the photograph to the point at which the error in ground co-ordinates is required.

Even if a large deflection of the buoy is assumed, say 20mm., at the 'worst' position in the frame, ie. the longest radial distance to the corner of the frame, the error in the ground co-ordinate is of the order of 1mm.; for small deflections the assumption is almost exactly correct. Even though this constant height assumption has to be made it can be seen that the errors generated will normally be insignificant,

146 During the course of this study no attempt was made to correct the measured co-ordinates of the buoy targets or the control points for lens distortion. Even though it was shown earlier that lens distrotion did occur this is not a serious limitation because the displacements deduced are obtained from images (the buoy targets on the negatives) whose relative positions in two frames hardly differ. If a correction factor were applied the improvement in the accuracy of the results, especially where small displacements occur, would be, minimal.

Although there are various possible sources of error in this method of velocity determination they are generally compensated for by the nature of the measurements required from the photographs ie. small relative movements. In general the accuracy of a displacement is likely to be of the order of ± 1mm. or less although if all the possible errors occurred simultaneously this figure would be somewhat higher. Since water flows are being measured any seriously erroneous results are liable to be exposed by an examination of the velocity.field. Clearly continuity must be satisfied and hence velocity vectors which completely contra- dict the trends of the surrounding flow are easily identified.

147 4.7 THE COMPLETE PROCESS

It is convenient to conclude this chapter with a simplified stage by stage summary of the steps taken in determining a velocity field for the hydraulic model:

(1)Photograph the model with no flow to establish the position of the buoys and targets at their 'zero' positions.

(2)Photograph the model with the internal flow that it is required to be measured to establish the position of the buoys and targets at their 'deflected' positions.

(3)Analyse each photograph (or set of photographs if the model has to be photographed from several camera stations) recording the (x, y) co-ordinates, relative to the analyser axes, of the buoy targets, the required number of photogrammetric control points to ensure accurate exterior orientation determination and eight edge points, two on each edge as demonstrated in figure

4-16.

(4)Using the eight edge point measurements for each photograph determine the co-ordinates of the centre of the photograph relative to the analyser axes.

(5)Using the control point measurements, their known ground co-ordinates from the triangulation survey and the photograph centre co-ordinates determined in (4) use a computer program, which performs a space resection of the camera position, to calculate the exterior orientation parameters for each photograph.

148 (6)With a knowledge of the exterior orientation of the camera

and the photograph centre co-ordinates the ground co-ordinates

for the buoy targets as they appear in each photograph (zero'

and 'deflected') can be found.

(7) By subtracting the ground co-ordinates of each buoy target

as determined for the 'zero' flow position photograph from the

ground co-ordinates for the same buoy as determined for the

;!deflected,'. position photograph the absolute deflection compon-

ents of the buoy in the x and y directions (relative to.the

arbitrary cartesian axes chosen when fixing the positions of the

photogrammetric control points) can be found.

(8)With•the absolute deflections in the x and y directions known

the velocity components inducing"these deflections can be found

by using the relationships of equations 4-1 and 4-2. Combining

the two component velocities gives the overall velocity magni-

tude and their ratio a measure of the direction of flow at each

tethered buoy position. The velocities at each grid point over

the area of the hydraulic model are presented as a velocity field

,(see Appendix A). A computer program has been used to undertake

the steps from (6) to (8) and the plotting of the velocity fields

and is listed, with a short explanation, in Appendix D..-

149 CHAPTER

THE JET-INDUCED CIRCULATION IN SEVERAL MODEL RESERVOIRS

150 INTRODUCTION

With a viable method for measuring the low water velocities occuring in hydraulic reservoir models in operation it was

possible for the research programme to investigate the jet- induced circulation in reservoirs, the parameters which are important for reservoir circulation and are most likely to influence it were identified by dimensional analysis.

151

5.1 DIMENSIONAL ANALYSIS OF RESERVOIR CIRCULATION

Throughflow induced circulation in a reservoir (eg. by a momentum

jet) can be assumed to depend on the following parameters (see

definition sketch Fig. .5-1):

Qo' the reservoir throughflow, [L13 [T]-1

Ko = Qo Vo, the kinematic jet momentum flux per unit 4 -2 density [L7 [T]

(where. V -is the average velocity at the point of entry of the o , throughflow into the reservoir),

f = 2 SLsin¢, the Coriolis parameter

(where SZ is the Earth's rotation rate anq is the latitude of

the reservoir),

L, a horizontal length typical of the reservoir (eg. the [L]

diameter in the case of a circular reservoir but, in

general, can'be defined as the reservoir circumference

divided by 'N ),

h, the average depth of the reservoir, [L] k s, the average bed roughness height, [L] P the ambient density of the homogenous reservoir IM] [L]-3 water, CL1-3 ap the density difference between the inflow and the [M] ambient density,

, the dynamic viscosity of the reservoir water, [MN-1[T]-1

g, the local gravitational acceleration, L ] [TJ -2 There are various additional geometric parameters which could

152 , buoyancy JD, mass density Inlet pK0 , momentum flux /,t, dynamic ,Qo, mass flux viscosity g, accn, due to SL s in gravity 96, latitude •

L 'diameter'

Figure 5-1

Definition sketch for dimensional analysis

153 have been included, particularly those associated with the inlet

and outlet geometry, however, they are considered to be of

secondary importance and can, in general, be non-dimensionalised

by inspection if required.

There are 10, variables considered with three independent primary

dimensions [MI, [L] and [T] so Buckingham's ]:theorem gives 7

dimensionless groups. K0, L and p were chosen as the most.

frequently recurring variables yielding the following non-dimen-

sional groups:

K K 7 L L 0 0 = fL2 2 - Q 3 h (5-1) I D J o K0 7( ~4 - 5 gL 6 L 7

A little algebraic manipulation transforms these non-dimensional

7: groupings intd more familiar dimensionless quantities: x 1 can be rewritten as 7C 17:2 hence

K 0 = Ro., a jet Rossby number. Qof

Sobey (1973a) has written 71 as 7(17 giving

(Ko/f2)*

which he has shown to represent the penetration of an axisym-

metric momentum jet into a rotating basin. 7( 2 can be left in its original form, representing the jet geometry. 7C 3 is the aspect ratio, the principal geometric reservoir

I 5L. parameter which, as will be shown later in this chapter, is very

important for hydraulic reservoir modelling.

J0 Q0 z 40 ° / ° v° / A° V° _ - _ - - Rej p p (where Ao is the area of the inflow entry ) is a jet Reynolds number.

A- 5 can be rewritten as (N hence 5)3 )i 7t 3 •Ko /Qo (X5 r3)q 7r 3 = (gh) Fr., a jet Froude number.

Recasting 7r 6 as 7C 67t3 gives

k s -K 67[3 , the relative roughness height or the bed = h resistance.

7(1 7 can be left in its original form to represent the jet buoyancy.

Table 5-1 gives some typical values of the aforementioned dimensionless parameters together with general geometric data for three reservoirs of the Thames Water Authority where the inflow is generally introduced through momentum jets. The groupings 16 and %(7 have been omitted from the table as no reliable information is available on their values for the three reservoirs; in any event they will.not be considered further in this study.

155 Table 5-1 Typical Figures for 3 Thames Valley Reservoirs

RESERVOIR QDN WRAYSBURY QUEEN ELIZABETH (II) MOTHER (1962) (1972) (1975)

Surface Area 128 202 192 (ha)

Volume 6 3 19.6 33.9 38.0 (x10 m; )

Mean Depth 15.3 16.8 19.8 h (M)

Max. Depth 17.8 21.6 22.9 (m-)

Typical Tnrroghflow 3.5 5.3 5.3 Q° (ba )

Typical Residence 65 76 83 Time Volume/go (days)

No. and size of 2 No. 2 No. 2 No. jets typically used 0.9m. dia. 1.1m. dia. 0.8m. dia. 1 No. 0.6m. dia.

Jet •Velocity 2.6 '2.9 4.3 - 4.4 Vom )

Vertical Orientation 0°, 223° 0°, 22 * 227°* of jets (angle to and 45° and 45° and 45° Horizontal)

Typical Length L (Km.)

156 Table 5-1 contd.

RESERVOIR QUEEN WRAYSBURY QUEEN ELIZABETH (II) MOTHER

(1962) (1972) (1975)

Jet Momentum Flux' 9.2 15.5 23 per uni t4dengity Ko (~ s. )

Ko Rop = 17.8 25.8 25.6 Qo fL

1 K0 L 1134 748 1367

L/h 85 6o 76

-1 ° Re. = 3 3.9 4.8. ()C1O )

K 0 Fr . = 'r 0.22 0.23 0.31 Qo (gh)7

Notes: (1) Orientation, * denotes that most commonly used.

(2)Latitude for the three reservoirs assumed to be 52°N -4 giving f = 1.14 x 10 rad sec-1.

(3)L defined by equating the perimeter of the reservoir to 7r L.

157 5.2 A DYNAMIC RESERVOIR MODEL

With the parameters identified which are most likely to affect throughflow induced circulation it was necessary to consider the modelling concept to be used and hence the dynamic and geometric parameters-and dimensions of a suitable reservoir model:

5.2.1. Dynamic Similarity

When dealing with the currents in relatively large water masses on the Earth's surface it is possible to make use of the so-called

'fixed' co-ordinate system in which the origin of the system is at the centre of the Earth and the axes pointed towards fixed- positions between the stars. The use of such a co-ordinate system, however, is not justified in this case since there is no advantage and in fact quite an obvious disadvantage in describing currents in 'absolute' terms rather than relative to the

Earth. It is therefore most useful to adopt a co-ordinate system, transformed from a fixed system, which rotates with the Earth.

This transformation results in the appearance of the Coriolis acceleration term, which represents the deflecting force per unit mass due to the Earth's rotation, in the equation of motion.

The customary way of writing the equation of motion for geo- physical problems is to use a left-handed rotating co-ordinate system, with the z-axis positive towards the centre of the

Earth and with the x-y plane tangential to the Earth's surface.

The equations of motion of an element of water in an unbounded body of water on a rotating planet must include the effects of gravitational forces, pressure gradients (which may be the

158 result of external forces eg. due to a surface wind stress or a source of momentum), the deflecting forces of the Earth's rotation and frictional forces. For an incompressible fluid and using the above co-ordinate system and writing the equation in vector form:

+ (q. 0 )q + 24)A + '`%p - g - Ep 2 q = 0 a t P (5-2) where q = velocity vector 9 ,iu + jv + kw,

p = pressure, 1 5 mass density of the fluid, E = kinematic eddy viscosity of the fluid (assumed independent of direction),

g = acceleration due to gravity = kg,

t = time,

W = -k sind

Q = vector operator = i āx + j— + kāz 1 az + + ?$2 D = scalor operator = a x2 ay2 z2 a

j and k are unit vectors in the x, y and z directions.

To determine the conditions of dynamic similitude for a model study it is useful to non-dimensionalise the equation of motion

(5-2) by the introduction of reference quantities which, for a given system, are constant. Using the reference quantities

Vo = reference velocity

Bo = reference length a group of dimensionless variables can be defined

159 _/ q p t I / q i — , p' = 2 , t = Vo JV o Bo/Vo (5-3) .. )2 2 2 ( V Bo , I = 0 B0 V 2 0 Substitution of (5-3) into equation (5-2) and dividing by B 0 gives:

d q ~ (5L sin) _ I r _ _I + (q • Q ) q:; + Bo (2k A q ) t V0

g Bo ( V )2 qi (5-4) /13/ (—T)V o - V B 0 0 0 Following established notation:

V B 0 0 Re = E , a turbulent Reynolds number,

70 Fr = , a Froude number, (g Bo)7 Vo and Ro = , a Rossby number. (51 sin) Bo

The turbulent Reynolds number, Re, is a ratio of inertia forces to forces due to turbulent shear stresses; Fr is a ratio of inertia forces to gravity forces; and Ro is a ratio of inertia forces to Coriolis force.

Equation (5-4) becomes after introduction of the dimensionless parameters Re, Fr and Ro.

- I

a q / _/ _ -- - + ) q + (2k Aq' ) + Q I p I I 2 k t i Ro Fr p 2 q f (5-5) Re

160 Figure 5-2

Components of the Coriolis force due to the Earth's rotation

161 Thus the unique condition for complete dynamic similarity between geometrically similar model and prototype reservoirs is the equality of Ro, Re and Fr for model and prototype. Equation (5-5) is not

exact since the frictional forces are only approximated by a

constant eddy viscosity and, furthermore, the eddy viscosity is

not a fluid property since it is dependent upon the nature of the

flow.

The components of the Coriolis forces result from the components

of the Earth's rotation vector. These components, for a point in the Northern Hemisphere, can be seen to be one perpendicular and one tangential to the Earth's surface (see figure 5-2). The important component here is the normal component (S1 sin4) which gives rise to the deflecting force on water particles in motion.

The tangential component ( cos() produces a vertical force,

however, this is not of sufficient magnitude to modify the gravitational force. In the Northern Hemisphere the Coriolis force arising from the rotational vector component ( A sino) will always act to the right of an observer looking in the direction of the relative flow. The absolute magnitude of the Coriolis force fc,

per unit mass, is given by:

fc = 2 5Z- sinli V (5-6) where V is the current velocity.

5.2.2 Model Considerations

In section 5.2.1 it was shown that the condition for complete

dynamic similarity between model and prototype is equality of

Ro, Re and Fr in model and prototype. Unfortunately, in practise,

162

it is virtually impossible for the Reynolds and Froude numbers

to be both equal and, since the type of flows considered are

with a free surface and dependent on the water surface slope

from inlet to outlet generated by the momentum of the inflow,

it is vital that the Froude number be equal in the model and the

prototype. Although the flow in the prototype is turbulent

(_by virtue of the large water depth rather than the magnitude

of the velocities) it is not greatly dependent on the Reynolds

number and, as will be shown later, to distort the vertical

scale or artificially roughen the model bed in order to preserve

Reynolds number equality or turbulent flow can lead to misleading results.

For a reservoir model where the currents are generated by the

throughflow it is desirable to use a modified Froude number

scaling which takes into account the balance between the inlet

momentum flux and the total mass of water in the reservoir since

it is obviously the action of one ow the other which dictates

the degree and type of circulation occuring in a reservoir.

Referring back to section 5.1 equation (5-1) we can make use of

three of the original dimensionless groups:

K L Ko 0 x2 = 7C 3 = h— -"It5 g L Qo Using these to form the new dimensionless groups:

K L2 A V 2 L2 L2 0 0 0 = (]r = 2)2 Qo2 (5-7) A02 Vol Ao

163

V 0 7C = -~ (5-8) 7C5 ( 5X 3 3 (gh)2

Therefore from (5-7) and (5-8):

( 1 )2 Vol Ao 5 (5-9) gh L2 7C21 Multiplying the numerator and the denominator of (5-9) byf :

(7C 5/ )2 p V02 Ao Kinematic Momentum Flux (5-10) r p h L2 g 7C2 J Mass of Water in Reservoir

A0 V0 2 Equality of in model and prototype will thus ensure h L2 g

a dynamically similar balance between the momentum flux and the

mass of water it circulates.

For an undistorted model (ie. equal horizontal and vertical scales):

A0 V0 2 A0 V02 (5-11) h L2 g MODEL h L2 g PROTOTYPE In developing the appropriate model relationships it is convenient

to use the subscript m for model quantities, p for prototype

quantities and r for the ratio of model to prototype.

(Lh)p (Ao)p (5-12) (A0)m (Lh)m. Lr hr

Therefore equation (5-11) becomes

(Vo2 )m gm 2 h - (5-13) .Lr r (V02 )p = Lr hr gp

Since gm = g equation (5-13) becomes

164 (V ) o = V = (Lr)4- (5-14) (Vo)p For an undistorted model L = h r r )m 1 (Vo Therefore = V = (h )7 (5-15) ) r r (Vo p

Equation (5-15) represents a typical Froude law scaling.

As far as the rotation of the model, to simulate the Coriolis force, is concerned the speed of rotation is governed by the time ratio T . r The time ratio can be obtained from the relationship

L L L r ✓ r Vr = — or T = — = (5-16) r Tr ✓ (h )7 ✓ r for a Froudian modelling.

As was pointed out in section 5.2.1, the Coriolis force is due to the normal component (R. sing) of the Earth's rotational velocity vector. To achieve rotational similitude, this component must be modelled. The angular velocity of the model, W m , is given by the angular velocity of the prototype divided by the time ratio:

1 ~) m CJr = - _ (5-17) T ( sinO) r p (.R sin~)p ( sin()p hr~ Therefore ū) m = _ (5-18) T L r r

The above relationships satisfy the equality of Rossby numbers, which is the dimensionless number of ten used where rotational similitude is desired. If horizontal length is taken as the characteristic dimension, the Rossby number is:

165 V r (R ) = o r (5-19) w r Lr

V r Since (,t) =i— = — from equation (5-16) : r T L r r

(Ro)r = 1 (5-20)

Thus the Rossby numbers of model and prototype will be equal if equation (5-18) is satisfied.

5.2.3 A Circular Reservoir Model

As was described in chapter 3_ the model available for experimental investigation of the circulation in reservoirs was circular and 5m. in diameter. A circular model is clearly an idealised shape to consider initially, however, if no restrictions were placed upon designers regarding the shape of reservoirs it is most likely that they would choose a circular shape for prototypes as this maximises the stored volume for a given area. It was also desirable to make maximum use of the 5m. horizontal dimension and to relate the 'idealised' model to a typical prototype reservoir through the inlet characteristics. A suitable reservoir which is almost circular in shape and for which the required data was readily available was the Queen Mother Reservoir at Datchet. From table 5-1 it can be seen that the typical horizontal dimension of this reservoir is 1.5km. which for the

5m. horizontal dimension of the model gives a horizontal scale of 1:300 (Lr = 0.0033), for an undistorted model the mean depth in the prototype of 19.8m. would be 66m; in the model. The velocity measuring buoys, however, were designed for a water

166 depth of 80mm. and hence the depth in the model was increased to this value representing a prototype depth of 24m. Since the maximum depth of water in the Queen Mother Reservoir is 23m. this was not thought to be an unreasonable or unrealistic increase in the model depth.

With an increased depth in the model it was necessary to use equation (5-11) to determine the inflow characteristics since the momentum/mass balance had been disturbed.

Recasting equation (5-11) gives:

I o L2 \ (Vo )m A h (5-21) (vo )p h L2 p Ao /m

Since in the prototype there are three inlets (A) was deter- 0 P mined by taking the overall momentum flux (K and dividing it 0 )p by (Vo)p (both from table 5-1) this gives a value of .(A0)p of

2 1.22m . (Ao)m is given by equation (5-12).

2 1.22 (8x1o_2x52x300 2)j Hence (V ) = o r 19.8 x 15002 1.22 therefore Vr = 0.064; this compares with r = 0.059 if no correction had been applied for the increased model depth.

The inlet characteristics for an undistorted model are thus:

Ao = 13.5mm2 (1.22m2 in prototype)

(3.7mm. diameter)

Vo = 27.8cros-1 (4.35ms 1 in prototype)

Q 3.8cm3s-1 (5.3m3s-1 in prototype) 0

Ko = 104cm4s-2 (23m4s-2 in prototype)

167 The model was set up with this inlet arrangement and preliminary tests undertaken to establish whether, with this momentum flux, it was possible to make velocity measurements over the majority of the model, no attempt was made to consider the form of circulation generated. Unfortunately it was soon evident that this jet momentum (Ko = 104cm4s 2) was not sufficient to generate currents of a measurable magnitude in the areas of particular interest ie. the slowest moving areas. With an increase in jet momentum clearly necessary the area of the nozzle was increased by using a jet 1.2cm. in diameter and then the efflux velocity determined which would just give measurable velocities. Although measurable results have been obtained with this jet size and an -1 efflux velocity of 9cms 1 (figure A-1) it was decided that 18cros was a more suitable lower value of Vo. These changes in the inlet characteristics resulted in a 252% increase in Ko, a 435% increase in Qo and a 737% increase in Ao; Vo was reduced by 35% from its previous value of 27.8cros 1.

168 5.3 INVESTIGATIONS OF A CIRCULAR RESERVOIR MODEL

With the geometric and dynamic characteristics of a circular

reservoir model established and a velocity measuring technique

developed the next step was to investigate the circulation in

the model and how various geometric and dynamic parameters affect

it.

Sobey and Savage (1974) have reported the results of a simple

theoretical model of the jet-forced circulation in a circular

reservoir. They hypothesised from their mathematical model

and some limited experimental results that the mixing in a specific

reservoir will be improved if the forced circulating flow (gc) is

maximised. In turn from the mathematical model they found that

Q was only dependent on L/h and k /h of the dimensionless para- c s meters derived in section 5.1 (their mathematical model was

homogenous and took no account of the influence of the Earth's

-rotation or of any density difference between the inflow and the

ambient reservoir density) and was independent of the jet geometry

parameter Ko7 L , the jet Reynolds number AoI Vo Q. 0 and the jet Froude number KC/Q 1 (gh)7

Earlier experimental work by Cooley and Harris (1954) had not

taken advantage of dimensional arguments as they arrived at an

empirical equation from their results:

169 Qc — = 0.12 R1 22 ō (5- ) h in which 0.12 has units of feet-land R1 was the radial distance.

An effective L for their experimental basin is 4.79m. (15.7feet).

Equation (5-22) then becomes:

Qc L -1 0.94_ (5-23) KL h

Equation (5-23) is consistent with the findings of Sobey and

Savage (1974), their experiments also indicated a significant reduction in Qc with an increase in the boundary resistance ks.

Although these previous studies have contributed to the know-

ledge on reservoir circulation and given a clue as to the para-

meters which are most likely to influence the throughflow-

induced circulation there was obviously scope for further inves-

tigations particularly in order to obtain more accurate and

useful experimental measurements. The question of field measure-

ments has also to be resolved but was not within the scope of

this present study.

Other factors possibly influencing the circulation in reservoirs

which could be studied included the position of inlet in relation

to the reservoir shape, its direction of discharge, its size

and shape, the momentum of the throughflow and, for the outlet,

its position in the reservoir, its type and finally the effects

of the Earth's rotation. Although many of these variables were

studied, it was not possible to include them all.

170 The model investigation undertaken was for a homogenous fluid- filled reservoir with a steady throughflow (ie. inflow rate outflow rate, mass extracted at the outlet but not momentum) and with no density difference between the inflow and the ambient density of the reservoir water. No account was taken of bed roughness effects and, for many of the experiments, the Coriolis effect.

5.3.1 Tangential Jet and Central Outlet Model

A circular model with a tangential jet inlet and an outlet located at the geometric centre of the model was considered by

Sobey and Savage (1974) and is the simplest case for a circular model since it is easy to measure the overall circulating flow

(which is not the case with jets issuing away from the boundary) and as such it represented a suitable starting point.

The 5m. diameter circular model was set up with an 80mm. depth of water and the velocity measuring buoys tethered at each inter= section point of a 0.4m. grid, symmetric about the orthogonal model centre-lines, covering the whole model area. A 12mm. diameter circular nozzle (made of a plastic material) was positioned on the model bed at the perimeter of the model and directed so that it would discharge around the boundary of the model as if it were exactly tangential. The delivery of water to the nozzle was through plastic tubing from a pump and via a flow-regulating needle-type valve and a rotameter as described in Chapter 3 and

,shown in figure. 3-7.

171 The outlet pipe was placed at the geometric centre of the model and acted as a sink. It was made two-dimensional by a series of holes drilled in the submerged section. The water depth was maintained at 80mm. by an adjustable overflow. weir. The outlet system was maintained as a primed siphon and hence the water level rise due to the inflow was compensated by a water level drop due to the outflow.

As described in Chapter 4 before experiments using the velocity measuring technique could begin it was necessary to record the

'zero' positions of all the buoy targets relative to the photogrammetric control points by photographing each of the four quadrants with no flow in the model ie. in stagnant water.

Then the inflow was switched on and the regulating valve adjusted to give the required flow rate, as measured by the rotameter.

It was then necessary to adjust the outlet conditions to maintain the constant depth of 80mm. This water depth had to be checked regularly until a steady state velocity field was established as the changing velocities in the model caused the water level to alter.

The model was left with a constant throughflow for four or five hours to ensure that the velocity field was in a steady state; once it had been decided that the reservoir flow was in relative equilibrium a further set of photographs of each of the quadrants could be taken to record the 'deflected' position of the buoy targets and hence the velocity field existing at the instant of film exposure. As only one Hasselblad camera and lens was available for this project it was necessary to transfer the

172 camera between the four stations quickly. As a result the velocity

fields were not recorded instantaneously, but since the

time taken to photograph the four quadrants of the model was of

the order of 2 to 3 minutes this was not considered a serious

limitation (this aspect was investigated and is reported in section 5.3.5).

For the arrangement with a tangential jet and central outlet in

the 5m. diameter model the velocity fields were photographed,

analysed and determined for six different values of the jet

momentum flux per unit density (K0). The nozzle was circular and

12mm. in diameter in all cases but the discharge and hence the

jet velocity was increased so: that the momentum increased in 1 roughly equal increments. The lowest jet velocity was 9crps 1 as compared with the value of 18cros proposed in section 5.2.3.

The six computed velocity fields are shown in figures A-1 to A-6

(appendix A). Its should be pointed out at this first reference

to velocity fields that firstly the point at which the velocity

was measured is at the base of the direction arm of the arrow

representing the velocity vector, secondly the position of the

inflow and its direction are given by the arrow outside the model

boundary projecting inward but not bisecting the boundary and

finally that the outlet position is indicated by the small circle

within the reservoir.

The essential features of this type of circulation are visible in

figures A-1 to A-6; the major flow is concentrated into a wall

jet around the circumference of the model, the growth rate of

this shear layer decreases with distance from the inlet and does

173 not appear to grow significantly beyond 7r radians from the inlet.

Lateral turbulent momentum transfer from the wall-jet region to the interior flow appears to be the major mechanism which induces forced circulation. The interior flow reaches a steady state when the angular momentum flux of the inflow balances the integrated friction torque from the bed and side boundary walls.

The interior flow-velocities are noticeably less than the maximum velocity at the same azimuth however, the total circulating flow is significant since the velocities, although small, act over a width considerably greater than the wall-jet region width.

Sabey (19238.).,for his mathematical model, used three parameters to describe satisfactorily the depth-averaged azimuthal velocity profiles. These can be summarised thus:-

(1)a characteristic azimuthal velocity for the jet

development

(2)a characteristic radial width for the jet develop-

ment

(3)a characteristic azimuthal velocity describing the

forced circulation.

The definition of this is shown in figure 5-3, however, from the results obtained from the six velocity fields together with further ' work by Ali et. al. (1978a) it is clear that the outer flow velocity profile is Gaussian conforming to an equation of the form

(y - ym)2 1 U = Um exp -0.69 (5-24) (y, as given by Newman (1969) for a wall jet flowing over a surface in still surroundings. Figure 5-4 identifies the velocity scale

174 Um and the length scales ym and yym. Very close to the wall the

conventional logarithmic law of the wall applies for the inner

boundary layer. Sample velocity profiles at a representative

azimuth for the velocity fields of figures A-2 and A-6 are shown

in figures 575 and 5-6 together with a Gaussian velocity dis-

tribution satisfying equation (5-24) and determined by using

values of m, ym, and estimated from the model measurements y1z m across the selected azimuth.

Although a general impression of the quantity of a particular

circulation pattern can be gained by a qualitative examination of

the velocity fields it is most useful to quantify or classify the

field by the suitable choice of a parameter or non-dimensional

quantity. Two such parameters have been used in this study and

non-dimensionalised by inspection:-

(1) A parameter often used in the past (White et. al., 1955;

Sobey and Savage, 1974; Ali et. al., 1978a) in.the description

of reservoir flows is the overall circulating flow or circulating

discharge (Qc) of a gyre or of a system of gyres. In a single

gyre system such as those shown in figures A-1 to A-6 for the

tangential jet inlet it is quite a simple operation to calculate

Q by depth (assumed constant) and radial integration of the c azimuthal velocity distribution along suitable radii of the gyre.

Ali et. al. (1978a) have given the circulating discharge as

Qc = U 1 h (5-25)

where U is the mean velocity at the section in question, and 1 and h are respectively the length and mean depth of the section.

175 1.. s(e)1

wall jet V(e)

forced circulation a(e) o •- central r r=jR r=R outlet

S — characteristic radial width of wall jet V — characteristic velocity in well jet a — characteristic velocity in forced circulation R —radius of model reservoir r— radial coordinate e —azimuthal coordinate

Figure 5-3

Definition sketch for azimuthal velocity profiles of Sobey (1973a)

176

Ua YMT ..._.«,n_...

Figure 5-4

Definition sketch for wall jet flowing over a surface in still surroundings (Newman, 1969)

177 U ms-1 (x163 )

o experimental points o--

o U = 0.005 exp (-0.69( y-0.2 )0.92 ) mss K m4 s-2 K. = 3.66 x166 e

cr c 0 a a)

r I 0.5 1 - 1.5 .2 2,5 Radial distance from centre of model y m, Figure 5-5

Sample velocity profile at a representative section, tangential jet inlet Ko = 3.66 x 10-6m4s2 -1 3 U ms x 10

o experimental points

• U = 0.015 exp (-0.69 ( y-0.15)2 0.222 ) m 15 KO = 28.25 x 1 Ō6 m4 s2

0.5 1.5 2 2.5 Radial distance from centre of model y m. Figure 5-6 Sample velocity profile at a representative section, tangential jet inlet, Ko = 28.25 x 10-6mks-2 Although the measurement is fairly easy with a single gyre system it becomes a little more complex when several gyres occur

(section 5.3.2).

The circulating flow or discharge of a reservoir is often based on the mean of several sections and can be non-dimensionalised by inspection considering the form of the grouping 7r2 of. the original dimnsional analysis in section 5.1 giving the parameter:

Qc for the circulating flow. K 7 L 0

(2) Another possible way of expressing the quality of a particular circulation from the velocity field is by a consideration of the mean velocity magnitude (m) and the standard deviation (Vs) from this mean of the velocities in the reservoir. This quantity has proved difficult to abstract from data in the past since the use of integrating floats does not provide the uniformity of coverage required to ensure a truly mean value. Measurements of velocity over a uniform grid by the velocity measuring buoys are as uniform in coverage as possible and hence a true mean value of the velocities in the reservoir can be calculated.

Although an increase in the jet momentum would be expected to increase the mean velocity magnitude in the reservoir it may not do so linearly and in doing so it may be that larger areas

of effectively stagnant water occur, this occurrence should be indicated by an increase in the standard deviation (Vs) relative

to the value of Vm. Vm and Vs are easily computed from the

velocity components derived by the computer program in deter-

mining the velocity field. Vm, the principle parameter can be

180

non-dimensionalised in the manner of ic 5 of section 5.3 to give the non-dimensionless group

V m , a reservoir Froude number, (g h)2

which represents the overall velocity magnitude in the reservoir.

It should be noted that although V /(gh)i is a Froude number it m should not be considered in the normal way as h has been assumed constant at 80mm. in calculating its magnitude.

The parameters Qc, Vm and Vs have been computed for the six velocity fields (A-1 to A-6) having the tangential jet inlet, central outlet arrangements. They are given in their non- dimensional form together with those dimensionless quantities derived in section 5.1 which are variable (L/h and L are Ko7

Qo constant, ō or (Ko/f2) are infinite since the model was

Qo fL L not rotating and bed friction and jet buoyancy were not included) in table 5-2. A third dimensionless parameter not derived in section 5.1 but which can be non-dimensionalised by inspection,

Q /Q , has been determined for each velocity field and represents o c the influence of the outlet on the reservoir circulation. The inference being that large Qo/Qc means that the outlet does affect the circulation and a small value that it is having very little effect.

181 Table 5-2"Experimental Results, Tangential Jet and Central Outlet, Large Model

V Qo/Qc -1 L /(gh)!. Rei . QQ ō ō Qc m s Qco Vm Fr 1 1 1 mas m4s`2 m2s m3a 1 me me 1 -4 -4 (x 10 ) (x lo6 ) (x 1o`3) (x 10 ) (x lo 3) (x 1ō3 ) (x lo-3)

0.102 0.92 0.96 3.35 .3.1 1.6 0,03. 0.07' 3.5 96o 0.10

0.203 3.66 1.91 4.5 3.3 2.5 0.045 0.047 3.7 1910 0.20

0.283 7.06 2.66 6.42 5.5 5.o 0.044 . 0.048 ._6.2 266o _ 0.28

0.396 13.84 3.72 6.89 5.3 5.6 0.057 0.037 6.o 3720 0.40

0.497 21.88 4.68 10.02 7.6 7.0 0.050 0.043 8.6 4680 0.50

0.565 28.25 5.32 10.75 7.5 7.7 0.053 0.040 8.5 532o 0.56

Notes: (1) Ao = 1.13'x 10-41112 (2) Ko L/Qo = 470.4 (3) L/h =-62.5 (4) g 9.81ms2 106m28 1 (5) c) (6) Rop = ao (7) °P and ~- (not included) h The dependence of the overall circulating flow Qc, and indeed

the mean velocity magnitude, on K01. can be seen in the graph

of figure 5-7 where both are shown to be directly related to

Ko It is noticeable, however, that the straight lines do

not pass through the origin, clearly this should be the case.

As will be seen later similar graphs of K against Q and V o c m for a irregularly shaped reservoir with an effectively tangential

jet do intercept the origin.

Sobey and Savage (1974) decided, on the basis of their computer experiments, that a variation in the jet Froude number or the jet Reynolds number had-no effect on the non-dimensional circu-

lating flow given by Qc/Ko3L for constant L/h and o1L/Qo.

Unfortunately their results were obtained from prototype reservoir data which was in some important aspects incorrect and therefore must be questioned especially as they were not verified experimentally. Ali and Hedges (1975) and All et. al. (1978a)agree with Sobey and Savage regarding the importance of the aspect ratio and also concluded, by comparing flow patterns at constant

L/h and varying Rej and Fr., that the jet Froude number and jet

Reynolds number are insignificant; a close examination of the flow patterns of Ali et. al. reveals some dissimilarities.

Ali and Hedges also investigated the influence of the jet geometry, K01L , and found the curve of Qc ' against KoIL to have KL Qo O QO a peculiar shape, Sobey and Savage dismissed this as being due to the irregular shape of the reservoir. Figures 5-8 (a) and (b) show the variation in Qc and m with Fr. and Rej at

7 - L (gh)2

183

PC 3 -1. m .s (1 Q4 ) 10 Qc points from table 5-2 Vm

- Qc ® points from table 5-3 9 4) Vm m s 1 3 8 8(x10 )

-6

-15

1 , 1 2 3 4 5 6 2 -1 -3 K m Figure 5-7 5 (x10 )

Graphs of Qc and Vm against K02, L/h = 62.5 and Qc against Kot, L/h = 18.75 for tangential inlets. 184 4

0.1 0.2 0.3 0.4 0.5 Fr. J 1 (a) Graphs of Qc/KC L and Vm/(gh)~ against Fr~ , L/1? = 62.5 and Qcō -L y

Fri, L/h =18.75 for tangential inlets. QC r h)1 (x103) 0.08 a

0.0E 6

Q04 4

0.02 2

1 2 3 4 5 Re. (x103) J (b) As (a) but against Rei Figure 5-8 185 L/h = 62.5. The curves of Qc against Frj and Rei do not

TI I 0 TA entirely agree with previous findings as the curves show clearly that Qc has a tendency to increase at lower values of Fr. and

Re.. The variation of the mean reservoir Froude number with Re. 3 and Frj can be seen to be linear. In table 5-2 the values of Q 0

Qc confirm the view, also noticed by examining the velocity fields, that the outlet has little or no effect on the reservoir cir-

culation. The values of Qo do not vary by much and are less than

0.06 in all six cases indicating that the flow through the outlet is less than 6% of the flow in the reservoir. Even with an overflow weir outlet the effect on the circulation of the outlet was small; Ali et. al. (1978a) confirm this finding.

Unfortunately the six experiments with a tangential jet were performed with a constant aspect ratio of 62.5 and consequently the influence of this parameter on either the circulating flow or the mean velocity magnitude was not tested.

A drawback of the velocity measuring technique at present, and which will need resolution for future work, is their inability to measure water velocities in depths other than 80mm. This disadvantage means that, if the velocities are to be measured in this way, the only way L/h can be varied is by varying L;

186 this clearly presents certain difficulties. In this present study the only way that the influence of L/h could be partially investigated was by making use of a smaller diameter circular model, used by Falconer (1976), having a diameter of 1.5m., which with a water depth of 80mm. gives an aspect ratio of

18.75. The model was timber-based with an aluminium vertical boundary wall and could be readily placed on the bottom of the large 5m. diameter model in one quadrant thereby making use of the established photogrammetric control. The inlet and outlet arrangements were as described for the 5m. diameter model and the model had a 100mm. grid drawn on its bed so that the buoys could be tethered at various grid intersection points. As can be seen from the velocity fields it was decided to place buoys where a velocity measurement was really necessary or possible (obviously the same values of jet momentum acting on a smaller water mass generates larger areas of higher velocity which are outside the measuring range of the buoys.

The velocity field was computed for the four lowest values of

Ko given in table 5-2 and these can be seen in figures A-7 to A-10. The general circulation patterns can be seen to be similar to those of the larger model (figures A-1 to A-6). The circulating flow (Qc) was measured over a suitable radial section, unfortunately the mean of several values of Qc at different sections could not be taken or the mean velocity magnitude calculated as the velocities were not known in several areas of the model.

The values of Qc found, together with the same non-dimensionless parameters as in table 5-2, are given in table 5-3.

187

Table 5-3 Experimental Results, Tangential Jet and Central Outlet, Small Model

L Rei Fr. Qo Ko ō Qc Qo/Qc Qc/ ol

mas 1 mks 2 m2s 1 m3s -6 (x 10-4) (x 10 ) (x 10-3) (x 10-4)

oo 0.102 0.92 0.96 1.34 0.076 0.093 960 0.10

0.203 3.66 1.91 1.89 0.107 0.066 1910 0.20

0.283 7.06 2.66 2.61 0.108 0.065 2660 0.28

0.396 13.84 3.72 3.73 0.106 0.067 3720 0.4

Notes: (1) Aō,1.13 x 10-4m2 (2) o3L/Qo = 470.4 (3) L/h = 18.75 6m2s.1 (4)g=9.81ms2 (5) (~=1Ō (6)Roj=00

(7) 4 and ke (not included) The variation of Qc with o can be seen to be linear in figure

5-7. Comparison of the values of Qc o1L in table 5-2 at L/h = 62.5 and at the four jet Froude numbers 0.1, 0.2, 0.28 and 0.4 with the corresponding values of Qc/ olL in table 5-3 at L/h = 18.75 confirms the hypothesis that the value of L/h has a direct

influence on the circulating flow for a constant jet Froude

number (or Reynolds number). This trend will be discussed further in section 5.5 for the irregularly shaped reservoir of Ali et. al.

(1978a). The graphs of Qc ō-1L against Frj and Rej are shown in figures 5-8(a) and (b) respectively and confirm the belief that

Frj and Rej may have some effect on the overall circulating flow and cannot be summarily dismissed. The graphs once more show that there is an initial fairly rapid decline in the circulating flow but that this reaches a threshold where there is no benefit gained by an increase in Frj. or Re.. Unfortunately it was not possible to determine the circulating flow at lower values of

Frj and Rej because of the measurement problems.

The values of Qo/Qc are, once more, small in no case being larger than 0.011 (less than 11% of the total reservoir flow through the outlet), they show an increase over the values when L/h = 62.5 but since from table 5-1 the prototype values of L/h for three typical reservoirs range from 60-85 it would seem that for a circular homogenous reservoir the outlet has little or no influence when the circulation is induced by a tangential jet.

With the simplest case of a circular model with a tangential

189 jet inlet investigated it was possible to continue and study

the effect of varying the direction of discharge of the jet

and the position of the outlet on the reservoir circulation.

5.3.2 Radial Jet and Diametric Outlet Model

The 5m. diameter model was set up as before with an 80mm. depth

of water and the buoys tethered at each intersection point of

the grid. The 12mm. diameter circular nozzle was placed on the

bed and directed to discharge along a diameter of the model.`

The outlet was positioned at the opposite side of the model and

on the same diameter as the direction of jet discharge with the

80mm. depth of water once more maintained.

The deflections of the buoy targets were determined in the

the previously described manner for four different values of

Ko, with the discharge through the nozzle being increased in

such a way that the momentum increased in roughly equal increments.

The four velocity fields computed are shown in figures A-11 to

A-14, the predominant features of this type of circulation induced

by a radial jet being clearly evident. The flow basically

consists of two gyres or swirls rotating in opposite senses and

separated at the reservoir centre by the relatively thin shear

layer of the inlet jet core. The gyres should, in theory, be

symmetric, however, a slight misalignment of the jet direction

away from the diameter, a slight assymmetry of the reservoir

'shape or an uneveness in the bed may cause the gyres to be

asymmetric and for the flow to be correspondingly biased. This

effect can also be induced by the Coriolis acceleration in a

prototype reservoir and hence a rotating model. The majority

190 of the flow in this type of circulation is restricted to the jet core and the thin side-wall boundary layers that complete the outer boundary of each gyre. The interior flow in each gyre is very much slower than the flow in the confining shear layers, this effect can be seen to be more pronounced in figures A-13 and A-14 where the jet momentum is much higher than for figures

A-11 and A-12. In theory a frictionless wall placed along the jet axis to intersect the diametrically opposite wall at the stagnation point would not affect the flow patterns but would merely isolate each gyre within a closed geometry boundary

(c.f. section 5.4). This fact will prove the basis for the determination of the overall circulating flow for the two-gyre systems.

As was mentioned earlier the determination of the circulating flow in the two-gyre or multi-gyre systems is not as straight- forward as the single gyre system of section 5.3.1. In this study the overall circulating flow in a two-gyre system has been computed by adding the flows in each gyre analysed separately.

The individual circulating flow in a gyre being determined as if it were a single gyre as in 5.3.1. The first stage of the determination involves finding the position of the centre of each gyre, this is defined as a point where the velocity magnitude tends to zero by virtue of the flow on either side of this point being in opposite directions and parallel to the jet direction. With this centre point known the velocity distribution can be integrated from the point to the model boundary over the radial distance and the depth (once more assumed constant at

80mm.). The overall circulating flow is then merely the sum of

191 the return flows of each gyre.

The overall circulating flow (Qc) and the mean velocity magnitude

(Vm) have been computed for the four velocity fields A-11 to

A-14 with the radial jet inlet and these, together with the other

dimensionless quantities are given in table 5-4.

plotted against Qc and Vm (Figure 5-9) again The graph of o4 demonstrates the dependence of these reservoir circulation

whilst the graphs of Qc and Vm against parameters on o1 1711= (gh)

Fr. and Re. are similar to those with the tangential jet/central outlet arrangement. The values of Qc are of similar magnitude

KL for a given value of Frj or Re. but the mean reservoir Froude number (Vm/(gh).1) is generally larger. Qo is small and indicates

that less than 6% of the circulating flow is leaving the reservoir; in order to physically test the validity of assuming that the

outlet was having no effect on the reservoir circulation the

experiments with the radial jet and four values of the momentum

were repeated but with the outlet repositioned asymmetrically on

the perimeter of the model on a diameter perpendicular to the

direction of jet discharge. The velocity fields measured are

shown in figures A-15 to A-18 and are comparable to figures A-11

to A-14. Clearly the velocity fields are as expected with the

outlet having no influence on the flow pattern in any of the four

192

Table 5-4 Experimental Results, Radial Jet and Diametric Outlet, Large Model

K V /(gh) Fr Qo o Ko Qc Vm Vs Qo/Qc Qc o4 L m Rei . m36-1 4 -2 2 -1 m35-1 -1 -1 ms ms ms ms -6 i (x 10-4) (x lo ) (x 10-3) (x 10-4) (x 10-3) (x lo-3) (x 10-3)

0.203 3.66 1.91 4.2 3.4 3.4 0.05 o.044 3.8 1910 0.20

0.283 7.06 2.66 7.0 3.8 3.4 0.04 0.052 4.3 2660 0.28 w -

0.396 13.84 3.77 8.o5 6.7 6 0.05 0.043 7.6 3720 0.40

0.565 28.25 5.32 9.9 8.2 7.4 0.06 0.037 9.3 5320 0.56

Notes: (1) Ao = 1.13 x 10-4m2 (2) KolL/Qo = 470.4 (3) L/h = 62.5 (4) g ,= 9.81ms 2 k6 (5) Ū = 10 6m2s1 (6) Rod = 00 (7) 11p and not included Table 5-5 Experimental Results, Radial Jet and Asymmetric Outlet, Large Model

Ko L V /(gh) Rei Fr Qo ō Qc Vm Vs Qo/Qc Qc ō m j -1 4 -2 -1 -1 -1 -1 mas m s mas mas ms ms -6 (x to 4) (x lo ) (x lo-3) (x 10-4) (x lo-3 ) (x 10-3) (x to 3)

0.203 3.66 1.91, 4.7 3.5 3.3 0.04 0.049 3.95 1910 0.20

0.283 7.06 2.66 6.45 4.1 4.7 0.04 0.048 4.6 2660 0.28

0.396 13.84 3.77 6.7 6.5 5.4 0.06 0.036 7.3 3720 0.40

0.565 28.25 5.32 10.7 9.8 7.9 0.05 0.040 11.1 5320 0.56

-4 Notes: (1) Ao = 1.13 x lo m2 (2) L/Qo = 470.4 (3) L/h = 62.5 (4) g = 9.81ms Ko 2 (5) 0 = 10 6m2s 1 (6) Ro = 00 (7) dP and ks not included. J ~O h m7-3-3 fix-71- 0-4

10 Qce points from table 5-4 Vm') Qco points from table 5-5 • 9 Vm 0 Vm m s-1 _3 diametric outlet (x10 ) 8 asymmetric outlet 8

7 7

6 / 6

5 /

4 4

3 3

1 2 3 4 5 6 Figure 5-9 Ka)2 m2s1(x103~ Graphs. of Qc and Vm against o4, L/h = 62.5, radial jet with diametric and asymmetric outlet.

195 0.08

0.06 G

0.04 4

Q02 2

0.1 02 0,3 0.4 0,5 Q6 Fr. (a) Graphs of Q /K lL and V /(gh) against Fr., L/h = 62.5, radial jet with diametric and asymmetri c outlet

Q~~L

0.08

0.06 6

0.04 4

0.02 2

1 2 3 4 5 6 3 (b) Rei (x10 ) As (a) but against Re. Figure 5-10 196

cases. The values of the overall circulating flow, mean velocity magnitude etc. have been calculated and are given in table 5-5 and plotted in parallel with the symmetric arrangement in figures

5-9 and 5-10. From the tabulated values and the graphs it is demonstrated that repositioning the outlet has had no effect.

With the radial jet arrangement it is interesting to consider the variation in the velocity along the expected jet axis and to compare-the experimental values with the theoretical prediction for an axisymmetric or circular turbulent momentum jet issuing into a motionless fluid. The (boundary layer) equations of motion for an axisymmetric turbulent jet (figure 5-11) are (Townsend,

1956)•-

{u2 r ~ U U~U VaII a - 2) 1 O (uv r) _ + + - r a r r ōr (5-38) II 1 (Vr) = 0 (5-39) x r a r and the flow is self-preserving or self-similar at a sufficiently large Reynolds number if (Townsend, 1956)

U = Uc f(r/Y,o) = i Uc f(9 ) (5-40)

Uc2 (5-41) u? = g11 (1/Lo) = Uc2 g11(9)

v 2 = Uc2 g22 (r/Lo) = Uc2 g22() ) (5-42) Uc2 ūv = Uc2• g12 (r/Lo) = g12() ) (5-43)

r/L . where 9 = 0 The functions.f and g are independent of x and r with the length scale Lo and velocity scale Uc being defined in figure 5-11.

197 (cylindrical Co ordinate)

Figure 5-11

Definition sketch for the equations of motion of an axisymmetric turbulent jet.

198

Using the continuity equation (5-39) to relate the two components of the mean velocity gives:-

-1 d(U L ) 9 V = 13 a,c 0 9fdi7 (5-44) o

Substituting equation (5-44) and equations (5-4o) - (5-43) into the momentum equation (5-38) gives:-

Lc dU f2 1 d(U L ) 1 df / 2L dU c _ c o _ / fd9 + o c (g11-g22) [ Uc dx Uc dx 7 d9 o / Uc dx di2 dLo 9 dg11 - dg22- + ( + g12 dx d9 del di}

d2fj Lo df Jc Uc (5-45) FUL l d dn 9 The terms included in square brackets are functions of x only and the remaining terms of . Since there are no square-

bracketed terms multiplying/the final inertial terms dg12 and

g12 in equation (5-45) self-preservation of the flow is only 9 possible if•the square-bracketed terms are all constant, also

at sufficiently large Reynolds numbers the viscous terms on the

right-hand side of equation (5-45) tend to zero and the conditions

for self-preservation are then:-

dL —° = constant 4 p( (5-46) dx a

L dU and —° o = constant (5-47) U dx 0

199 Equation (5-46) indicates that the jet width (L0) increases linearly with x, the growth rate oC has been shown to be a -= approximately 0.091 from experiments. If the momentum equation (5-38) is integrated, neglecting the normal Reynolds stress terms (u2 and v 2), across the jet from r =,,00 to 0 an additional relationship is derived:

a .5. +00 dK 27CrU2 dr = ° = 0 (5-48) dx o dx Therefore the kinematic jet momentum flux per unit length is given by:-

00 K = 27C r U2 dr = constant 0 (5-49) 0 Substitution of the similarity form U = Uc f(r/L0 ) into equation (5-49) gives:-

Ob Ko = 27( r Uc 2 f 2 (r/L0 ) dr (5-50) 0 Now 9 = r/L0 therefore dr} = dr L 0 Substitution of these into (5-50) gives:-

Uc21,02 (5-51 ) Ko = 2 9 f2( / ) d9 0 Solving equations (5-46) and (5-51) gives:-

Lo = oC a (x - x0 ) (5-52)

and Uc = K0 (x _ x0 )-1 (5-53) 2r ( ‚2a cC a

200 oū where I = f2(()) d7 a I 0 equation (5-53) implies that the velocity scale decays with the inverse of x.

The lateral velocity is given by:-

1 V = oC a Uc J f (9 ) - / f(9 ) d (5-54) 0 indicating that the entrainment velocity, although an order of magnitude smaller, decays with the velocity scale U.

The functional form of the similarity profile f(r)) is indeter- minant .and various assumptions have been made in the past to obtain a useful form. The most popular approach is to assume a

Gaussian form for the profile, ie.

f(9 ) T exp (- c9 2) (5-55)

where c = loge 2 = /0.693. With a Gaussian profile assumption the jet velocity becomes 2 2 U = Uc exp (-0.69 r /1, ) (5-56)

where the velocity scale is given by:- 1 IIc z 7.3 Kot (x - xo)-1 (5-57) and the length scale by:- Lo = 0.091 (x - xo) (5-58)

The values of U predicted by equation (5-57) can now be compared

with the centre-line velocity as determined in the experiments

with a radially-directed jet inlet. With the two highest values

of jet momentum (K0 = 13.84 and 28.25 x 10 6m4s-2) the centre- line velocities were, for the most part, out of the range of the

velocity measuring buoys and hence the centre-line velocities

can only be compared for the values of Ko of 3.66 and 7.06 x

201 Table 5-6 Experimental and Theoretical Centreline Velocities for a radial jet

Kr=2.66x 10-3 K 7 = 1.9x1Ō3 ° 2 -1 ° 2 -1 m s m s

(x-x.) (x- xo)-1 Uc IIc uc IIc (m)(m 1 ) (eq.5-58) (exp.ms1 ) (eq.5-58) (exp.ms1 )

0.66 1.52 0.0294 0.0210 -

1.07 0.93 . 0.0183 0.0132 0.016

1.47 0.68 0.0133 0.0156 0.0095 0.0084

1.87 0.53 0.0104 0.0122 0.0075 0.0059

2.27 o.44 0.0086 0.0113 0.0062 0.0061

2.67 0.37 0.0073 0.0079 0.0052 0.0052

3.07 0.33 0.0063 0.0069 0.0046 0.0049

3.47 0.29 0.0056 0.0066 0.0040 0.0038

3.87 0.26 0.0050 0.0046 0.0036 0.0027

4.27 0.23 0.0046 0.0037 0.0033 0.0022

4.67 0.21 0.0042 0.0023 0.0030 0.001

202 6m4s 10 2. The theoretical and experimental values of Uc are given in table 5-6 for the two values of jet momentum together with (x - xo) and (x - xo)-1. In solving equation (5-57) xo was taken as 10mm. (0.83 x the jet diameter of 12mm.) based on

Townsend (1956) who measured the distance to the virtual origin as between 0.5 and 1.5 diameters downstream of the orifice.

A graph of (x - x0)-1 can be plotted against Uc for the theoretical and experimental values (Figure 5-12) of Uc for each of the jet momentum values. The agreement between theory and experiment is in general quite good showing that the large model approximates to an infinite environment. The agreement is poor where one might expect, firstly close to the jet where there is a strong influence of the entrainment velocity and the proximity of the model boundaries, the velocity profile close to an axisymmetric jet is also well known to be three-dimensional. In the centre of the reservoir, away from the model boundaries, the agreement is good and co-flowing stream quite weak whereas at the opposite side of the reservoir close to the stagnation point where a comparison shows the experimental values to be lower than the theoretical, this is probably because the velocities are stongly influenced by the boundary intercepting the flow and causing an adverse pressure gradient to occur, the velocities are consequently reduced. Another possible source of error is the assumption that the maximum velocity actually occurs on the jet centre-line; this is not always true as can be seen in the velocity fields.

The good agreement between the theory and experiment is a further confirmation of the velocity measuring technique.

203 0,5 1 1,5 -1 (x -x Figure 5-12 rn`.1 Graphs of Uc against (x-x0) for experimental and theoretical results 5.3.3 Asymmetric Jet and Outlet Model In order to determine the influence of an asymmetric inflow direction the jet was re-directed so as to have a 20° eccentricity away from the radial direction of section 5.3.2. The outlet was kept in the same position as for the latter experiments of the previous section with an asymmetric outlet. The experimental procedure for velocity field determination was undertaken as before with the four principle values of jet momentum used, the resulting velocity fields are shown in figures A-19 to A-22.

The main features of the circulation are as for the radial jet but with the gyres being distinctly asymmetric and having a vastly different angular velocity. The thin shear layer of the jet inlet core is retained but with a much shorter length and the majortiy of the flow, particularly for the larger and slower moving gyre is confined to the thin side-wall boundary layer.

The overall circulating flow (Qc), for this asymmetric circulation was calculated in the same way as for the symmetric situation of section 5.3.2 ie. the individual gyre flows were added to give the overall circulation. The values of Qc, Vm and other quantities are given in table 5-7 and the graphs of Qc and Vm against ā1- and ōI L and m/(gh) against Frj and Re are shown in figures Qc/ 5-13 and 5-14. Once again Qc and Vm vary linearly with Ko , this will be discussed in comparison with the previous cases in 1 1 section 5.3.4. The graphs of Qc/Ko2 L and V/(gh)2 against Re. and Frj are inconclusive and in fact for the circulating flow shows a difference from the results found for the tangential

205

Vm m 3s~ ms~ (x10_4 0 Qc o points from table 5-7 000'3 ) 10 Vm 10 0

9 9

7 7

5 5

4 4

3 3

2 2

1 1

1 2 3 4 5 G Figure 5-13 K,z m2s1 (x10 3 ) Graphs of Qc and Vm against K02, L/h=62.5, asymmetric jet and outlet.

206 Vg h)

Qr~ o points from table 10 (x10-3] 5-7 ~~

r J 0.08 Vmxgo 8

0.06 6 Q

0.04 4

0.02 2

0.1 0.2 0.3 0.4 0.5 0.6 (a) Fr~ Graphs of against Fr., L/h = 62.5, asymmetric jet and outlet. V

9h)~ 10 (x10 3

1 2 3 4 5 36 As (a) but against Rei Rei (x 10 ) Figure 5-14 ^inn Table 5-7 Experimental Results, Asymmetric Jet and Outlet, Large Model

Q K K Q V v Q /Q Q /K lL V /(gh)i Rei Frj 0 0 o c m s o c c o m -2 -1 s -1 -1 -1 m3s-1 m4s m2s m3 ms ms (x 10-4) (x 10-6) (x 10-3) (x 10-3) (x 10-3) (x 10-3) (x 10-3)

0.203 3.66 1.91 3.66 2.8 2.7 0.05 0.038 3.16 1910 0.20

0.283 7.06 2.66 7..06 3.5 3.0 0.04 0.053 3.95 2660 0.28 00ō

0.396 13.84 3.77 9.66 6.6 5.2 0.04 0.052 7.45 3720 0.40

0.565 28.25 5.32 10.45 9.5 8.2 0.05 0.039 10.72 5320 0.56

-4 2 Notes: (1) Ao = 1.13 x lo m (2) KolL/Q o = 470.4 (3) L/h = 62.5 (4) g = 9.81ms2 10 6m2s1 (5) (~ = (6) Rod = 00 (7) AY and ks not included. and radial inlets, this may be due to the asymmetry of the jet

direction or to an anomaly in the calculation of Qc. This

aspect evidently requires some further investigation. Qo /Q c once again indicates that the outflow constitutes less than 5%

of the total circulating flow.

5.3.4 Comparison of the three types of circulation

Since velocity fields were computed for at least four values of

the jet momentum for the three different jet directions it is

useful to compare the relative merits and efficiency of each

type of circulation.

Figures 5-15 and 5-16 show the variation of circulating flow (Qc)

and mean velocity magnitude (Vm) with square root of the jet

momentum flux per unit density (K0 ) for the three jet directions,

the lines of 'best fit' have been determined by a linear re-

gression analysis. The variation of Qc with Kot is consistent

for each of the jet directions with a finite intercept on the Qc axis (in theory the lines should pass through the origin). For

the Qc v Vm graph there is a variation in the slopes and intercepts

of the regression lines, as might be anticipated the radial and

200 jet directions are in fairly good agreement, however, the tangential inlet shows higher values of Vm at low values of o

(below about 3.75 x 10- 3m2a1 ) and lower values for values of -3 2 -1 Ko1- above 3. 75 x 10 m s A possible explanation for this is that at the higher values of jet momentum the wall jet shear layer is very strong and momentum transfer to the interior is by shear

alone, whereas for the lower values of Ko the wall jet has a chance to spread and detach and generate velocities of a finite

209 o tangential jet o radial jet 0 — — — — 0 20 jet

1 2 3 4 5 6

Figure 5-15 1

Graphs of Qc against Kot for tangential, radial and asymmetric

jet direction, L/h = 62.5. 210 o tangential jet — • a radial jet 0 20° jet

3 . 4 5 6 -3 Figure 5-16 K0 m2s 1 (x10 ) Graphs of Vm against Kot for tangential, radial and asymmetric jet directions, L/h = 62.5.

211 magnitude over a wider area of the reservoir. For all three cases the variation of Vs does not supply any useful information being of the same order as m.

From the preceding results in sections 5.3.1, 5.3.2 and 5.3.4 it'would seem possible that for a given jet momentum and a circular reservoir the jet direction has no significant effect on-the overall circulating flow and the mean velocity magnitude in the reservoir. The question of"the suitability of an asymmetric jet needs further investigation as it may be that a particular direction, not studied here, is beneficial. It may be preferential to have an asymmetric jet as opposed to a symmetric radial jet where the jet has been seen to become unstable and generate an unpredicted asymmetric circulation.

5.3.5 Unsteady Flows As mentioned in section 5.3.1 the camera was quickly transferred between the four camera stations when a velocity field was being photographed. This meant that the velocity fields were not instantaneously recorded and that there may have been some small variations with time of the measured velocities. It was therefore desirable to investigate the variation of the velocity fields with time and determine if any short term or long term unsteadiness of the flow existed.

5.3.5.1 Short Term Stability As a check on the short term stability of a velocity field one of the quadrants of the model most strongly affected by the jet,

212 was monitored for a total period of one hour from the initial entry of the jet. The velocity field in the quadrant was determined from photographs taken at five minute intervals. This radial jet and diametric outlet arrangement corresponds to figure A-13 discussed in section 5.3.2. The velocity fields in the quadrant at each time are shown in figures A-23 to A-34 where it can be seen that the shape of the velocity field was becoming established 20-25 minutes after the initial jet efflux, from this point in time on it was the velocity magnitudes that varied, particularly close to the model boundary where the side- wall boundary layer was concentrated. This gradual increase in velocity magnitude continued until 40-45 minutes when the velocity field in this quadrant was well-established and similar to the final velocity field taken four hours later and shown in figure

A-35 (this final field is a repeat of the experiment giving velocity field A-13, the similarities are easily seen).

It is possible to demonstrate how quickly the velocity field became established by considering the magnitude of the circulating flow (Qc) through the quadrant and the mean velocity magnitude

(Vm) in the quadrant for each velocity field and hence at the particular time. The variation in these quantities with time is shown in figure 5-17. For the variation of Qc with time it can be seen that the circulating flow reached a relatively stable value after about 20 minutes, for the next 30 minutes there was -4 some fluctuation about this value (approximately Qc = 3 x 10 3 m s-1 ), however, after about 50 minutes there was a unexpected sudden increase in Qc to 4 x 10- 4m3s 1 which lasted for an unknown length of time. The final value, some four hours later

213 m3 1(x1Ō4) vmsm ~ I G (10 3

Final value Vm=5.5

0 4

Final value Q =2,93 3

Qc o Vm

10 20 30 40 50 60

minutes after initial jet efflux

Figure 5-17

Variation of Qc and Vm with time for test on short term stability of a velocity field.

211+ was 2.93 x 1Ō m 3s1 which is close to the earlier stable value.

The results suggested that for a typical jet efflux into the model reservoir a stable circulation was quite quickly attained, the final velocity field suggested that this circulation was quite stable over a longer period. In view of the results it was not thought that moving the camera to photograph each quadrant in turn was going to affect the velocity field measurements.

5.3.5.2 Long Term Stability With some idea of the degree of short term stability of a velocity field a test was undertaken to determine the long term stability of a typical velocity field. The 5m. diameter circular reservoir model was used with an 80mm. depth of water. Since in this case the investigation was concerned with long term stability (over a period of one hour say) it was possible to monitor the whole model (all four quadrants) during the test; the small amount of time expiring between the photographs of the quadrants due to moving the camera between the stations was considered insignificant in comparison with the time between photographs of the — 6 s same quadrant. The jet momentum of Ko = 7.06 x 10 m4 2 was different to the short term stability test described in section 6m4s 5.3.5.1 (lco = 13.84 x 10 2) and velocity fields were determined at one hour intervals from the initial jet efflux to a final time of five hours. The velocity fields obtained are shown in figures A-36 to A-40 and clearly show that there was very little change in the form of the velocity field over this five hour period. Once again the values of Qc and m have been computed for each field and the values are given in table 5-8.

215 Table 5-8 Experimental Results - Long Term Stability

Time Qc Vm Vs - Hours mas1 (x 10-4 ) ms 1(x 10 3) ms1 (x 10-3)

1 4.9 3.5 3.3

2 5.5 3.6 3.1

3 5.5 3.7 3.1

4 5.6 3.9 3.0

5 6.0 3.8 • 3.4

It can be seen from the respective values that there is a gradual

increase in both Qc and Vm over the five hour period with Vs

also fluctuating slightly. The results indicate the necessity

of allowing a considerable period to elapse between the 'zero'

position photographs and the 'deflected' position photographs

of the buoy target positions to ensure that the velocity field

has had sufficient time to attain relatively stable values of

Qc and Vm.

5.3.5.3 Decay of a Velocity Field Another form of flow unsteadiness that has been studied is the

decay of a velocity field from its steady state stable circulation

after the jet efflux has instantaneously been ceased. This

might represent a real situation, during the summer, when the

weather is relatively calm (ie. insufficient wind-induced mixing and circulation) and the reservoir is relying on the momentum

216 of the throughflow (possibly via jet inlets) to circulate and mix the vulnerable water mass. A toxic pollutant may suddenly be detected in the river and the inflow has to be switched off, for this case it is of interest to the reservoir manager to know for how long a significant velocity field is sustained by the momentum already existing in the reservoir. Unfortunately, as with much of the work involving time variations, the number of photographs required proved prohibitive to the degree of investigation possible with the limited time available, however, it was possible to obtain a preliminary idea of how a velocity field decays which might be used as a basis for future work in this area.

The first step was to record the velocity field as it existed before the jet inflow ceased as the initial condition for the decay of the velocity field. This was done in the routine way with the model being left for four to five hours to ensure a well- established velocity field, the field is shown in figure A-41

(Ko = 13.84 x 10 6mks-2) and is comparable with figures A-13 and A-35 for the same situation.

When the jet inflow had been switched off the quadrants were recorded with ten minute intervals between corresponding quadrants, one minute intervals between successive anti-clockwise quadrants in the group of four commencing with the first quadrant 5 minutes after the ceasure of the inflow. The sequence was thus 5, 6, 7 and 8 minutes, 15, 16, 17 and 18 minutes, 25, 26, 27 and 28 minutes etc. with the mean for the velocity field being taken as

6.5, 16.5, 26.5 etc. minutes with there being a maximum of 1.5

217 minutes variation from this mean of a quadrant being photographed

in respective sequence. The velocity fields are shown in figures

A-42 to A-46, the effects of stopping the inflow momentum are

plain, the jet centre-line velocity decays very quickly with

the peripheral velocities being much slower to decay. The overall

circulating flow (Qc) and the mean velocity magnitude (V ) have m been calculated for the complete velocity field for each of the

five mean times, the results are listed in table 5-9i_'and plotted

against the mean times in figure 5-18.

Table 5-9 Experimental Results - Decay of Velocity Field

Mean V Qc ml Time m3s 1 ms (min..) (x 10-4) (x 10-3)

0 7.36 5.8

6.5 6.75 4.2

16.5 5.29 3.3

26.5 5.81 3.0

36.5 4.97 2.8

46.5 3.26 2.8

The graphs.in figure 5-18 indicate that there is a possible

difference in the way Qc and Vm decay with time after the jet

has been switched off. Qc appears to decay linearly and with a continued linear rate of decay there would be minimal circulation

218 10 20 30 40 50 minutes after jet switched off

Figure 5-18

Graphs of Qc and V against time to test the decay of a velocity field.

219 in the model after about 70 minutes, approximately 20 hours prototype (assuming that the time scale is 1:). On the other hand Vm, the mean velocity magnitude in the reservoir, decayed fairly rapidly over the first 10 minutes but appeared to reach a stable value at around 2.8mms1 a little later where it remained for the rest of the time of study. Such effects would certainly need further substantiation, however, it seems that although the circulating flow may decay at a constant rate the mean velocity magnitude (and quite probably the velocity gradients) in the reservoir may remain significant for some time.

5.3.6 The Influence of the Earth's Coriolis Acceleration

Since the Earth's rotation may affect the circulation and mixing in prototype reservoirs it is desirable, if not necessary, to rotate a model of a reservoir at a rate determined by the time scale (hence the length and velocity scales) so that these effects due to the Coriolis acceleration are reproduced in the model.

In this study since there was no interest in a particular reservoir and since the rotation facility was not always available as it was being overhauled it was decided that only a limited experimental programme where rotation of the model was included would be undertaken to determine whether rotation was really necessary.

In section 5.2.2 consideration was given to the scales of a reservoir model and in section 5.2.1 it was shown that the

Coriolis force is due to the normal component (Q sino) of the

Earth's rotational velocity vector. The angular velocity of the

220

model Wm was given in terms of the angular velocity of the prototype in equation (5-18):

(JL sin4)p ( JL sinpp hri W m = (5-18) T h

Now (1i, sinf)p for the three reservoirs in table 5-1 is given by:

(fi, sint)p = sin52° rev/day (5-58)

Therefore for an undistorted 1:300 scale model

0.788 300 6) m = = 0.57 rev/hr. 24 3002. Therefore a 1:300 undistorted hydraulic reservoir model would need to be rotated at 0.57 rev/hr to be dynamically similar to the prototype with respect to the Coriolis acceleration. However the 5m. diameter model used for the experiments in sections

5.3.1 - 5.3.3 was not completely similar since the jet momentum had been increased to ensure a measurable velocity field.

For an undistorted model:

V (V )- m 1 o m- V = — = hr = (V ) r . = o r (5-59) V (Vo By increasing the jet momentum the jet velocity ratio (Vo)r has been altered and hence the velocity scale. With a different velocity scale the time scale must also have altered and hence the rate of rotation of the model need to be re-calculated.

For the two experiments with rotation of the circular model, with a radially directed jet inlet, it was decided to use the two higher values of jet momentum, ie. Ko = 13.84 and 28.25

221 6m4s x 10 2, which meant that the jet velocities were Vo = 0.35 -1 and 0.5 ms respectively. From table 5-1 a typical jet velocity in a prototype reservoir such as the Queen Mother Reservoir at

Datchet is 4.3ms 1, this gives for the two model jet velocities values of (Vo) of 0.08 and 0.12 respectively, the horizontal length ratio Lr is 0.0033 and hence Wm = 0.8 and 1.2 rev/hr.

The experiments were performed on the circular model with a radial jet and diametric outlet as before but with slight variations in the photographic sequence due to the rotation of the model.

The'zero' positions of the buoy targets were recorded in the usual way (section 5.3.1) with the model stationary and with no flow.

The model was then spun at the required rate with no jet discharge until the water was spun-up to solid body rotation (see Appendix A).

After this the jet discharge was begun and the reservoir model left for four to five hours to ensure a stable velocity field, the deflected positions of the buoys were then photographed with the model rotating. Since the image was effectively moving at the time of exposure it was necessary to use a very short exposure time so that the movement of the buoy targets at the edge of the model were very small in comparison with deflections of the order of 1-2mm. Unfortunately with the model rotating the quadrants could not be photographed in the same sequence as before; two photographs could be taken at one position of the model but it was necessary to wait for the model to complete another quarter of a revolution before the other two quadrants could be recorded.

This photographic sequence meant that the quadrants were not photographed from the same camera stations as normal, this did not pose any problems provided care was taken to ensure that the

222 photograph axes were correctly orientated when the negatives were analysed. The velocity fields found for these two experiments with rotation are shown in figures A-47 and A-48 and correspond to the non-rotating cases of figures A-13 and A-14.

Examination and comparison of these velocity fields reveals that rotation of.the model has influenced the velocity field. The angular velocity of the clockwise gyre is clearly increased and the jet direction deflected towards the anti-clockwise gyre in both cases where rotation of the model is included. The overall circulating flow and the mean velocity magnitudes have been computed for these fields and are given, together with the mean values of these quantities for the same situation without model rotation, in table 5-10.

Table 5-10 Effect of Model Rotation

ō = 13.84 x Ko = 28.25 x 2 10 6m4s 2 106m4s

V Qc Qc Vm -1 m3s1 ms m3s 1 ms-1 (x 10-4) (x 10-3) (x 10-4) (x 10-3)

Model 8.1 6.3 10.3 9.0 Stationary

Model 5.1 5.6 9.5 8.1 Rotating

223 The most apparent feature of the figures given in table 5-10 is the reduction in Qc and Vm when the model is rotated. The differences between the figures are significant enough to suggest that models of large bodies of water should be rotated in order to take into account the influence of the Coriolis acceleration; a further experiment with rotation was conducted for the irregularly-shaped reservoir 4 discussed in section 5.5.

Savage and Sobey (1975) have analysed the situation where a turbulent jet issuing from a circular orifice into a large rotating basin have predicted that for deep water the path of the jet is a clotoid having a length scale given by

(5-6o) where is the growth rate of the jet (approximately 0.095) and ā .52,1 is the rate of rotation-of the basin. They found that the predicted path agreed closely with the experimentally measured paths when the non-dimensional depth h/Lc was large (h/Lc 0.21) and when the depth was small (h/Lc E 0.024) that the confining effects of the upper free surface and the basin bed tended to make the mean flow two-dimensional and hence the jet path-was straight. For the two cases studied here the values of

Lc were 5.4m. and 5.3m. when the speeds of rotation were 0.8 and 1.2 revolution 1 hour respectively. In both cases the depth of water h was 80mm. giving a value of h/Lc of 0.015 in each case.

These can be compared with a value of 0.029 given by Savage and

Sobey for Datchet Reservoir.

221+ On the basis of the findings of Savage and Sobey rotation of the

hydraulic model should have had no effect on the jet path and

hence on the circulation, however, as was shown earlier in the

comparison of figures A-47 and A-48 with figures A-13 and A-14

this was not the case with these experiments. A possible ex-

planation is that the jet was not discharging symmetrically

about the diameter of the circular model and that this slight

asymmetry caused the particular differences described. If this

is the case then a theoretical analysis of this type has a limited

value since it is unlikely that a completely symmetric reservoir

situation would ever be encountered in practise.

One .point emphasised by Savage and Sobey (1975) is on the question

of distorted vertical scale models and their rotation. They

point out that typical prototype values of h/Lc are less than

0.024 and hence, on the basis of their work, rotation of the

model would be expected to have no effect on the jet path.

. However, the practise of distorting the vertical scale of reservoir

models to preserve turbulent flow is common and this distortion .

could influence the value of h/Y,c sufficiently for it to be in

the region (y 0.21) where rotation could have an appreciable

effect on the flow patterns and hence give misleading results.

5.3.7 Influence of Bed Topography It was intended when the research programme commenced to include

a study of the influence of bed topography as a major part of

the total investigation however, lack of time and a change of

emphasis resulted in this aspect being neglected with only two

experiments being performed with a model bottom other than flat.

225 Since the velocity measuring technique, at the moment, only works for an 80mm. depth of water it was decided that as a preliminary investigation it would be of interest to form a sloped bed in one half of the model and study the velocity field in the other half with a constant 80mm. depth of water. The slope was constructed of well-compacted sand of nominally Imm. diameter and reduced the maximum water depth on a diameter uniformly to zero

(based on 80mm. of water) where the sand slope met the circular model boundary. The tethered buoys were on the 0.4m. grid in the constant depth part of the model. Experiments were undertaken in the normal way with the inlet characteristics as for the velocity fields of figures A-17, A-35 and A-41 and the direction of discharge along the diameter at the base of the slope. The outlet was positioned as for the asymmetric outlet experiments of section 5.3.2. Photographs were taken of the half of the model with the buoys and the velocity fields determined for the model static and rotating and are shown in figures A-49 and A-50 and are comparable to figures A-17, A-35 and A-41 for the static model and A-47 for the rotating model.

Since such a very limited experimental programme of debatable validity was performed it is appropriate to discuss these velocity fields very briefly and on a qualitative basis. The velocity field in figure A-49 is very similar in magnitudes and direction of velocities to the figures A-17, A-35 and A-41 with no slope in the other half; it might have been expected that the reduced depth in one half would increase the circulating flow in the other or due to the asymmetry deflect the jet away from the

226 diameter, this is seemingly not so; the apparent loss of cir-

culating flow cannot be verified as the velocities in the reduced

depth half may have been increased to compensate and of course

there are also the complex frictional effects most likely to occur in this half.

The velocity field with rotation of the model, figure A-50, shows the same effect as seen with figure A-47 where the angular velocity of the clockwise gyre was increased. The results given here are of limited significance, a more extensive analysis of the effects of bed topography is required.

227 5.4 INVESTIGATION OF A MODEL OF TURRIFF RESERVOIR

A large void exists in the information available on reservoir circulation as to the actual behaviour and magnitude and direction of currents in operating reservoirs. As was discussed in chapter

2 there are major difficulties confronting any attempts to make- field measurements of currents in large reservoirs, these difficulties are in fact an amplification of many of those measurement problems encountered in hydraulic models and discussed earlier. The currents are small and hence errors can be highly significant; drogues, which are most often used, rarely measure the currents exactly where they are desired. Drogues are strongly influenced by the wind making it difficult to decide whether the measured current is due to the throughflow, the wind or, most likely, an interaction between them.

With this lack of information in mind, the author was pleased to have access to field measurements (private communication) made by members of the Water Research Centre on Turriff Reservoir.

This reservoir is small in comparison with the typical reservoir envisaged in the original 1:300 scale model being circular and of 195m. diameter. At. the time of the field measurements the reservoir water level was drawn down below top water level and- as such had a mean depth of 3.07m.; the reservoir being essentially flat-bottomed. The normal mean depth is 6.55m. giving a stored water volume of 1.75 x 105m3. The main purpose of the reservoir is to act as a bankside storage reservoir which means that with its 7 day nominal retention time its storage capacity is intended not as a supply reserve against periods of low river

228 flow nor as a primary treatment process but as a 'buffer' between the river and treatment works in the case of an unexpected pollutant appearing in the river.

The physical dimensions of the reservoir with a 3.07m. mean 3 water depth (Volume 8.55 x 10m ) were very convenient for a hydraulic model utilising the existing 5m. diameter basin. The

5m. diameter of the circular model represents the 195m. of

Turriff Reservoir at a scale of 1:39 with the 80mm. depth of water representing 3.12m. in the prototype giving an undistorted model. The throughflow in the prototype, 0.13m3s1 whilst the reservoir was at its low water level, is not introduced as a momentum source but in such a way that the momentum is minimised

(the reason for this is not known). The incoming water flows along a narrow channel and out into the reservoir through a series of gaps in one of the constraining channel walls (inlet ports) and falls down the side of the inlet structure into the reservoir.

There is obviously a little momentum generated when the vertically flowing water encounters the reservoir bed and is deflected horizontally, however, it is not possible to quantify this small amount of momentum. With the model being an undistorted 1 :39 scale model the velocity scale is 1:6.24 (h 39) and hence the discharge scale is 1:9499 giving a discharge into the model of 1.37 x 10-5m3s1 (821 cm3/min). The discharge was set to this value for the first experiments (steady state inflow = outflow, outlet positioned on the 'no inflow' side of the inlet) but it soon became clear that this flow rate was not going to generate currents in the model of sufficient magnitude for the tethered buoys to measure them, as a result the flow rate was increased

229 to 3.33 x 105m3s 1 (2000cm3min1 ) which produced significant

currents in the model.

The model inlet was constructed of timber and geometrically

similar to the existing inlet at Turriff Reservoir. The ex-

periments were conducted as before with there being a four to

five hour time gap between the photographs for the 'zero'. and

'deflected' positions of the buoy targets. Velocity fields were computed for different throughflows and barrier lengths.

The barrier length as constructed in the real reservoir is as shown in figures A-51 and A-52 where the velocity fields are computed for thronghflows of 2000cm3min 1 and 3000cm3min1 respectively. Unfortunately since the effective momentum of the throughflow is unknown it is not possible to relate inlet characteristics to a reservoir circulating flow or a mean velocity magnitude and hence only a purely qualitative discussion is possible.

The prototype field velocity measurements comparable to figures

A-51 and A-52 are shown in figure 5-19 and represent drogue field paths on a representative day. The magnitudes of the currents in figure 5-19 are not significant here as there was a wind acting on the surface of the reservoir, however, the direction of flow is of interest since the field measurements were. made when the reservoir was subject to the influences of throughflow and wind stress at the water surface whereas the model currents were purely throughflow-induced. A comparison of the flow patterns may indicate the relative contribution of the throughflow (it was considered by the operators of the

230 Figure 5-19

Field velocity measurements for Turriff Reservoir (Courtesy of the Water Research Centre)

• 231 reservoir that all the mixing and horizontal circulation was wind induced. Figures A-51 and A-52 showing the model velocity fields are clearly similar, close to the inlet the throughflow induces a small scale clockwise eddy with the majority of the flow in the southern half of the reservoir being forced around the boundary edge as a wall jet in the manner of the tangential jet of section 5.3.1. This wall jet can be clearly seen to detach from the boundary at a position approximately one eigth of a circumference from the position where the barrier centre- line would intersect the boundary in both cases. The main flow then moves in a gradual arc partially entering the NE quadrant before returning along the inlet side of the barrier. This irregular kidney-shaped gyre is the principle circulation occuring in the reservoir, the NW quadrant where the outlet is situated having very little flow induced with the mechanism whereby water leaves the reservoir being by very gradual transfer between the very slow small-scale eddies generated in the northern half of the reservoir. Comparison of figures A-51 and A-52 with the field measurements shown in figure 5-19 shows that there is a considerable similarity between many of the features of the flow patterns. Particularly noticeable is the wall-jet type flow along the boundary edge in the southern half of the reservoir which occurs in both model and prototype, however, the boundary detachment of this flow does not seem to occur in the prototype and also the circulation in the northern half and particularly the NE quadrant is more pronounced in the field measurements.

Once more it is clear that the outlet does not influence the overall circulation patterns, this being evident from both the field and laboratory measurements. The similarity between the

232 flow patterns in the prototype, supposedly wind-induced, and

the model, undoubtedly throughflow induced, brings into question the interaction mentioned by Steel (1972) and discussed in chapter. 2. The idea that the circulation is primarily wind- induced has presumably been the result of operating the reservoir under various conditions, calm, windy and with and without throughflow, however, the similarity between the flow patterns indicates that an agency action between the wind and throughflow is possibly occuring.

With a model of Turriff Reservoir operational it was a useful exercise to extend the experimental investigation of this reservoir by altering the length of the barrier and determine the effect of this on the velocity fields. Figure A-53 shows the velocity field for an inflow of 3000cm3min 1 and for the shortest length of barrier used, the pattern is similar to figures A-51 and A-52 but includes a larger area of the reservoir in the major gyre. Figures A-54 and A-55 for throughflows of

2000 and 3000cm3min 1 respectively and a slightly longer barrier than in figure A-53 but shorter than in the prototype demonstrates similar effects. Figures A-56 and A-57 where the barriers were very much longer and,in the case of A-57 the barrier almost completely bisects the reservoir,with the gap between the barrier and the reservoir boundary being less than 23% of the diameter, had throughflows of 3000cm3min1 . The velocity fields are clearly different to those before. The barrier imposes a restrain- ing action on the flow such that the two halves of the reservoir essentially having their own circulation systems. However, sufficient momentum is transferred across the gap for a sig-

233 nificant circulation to be generated in the northern half, the

southern half having a quite strong circulation similar to those

seen earlier for a tangential inlet. Figure A-58 is the same as the situation in figure A-57 but with the throughflow increased

to 3400cm3min1 , the pattern is essentially unchanged. !rom the velocity fields with different lengths of barrier it seems that

any form of barrier with the existing inlet and outlet arrange-

ments tends to suppress the throughflow induced circulation in

the reservoir; its effects on the wind-forced currents must be

extremely complex.

231+ 5.5 INVESTIGATION OF A MODEL OF RESERVOIR 4 (ALI, HEDGES AND WHITTINGTON, 1978a).

Ali, Hedges and Whittington (1978a)have described in some detail an experimental investigation of water circulation in two reservoirs with extensive experiments being conducted for various operating conditions. These experiments were performed on models of three different horizontal scales and with various vertical scales and single and twin inlet nozzles of different sizes and at different locations in the models.

The two reservoirs described in their paper form part of the water supply system serving Liverpool, their combined capacity is approximately 1.6 x 106m3 which provides a reserve of storage between long supply aqueducts and the distribution system. The reservoirs have quite a large surface area but are relatively shallow having a mean depth of approximately 4m. as such they have very little problem with thermal stratification during the summer months.

The two reservoirs are interconnected by several large diameter pipes but their individual circulation is independent of the other reservoir as the interconnecting pipes merely maintain a constant water level.

As with the large reservoirs of the Thames Water Authority

(table 5-1) the wind can, in general, be relied upon to cause substantial mixing and circulation, however, with long periods of relatively calm weather jetted inlets offer a powerful means of improving circulation in the reservoirs. Ali et. al. made the basic assumption that since inadequate mixing was only likely to

235 occur when there was very little wind the models could be tested in a windless state..

One of the two reservoirs modelled by All (called reservoir 4) was of particular interest in the light of this study being roughly circular in shape and hence fairly easy to model by adapting the shape of the 5m. diameter turntable model. They had made extensive measurements in their models by using dye photography, surface float tracking, fluorimetry and electrical conductivity techniques. This provided an excellent opportunity to check the validity of the tethered buoy velocity measuring technique and to supplement Ali's results with further, more uniform, velocity measurements; it was the surface float tracking which was of particular interest.

The model of reservoir 4 constructed in the 5m. diameter basin with an 80mm. depth of water (a requirement of measurements with the tethered buoys) had the same vertical scale (1:50) as the main model described in the paper but a horizontal scale of 1:116 as opposed to 1:250. The outer vertical boundaries of the model were lengths of aliminium sheet placed inside the basin and forming the required shape.. The main prototype inlet for this reservoir was a 0.61m. (24") nozzle positioned 0.76m. above the reservoir bed (it was positioned on the floor in the model) having an area of 0.29m2, hence, for the model with scales of

1:116 horizontal and 1:50 vertical the area of the model inlet should be 50.4mm2 or for a circular nozzle 8mm. in diameter and the model throughflow (59Mi/day in the prototype) 1.67 x 10-5m3 -1 s (1000cm3min 1).

236 The reservoir was set up with this inlet arrangement positioned in the same planimetric position and with the same direction of discharge, and the outlet pipe positioned on the boundary of the model in approximately the same position as they occur in the prototype. The velocity field was computed for this situation in the normal way by photographing the tethered buoys and is shown in figure A-59, this corresponds to figure 5-20(a) which shows the flow pattern as determined by Ali with the appropriate inlet characteristics.

The nozzle, size was subsequently reduced, thereby increasing the jet momentum for the same throughflow, and the velocity fields determined for flow rates of 1000cm3min1 issuing from circular nozzles of 6 and 4.5mm. diameter, 900cm3min-1 issuing from a 2.44mm. diameter nozzle and 650cm3min 1 issuing from a

1.51mm. diameter nozzle. The velocity fields are shown in figures A-60, A-61, A-63 and A-64 respectively, with figures

A-61 and A-64 corresponding to figures 5-20(b) and (c) showing the flow patterns as found by Ali et. al. The flow patterns for the two models show a gratifying similarity, the flow is split into two gyres of very different sizes, the gyre in the NE part of the models, very close to the jet inlets although small in size has a high angular velocity whereas the major gyre covering most of the reservoir is much slower moving with the water close to its centre hardly moving at all. The major flow in this gyre is concentrated as a wall jet around the boundary edges, however, the wall jet does detach in the SW corner of the model. The horizontal velocity profile of the wall jet can once more be seen to be Gaussian. In both cases the area of secondary

237 De pt h=80mm Nozzle Diam.=5.45mm Flow Rate=464cm~min-1 Surface Velocities in cm IIlJ.n. -1

(a)

Depth=80mm Nozzle Diam.=3.18mm 3 -1 Flow Rate=464cm min

(b)

Depth=8Omm Nozzle Diam.=1.59mm 1 Flow Rate=464cm3min-

(c) Figure 5-20 Flow patterns measured by Ali et. al. (1978a) (reproduced by permission of the Institution of Civil Engineers) circulation increases with increasing jet momentum although the

1:116 model always indicates a rather larger area than is shown

by the flow patterns of figure 5-20. This slight discrepancy

could be due to the difference in the velocity measuring methods

ie. a form of depth-averaged velocity measured by the tethered

buoys as opposed to a surface velocity as measured by floats,

however, it more likely to be due to the difference in vertical

exaggeration of the models since it has been suggested (Sobey and

Savage, 1974) that such reservoir models are sensitive to exag-

geration of the vertical scale; the 1:116 model has approximately

half the vertical exaggeration of the 1:250 model of Ali et. al.

This type of scale effect is also apparent when comparing the

circulating flows in the models, the values of Qc for the 1:250

'model were given by Ali et. al. in their paper and values for the

1:116 model calculated from the velocity fields and based on the

mean of four values taken at orthogonal sections. The mean

velocity magnitude (V ) was also calculated for the 1:116 model m velocity fields and this together with the circulating flows and

the non-dimensional parameters previously calculated are given

in table 5-11; the values for the 1:250 model are given in table

5-12, the mean velocity magnitudes were not-calculated for this

model. Figure 5-21 shows a comparison of the circulating flows

for the two models where Qc is plotted against Ko ; the mean

velocity magnitude VV is also plotted against K07 for the 1:116

model.

As for the earlier experiments with the circular model it can be

clearly seen that both Qc and Vm vary directly with Koz. From

experiments on their 1:250 model of reservoir 4 with a constant

239 0c 8 mas ( x10-4 ) 7

6 Vm ms (x1Ō3 ) 5 10 rN

3 6

Qmā5 points ;Rom -table 5-11 1 Coco points from table 5-12 2 3 4 5 6 7 y .8 9 Figure'5-21' - • K„" mzs'(x10'3) a Graphs of Qo and Vm against Kot, L/h = 58 and Qe against Kt, L/h = 27, Reservoir 4. 'I.

Qc k c3L 0.08 L L 4) =27

0,06 -Q o points from table 5-11 ~, ' IA = 58 -12 $- KaL • M 5-12 Vm/(( +t 5-11 A A V h) 004 8

• 0,02 • a 1-4=58 o 4 • Qo/L

1 2 3 4 5 6 7 Fr, Figure 5-22 J 1 Graphs of Qo/KolL and Vm/(gh)2 against Fri, L/h = 58 and Qe/Ko L against Ko3, L/h = 27, Reservoir 4 Table 5-11 Experimental Results Reservoir 4,1.:116. Model

A K K Q V V Q /Q Q V /(gh)3 Re. Fr, KL 0 Q0 0 o c m s o c c m o 2 -1 4 -2 2 -1 3 -1 3 -1 3 -1 m m 3s m s m s m s m s m s KL -3 Q (x 10-5 ) (x 10-5 ) (x 10-6 ) (x 10-3 ) (x 10-4 ) (x 10-3 ) (x lo3 ) ° (x 10 ) °

5.02 1.67 5.6 2.37 1.96 2.4 1.8 0.085 0.018 2.7 2370 0.38 662

2.83 1.67 9.85 3.14 1.98 3.3 2.6 0.084 0.014 3.7 3140 0.67 877

1.59 1.67 17.53 4.19 2.96 4.1 3.7 0.056 0.015 4.6 41go 1.19 1170

0.47 1.5 48.37 6.95 4.07 7.5 7.7 0.037 0.013 8.5 6950 3.60 2161

0.18 1.08 65.87 8.12 5.92 9.9 9.2 0.018 0.016 11.2 8120 6.77 3507

1.59 1.67 17.53 4.19 2.61 4.6 3.7 0.064 0.013 5.2 4190 1.19 1170

Notes: (1) L/h = 58 (2) = 10 6m2s1 (3) Rod = 0O except* where Rod = 98.2

(4) hp and ks not included. Table 5-12 .Experimental Results Reservoir 4,1::250 Model All et. al. (19784)

A Q K K Q Q /Q Q /K 1 L Re . Fr: K 1 L/Q 0 0 0 o c o C c 0 ~ ~ 0 0 m2 m3 s 1 m4s2 m2s1 mas 1 (x 10-5) (x 10-5) (x 10-6) (x 10-3) (x 10-4)

2.33 0.773 2.56 1.6 2.89 0.027 0.083 1600 0.37 448

0.79 0.773 7.54 2.75 ' 4.41. 0.018 0.074 2750 1.10 770

0.2 0.773 30.18 5.49 8.03 0.010 0.068 54go 4.41 1538

Notes: (1) L/h = 27 (2) 0 = 10- 6m2s1 (3) Rod = 00 (4) and not included. jet Froude number and a varying aspect ratio (L/h) Ali et. al.

found that the non-dimensional circulating flow Q /K 114 was c o strongly dependent on L/h. This result is confirmed in figure

5-21 where for a given momentum the circulating flow is lower

for L/h = 58 than for L/h = 27 and by comparing the values of

Qc/KolL from tables 5-11 and 5-12 for similar values of the jet

Froude number. The most likely reason for the decrease in

Qc/ ō4L with increasing L/h is due to the fact that the bottom boundary area has increased as L/h has increased and therefore

the losses due to friction at this boundary are greater subse-

quently reducing the circulating flow.

All et. al., for their reservoir 4, reproduced the experiments of Sobey and Savage (1974) with a tangential two-dimensional slot

inlet and a central outlet. They concluded by comparing the

circulation patterns in two models rune&.ng at the same jet Froude

number and aspect ratio that since the patterns were similar the

Reynolds number was of secondary importance (the Reynolds number

of the larger model was 2.8 times that of the smaller one)

although this may be true it is clearly not a very satisfactory

basis for making such a statement. In general Ali et. al. agreed

with the findings of Sobey and Savage but found the jet geometry

group to be important (their proposed curve for this when plotted

against Qc olL is shown in figure 5-24). The author has also

computed some of these parameters for the experiments in the 1:116

model. Figures 5-22 and 5-23 show the variation of Qc/Kok and

V /(ghA with the jet Froude number and the jet Reynolds number m together with the values from comparable experiments by Ali.

For the 1:116 model the results confirm the belief that Fr. and

244 ~ Kō L 0,0 8

0.06

0.04 8

002 4

1 2 3 4 5 6 78 3 9 Rel ( x10 ) Figure 5-23

Graphs of Qo/KolL and Vm/(gh)2 against Re, L/h = 58 and Qo/Ko1L against o2, L/h =:27, Reservoir 4 o point from table 5-11 • 5-12 O Ali et al (1978a) tangential inlet

0,5 1 1,5 2 25 3 3,5 3 K j (x10 ) Figure 5-24 Qo 1 Graphs of Qo/KolL against Ko2L/Qo for L/h = 58 and 27 Rej are unimportant for reservoir circulation, however, the results of Ali show a similar curve to those found earlier for the circular model. An interesting point to note concerns the mean reservoir Froude number variation with Rej and Fr., when plotted against Rej in figure 5-23 it shows the linear form seen earlier in section 5.3, however, when plotted against Frj the graph is curved as opposed to linear earlier;. this effect may be connected with the variable K0 L/Qo as opposed to a constant Ko1L/Qo with the circular model.

On the subject of the importance of the jet geometry group the variation with Qc o3L is shown in figure 5-24 for L/h = 58 and

27 together with Ali's results for a tangential inlet and his proposed curve. Once more the possible effect of the vertical exaggeration can be seen as at L/h = 58 (the least distortion of the vertical scale) the increase of the jet geometry group from 500 to 3500 has practically no effect on the non-dimensional circulating flow whereas at L/h = 27 it clearly does, although the form of the curve suggested by All et. al. looks a little strange. From the results obtained originally by Ali et. al. and later by the author for reservoir 4 it seems clear that the distortion of the vertical scale has a profound effect on the circulating flow, the effect of other jet parameters on this is not easy to discern and these effects may be complicated by the vertically exaggerated models. A full scale study as to the importance of these parameters is obviously required possibly making use of the velocity measuring technique, which almost certainly gives more satisfactory results over the-whole reservoir area.

247 One other factor possibly influencing the motion of large masses

of water discussed in section 5.3.6 not considered by Ali et. al.

is that of the Earth's rotation ie. the Coriolis acceleration.

In the case of the prototype reservoir 4 velocities of the order

of lcros1 could be induced and calculations using the concept of

Ekman (1905) and the solution by Welander (1957) given by Ali et.

al. indicated that geostrophic deflections of currents might

range from 0.5 to 3° according to the value of eddy viscosity

assumed. Ali et. al. have avoided the issue by citing 'the

profound effect of vertical exaggeration on the circulation

patterns' as their reason. With the rotation facility available

and a reservoir 4 model in use it seemed sensible to undergo one

of the previous static tests with rotation to see if there were

any noticeable effects.

As was discussed in section 5.2.2 the rate of rotation of the

model is determined by the time scale which is in turn determined

by the length and velocity scales. For the 1:116/1:50 model with

the jet inlet conditions as used to determine the velocity field

in figure A-59 (the true prototype model) the required rate of

rotation of the model for complete dynamic similarity (assuming

= 52°) would be:-

0.788 116 x--'- = 0.54 rev hr1 . m 24 502'

Unfortunately the control of the rotation of the model is not

sufficiently sensitive to ensure a uniform rotation at such a

low rate. It was therefore necessary to increase the rotation

rate if the effects of Coriolis acceleration were to be examined,

248 this could only be justified where the time scale had been

artificially decreased by decreasing the velocity scale. This

was true where the jet momentum had been increased increasing

Vo, the discharge velocity. For the prototype the mean jet velocity

was 2.4ms 1 and for the model, with 4.5mm. diameter nozzle

(15.9mm2 cross-sectional area) and a throughflow of 1000cm3mm 1, 1 1.05m giving a velocity scale of 1:2.3 and for a horizontal length scale of 1:116 a time scale of 1:50. The rotation -1 rate required for this case was hence 1.65 rev hr . An experi-

ment was performed with this arrangement with the experimental

details of'the photography with model rotation as described in

section 5.3.6. The velocity field is shown in figure A-62 and

corresponds to the non-rotating case of figure A-61. As might

be expected the effect of the rotation is small in the regions

of high velocity but appears to have important consequences in

regions of low velocity such as the SE corner. The value of Q c is slightly decreased whereas the value of Vm is increased, this

can be compared with the result of rotating the circular model

which was found to decrease both Qc and Vm, however, the radial

jet in this case was probably more susceptible to a disturbance.

In section 5.3.6 the value of Lc (equation 5-60) was computed for the model and the value discussed in the light of the findings

of Savage and Sobey (19?5). For reservoir 4 the prototype value

of Lc is 600m. which, since the water is 4m. deep,gives a value of h/Lc of 0.007 and for the rotated model Lc is 4m. and hence

h/Lc = 0.02. Both of these values are below 0.024 which was given by Savage and Sobey for there to be no effects due to rotation, there clearly was an effect in the model. This calls into question the

~49 suitability of applying the limits of the h/Lc value to help predict the effects of rotation in asymmetric models.

250 CHAPTER 6

CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH

251 6.1 CONCLUSIONS

The measurement of low water velocities in hydraulic reservoir models has posed considerable difficulties in the past. However, the development and use of a simple but effective technique for measuring water velocities in the range 1-25mms-1 simultaneously and over an area of several square metres using a tethered buoy and photogrammetry has been described in chapter 4.

It has been shown that, due to the complexities of the surface tension forces on a rod piercing a liquid surface asymmetrically, the deflection-velocity relationship of the buoy cannot be derived theoretically; however, an experimental batch calibration proved an adequate and successful alternative.

A non-metric camera (Hasselblad 500 ELM) has been used to determine the relative deflections of the buoys in various hydraulic models photogrammetrically and the errors generated, although significant in absolute terms, have been shown to become small when the measurements from two photographs, where the images concerned appear at almost exactly the same position in each photograph, • are subtracted to give the deflection components.

A drawback of the method has been the constant depth (80mm.) restriction that they have imposed. However, since the method is well-established it would not involve too much extra effort to design and calibrate a series of buoys of different geometries for use in a number of water depths.

252 The velocity measuring technique has been successfully applied to

a number of model reservoirs and the velocity fields determined

for various geometric and dynamic parameters (Appendix A). The

velocity fields were initially used to investigate the three

different types of circulation possible in a circular model with

a single inlet of variable direction:

(1)With a tangentially-directed jet inlet the major flow is

concentrated into a wall-jet around the circumference, with the

growth rate of this shear layer decreasing with distance from

the inlet. The slower-moving interior flow appears to be induced

by lateral momentum transfer from the wall jet with the velocities

in the interior being noticeably smaller than the maximum velocity

in the shear layer at the same radial section.

The azimuthal velocity profile determined at various sections for

several of the velocity fields was shown to be Gaussian, conforming

to an equation of the form

2 U = m exp (-0.69 (y - ym) (5-24)

(y1 - ym) 2

The inner boundary layer, very close to the model wall, was not

investigated but is thought to conform to the well-known logar-

ithmic law of the wall.

(2)When the inlet is directed radially into a circular reservoir

model the overall flow basically consists of two gyres or swirls

which rotate in opposite senses and are separated by the relatively

253 thin shear layer of the inlet jet core. The majority of the flow is restricted to this jet core and the comparably thin side-wall boundary layers of the gyres. At the centre of each gyre the flow is quite slow in comparison with the flow in the confining shear layers and these areas of low velocity are potential stagnation zones. It is noticeable that the areas of slow move-

_went are more pronounced when the jet momentum is increased.

In theory the gyres should be symmetric but in practise they are very often asymmetric, either in shape or by virtue of their different angular velocities. It is thought that this asymmetric effect is caused by either a slight misalignment of the radial jet direction or an uneven bed topography in the model. However, if such effects are significant in a model they may become more pronounced in the prototype and it might therefore be desirable to introduce a natural geometric asymmetry into the system.

The variation of the centre-line velocity, as found from several velocity fields, was found to agree well with the theoretical prediction for an axisymmetric jet issuing into a motionless environment (figure 5-12). Naturally enough the largest discrepancies occurred in areas of strong pressure gradients, eg. close to the jet orifice and in the region where the jet stream encountered the model boundary. Of the two velocity fields analysed that where the jet momentum was largest showed the best agreement.

(3) The introduction of the inflow in an asymmetric direction

(a particular eccentricity of 20° was studied) produced a flow

254 pattern analogous with that described in (2). Two gyres were

again induced but they were distinctly asymmetric, with the

larger gyre having a considerably lower angular velocity. The

thin shear layer of the jet inlet core was shorter in length

but the major flow, particularly for the larger gyre, was con-

fined to the side-wall boundary layers.

For the different reservoir models studied the overall forced

circulating flow (Qc) was used as the principal parameter des-

cribing the circulation and was determined by depth and radial integration of the azimuthal velocity distribution along a suitable radius of a gyre; for multiple gyre systems the overall circulating flow is the sum of the individual gyre values of Qc. Qc can be non-dimensionalised as Qc~4- L. Another parameter used,

of somewhat less significance, was the mean velocity magnitude

(m). It is convenient to summarise the influence of the different geometric and dynamic variables on the circulation by describing briefly their effect on the two circulation parameters.

(1) 5m. diameter Circular Model (approx. 1:300 scale of a real

reservoir)

Jet Momentum (K0)

When the jet momentum (Ko) was varied Qc and m were found to vary linearly with the value of whatever the jet direction. For 0 the particular case of a tangential jet direction a smaller

(1.5m. diameter) model was also investigated giving the same result.

255 Jet Direction

Three directions were studied for the 5m. diameter cirular model

with an inlet discharging tangentially, radially and. ymmetrically

(20o eccentricity) and the circulating flow was found to be

independent of the jet direction. For the values of Vm the radial

and asymmetric jet hardly showed any difference but the tangential

inlet had a higher value at the two lower values of jet momentum

and vice versa. This may be due to the fact that at higher values

of jet momentum the wall-jet shear layer is so strong that momentum transfer to the interior flow is by shear alone. However, in general, it can be accepted that the circulation is independent of the jet direction.

Aspect Ratio (L/h)

Unfortunately it was not possible to investigate the influence of this variable thoroughly due to experimental constraints imposed by the velocity measuring technique, ie. requiring a constant depth. However, the trends predicted by Sobey and Savage (1974) were confirmed; comparison of Qc/Ko7L for the different values of Ko and Qc from the two circular reservoirs, where L = 5m. and

1.5m. (L/h = 62.5 and 18.75), showed that Qc ō L increased significantly with a decrease in L/h. This is most likely to be due to the decrease in the effect of the bottom boundary resistance as L/h decreases.

Jet Froude and Reynolds Numbers (Fr. and Red)

Sobey and Savage (1974) suggested that Qc/K02L was independent of the values of Fr. and Re., however, the experiments with a circular model have not confirmed this and have tentatively suggested

256 a limited influence at low values of F'r. and Re.. The results J J are not entirely conclusive and further investigation is required.

Position of the Outlet (Qo/Qc)

Examination of the velocity fields, and in particular those of a

series of four experiments with a radial jet where the outlet

was re-positioned, have indicated that the outlet has little or

no effect on the overall circulation. This conclusion has been

confirmed by the values of the non-dimensional parameter Q /Q o c which show that for all circulations the outflow never exceeded

6% of the total circulating flow.

Rotation of the Model

A limited number of experiments with anti-clockwise rotation of

the circular model, at a rate derived from the jet velocity scale

and the length scale and with a radial jet direction, have shown

that the flow patterns are affected noticeably. The clockwise

gyre showed an increase in angular velocity and the flow in the

jet stream was deflected towards the anti-clockwise gyre. A

comparison of the values of Qc and Vm for the rotating cases with

the corresponding static tests indicated that their values were

decreased by rotation.

A criteria for predicting the influence of rotation derived by

Savage and Sobey (1975) was contradicted. However, since it was

shown earlier that an effectively radial jet can give a noticeable asymmetry it would be wrong to make gross assumptions about the validity of their criteria and hence it is difficult to draw

definite conclusions.

257 Other Experiments

A series of experiments studying the variation of a velocity

field with time (steady throughflow) showed that there was little

change in the circulating flow or the mean velocity magnitude

once the time taken to establish the circulation had been reached.

A study of the decay of a velocity field with time has tentatively

suggested that the mean velocity magnitude decreases linearly

whereas the circulating flow tends to decay exponentially, however, further investigation would be necessary to firmly establish this conclusion. Two experiments where variable bed topography was included as a uniform slope in one half of the model were inconclusive.

(2) Reservoir 4 Model (1:116 horizontal, 1:50 vertical)

The investigation of an irregularly-shaped model reservoir was reported by Ali et. al. (1978a) a model of which was constructed for this project having the same vertical scale (1:50) but having a horizontal scale of 1:116 (as opposed to 1 :250). A comparison of the flow patterns in the models showed that they were very similar, further confirming the validity of the tethered buoy velocity measuring technique. The only significant differences that occurred were in the small areas of secondary circulation and were more likely to be due to the difference in vertical exaggeration than the differences in method and type of velocity measurement and is a further indication of the significance of L/h.

As with the circular model it was possible to determine the overall circulating flow and the mean velocity magnitude for the 1:116

258 and, for Qc, the values could be compared with those found by

Ali et. al.

Jet Momentum (Ko)

As for the circular model Qc and Vm varied directly with Koh; a replotting of the results of All et. al. showed the same trend

(figure 5-21).

Aspect Ratio (L/h)

Although the values of Qco L from the two models cannot be directly compared their trend indicates the importance of the aspect ratio. The 1:16 model was approximately twice the size 1 of-the 1:250 model yet, for comparable values of 0 L/Qo of 877 -6 4 -2 and 770 respectively (Ko = 9.85 and 7.54 x 10 m s ), Qc/Ko2L was 0.014 and 0.074 for the aspect ratios of 58 and 27.

It is clear from the results here and previously that the aspect ratio is extremely important in reservoir models and is a warning against the use of exaggerated vertical scale models. In many cases the model is distorted to preserve turbulent flow and since this distortion is clearly undesirable it might be tempting to artificially roughen the bed as an alternative. In this case it should be noted that Sobey and Savage (1974) found, from a mathematical model, that ks/h (the bed resistance) also significantly effects Qc/KoiL.

Jet Froude and Reynolds Numbers (Fri and Rei)

The variation of Qc/K~L with Fri and Rei was small for both models, on this occasion confirming the views of Sobey and Savage(1974).

259 1 Jet Geometry (K0L/Q0)

The influence of Ko1L/Qo on Qc/Ko was also found to be small

for both models although when Ali et. al. investigated a completely

tangential inlet they found 0/Q0 to be important. This para-

meter clearly needs further investigation.

Position of Outlet (Qo/Qc)

An examination of the flow patterns once more revealed that the

outlet was not affecting the overall circulation. The values of

Qo/Qc show that for the 1:116 model the outflow constituted less

than 9% of Qc and for the 1:250 model less than 3%. It can

, therefore be concluded that the circulation is independent of the

position of the outlet; the outlet can therefore be placed in a

position where it minimises the likelihood of short-circuiting.

Rotation of the Model

The effect of rotating the 1:1 6 model anti-clockwise was only

evident in areas of low velocity and in particular-in the small

area of secondary circulation. The value of Qc was decreased

and Vm increased relative to the values for a corresponding test

with no rotation. As with the circular model the criteria of

Savage and Sobey (1975) was not verified, however, it would seem,

from a consideration of the findings of this study, that it is

desirable, if not necessary, to include rotation of the model as

a variable.

(3) Turriff Reservoir Model (1:39)

The velocity fields determined for a 1:39 scale model of Turriff

Reservoir were similar in certain respects to field measurements

260 available for this reservoir. Since the field measurements were thought to be wind-induced the possibility of wind/throughflow interaction was suggested, a further confirmation of the hypothesis of Steel (11972 and 1975).

Further experiments with the barrier length varied indicated that the practise of dividing a circular reservoir, with a tangential inlet, by a diametric barrier may be detrimental to reservoir circulation restricting momentum transfer and thereby suppressing circulation in certain remote areas of the reservoir.

261 6.2 SUGGESTIONS FOR FURTHER RESEARCH

As a result of the findings of this research many areas have emerged where further investigation might prove worthwhile and the more important are listed below:

(1)Further development of the velocity measuring technique,

particularly with regard to the geometry of the buoy, so

that the method might be successfully applied to hydraulic

models over a range of water depths.

(2)The depth-averaging effect of the buoy has not been

investigated; it may be of interest to experiment with

composite buoys which sample the velocity at a particular

depth and then to use them to establish the velocity profile.

(3)For the steady flow case the importance of the aspect

ratio, L/h has been established, however, the significance

of other parameters, eg. Rei, Fri, Ko7L/Qo etc., remains

rather vague and in particular the influence of k /h has s only been established mathematically. If recommendation (1)

could be satisfied then it would be extremely useful to

undertake an extensive programme to consider the influence

of the various variables on the circulation and mixing.

(4)The forced circulating flow (Qc) has been consistently

used as the parameter describing the circulation with it

generally being inferred that maximising Qc for given

conditions also māximises the mixing. This contention is

262 largely unproven and should either be established or a more

appropriate parameter found which describes the quality of

circulation and mixing.

(5)A considerable amount of experimental work remains to

be done on unsteady flows, stratified reservoirs, density

differences and bed topography, however, these should not be

tackled until the steady flow homogenous case is completely

understood.

(6)In this study the influence of the Coriolis acceleration

appeared to be significant; unfortunately the rotation rate

was increased by considering a model with a distorted jet

velocity scale and it would therefore be desirable to

establish the significance of model rotation at a true scale.

(7)The investigation of the decay of a velocity field

produced interesting results but more detailed study is

required. The results of such a study could prove beneficial

to reservoir operation and design particularly if they

suggest the intermittent use of several inlets.

(8)An area requiring considerable attention is that of field studies. The data available for verification of

hydraulic and mathematical models is minimal and is often

confused by complex wind-throughflow interaction. It would be

most useful if field data under calm as well as windy conditions

could be obtained and with and without throughflow.

263 ,(9) The question of the mode: of operation of the small

bankside storage reservoirs (ie. 'plug' or 'completely

mixed' flow) is in need of resolution and carefully

considered operating ideals will need to be specified to

enable such a decision to be made.

(10)Further work is required on the mathematical models

of reservoir circulation (eg. Falconer, 1976). Restrictions

have prevented the study of stratification effects, density

differences and the influence of secondary currents. More

sophisticated turbulence models could be incorporated and

the unrealistic inlet and outlet dimensions made more

representative of the real situation. Incorporation of the

convective-diffusion equation would also allow the inclusion

of water quality parameters such as temperature and tracer

concentration.

(11)To aid the study of large storage reservoirs where

artificial circulation is induced by momentum jet inflow

the current vague operating ideals should be clarified to

enable particular areas requiring special investigation to

be identified.

264 APPENDIX A

VELOCITY FIELDS FOR '1't1E RESERVOIR MODELS

265 • JET C I RCULRT I CN IN ! RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 C1 VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CH JET INLET ST BED JET VELOCITY 9 CM/S CIRCULAR JET 1.2C+1 DIAM Figure A-1

Tangential Jet and Central Outlet, Ko = 0.92 x 10- 6m4s 2

266 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S

INITIRL DEPTH 8.0 CM JET INLET AT BED

JET VELOCITY 18 CM/S CIRCULAR JET 1.2CM DIAM Figure A-2

Tangential Jet and Central Outlet, Ko = 3.66 x 106 m4s2

267 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL CEPTH 8.0 CM JET INLET RT BED JET VELOCITY 25 CM/S CIRCULAR JET I.2CM DIRM Figure A-3

Tangential Jet and Central Outlet, Ko = 7.06 x 10-6m4s 2

268

JET CIRCULATION IN A RESERVOIR

EXPERIMENTRL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH- 8.0 CM JET INLET AT BED JET VELOCITY 35 CM/S CIRCULAR JET 1.2CM DIAM

Figure A-4-. 2' Tangential Jet and Central Outlet, Ko = 13.84 x 10 6m4s

269 JET CIRCULATION IN *A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCALE 8.0 MM/S

INITIAL DEPTH 8.0 CM JET INLET AT BED

JET VELOCITY 44 CM/S CIRCULAR JET 1.2CM DIAM Figure A-5 6m4s 2 Tangential Jet and (bntral Outlet, Ko = 21.88 x 10

270 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

/I/ / /V' 7 1 1 1 T 1' t

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIRL DEPTH 8.0 CH JET INLET AT RED JET VELOCITY 50 CM/S CIRCULAR JET 1.2CM DIAM Figure A-6

Tangential Jet and Central Outlet, Ko = 28.25 x 10 6m4s 2

271 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 10.0 CM VELOCITY SCALE 3.0 MM/S INITIAL DEPTH 8.0 CM SMALL TANK JET VELOCITY 9 CM/S CIRCULAR JET 1.2CM. DIAM. Figure A-7 6m4s-2 Tangential Jet and Central Outlet, Ko= 0.92 x 10

272

JET CIRCULRTION IU R RESERVOIR

EXPERIMENTAL RESULTS

N L \ / / , T N \ \ L 1 ~ 1 1 i r 1 1 L 1 \ i I 1 1 1

1 \a \ .-

LENGTH SCALE 10.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM SMALL TANK JET VELOCITY 18CM/S CIRCULAR JET 1.2CM. DIAM. Figure A-8

Tangential Jet and Central Outlet, Ko = 3.66 x 10 6m4s 2

273 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 10.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM SMALL TANK JET VELOCITY 25CM/S CIRCULAR JET 1.2CM. DIRM. Figure A-9

Tangential Jet and Central Outlet, Ko = 7.06 x 10 6m4s2

271+ JET CIRCULATION Ī N. A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 10.0 CM VELOCITY SCALE 0.0 MM/S INITIAL DEPTH 8.0 CM SMALL TfNK JET VELOCITY 35CM/S CIRCULAR JET 1.2CM. DIAt1. Figure A-10

Tangential Jet and Central Outlet, Ko = 13.84 x 10 6m4s-2

275 JET CIRCULATION IN A RESERVOIR

EXPERIMEMTPL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCRLE 0.0 MM/S

INITIPL DEPTH 8.0 CM JET INLET RT BED

JET VELOCITY 18 CM/S CIRCULAR JET 1.2CM DIRM Figure A-11 6m4s 2 Radial Jet and Diametric Outlet, Ko = 3.66 x 10-

276 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIRL DEPTH 8.0 CM JET INLEī AT BED JET VELOCITY 25 CM/S CIRCULAR JET 1.2CM DIRM Figure A-12

Radial Jet and Diametric Outlet, Ko = 7.06 x 10 6m4s-2

277 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY B5 CM/S CIRCULAR JET 1.2CM DIAM Figure A-13

Radial Jet and Diametric Outlet, Ko = 13.84 x 10 6mks-2

278 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIRL DEPTH 8.0 CM JET INLET RT BED JET VELOCITY 50 CM/S CIRCULRR JET 1.2CM DIRM Figure A-14

Radial Jet and Diametric Outlet, Ko = 28.25 x 10 6m4s-2

279

JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET lT BED JET VELOCITY 18 CM/S CIRCULAR JET I.2CM DIRM Figure A-15

Radial Jet and Asymmetric Outlet, Ko = 3.66 x 10-6m4s 2 280

JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 25 CM/S CIRCULAR JET I.2CM DIAM Figure A-16 6m4s Radial Jet and Asymmetric Outlet, Ko = 7.06 x 10 2

281 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 3SCM/S CIRCULAR JET 1.2CM DIAM Figure A-17

Radial Jet and Asymmetric Outlet, Ko = 13.84 x 10 6m4s-2

282 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIRL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 50CM/S CIRCULAR JET 1.2CM DIAM Figure A-18

Radial Jet and Asymmetric Outlet, Ko = 28.25 x 1Ō6 m4s2

283 JET CIRCULATICN IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 18 CM/S CIRCULAR JET 1.2CM DIRM Figure A-19 6 -2 20° Jet Asymmetry and Asymmetric Outlet, K° = 3.66 x 10 m 4s

284 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCRLE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 25CM/S CIRCULAR JET 1.2CM DIRT Figure A-20

20° Jet Asymmetry and Asymmetric Outlet, Ko = 7.06 x 10 6m4s-2

285

JET CIRCULATīON IN R RESERVOIR

EXPERIMENTRL RESULTS

~ 1 r

LENGTH SCALE 40.0 CM VELOCITY SCRLE 8.0 MM/S INITIRL DEPTH 8.0 CM JET INLET PT BED JET VELOCITY 35 CM/S CIRCULRR JET 1.2CM DIRT Figure A-21 6m4s 2 20°Jet Asymmetry and Asymmetric Outlet, K°= 13.84 x 10

286 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCALE 8.0, MM/S IHIīIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 50 CM/S CIRCULAR JET 1 .2CM pmAM Figure A-22

20° Jet Asymmetry and Asymmetric Outlet, IL° = 28.25 x 10- 6m s-2

287 JET C I RCULRT I Old IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET RT BED JET VELOCITY 35 CM/S CIRCULAR JET 1.2CH DIRM Figure A-23

Short Term Stability Test, Ko = 13.84 x 1Ō6 m4s2 , 5 mins. after initial jet entry 288 JET CIRCULATION IN P, 'RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT 8E0 JET VELCCITY 35 CM/S CIRCULAR JET 1.2CM DIAM Figure A-24

10 mins. after initial jet entry

289 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET RT DED JET VELOCITY 35 CM/S CIRCULAR JET 1.2Ci1 DIAM

Figure A-25

15 mins. after initial jet entry

290 JET CIRCULATION I N A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S

INITIAL DEPTH 8.0 CM JET INLET AT BED

JET VELOCITY 35 CM/S CIRCULAR JET 1.2CM DIAM Figure A-26 20 mins. after initial jet entry

291 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCRLE 8.0 MM/S

INITIAL DEPTH 8.0 CM JET INLET RI 6E0

JET VELOCITY 35 CM/S CIRCULRR JET 1.2CM DIRM Figure A-27

25 mins. after initial jet entry

292 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 6.0 CM JET INLET AT BED JET VELOCITY 35 CM/S CIRCULAR JET 1.2CM DIAM Figure A-28 30 mins. after initial jet entry

293 JET CIRCULATION I N A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCRLE 8.0 MM/S

INITIRL DEPTH 8.0 CM JET INLET RT BED

JET VELOCITY 35 CM/S CIRCULSR JET 1.2CM DIRM Figure A-29

35 mins. after initial jet entry

294 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 Ci'! VELOCITY SCALE 8.0 MM/S

INITIAL DEPTH 8.0 C!1 JET INLET AT BED

JET VELOCITY 35 CM/S CIRCULAR JET I.2CM (IAM Figure A-30

40 mins. after initial jet entry 295

JET CIRCULATION IN A. RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CFS VELOCITY SCALE 3.0 MM/S INITIAL DEPTH 3.0 CM JET INLET AT BED JET VELOCITY 35 CM/S CIRCULAR JET 1.2CM DIRM Figure A-31

45 mins. after initial jet entry

296 JET CIRCULATION I N R RESERVOIR

EXPERIMENTRL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCRLE 6.0 MM/S INITIAL DEPTH 6.0 CM JET INLET RT BED JET VELOCITY 35 CM/S CIRCULP.R JET 1.2CM DIRM Figure A-32

50 mins. after initial jet entry

297 JET CIRCULATION IN A RESERVOIR

EXPERrMENTAL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCRLE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 35 CM/S CIRCULAR JET 1.2CM DIAM Figure A-33

55 mins. after initial jet entry 298 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S

INITIAL DEPTH 8.0 CM JET INLET AT BED

JET VELOCITY 35 CM/5 CIRCULAR JET 1.2CM DIAM Figure A-34

60 mires. after initial jet entry 299 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCRLE 40'.0 CM VELOCITY SCALE 8.0 MM/S

INITIAL DEPTH 8.0 CM JET INLET AT BED

JET VELOCITY 35 CM/S CIRCULRR JET 1.2CM DIRM Figure A-35

Short Term Stability Test, Ko = 13.84 x 10 6m4s-2, final velocity field 300 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCRLE 40.0 CM VELOCITY SCRLE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET RT BED JET VELOCITY 25 CM/S CIRCULRR JET 1.2CM DIRM Figure A-36

Long Term Stability Test, Ko = 7.06 x 10- 6m4s 2, 1 hour after initial jet entry.

301 JET CIRCULRIICN IN Ff RESERVOIR •

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIFAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 25 CM/S CIRCULAR JET 1.2C,`1 DIRM Figure A-37 2 hours after initial jet entry

302

JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

/ .. y S t r

I i i 1 ti c—F 1 , I t r I I I

\ I \ \ /A / Ii \

\ ic r I l i d I 1\ \ \ \ \i , %`

\ Y. 4= 17 I \r \ '. / /

LENGTH SCALE 40.0 CM VELOCITY SCALE 6.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 25 CM/S CIRCULAR JET 1.2CM DIAM Figure A-38

3 hours after initial jet entry 303 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

11.

/ 1 , f t . I

/ 7 1 ? ~' s r I \

, T I i ti .. t 't I

T 1 1 / 1 , l

1 1 / _. I i \„ . ' 1 , ,~ , %' d~ 1 , v .i 1\7 \\'' C

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 25 CM/S CIRCULAR JET 1.2CM DIRM Figure A-39

4 hours after initial jet entry

304 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCRLE 8.0 MM/S

INITIAL DEPTH 8.0 CM JET INLET AT BED

JET VELOCITY 25 CM/S CIRCULAR JET 1.2CM DIRM Figure A-40

5 hours after initial jet entry 305 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 6.0 nl?/S I?IITIii. DEPTH 8.0 CM JET I'!LET RT BED JET VELOCITY 35 CM/S CIRCULRR JET 1.2CM DIRM DECRY OF VELOCITY FIELD-IrITIRL i:ONOITIOM

Figure A-41

306 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

•

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INiTIRL DEPTH 8.0 CM JET INLET RT BED JET VELOCITY 35 CM/S CIRCULAR JET 1.2CM DIRM DECRY OF VELOCITY FIELD-RYE. 6.5MINS. AFTER JET OFF

Figure A-42

307 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

i /` l 1 1 1 _ r t 1 \ % l r 1 j r r r I

l 7 I

r t

LENGTH SCALE 40.0 CM VELOCITY SCALE 3.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 35 CM/S CIRCULAR JET 1 .2Ct1 DIRM DECRY OF VELOCITY FIELD-AVE. 1S.SHINS. AFTER JET OFF

Figure A-43

308 JET C I RCULRT I CN IN R RESERVOIR

EXPERIMENTRL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 35 CM/S CIRCULAR JET I.2CM DIAM DECRY OF VELOCITY FIELD-AVE. 26.5MINS. AFTER JET OFF

Figure A-44

309 JET CIRCULATION IN A RESERVOIR .

EXFERIMENTRL RESULTS.

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET RT BED JET VELOCITY 35 CM/S CIRCULAR JET 1.2CM DIRM DECRY OF VELOCITY FIELD-RVE. 36.SMINS. AFTER JET OFF Figure A-45

310 JET CIRCULATION I N A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 55 CPS/S CIRCULAR JET 1.2CM DIAM DECAY OF VELOCITY FIELD-AVE. 46 . Sit I NS . AFTER JET OFF

Figure A-1+6 311 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY tis CtI/S CIRCULAR JET 1.2CM DIAM RESERVOIR ROTATING.0.8 REV/HR ANiīI-CLOCKWISE Figure A-47 Radial Jet and Diametric Outlet, Ko = 13.84 x 10-6m4s-2, with rotation of the model 312 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS.

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT SED JET VELOCITY 50 CM/S CIRCULAR JET 1.2CM DIAM RESERVOIR ROTRTING,1.2 REV/HR ANīI-CLOCKWISE Figure A-48 _ 6 Radial Jet and Diametric Outlet, Ko = 28.25 x 10 m4s 2, with rotation of the model 313 JET CIRCULATION IN P RESERVOIR

EXPERIMENTRL RESULTS.

SLOPING SAND BED (31.5-11

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8:0 CM JET INLET AT BED JET VELOCITY 35CM/S CIRCULAR JET 1.2CM DIAM Figure A-49 -6 4 -2 Radial Jet and Diametric Outlet, Ko = 13.84 x 10 m s uniform slope in one half of the model.

314 JET CIRCULRTION IN R RESERVOIR

EXPERIMENTAL RESULTS

SLOPING SAND BED 131.5-11

LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM JET INLET AT BED JET VELOCITY 35CM/S CIRCULAR JET 1.2CU DIRM RESERVOIR ROTATINL,0.8 REV/MR RNTI-CLOCKHISE Figure A-50 As figure A-49 with model rotation 315 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF TURRIFF RESERVOIR SCALE-39-1 LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM INFLOW VIP OVERSPILL PORTS

JNFLOW=2C00 CC/MIN. OUTFLOW-2000 CC/MIN. Figure A-51 Turriff Reservoir Model 316 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF TURRIFF RESER'?OIR SCRLE=39-1 LENGTH SCALE 40.0 CM VELOCITY SCRLE 8.0 MM/S INITIfL DEPTH 8.0 CM INFLOW VIA OVERSPILL PORTS INFLOW=3000 CC/MIN. OUTFLON=3000 CC/MIN. Figure A-52

317 JET CIRCULATION IN A RESERVOIR

EXPERIME? TRL RESULTS

✓

t ..

•`.

.• ,- ... ! ` 1 1

• l , .01

MODEL OF TURRIFF RESERVOIR SCALE=39-1 LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM INFLOW VIA OVERSPILL PORTS INFLOW=3300 CC/MIN. OUTFLOW=3000 CC/MIN. Figure A-53

318 JET C I RCULR ī I ON I N? A RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF TURRIFF RESERVOIR SCALE=39-1 LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH '8.0 CM INFLOW VIA OVERSPILL PORTS INFLOW=200O CC/MIN. OUTFLOW=200Q CC/MIN.

Figure A-54

319 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

t N. 4—

r 1 1 f l

r l I 1 1 ,._. 4,- Z r

r .... I L i - 4-- +-- 1

t / t \ ! l le"'. 1 +-- ,i 1 1 46... s. % 1 1 1% r t I / 1,

% l l / I

- •, N 1 1 •` 1 /

f ` t 4__ .... %'

--•

MODEL OF TURRIFF RESERVOIR SCALE-39-1 LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL OEPTH 8.0 CM INFLOW VIA OVERSPILL PORTS INFLOW=3000 CC/MIP. OUTFLOW-3000 CC/MIN. Figure A-55

320 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF īURRIFF RESERVOIR SCRLE=39-1 LENGTH SCRLE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIFL DEPTH 8.0 CM INFLOW VIR OVERSPILL PORTS INFLOW=30OO CC/MIN. OUTFLOW-3000 CC/MIN. Figure A-56

321

JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

ti 1 e e e 1

t f .— . — ... .- -- 1 I

t —. r .— r r 1 1 t - r ✓ r r ,r 1

t '. • 4.. «- / ... r r r 1 1

r~ 1

= .. r i ...) ▪ r T

— r 1 / t r I f I

I _ 1 r r / r i /

MODEL OF TURRIFF RESERVOIR SCALE=39-1 LENGTH SCRLE, 40.0 CM VELOCITY SCRLE 8.0 MM/S INITIRL DEPTH 6.0 CM INFLOW VIA OVERSPILL PORTS INFLOW=3000 CC/MIN. OUTFLOW=3000 CC/MIN. Figure A-57

322 JET C I RCULRT I Uhr IN R RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF TURRIFF RESERVOIR SCALE=39-I LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITItL DEPTH 2.0 CM INFLOW VIA OVERSPILL PORTS INFLOW=3400 CC/MIN. OUTFLOW=3400 CC/MIN.

Figure A-58

323 JET C I RCJLAT1 CN IN A RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF RESERVOIR 4 SCALES.HORIZ=116-1.VERT=50-1 LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM 8.0MM. DIAM.NOZZLE INFLOW=1000 CC/MIN. OUTFLOW-1000 CC/MIN. Figure A-59 Reservoir 4 Model, Ko = 5.6 x 10-6m4s-2 324 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF RESERVOIR 4 SCRLES.HORIZ=116-1.VERT=50-1

LENGTH SCALE 40.0 CM VELOCITY SCALE 0.0 MM/S INITIRL DEPTH 8.0 CM 6.0MM. DIRM.NOZZLE INFLOW=1000 CC/MIN. OUTFLOW=1000 CC/MIN. Figure A-60

Reservoir 4 Model, Ko = 9.85 x 10-6m4s-2 325 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF RESERVOIR 4 SCALES.HORIZ=116-I.VERT=50-1 LENGTH SCRLE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL OUTH 8.0 CM 4.5MM. DiRM.NOZZLE INFLCW i000 CC/MIN. OUTFLOW-1000 CC/MIN. Figure A-61

Reservoir 4 Model, Ko = 17.53 x 10 -6m's-2 326 JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF RESERVOIR 4 SCALES.HORIZ=116-1,VERT=50-1

LENGTH SCALE 40.0 CM VELOCITY SCRLE 8.0 MM/S INITIfiL DEPTH 8.0 CM 4.5 MM. DIRM.NOZZLE INFLOW=10O0 CC/MIN. OUTFLOW=1000 CC/MIN. RESERVOIR ROTATING 1,S5,REV,/HR. Figure A-62

As Figure A-61 with model rotation JET CIRCULATION IN A RESERVOIR

EXPERIMENTAL RESULTS.

MODEL OF RESERVOIR 4 SCALES.HORIZ=116-1.VERT=50-1 LENGTH SCRLE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM 2.44MM. DIRM.NOZZLE INFLOW 900 CC/MIN. OUTFLOW= 900 CC/MIN. Figure A-63 6m4s Reservoir 4 Model, Ko = 48.37 x 10 2 328 JET CIRCULATION IN R RESERVOIR

EXPERIMENTAL RESULTS

MODEL OF RESERVOIR 4 SCALES.HORIE=116-1.VERT=50-1 LENGTH SCALE 40.0 CM VELOCITY SCALE 8.0 MM/S INITIAL DEPTH 8.0 CM 1.51MM. OIAM.NOEZLE INFLOW= CSO CC/MIN. OUTFLOW= 650 CC/MIN. Figure A-64 Reservoir 4 Model, Ko = 65.87 x 10-6m4s-2 329 APPENDIX B

THE SPIN-UTP TIME OF A SHALLOW CIRCULAR BASIN

330 Consider a right circular cylinder containing a homogenous fluid

with the cylinder and fluid rotating together at a uniform angular

velocity about the axis of symmetry and that this uniform angular

velocity is suddenly increased and maintained at a new constant

value. The bulk of the fluid away from the container boundaries

is, at first, unaffected and the equilibrium balance between

the pressure gradient and the centrifugal force persists. However,

close to the horizontal boundaries the viscous stress induces layers of fluid to rotate at a faster rate and with the increased centrifugal force being too great for the prevailing pressure gradient, the fluid in the horizontal layers is driven radially outwards. This induces a flow of the bulk interior fluid, downwards into the boundary layers generating secondary circulations.

The secondary circulation convects the interior fluid towards the axis of rotation and, with the absence of appreciable interior torques, the convected fluid behaves as a free vortex. This complete process, which continues until the whole fluid is rotating with the cylinder as a forced vortex at the new angular speed, is known as spin-up and the time for it to be reached is known as the spin-up time. The process just described is the simplest type of spin-up being linear and for a homogenous fluid. This study has marginal interest in non-linear spin-up (from rest) of a homogenous fluid.

The basic concepts of spin-up have been known since 1905 when

Ekman discussed the fundamental dynamics of this type of flow near the horizontal boundaries in an analysis of his now classic

Ekman boundary layer. Much of the more recent interest, however, was stimulated by the study of Greenspan and Howard (1963).

331 The interest of this study in spin-up relates to experiments

where the Coriolis forces are scaled by rotating the hydraulic

model. Before any experiments studying the effects of through-

flow can begin the water in the model must be spun-up to solid

body rotation. This is achieved by rotating the turntable at the required rate for a length of time at least equal to the spin-up time. This ensures that the complex processes taking

place during spin-up do not interfere with the model flows.

Sobey (1973) summarised the literature on spin-up and commented on several marked discrepancies when comparing actual and theoretical spin-up times for the laboratory turntable.

The theoretical spin-up time (from rest) for a shallow rotating circular basin is given by Greenspan and Howard (1963) as:

T1 = L (cJI0 )4 (B-1) where L is a characteristic length, usually the depth of fluid in the container, O is the characteristic angular speed of the basin and 0 is the kinematic viscosity of the fluid. For the

5m. diameter laboratory model with a water depth of 80mm. and a rate of rotation of 1 revolution hour-1 the theoretical spin-up time T1 (equation B-1) would be 32 minutes; this can be compared with the actual time of approximately 120 minutes to attain this state of solid body rotation. This order of discrepancy has been noted by other authors using rotating hydraulic models eg.

Ruiner (1970) noticed that his actual spin-up time of 10 minutes was longer than the theoretical time of 3.4 minutes. Such discrepancies are in need of explanation, however, the solution is not to be found in the literature where good agreement between

332 theory and experiment is frequently reported. Sobey (1973) suspected that the difference in, aspect ratios between the purely

academic studies (order of unity) and the hydraulic models (for this study 62.5) may be the reason for such good agreement.

Benton and Clark (1974) have given an excellent review of modern developments in spin-up theory and experiments including a section on non-linear spin-up of homogenous fluids which is of particular interest in the context of this study.

Strongly non-linear spin-up in a circular cylinder is unique by virtue of the unusual phenomenon that occurs. Wedemeyer (1964) studied the ultimate non-linear case of spin-up from rest in connection with problems involving liquid-filled projectiles by using a linearised Ekman compatibility condition. In essence

Wedemeyer showed that the spin-up from rest in a circular cylinder is characterised by a cylindrical wavefront (a discontinuity in shear) which propogates from the cylinder walls to the central axis. During this time the entire volume of fluid ahead of the wavefront is eventually pumped through the Ekman layers and ejected into the region behind. Although the speed of propo- gation is predicted to be an exponentially decreasing function of time, so that in theory it never reaches the cylinder axis, he found that the fluid had been substantially spun-up by a few

Ekman times (T1). This flow pattern has been verified experimentally by Goller and Ranov (1968) and Greenspan (1969) who prints a photograph showing the basic picture of the flow. A method using the results of Rogers and Tarice (1960) to couple the influence of the Ekman boundary layers has recently been shown to be invalid by Benton (1977) who states that only the

333 Wedemeyer model is valid.

Weidman (1976) has conducted a large scale study of spin-up and

has generalised the Wedemeyer model to include spin-down (not

considered here). He found that inclusion of the full non-linear

Ekman suction in a Wedemeyer-type analysis of spin-up from rest

resulted in a velocity discontinuity for the inviscid model,

experimental measurements made using a laser doppler velocimeter

confirmed the adoption of his theory.

A growing body of recent literature has been concerned with

experimental observations of spin-up and confirmation of various

theoretical aspects of the subject. Examples include Goller and

Ranov (1968) who studied the effect of a free surface on impul-

sively started spin-up from rest in a circular cylinder, Weidman

(1977) mentioned previously and Warn-Varras (1978) who in common with Weidman used a laser doppler velocimeter as the basic measuring device, this device is well-suited to the study of such flows.

Watkins and Hussey (1977) have lately published a report of a study of spin-up from rest in a circular cylinder. They have shown that the shape of the velocity profiles depends on a parameter 3o defined as

o = h (ū/0))7 (B-2) p 2 a

Where h is the depth of fluid in the cylinder and a is the radius of the cylinder.

334 When g was less than 0.005 the velocity profiles agreed well

and the fractional spin-up time to solid body rotation at a

particular radial position r depends on the geometrical, kine-

matical and fluid parameters through the Ekman spin-up time T1

(equation B-1) regardless of the value of R = r/a and the angular

velocity 6). For values of greater than 1 the fractional 3 spin-up time is proportional to a2/ID . A typical value of R !- o for the turntable is 0.0003 implying that the former fractional

spin-up time will apply, this result was found to be consistent

with measurements taken for this project.

In order to check the spin-up time to solid body rotation for the

5m. diameter hydraulic model use was made of the low velocity

measuring technique described in..-chapter 4. One quadrant of the

model had buoys placed at each grid point and was continually

monitored by taking photographs of the quadrant, and hence the

buoy targets, at regular intervals (eg. 5 minutes) for each

rotation rate until the spin-up time was reached. In order that

the photographs were taken at the expected spin-up time and to

minimise analysis it was necessary to gain an approximate idea of

the time taken to attain solid body rotation for each speed

beforehand. This was achieved by noting the time taken to minimise the flow-induced dispersion of potassium permanganate dye

crystals. Analysis of the photographs and computation of the velocity vectors, in the way described in chapter 4, enabled the time to be determined at which the deflection of each buoy became zero once more (corresponding to the zero position). This time was the experimental spin-up time and is compared with the theoretical Ekman spin-up times (T1) for each rotation rate in figure

335 B-1. The figure shows the experimental spin-up times for rota-

tional speeds of 3 revolution hour 1 and upwards are directly

related to the theoretical spin-up times being typically 2.5 to

3 times larger than the Ekman spin-up time. This confirms the

theory of Watkins and Hussey (1977) concerning fractional spin-

up times.

As was mentioned earlier in this Appendix a possible reason for the discrepancy between some theoretical and experimental spin- up times is the contrast in aspect ratios of those experiments used to verify the theories and the model aspect ratio, the

Reynolds number and Froude number being comparable. The influence of aspect ratio could be connected with the side-wall boundary layer effect which is generally neglected in the theoretical analyses.

336 SPIN UP TIME(min.) T1 -120

10 h=80mm. =1 mm2s-' 80

60 Experimental

40 Theoretical T1=h(ca Y . 20

1 2 3 4 5 6 -1 7 8 9 10 REVOLUTION HOUR (C /2c ) Figure B-1 Spin-up times for the 5m. diameter turntable. APPENDIX C

PHOTOGRANMETRI C CONTROL DATA

338 As was described in chapters 3 and 4 it was necessary to provide

good photogrammetric ground control over the whole area of the

5m. diameter model. This ground control was needed to determine

the exterior orientation of the camera and then, from this data,

the ground co-ordinates of the buoy targets at their positions

in the model. The photogrammetric ground control was provided

by points on the edges of nominally horizontal white-faced steel

surveying tape suspended across the model basin at two levels

under tension, and by the centre of a cross on each of four

plates projecting from the model wall into the interior of the

circular model. A control point on a tape was defined as the intersection of a selected graduation line with an edge of the tape. The space co-ordinates of 34 control points (including the four crosses on the plates) were determined from a triangulation survey and the choice of a suitable arbitary 3-dimensional co-ordinate system.

The co-ordinates of the 34 control points as determined from the survey and as used for the velocity measurements are given in table C-1 with the approximate position of the control points being as defined in figure C-1.

339 Table C-1 Control Point Space Co-ordinates

Control Point x co-ord. y co-ord. z co-ord. (mm.) (mm.) (mm.)

Ti 4686.0 1310.6 96.7 T2 4694.7 4806.7 96.3 T3 1113.1 4726.7 96.1 T4 1285.8 1246.1 96.1 CrA (1) 586.7 2874.9 399.75 CTA (5) 1805.7 2923.1 393.85 CTA (8.3) 2810.9 2963.7 390.45 CTA (12) 3938.1 3008.9 387.75 CTA (16) 5156.8 3058.4 386.2 CTB (2) 2790.4 5081.8 398.0 CTB (6) 2790.1 3861.8 393.o CTB (9) 2790.3 2947.3 390.3 CTB (12) 2790.6 2032.7 388.15 CTB (16) 2790.6 813.1 386.15 LT1 (34) 5192.3 2815.4 97.9 LT1 (37) 4546.9 2167.3 98.5 LT1 (39) 4116.6 1735.3 99.6 LT1 (41) 3686.2 1303.2 101.2 LT1 ,(43) 3256.0 871.6 103.25 LT2 (21) 5152.4 3400.1 96.1 LT2 (24) 4504.1 4045.2 94.6 LT2 (26) 4071.8 4474.8 94.8 LT2 (28) 3639.5 4904.9 96.0 LT2 (31) 2988.3 5546.9 98.5 LT3 (19) 530.9 3020.o 96.5 LT3 (23) 1364.4 3911.7 95.7 LT3 (25) 1780.7 4355.9 96.3 LT3 (27) 2197.4 4801.7 97.2 LT3 (29.5) 2716.8 5358.7 99.o LT4 (51) 2858.9 651.o 1o4.5 LT4 (54) 2184.8 1269.0 99.75 LT4 (56) 1735.6 1680.7 97.8 LT4 (58) 1286.1 2093.1 97.o L24 (61) 611.5 2710.2 97.6

340 \xvi Steel tape Control o 100 mm. above bed points o 400 mm.above bed A Theodolite stations

Figure C-1

Photogrammetric control

341 APPENDIX D

COMPUTER PROGRAM USED TO DETERMINE THE VELOCITY FIELDS

3142 •

A computer program was used to determine the velocity field

(or fields) for each particular model configuration. The data

required for the program was as follows:-

(1)Measured (x, y) co-ordinates of the position of each buoy

target for the 'zero' photographs.

(2)Measured (x, y) co-ordinates of the position of each buoy

target for the 'deflected' photographs.

(3)Exterior orientation data for each photograph; this was determined as described in chapter 4.

A listing of the program is given in the following pages and

contains explanatory comments at each stage of the program, however, to aid clarity the exterior orientation data is outlined

below:-

'ZERO' 'DEFLECTED'

Negative Centre Co-ords. (ZNCCX, ZPCCY) (DNCCX, DNCCY)

Flying Height FH FHD

Tilt about 'x' axis TY TYD

Tilt about 'y' axis TX `XD

Swing S SD Perspective Centre Co-ords. (ZPCCX, ZPCCY) (DPCCX, DPCCY)

343 ++f+i.##*#*.►f..i++R+..f.$++**+++*++*#+***..**i+.>**++++++***+**** PROGRAM TO DETERMINE THE .JET—FOī:CED VELOCITY FIELD IN A H•rtIRAi'L IC MODEL FROM PHOTOGRAPH MEASUREMENTS UF THE P❑=ITION:_: OF TETHERED BU0•; _•• E;:;TERI❑R ORIENTATION DATA AND CALIBRATION OF BUOYS

♦+.♦ff4.# ff #**f+++++++f++*#++*+**♦*+**++.++++**++f+*+** ***..-i.ff s DIMENSION UU(15,15), V'V(15,15) 5 I r1EN:_ 10N . PAGE(15)1 PAGE (15) . ī_r•1EN:SI❑N VVr9A'3(15, i5), VVDIR(15, 15) 'A.' '1 (16 =:Or•ir1❑r1 ❑ (16, 16s) , O (16, 16) , ;;D (16, 16) , YD <: 16, 16) , ::M (16, 1 ) , `r 1,15),Lt:16, 16),V(16, 16),VMAG(16, 16),VDIR(16, 16) COMMON /B! zNCCx:, ZNCCY, FH, TV, TX, s, ZPCCX, EP'CcY, DNCG:;, DNCCY, FHD, TYD, 1 T4D, SD, DP CCX, DPCCY COMMON 'C. ZCPX, ZCPY, DCPX, DCPY PI = :3. 1416 NMAX = 16 MMA`S = 16 i+i***.f*.fff i#♦*f♦+++fffif+*+a#f++++ff f+#**f++*f LETS ARRAYS TO ZERO AND READS ZERO—PHOTO MEASURED CO—GRDS AND AVERAGES THEM

DO 10 M = i , Mr•1AX DO 10 N = 1, NMAX ❑(N,rt) = 0.0 (O(M,M) = 0.0 : D(N,M) = 0.0 'VD (N, M) = rfi. 0 . ri(N,M) = 0. 0 - YM (N, M) = 0. 0 tJ (N, M) = 0. t) V (N,P1) = 0. 0 WAG (r! , r•1) = 0. 0 VDIR(N, M) = 0. 0 READ (5,1000) XO 1, O2, X❑3, Y'❑ 1, (02, ?03 XO (N, M) = 01+X02+X03) %3. 0 Y❑'SN, M) = (V01+Y02+Y03) /3. 0 10 C❑NTINUE +++*+4>*ffi**+f*+44**ff***##+f*****#+*#**+.*+*f*+*f* ♦#*f**++♦+#*f

READS DEFLECTED—PHOTO MEASURED CO—OROS AND AVERAGES THEM

DO 20 M = 1, lir1AX DO 30 N = 1, NMAX READ (5,1010) XD1, XD2, :D3, YD1, YD2, YD:3 XD (N, M) = ( D 1+xD2+ ;D:3) , 3. 0 r'D (N, P1) = (YD1+YD2+Y'D3) f3. 0 2') CONTINUE #f.*ff+*++.#f*f#*ff *f#f♦**#+##*f.+4 f+.*♦+*+**++f+++*++ DO 30 M = 1,15 DO ::0 rI = 1,15 UU (N,ru = O. 0 '•1'1 ;N, M) = 0.0 :?0 CONTINUE J = 0 EXTERIOR ORIENTATION DATA 17!UADRANT 1 ZNCCX = 026360. ZNCCY = —019550 FH = 4467.9 TY = —1.1:370 TX = —0.1774 _ —2.6202 ZPCCX = 4173.0 ZPCCY = 1927.8 DrICCX = 025610. DNCCY = —021070 FHD = 4467.6 = —1.01)4n T iD = —ii. (993 3D = 0.669,_: DPCC; = 4183.9 DPCCY = 1926.4

344 40 J = J+1 ******* ***** 4041.41.440.44,4444-11.44.41-04#441.444.04411.4.**4* REFIE; AND AVERAGES PHOTO ORIGIN CORRECTIONS

READ (5:1010? ZCPX1:217PX2::CPX35,7CPY1:ZCPY2:2CPY3 READ (5:1010) DCPX1:DCPX2:DCRX3:DCPY1IDCPY2:ACRY3 -CPX = (2CPX1+ZCPX2+7CRX3)/3.0 7CPY = (ZrRY1+ZCAY2+7CRY3)/3.0 DCPX = (DCPX1+DCPX2+DrPX3?/3.0 DCPY = (DCPY1+DCRY2+DCAY3)/3.0 +++++ -04-0-.4.40.0.4.+++.41.4.4.4.4.44.4.4.4.4.4.444.444.4.44444.444.44.4.4.4.44.0-41.+ IF (J.EQ.2) GO TO 60 IF (J.EQ.3) GO TO :30 IF (J.E0.4) GO TO 100

CALL. UBROUTINE JANE TO DETN. '/EL.. IN 6UADRA1iT 1

DO 50 M = 1:8 DO 50 N = 18 CALL JANE (b P1) 50 CONTINUE

4,41.4.4.4.41.4.44-0.04-4■40404.4411-04-10.4.+41.. +++++ 41-11-4.11.44.4.41,41-040.40-04.4.41.44. +++++ 4.41,11-0.104-40,14-40.** EXTERIOR ORIENTATION DATA QUADRANT 2

2MCCX = 026260. ZNCCY = 022630. FH = 4478.2 TY = -0.2127 TX = 0.0938 3 = 0.7067 2PCCX = 4131.2 ZPCCY = 4109.4 DNCCX = 024880. DMCCY = 023620. FHD = 4477.9 TYD = -0.5631 TXD = 0.0856 3D = -1.4683 DPCCX = 41:32.6 DPCCY = 4112.2

41040440040.444.444441 14 111410*-04.441144-04-4044444.1 414,4104.41.-0404411.4.441 0444,4-41-40+441044+41.44.

GO TO 40 ++++++++ *404-11-044-44.41144.4144.44444.414-011,41.44.4. +++++ 41-0441.4144. CALLS SUBROUTINE JANE TO DETN. VELS. IN QUADRANT 2

60 DO 70 M DO 70 N = 1,a CALL JANE (N: M) 70 CONTINUE 41.444.44.4.44.4.44.4.44.444.4.4.4.4.4 ++++++++++ 4.44.444++++++ +++++ EXTERIOR ORIENTATION DATA QUADRANT 3

ZNCCX = -022050. ZNCCY = 023320. PH = 4481.1 TY = 0.3292 TX = 0.8869 S = 1.1216 2PCCX = 1661.4 ZPCCY = 4046.0 DNCCX = -021970. DNCCY = 023310. FHD = 4480.3 TYD = 0.4203 TXD = 0.9223 SD = 1.1289 DPCCX = 1658. DPCCY = 4043.0

345 4+a-t*tttt ▪ .ts.:.++..+ato44 4∎4.4.84 6>*,$+++s+a.4++++4+ **t!4t4Od>t t4+

G❑ Il 40 • ...ta.+..t+...+.+.a+t.+t+.+.++t+.tt+a++tt.t.+.+.t..+a+*t.++ CALL •SUBROUTINE JANE TO DETN. VELS. IN QUADRANT 3

0 DO 90 1'l = 9,16 • D❑ 90 N = 9,16 CALL JANE (N,M) 90 ' CONTINUE tt..tt++♦a+ttt♦♦♦...t.♦ttt+...tt.t..+a++tt++.t..t.t.ttt++t...tt. EXTERIOR ❑RIENTATI❑N DATA QUADRANT 4

ZNCC ; = — 019040. ENCCY = — 025730. FH = 4485.1 TY = 0.3113 TX = 1. 09:35 S = —2.6334 zPCCX = 1761.3 Z CCY = 1623.O DNCC; = -020180. DNCCY = — 024560. FHD = 44:34.2 TYD = 0.5273 TXD = 1.1041 SD = 0.7154 DPCC> = 1766.5 DPCCY = 1626.1

+ ++++e++++++++++++t+..+.♦++++++++t+++..t++t+.+++.ttt+++++tet+.++.

GO TO 40 t +++t+t.+++++t++++.+.+++.++++..t.♦++.t.t++.t+t+.++++t.+ttt+.+t♦+. CALLS SUBROUTINE JANE TO DETN. VELS. IN QUADRANT 4

100 DO 110 M = 1,8 DO 110 N = 9,16 CALL JANE (N,1'1) 110 CONTINUE ♦ ++++++.++.+++++++++9t+tt+t++...++++++++.+.+♦+...... ++++t.+t+t.++ .+.+++.++.t++t.+.t..+.++..t.+.t..ttst+...t...+.++tttt.t.t+..+++.. I,JRITES OUT <:<,Y) GROUND CO-iJP.DS FOR ZERO AND DEFLECTED POSNS BUOY

WRITE (6,1130) .IJRITE (6,1020) 4JRITE (6, 1060) DO 120 P1 = 1, r'1MAX 1~1R ITE (6, 1050 M (XD (N,(1) , N=1,NNr1A ) 120 C❑NTINUE WRITEITE (6, 1130) WRITE (6,1020) WRITE (6,1060) DO 130 P = 1,MMAX WRITE (A.1050) r1, (::D(N,M),N=1,NMA -<) 130 CONTINUE WRITE (6,1130) WRITE (6, 10.30) WRITE (6,1060) DO 140 M = 1, mm AX !AIR ITE (6, 1050) M, .'r ❑ (N,11) , N=1, NMAX) 140 C❑NTINUE WRITE (6,1130) WRITE (6,1030) WRITE (6, 1060) DO 150 M = 1, Mr1AX 'JRITE (5, 1050) M, (YD(N,M),N=1,NMA .:) 150 C❑NTINUE

3146

++r~°*++~+°++°~+°++~*+*°+*^++++~°+~+°++^~°^+++++~~~~°~~+°~°p~.~~+ TAKES VELOCITIES ON '~16X16) GRID AND AVERAGES T~OSE VELOCITIES OF TETHERED BUOYS ON COMMON CENTRELINES THEN TRANSFORMS TO (15X15) GRID

DO160M=1,7 UU(8,M) = (U(8,,)+Ur9,M)//2.0 YV(8, M/ = (Y(G,~/+V~9,M/)/2.0 IF (U(8,M).EQ.0.0.OR.U(9,M`.EQ'0.V/ UU(O,M` = 2.0°UU(8,M) . IF (V(8,M/.EQ.0.0.OR.V~9,M/.EQ.0.o) YV(8,M/ = 2.0~VV(8,M/ ' 160 CONTINUE DO 170 M = 10,16 MM = M-1 UU(O,MM) = V](O,M)+U(9,M))/2.0 VV(8,MM) = (V(8,M)+V(9,M))/2.0 IF 01(8,M).EQ.0.0.OR.U(9,M).EQ.0.0) UU(8,MM) = 2.0+00(8,MM) IF (V(S,M). EQ. 0.0. OR. Y(9,M).EQ.0.o) V./ (8. = 2.0~Vv(8,MM) 170 CONTINUE DO 180 M = 1,7 dU(H,8) = (U(M,8)+U(M , 9) )/2.0 VV (M,8) = (V(M,8)+V(M,9))/2.0 IF (U(M,O). EQ. 0.V.OR.U(N"9). EQ. 0.O/ UU (~,8) = 2. 04.0_1(N. . IF (Y(M,8).EQ.0.0.OR.V(M,9/.EQ~0.0) YV(M,8) = 2.0*VV(M,8) 180 CONTINUE DO 190 N = 10,16 MM = M-1 UU(MM,8) = (U(M,8)+U(M,9))/2,0' VV (MN,8) = (V(M,8)+v(M,9))/2. 0 IF (U(M,8).EQ.O.0.OR.U(M,9).EQ.0.0) UU(NH,8) = 2.0+UU(MH,8) IF (V(M,8).EQ.0.0.OR.V (Mr 9).EQ.O.0~ VV (NM. = 2.0~vY CNN. 8/ 190 CONTINUE UU(8,8) = (U(8,8/ +U(8,9) +U(9,~~+U(9,9))/4.0 VV(8,8) = (V(8,8)+V(8,9)+V(9,8)+V(9,9))/4.0 DO 200 M = 1,7 DO 200 N = 1,7 UU(M,M) = UMW) YY(M,M) = V(M,M! 200 CONTINUE DO 210 M = 10,16 DO 210 N = 1,7 MM = M-1 UU(M,MM) = U(M,M) VV(M,MM) = V(M,M) 210 CONTINUE DO 220 M = 1,7 DO 220 M = 10,16 NM = M-1 UU(dM,M) = U(M,M/ YV(MM,M) = V (M,M) 220 CONTINUE DO 230 M = 10,16 DO 230 N = 10,16 MM = M-1 MM = M-1 UU(MH,MM/ = U(N,M) Y;(MM,MM) = ;(M,m/ 230 CONTINUE °~.+*.++°**+*+~°+*°••°••°••+++++•*++~4+++°+°^+*++°°++.+~++++°+4++ WRITES OUT VELOCITY COMPS (15X15) GRID WRITE (6,1130) WRITE (6,1070) WRITE (6,1120) DO 240 M = 1,15 HRlTE (6,1110) M,(UU(M,M),M=1, 15) 240 CONTINUE WRITE (6,1130/ WRITE (6,1080) ~/RlTE .A,1120) DO 250 M = 1,15 ~RITE (6,1110) M,(VV(M,M),N=1,15) 250 CONTINUE ~~r4°0+*°~++0+~°+~+*~4~+^**°+~°~*°~+~++4++°+^+^°°*+*°~++~°0+4°°++

~~ +++++14+++4++++++++++♦++1++++++4++4++++++++++++++++4++++++++++♦++ 4ipI TE: OUT VEL. C❑I1P:,= MAG= ",DIEN:, '.16 :16) GRID WRITE (6,1130) 'IR.ITE (6,1070) WRITE (5, 1050) DO 260 M = 1, r1MAX . t.dr? I TE (6,1050) r'19 (.1_1(11 , N=1, NMAX) 260 CONTINUE wRI rE (6,1130) WRITE (6,1080) WRITE (6, 1060) DO 270 M = 1, r1MA> WRITE (6,1050) h1,(V

PLOT 3UEROUTINE

CALL °TART (2) PI = 3.1416 DELTA:; = 0.62 DELTAY = 0.62 VEL3CAL = DELTAX,3.0 F'L❑TMA< = DELTA?,+4.0 AS OEMELE AND PLOT GRID

DO 390 N = 1,15 D❑ 300 M = 1,15 FM = FLOAT(M) RN = FLOAT Shi) ;PALE (M) = 75+ (RM+2. 0) +DELTAX ePAGE (N) = 0.6+ (RN+1.5) +DELTA'r 300 C❑NTINUE

WIDTH = 0.01 DO '310 N = 1115 DO 310 M = 1,15 IF (UU'N,M).EQ.0.0) GO TO 310 IF ( Vthi,M).EQ.0.0) GO TO 310 CALL :`Ir1E❑L (XPAGE (M) , YPAGE (N) , W I L1 H, 1 H+, 9 0. 0, 1) 310 C❑NTINUE

31+8 PLOT OF VELOCITY VECTORS

DO 320 1 = 1,15 0] 320 M = 1,15 IF (UU(M,M).BQ.0.0,AMD.VV(H,M).B~.0.0) 5O TO 320 YVMH6(M`M} = :QRT((UU/M,M/^w2.0)+(VV(M,M)++2.0) ) V;U[R/N"M/ = 8THM2(VV ~N,0/,UU(M,M)) HEIGHT = y;MH5/N,w/°;E~3CHL IF (HEl5HT.EQ.0.0' GO TO 320 IF 0HElGHT,5T.PLOTMHX0 GO TO 320 VVDIP(M,M) = ;VDIR0H,M/+180.0/P[ ANGLE = -VVDlR(H,M/+270.0 ANGLE = Hn6LE+PI/180.0 XCORD = XPH6E(M)-~EI6HT+COC(RM5LE)°0.28 YCORD = YPA5E(M)-HEl6HT*SlM(HM6LE)°0.28 ANGLE = HM6LE+180.0/PI CALL SYMBOL (XCORD,YCORD,HEI6HT,83,HM5LE,-1) 320 CONTINUE 4444444-0~4~~44~444444444~04~444~44~~444-44-4 +++++ 4-44444444444-444444 °+4°*++*++++~444+4444°*444°44°°+~4+~++4+++°*+^4 ++ 44.404.44.44-4444.4144.4, PLOTS CIRCULAR MODEL BOUNDARY AND POSITION OF OUTLET

K = MMAX/2 c = NMAX/2 XCIROR = .:PAGE(K) YCIROR = YPH5E(L) DI8M = (FLOHT(MMHX-4)+DELTHX)+DELTHX/2.0+0.25 RADIUS = 0.5+DIAM CALL CIRKL (XCIROR,YClROR,RHDIUC) CALL CIRKL (XCIROR,YCIROR,0.15)

PLOTS DIRECTION OF DISCHARGE ARROW

DIRX = XClROR-RADIUS-0.42 DIRY = yClROR-1.5 CALL SYMBOL (DI~X,UIRY,1.5,O3,0.0,-1) +~+++.++~°°°~~~+~+^~~^~**~~~~+~~+^++~+++*++°°°++++++4~°*+*+~~°+°+ WRITES TITLES AND DATA VALUES FOR JET PARAMETERS

WlDCCRL = 0.12 yIDTlTL = 0.2 CORDY = YCIROR-14.5+WIDTITL CORDX = 1.5 CALL SYMBOL (CORDX,CORDY,HlDTITL,30HJET CIRCULATION IN H RESERVOIR 1,90.0,30) CORDx'~ = YCIPOR+1.0 CORD-1i = CORDX+0.5 CALL SYMBOL (CORDXM,CORDYM,WlDCCHL,20HEXPERIMEMTHL RECULTS,90.0.20 1) CORDXEM = XPH5E(15/+(DELTHX/2.0)+ulDSCHL+0.6 CORDYEM = YPH5E(1)+2.5+DELTHY CALL SYMBOL (CORDXEH,CORDYEM,WIDCCHL,51HLEM5TH :CALE 40.0 CM 1 VELOCITY SCALE 8.0 MM/S,90.0,51) CBRDXEM = CORDXEM+3.0+WIDCCHL CALL SYMBOL (CORDXEN,LOPDYEM,WlDCCHL,51HlMITIHL DEPTH 8.0 CM 1 JET INLET AT BED ,9O.0,51) CORDXEN = COPDXEM+3.0+HIDSCHL CALL SYMBOL (CORDXEN,CORDYEM,@lDSCHL,51H JET YELOCI7Y 35 CM/C 1 CLPCULHR JET 1.2CM DIHM,90.0,51/ CORD. EN = CORDXEH+5,O°VIDCCAL CALL SYMBOL (CORDXEM,CORDYEM,WIDTlTL,27HDHTE OF TEST 9 NOV. 1976 1 ,90.0,27)

PLOTC BORDER OF VELOCITY FIELD TO H2 :lZE

BORDEX = 16.53+DELTHX YORI5IM = 0.0 EJPDEY = 11.69 CALL PLOT (DELTHX,YORI5IM,3) CALL PLOT (DELTRX,8ORDEY,2) CALL PLOT (BORDEX,IORDEY,2) CALL PLOT (BORDEX,\~O~I5IM,2) CALL. PLOT (DELTHX,YORl6IM,2)

~~ +++...444.+..+s.+++++++4e++++4++++4+++*+3++++ass+s4+++++++++++++++ CALL Et- PLOT ♦..+...++...+.++..++++.A..i+..+e+♦+4.+4.+..+.+..e..+++++...++++...♦ STOP

1000 FORMAT (6F10.0) 1010 FORMAT (r.F 10. 0) 1020 FORMAT (1H ,//,,10';::,"X GROUND CO-OPD 1NATE:_'••" , ////) 1030 FCF:riAT (1H ,r .'', 10:', "Y GROUML: CO-ORDINATE=:", ,'///) 1040 'FORMAT (1H1, //) 1 050 FORMAT X, I2, 16F~. 1, ///) 1060 FORMAT (/4 ;, "r1 1 E :3 4 5 6 1 7 9 10 11 12 1.3 14 215 15"//) 1070 FORMAT (1H , 10 <:, "U VELOCITY COMPONENT (MM/:3) ' 2080 FORMAT (1H , IOX, "'v ?ELOCITY COMPONENT (MM/S) ") 1090 FORMAT (1H '10X," VELOCITY MAGNITUDE (MM/') ") 1100 FORMAT (1H 110x!" VELOCITY ANGLES IN DEGREES COUNTERCLOCKWISE") 1110 FORMAT (3X, 12, 15F8. 1,/,`/,) 1120 FORMAT (/4:;, .M 1 2 ❑ 4 S 5 1 7 8 9 10 11 12 13 14 • 215"//) 11:30 FORMAT (IH1,//,1OX,"9 NOVEMBER 1976 CIRC./•JET/CENTRRAL OUTLET VEL 1: =:3 5Crl/ S'•. ///) 1140 FORMAT (1HS) END

++ +++.++++s...... +e+.++4..4...... 4.+++41+4.4.e.+4..4..+.4..4.4..+++t4.+4.4.404 +++e4.40++40++++4.4.++*++.+t4 44+•11.+++940...+.*.+...++++*+++4.4.++++.+e4.++++ SUBROUTINE JANE cr•i,N) .e.s..o..++.+e40e+.+.+e.e++e4.+4..+4.4...+.4.4.4.4..+...... ee...... e..... COMMON /A/ X0(16, 16) , y0 (16, 16) , XD (1':., 16) , YD ( i 6, 15) , : M (16, 1 6) , YM::15 1,16) ,1_1(16! 16) ,V(16, 16),'v'MA'3(16, 16),'VD1R(16, 16) COMMON 'D/ Zr1CCX, ZNCCY, FH, TV, TX, S,I_PCC:t, ZF•CCY, Dr1CCX, DNCCY, FHD, TYD, 1 TXD, 'D, DFCCXI DF.CCY COMMON /C/ ZCPX,2CPY,DCPX,DCPY PI = 3.1416 4#44-s444804 ,40.4..4>04.304>.•4+0Oe.e.•Sfe4!>0•>>e»>0. >aO; >>c a •:aaoo+ PRINCIPLE DISTANCE (VALUE ASSSUMED)

FL = ?1.43 ++....++4.++++4.+.++++e++++4.4.e4.4...... e.+4.eee4..+....*e...+..4..ee4.ee HEIGHT OF BUOY TARGET: ABOVE MODEL BED (VALUE ASSUMED)

HF = '93.0 +++++++..+.++4++++++441++4.++++e++++++.++++.0..+++.+.+4+.+*..+.**+. CONVERTS CAMERA TILT ANGLES FROM DEGF:EE`3 TO RADIANS

THETAZ .= :3 THETAD = :3D THETAZR. = (THETAZ+PI) /180. 0 THETADR = (THETAD+PI)/180.0 TXR = (TX+PI) /180. 0 T'•I'R = (TY.P I) / 1'r 0. 0 ī:'.DR = cT:>:D+PI) 130.0 TYDF: = (TYD+PI)/180.0 ♦..+eee..e...e+.....4...+++.e...4...... 4.+.4.e...4..e4..+..:.....4.4.4..4..

IF r;0gN,M).E0.0.0;' G❑ TO 40 IF ('YJ (N, M) . EQ. 0. 0) GO TO 40 IF (;:D (N, M....E'?. 0. 0) GO TO 40 IF c'•i'D

X❑ Cr1,N = >::O(N,r•t)-:CP:>: YG (ri . rq) = '•(0 '..11 • M) -ZCPY >;D(r1,r1) = .:Gi.y.r:,-L„_F.• ` D(N,M) = YD(N,r'1)-DCR?

350 •+++++*++4++41.4++.4444.+4+4*444+44•.+4.4++.++4+44.4+4.++++.++*4444 PHOTO CO—ORDS RELATIVE TO PLATE CENTRE

. O(N,M) = X❑ N.M)—Zr1CC.. P4 • PP 1''❑ ': N • P{ = `i'G ', — NC C`i' xD':N, r1) = : ~

❑CN01' = ::GcN.r1)'10.+2.0 Y❑ (N. M) = VO (N, r•1)'' 10.+'3. 0 XD(N.M) = ::DCN,r1:'.'1£'++3. 0 YD (N, N) = YD (N, MI / 1 04.:3. 0

TRANSFORMS PHOTO CO—ORDS TO GROUND CO—ORDS USING FORMULAE OF . HIRV❑NEN ZKA't' = (F H—HF) / C CFL•COC (TYR) .COC :TxR:l) — (X❑ (P1. P1) +:• I N ( TYR) +C❑S(T ;R) ) 1—(YO(N,P1)+SINCTXR))) ti ' r H ZF.;)~' + .I~ ~~~3 '•:T~N.YP; aC~0'_ ( t;Rr; = :C:❑ :N,rt:...0:IrI ~T'~i'R: +:_It•itT:v:)+'.:IP'~T. ETr. IHETAZR))))—('iDrrl,M).CGSGT:>:P +:Ir''THETA R,::')+'::FL*C C3IN(T•'i'F:) •CC.S(THET 2AZ R)) — CCO.s ( TYR) +S I N CT P) . I N ( THE T A_R):') •:' ) BR ''i' = C Ci:O (ri. M) • C CCOS ( TYR) S I N (THETAZF:)) — (: I N CTYR) . _.I N (TXR) +COC CT I HETAZR)) :,) + ('(0 (N. M) +i_ OS (TxR) •COC ' THETAZR) + (FL+ C CS I N',TYR) +:' I ri ( THET AEIP:)) +CO.3 ( TYR) +S I N (TXR) •C❑•5 CTHETAZR)))) ) :c = C CXD (r'', M) + C( s I N tT'YL:R) +:3I N (T%;DR) +5I r'1': THE T A_DR)) + cCO.S ( TYDR) +C❑ 1:3 (THETADR)))) —'::'i'D(r'. M) •COS(TXDF:) +:31r•, rTHETADF')) .►. r:FL+': r.:3IN (TYDR) +CO: E (THETADR)) — (CO:O CTYDP l 4= IN (TXDR) +SIN':THETADR)))) ) BRD`i = C':::iD(r•{.r1)+C'::COS(TYDR)+SIr1(THETrDP)? —(SIN'::T`iDR) 4:_IN''TXDR*C0 1 S (THETADP)) l) +('r'D'::r', M:' +CO.3 CT?:DP) +CO _:':.THETADP)) + r.FL. f CC IN' TY'DR) +O IN (THETADR)) + CC❑O (TYDR) + IN(T:>:I'R) .0❑S .THETADR) )) ) YD(N. P1) = ;,F'CCY+ (ZK:A'•i+BRZY) XD (N. M) = Dr CCX+ (DKAY+BRDX) YD (N, P1) = DPCCY+ CDFCAY+BRDY)

SUBTRACTS ZERO AND DEF. X AND `i' CO—ORDS TO GIVE COMPS OF DEFLECT.

;iN (N, PD = :D (N, M) —X❑ ( N' rl) ''iPl CN, M) = `iD (N, rl) —Y❑ r:N. M) - CONVERTS DEF. COMP: TO VEL. COMP: U:3IMG CALIB. PELATION:-:HIPS

TOTDEF = 5 .PT { (xM (N' r') ++c. 0) +'xri tN. N) •+E. o:i ; :'DIR CN. Pl) = ATAN2 C:'M':rl. M) , Y'P1'N, M) ) IF CTGTi'EF. LE.:l. a:l GO TO 20 IF (T❑ TDEF. LE. 1S. 0) GO TO 10 Vr1AG CN, N.' = 1. 079 6+ (T❑TDEF+• 0. 83) GO TO 30 10 VMAG (rl, M) = 1. 707 3. CT❑TDEF++0. 6491) GO TO 0j 20 VMAG CN, rl) = 0. 0 0 IF r:Vr1Ai5':N. r'1) . EO. 0. 0. GO TO 40 UCrf,PD = ?MAG(N,N)+(CO (VDIF:(:N,P1)) ) ) V (N, P1) = ('VMAG (N, P1) + r: -: I N CVD I R (:N. N))) ) VDIReN.M) = 'VDIR(N 11)+1 0.0!PI 40 CONTINUE

RETURN END

351 REFERENCES

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ALI, K. H. M. and HEDGES, T. S. 1975 Discussion to Sobey and Savage (1974) in, J. Hyd. Div. Am. Soc. Civ. Engrs., 101, HY12, 1543-1545.

ALI, K. H. M., HEDGES, T. S., and WHITTINGTON, R. B. 1978a A scale model investigation of the circulation in reservoirs. Proc. Instn. Civ. Engrs.; 65, 2, 129-161.

ALI, K. H. M., HEDGES, T. S. and WHITTINGTON, R. B. 1978b Closure of discussion to Ali, Hedges and Whittington (1978a). Proc. Instn. Civ. Engrs., 65, 2, 712-717.

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353 BENTON, E. R. and CLARK, A. 1974 Spin-up. Ann. Rev. Fluid Mech., 6, 257-280.

BIRCHFIELD, G. E. 1969 Response of a Circular Model Great Lake to a suddenly imposed wind stress. J. Geophys. Res., 74, 5547-5554.

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BONHAM-CARTER, G. and THOMAS, J. H. 1973 Numerical Calculation of Steady Wind-Driven Currents in Lake Ontario and the Rochester Embayment. Proc. 16th Conf. Great Lakes Res., 640-662, Intern. Ass. Great Lakes Res., Ann Arbor.

BOSANQIIET, C. H., HORN, G. and THRING, M. W. 1961 The effects of density differences on the paths of jets. Proc. Roy. Soc., A263, 1314, 340-352.

BROOKS, N. H. and KOH, R. C. 1969 Selective withdrawal from density-stratified reservoirs. J. Hyd. Div. Am. Soc. Civ. Engrs., 95, HY4, 1369-1397.

BRYANT, P. J. and WOOD, I. R. 1976 Selective withdrawal from a layered fluid. J. Fluid Mech., ?7, 3, 581-591.

BYE, J. A. T. 1965 Wind Driven Circulation in Unstratified Lakes. Limn. and Ocean., 10, 3, 451-458.

CHENG, R. T. and TUNG, C. 1970 Wind Driven Lake Circulation by the Finite Element Method. Proc. 13th. Conf. Great Lakes Res., .891-903, Intern. Ass. Great Lakes Res., Ann Arbor.

• 351+ COOLEY, P. 1970 Wraysbury Reservoir: prevention of thermal stratification. Proc. 2nd. Conf. Water Qual. Tech., Budapest.

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365 Corrigenda - S. J. Robinson M. Phil. D. I. C. 1979

Page Line Passage reading Should read xiv Table 5-2 restults results xv List of Symbols FM Fm 11 11 F , buoyancy force insert s on glass rod. xvii It insert xd, x co-ordinate of buoy 'deflected' position. 1 11 tl xG, horizontal distance of line of action of FG from hinge point. 38 4 delete His 39 1 surface Surface 11 14 depth-average depth-averaged 40 25 from for 45 units of. ,l are gm cm-3 73 6 reduces reduce 11 7 minimises minimise 93 3 occuring occurring 11 22 futher further 94 2 occuring occurring It 17 11 ti It 95 4 ,' 99 14 a an 106 2 a ppeared appeared 119 1 Eq. 4-22 1 < R < 10 D < 0.6 mm 11 7 Eq. 4-24 10 R ` 100 D0.6 mm 125 5 staisfactory satisfactory 133 27 occuring occurring 137 4 non-metric metric 140 Table 4-2 contd. 'DEFELCTED' 'DEFLECTED' II 11 ' DEFET,F CT ION ' 'DEFLECTION' 143 4 a an 147 4 distrotion distortion 151 2 occuring occurring 154 7 p I Corrigenda Contd. - S. J. Robinson M. Phil. D. I. C. 1D79W

Page Line Passage reading Should read

159 16 scalor scalar Eq. 5-4) 160 should = 0 Eq. 5-5) 163 18 occuring occurring 165 18 •of ten often 166 13 area perimeter 173 End of Para 2 Insert: Note that the velocity scale corresponds to a velocity vectōr (arrow) which has a length equal to the grid spacing and that the length scale is equal to the grid spacing. 1U0 7 dimnsional dimensional 183 7 a an 187 17 of the buoys. of the buoys). 189 19 0.011 0.11 190 13 delete the ?03 18 delete where 11 20 stongly strongly 205 12 majortiy majority

209 18 Qc v Vm Vm 1 1 221 15 Eq. 5-59 hr2 hr2 ►i 19 need needs 224 22 revolution 1 hour revolution/hour 233 9 occuring occurring 236 19 aluminium aluminium 256 1 cirular circular It 2 ymmetrically asymmetrically 259 8 1:16 1:116 260 12 1:1 6 1:116