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APPENDIX 7: DISCRETE VORTEX MODELLING METHODS 1 THE POINT VORTEX APPROXIMATION PROBLEM The DVM relies on concentrating vorticity at points and allowing these points to interact with time using the Biot-Savart law (Batchelor, 1967) . This law gives the velocity induced by one vortex on another as proportional to the vorticity and inversely proportional to their separation. Close to the vortex these induced velocities are large thus requiring a very small time interval to prevent unrealistic inter-vortex accelerations. This problem has been solved using a variety of methods as detailed below. 1.1 Vortex Amalgamation The method of vortex amalgamation is often used to stop mutual vortex orbit, to limit high mutually induced velocities, to limit computational time and to simulate a naturally occurring merging process. The process is again ad hoc since there are no rules to govern the amalgamation action, but one common method is to place the new vortex at the centre of the vorticity of the pairing couple and conserve the total circulation. 1.2 Discrete Vortex in Cell Method Other pseudo amalgamation methods are based on the temporary amalgamation of clusters of vortices that are close to each other A7-1
with respect to the rest of the vorticity field. The Discrete Vortex In Cell method of Tiemroth (1986) replaced a large collection of vortices by a single vortex, relying on the dominance of the first order term in the interaction between members of the cluster and vortices away from the cluster. In this situation the inter-cluster spacing should be large with respect to the intra-cluster inter-vortex spacing. 1.3 Dipole in Cell Method Ghoniem et al (1986) used the dipole in cell method. Vortices were combined, if they appeared in the same square on a superimposed grid, by conservation of their first moments into two new vortices dependent upon the sign of their vorticity. The far field was then computed from this dipole whilst the near field was computed directly- Again, this was a time saving device used when many vortices had formed into large scale coherent structures. Other cut-off methods exist, such as Anderson (1986) and Greengard & Rochlin (1987), where the far field interactions were approximated using Taylor series expansions and finite difference meshes. 1.4 Rediscretisation The rediscretisation method of Fink & Soh (1974) overcame the problem of vortex proximity by rediscretising the sheet of A7-2
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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
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Chapter 3 C,;: Figure 3a-j. The pla
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Q CO 23 Figure 6. Distribution of p
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Chapter 3 X(cm) 1 2 3 4 5 6 7 8 9 1
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Chapter 3 U(m/s) V(m/s) U/V M1+M2 M
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O I r+ 21 CO 2 5 8 11 14 17 20 23 >
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8 11 14 17 20 23 25 + Position Figu
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Chapter 3 U/V= B5.7 Q9.3 BlO.7^12.8
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Chapter 3 Band 1 2 3 4 U(m/s) 0.2 0
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1 INTRODUCTION In chapter 3 we cons
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velocity for a maximum of 40% of th
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layer vortices extending over the d
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2.2 Methods for measuring Coanda-fl
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Section 3.4 considers suspension fr
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This same flow structure has also b
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3.2b) Effect of varying inter-crest
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those with fall speed 0.060m/s for
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3.4 Particle capture from stoss slo
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are discussed in the context of est
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jetted grains settling to the lee s
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Chapter 4 ENTRAINED FLOW Figure 1.
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Figure 3a-j. The plates show partic
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Chapter 4 Percentage occurrence •
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Chapter 4 Interval (s) 60 80 100 12
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Chapter 4 0.024m Conc=8% Figure 12.
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Chapter 4 Figure 16. A cloud of par
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1 INTRODUCTION We use a Rankine vor
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effective particle fall speed in th
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particle velocity Vx=0. The initial
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the vortex axis. Vortex circulation
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dimensional fall times T^ for diffe
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horizontal component lift force (fi
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corresponds to the trajectory trave
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stepping the calculation in VT . Do
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averaged exclusion of the coherent
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C from figure 11 (ie 2.2) then we h
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PflRTICLE DENSITY=1000.000 Kg/n3 LI
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Chapter 5 ft) J k I Figure 4. Varyi
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Chapter 5 Relative Fall Time va Sta
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Chapter 5 Figure 7a. Trajectories o
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Chapter 5 Velocity X —Inertia X L
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Chapter 5 Velocity X —Velocity Y
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Chapter 5 Velocity Y Inertia Y Lift
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Chapter 5 DENSITY PARTICLE=2850.000
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Chapter 5 Rad = 0.006 Rad=0.003 Rad
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Chapter 5 H3 Y=0 Oy=i m Y=2 D-y=3
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CHAPTER 6: NUMERICAL SIMULATION OF
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next. If we now adopt the classical
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st = (4) Where p p was the particle
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3 RESULTS 3.1 Specified 6 and V Fig
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the blue squares to double loops an
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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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Page 241 and 242: We have shown Coanda-flapping makes
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Page 247 and 248: Figure la-g. The seven trajectory m
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Page 251 and 252: Chapter 7 0.0351 0.03 " 03 0.025 0.
Page 253 and 254: Flow 16 (1) pp 19-34. Bernal LP (19
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Page 261 and 262: Spalding (1961). A single formula f
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Page 265 and 266: APPENDIX 1: BOUNDARY LAYER STUDIES
Page 267 and 268: (1967) used hydrogen bubble visuali
Page 269 and 270: The burst-sweep dynamics study of U
Page 271 and 272: Horseshoe vortex Vortex filament Ho
Page 273 and 274: APPENDIX 2: FAST9003.HR(RW)/NHT - P
Page 275 and 276: of the vorticity shed into the near
Page 277 and 278: transport dynamics of multiphase fl
Page 279 and 280: APPENDIX 3: FUNDAMENTAL NAVIER-STOK
Page 281 and 282: APPENDIX 4: CLOSURE METHODS 1 ZERO
Page 283 and 284: where cr k and CD are empirical con
Page 285 and 286: K = CDe^ 2 p 2 \U2 -U1 \=e 1e 2 c (
Page 287 and 288: APPENDIX 6: K-e MODEL OF JOHNS ET A
Page 289: They then defined a hydrodynamic co
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Page 299 and 300: i——COMPARISON WITH PREVIOUS WOR
Page 301 and 302: width speed (metres) ^(cm/s) 11 Fig
Page 303 and 304: U (33) = 1.16ms-' Depth = 2.65 m Fi
Page 305 and 306: APPENDIX 9 : DERIVATION OF FORCES O
Page 307 and 308: APPENDIX 10 : MODEL CODE FOR CALCUL
Page 309 and 310: APPENDIX 11: MODEL CODE FOR GROWING
Page 311: 105 if (scancode .eq. 57) stop GOTO
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