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sediment suspension during a boundary layer burst but supposed a Gaussian sediment distribution within the core rather than considering the interesting and complicated dynamical interactions in detail. Nielsen (1984) gave an analytical solution to the problem of sand interaction within a Rankine vortex, showing that particles could be suspended on closed trajectories in the core region. He noticed that particles could only achieve this closed trajectory if they happened to be there initially, or were entrained during an undefined vortex growth period. An asymptotically correct formulation developed by Auton (1984) for the force law of discrete elements in weak shear flows at large Reynolds numbers has been used by Thomas et al (1983) as a diagnostic tool to analyse the modes of particle-vortex interactions. The Basset term was not included in this analysis, consistent with the high Reynolds number approximation. Also neglected was the Magnus spin-lift force which is important in such flows as gas-solid pipe flow where the wall impacts are known to induce large spin momentum to the particles (as discussed by Tsuji et al, 1987) . We discuss this further in chapter 6. Below we consider particle interactions with Rankine vortex cores and highlight the sensitivity to initial conditions of the resulting trajectories. We identify the dominant forces on the particle throughout its trajectory, estimate the particle relaxation time to the changing velocity field and assess the 5-3
effective particle fall speed in these flows. The results relating to particles falling into or gathered up by the vortex flow are then used to infer likely interactions occurring in sediment transport from sandwave crests. More specifically, we identify a criterion which defines the limiting fall speed for which a particle can be captured by the vortex. 2 COMPUTATIONAL PROCEDURE The particle equation proposed by Auton (1984) incorporated individual contributions to the body force F from pressure gradient (P) , acceleration (I) , vorticity-lif t (L) and drag (D) , for which the resulting equation (see appendix 9) of particle motion can be written dt cvt cm f» L where v is the particle velocity, u the liquid velocity, u.Vu the (steady) flow acceleration, VT the terminal fall speed (quiescent liquid) , w=v-u the slip speed, w the vorticity vector, p p/L the particle/liquid density, Ap=p p -p L , g gravitational acceleration and t time. Also, the coefficients of virtual mass and lift are assigned their asymptotic quasi-steady values: 0^=0.5, C L=0.5. The equation for the fluid field is described in appendix 9. We solved this equation numerically for the prescribed vortex flow to obtain the acceleration of a particle at discretised time 5-4
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FLUID-PARTICLE TRANSPORT DYNAMICS O
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SYNOPSIS The local dynamics of sand
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TABLE OF CONTENTS LIST OF FIGURES C
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CHAPTER 3: JETTING EXPERIMENTS Summ
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2 Numerical scheme 6-5 3 Results 6-
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LIST OF FIGURES Chapter 1 Figure 1.
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U=830mm/s. Figure 16. Thomas et al
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in the boundary layer. a) The forma
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Chapter 3 Figure 1. After Alien (19
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at various values of U/V from 2.7 t
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y Coanda-flapping of the free shear
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of a sandwave crest in the estuary
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Figure 7h-i. The temporal fluctuati
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inclined downwards whereas the time
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starting position (X,Y) at zero ini
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d Sand grain diameter (m) Fi Number
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G Vortex core growth rate (tn/s) ki
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1 INTRODUCTION Sand forms the botto
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where U is the depth-averaged veloc
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The wavelengths of dunes primarily
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detached and then confined within a
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The vortex radius rn and apparent w
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Recent work by Nezu & Nakagawa (199
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found in the outer streaks. 4 MODEL
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(1987), for example, modelled gas-s
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Profiles of shear stress and turbul
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6ft v "' ' Re In the limit of large
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that the particle radius was much s
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and travels downstream at speed }*U
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capture by transient large eddies s
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Chapter 1 Figure 3. Alien (1984). L
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Chapter 1 10 8 e 4 I 08 •- 02 Yal
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Chapter 1 bw speed fluid Figure 11.
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Chapter 1 Figure 16. Thomas et al (
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Chapter 1 StCMl STRESS SCALE « K /
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Chapter 1 Haiht Vdodty Figure 22. S
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Chapter 1 Time = 1.2 2.4 ' ' :•
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1 INTRODUCTION AND DESIGN CONSIDERA
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Initial costing for a flume of the
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flanges and a maximum flow rate of
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could be wound above or below the l
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4 EXPERIMENTAL SETUP AND METHODS He
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4.2 Flow visualisation and model pa
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any divergence from the beam and to
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even flow over 0.45m width and up t
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Chapter 2 Opt 1 h(m) 0.01 Max D(m)
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Chapter 2 Part of side section Putt
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Chapter 2 Suction tank Bypass Disch
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Chapter 2 QUncorrectedflow £ Corre
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Chapter 2 60 r 501 40? 30r 20 r lOr
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1_ INTRODUCTION The main features o
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2 EXPERIMENTAL METHODS A detailed d
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the relationship is then simply ^ -
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number about 0.14. They were taken
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along the stoss slope. Figures 8 an
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3.3a) Jetting trajectories when U/V
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errors increase substantially due t
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likely distributions. 4 DISCUSSION
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Our suggestion here is akin to that
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not accommodate the differing migra
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considered by proponents of numeric
Page 123 and 124: Chapter 3 C,;: Figure 3a-j. The pla
Page 125 and 126: Q CO 23 Figure 6. Distribution of p
Page 127 and 128: Chapter 3 X(cm) 1 2 3 4 5 6 7 8 9 1
Page 129 and 130: Chapter 3 U(m/s) V(m/s) U/V M1+M2 M
Page 131 and 132: O I r+ 21 CO 2 5 8 11 14 17 20 23 >
Page 133 and 134: 8 11 14 17 20 23 25 + Position Figu
Page 135 and 136: Chapter 3 U/V= B5.7 Q9.3 BlO.7^12.8
Page 137 and 138: Chapter 3 Band 1 2 3 4 U(m/s) 0.2 0
Page 139 and 140: 1 INTRODUCTION In chapter 3 we cons
Page 141 and 142: velocity for a maximum of 40% of th
Page 143 and 144: layer vortices extending over the d
Page 145 and 146: 2.2 Methods for measuring Coanda-fl
Page 147 and 148: Section 3.4 considers suspension fr
Page 149 and 150: This same flow structure has also b
Page 151 and 152: 3.2b) Effect of varying inter-crest
Page 153 and 154: those with fall speed 0.060m/s for
Page 155 and 156: 3.4 Particle capture from stoss slo
Page 157 and 158: are discussed in the context of est
Page 159 and 160: jetted grains settling to the lee s
Page 161 and 162: Chapter 4 ENTRAINED FLOW Figure 1.
Page 163 and 164: Figure 3a-j. The plates show partic
Page 165 and 166: Chapter 4 Percentage occurrence •
Page 167 and 168: Chapter 4 Interval (s) 60 80 100 12
Page 169 and 170: Chapter 4 0.024m Conc=8% Figure 12.
Page 171 and 172: Chapter 4 Figure 16. A cloud of par
Page 173: 1 INTRODUCTION We use a Rankine vor
Page 177 and 178: particle velocity Vx=0. The initial
Page 179 and 180: the vortex axis. Vortex circulation
Page 181 and 182: dimensional fall times T^ for diffe
Page 183 and 184: horizontal component lift force (fi
Page 185 and 186: corresponds to the trajectory trave
Page 187 and 188: stepping the calculation in VT . Do
Page 189 and 190: averaged exclusion of the coherent
Page 191 and 192: C from figure 11 (ie 2.2) then we h
Page 193 and 194: PflRTICLE DENSITY=1000.000 Kg/n3 LI
Page 195 and 196: Chapter 5 ft) J k I Figure 4. Varyi
Page 197 and 198: Chapter 5 Relative Fall Time va Sta
Page 199 and 200: Chapter 5 Figure 7a. Trajectories o
Page 201 and 202: Chapter 5 Velocity X —Inertia X L
Page 203 and 204: Chapter 5 Velocity X —Velocity Y
Page 205 and 206: Chapter 5 Velocity Y Inertia Y Lift
Page 207 and 208: Chapter 5 DENSITY PARTICLE=2850.000
Page 209 and 210: Chapter 5 Rad = 0.006 Rad=0.003 Rad
Page 211 and 212: Chapter 5 H3 Y=0 Oy=i m Y=2 D-y=3
Page 213 and 214: CHAPTER 6: NUMERICAL SIMULATION OF
Page 215 and 216: next. If we now adopt the classical
Page 217 and 218: st = (4) Where p p was the particle
Page 219 and 220: 3 RESULTS 3.1 Specified 6 and V Fig
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Page 223 and 224: A0fl0 = 1.25V-0.33G-0.005 (7) and f
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Conversely, the upper line of the p
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equations. Such a discrete vortex m
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Figure 1. Schematic of the shear la
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Chapter 6 J G H K M Figure 3. Seven
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Chapter 6 1 T V=0.161 0.1 -- V41185
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Chapter 6 - DataV=0.0335, G=0.01
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CHAPTER 7: RECAPITULATION AND RECOM
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were shown to compare well with pre
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We have shown Coanda-flapping makes
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figure 5, was critically dependant
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stoss slope suspension studies of c
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Figure la-g. The seven trajectory m
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Chapter 7
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Chapter 7 0.0351 0.03 " 03 0.025 0.
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Flow 16 (1) pp 19-34. Bernal LP (19
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Mechanics of Sediment Transport. Is
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turbulence near a smooth wall in a
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moderate Reynolds number shear laye
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Spalding (1961). A single formula f
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USA. 1-12
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APPENDIX 1: BOUNDARY LAYER STUDIES
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(1967) used hydrogen bubble visuali
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The burst-sweep dynamics study of U
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Horseshoe vortex Vortex filament Ho
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APPENDIX 2: FAST9003.HR(RW)/NHT - P
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of the vorticity shed into the near
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transport dynamics of multiphase fl
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APPENDIX 3: FUNDAMENTAL NAVIER-STOK
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APPENDIX 4: CLOSURE METHODS 1 ZERO
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where cr k and CD are empirical con
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K = CDe^ 2 p 2 \U2 -U1 \=e 1e 2 c (
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APPENDIX 6: K-e MODEL OF JOHNS ET A
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They then defined a hydrodynamic co
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with respect to the rest of the vor
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The vorticity equation can be split
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around the crest and the motion of
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micrometers, with mean of 80 microm
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i——COMPARISON WITH PREVIOUS WOR
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width speed (metres) ^(cm/s) 11 Fig
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U (33) = 1.16ms-' Depth = 2.65 m Fi
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APPENDIX 9 : DERIVATION OF FORCES O
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APPENDIX 10 : MODEL CODE FOR CALCUL
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APPENDIX 11: MODEL CODE FOR GROWING
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105 if (scancode .eq. 57) stop GOTO
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