Chapter 8 Configuring Fluidity - The Applied Modelling and ...

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xii LIST OF FIGURES 3.10 Calculation of the upwind value, cuk , on an unstructured simplex mesh (a) internally and (b) on a boundary. The control volume mesh is shown (solid lines) around the nodes (black circles) which are co-located with the solution nodes of the parent piecewise linear continuous finite element mesh (dashed lines). An initial estimate of the upwind value, c∗ uk , is found by interpolation within the upwind parent element. The point-wise gradient between this estimate and the donor node, cck , is then extrapolated the same distance between the donor and downwind, cdk , to the upwind value, cuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.11 The Sweby (a) and ULTIMATE (b) limiters (shaded regions) represented on a normalised variable diagram (NVD). For comparison the trapezoidal face value scheme is plotted as a dashed line in both diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.12 The HyperC face value scheme represented on a normalised variable diagram (NVD). 45 3.13 The UltraC face value scheme represented on (a) a normalised variable diagram (NVD) and (b) a modified normalised variable diagram. For comparison (a) also shows the HyperC face value scheme as a dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.14 The coupled limiter for the field cI represented by the grey shaded area on a normalised variable diagram (NVD). Labels in the upper left blue region refer to the case when the difference between the parent � sum downwind and upwind values has the � � � � same sign as the limited field, sign c I − c I = sign . Similarly, labels dk uk � c I dk − cI uk in the lower right � yellow region refer to the case when the signs of the slopes are � � � � � opposite, sign c I − c I �= sign . The regions are separated by the up- dk uk � I−1 ck cI dk − cIuk � I−1 winding line, ¯ ĉI f = ¯cI ck + ¯c − ¯c f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Outline of the principal steps in the nonlinear iteration sequence. . . . . . . . . . . . . 3.16 A sketch representing the bisection method used in combination with a second order 48 50 Runge-Kutta algorithm to advect the Lagrangian detectors with the flow. . . . . . . . . 62 5.1 The fluxes between the biological tracers. Grazing refers to the feeding activity of zooplankton on phytoplankton and detritus. . . . . . . . . . . . . . . . . . . . . . . . . 71 8.1 A Diamond screenshot showing the /geometry/dimension option. Note that the option path is displayed at the bottom of the diamond window . . . . . . . . . . . . . 94 8.2 A Diamond screenshot showing the /geometry/mesh::Coordinate option. Note that the name is shown in brackets in the main Diamond window but after double colons in the path in the bottom bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.3 Periodic unit square with surface IDs 1-4 shown. . . . . . . . . . . . . . . . . . . . . . . 104 9.1 Configuration of a diagnostic field using a diagnostic algortithm in Diamond. Here a pressure gradient diagnostic is defined. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.2 Visualisation of a heat flux diagnostic in a 2D cavity convection simulation using statplot.155 9.3 Example configuration of the stat file for .../vector field(Velocity). . . . . . 156 10.1 Initial condition and numerical solutions after 100 s, for the 1D top hat tracer advection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.2 Conservation and bounds checking using statplot. . . . . . . . . . . . . . . . . . . . . . 170 10.3 Lock-exchange temperature distribution (colour) with meshes, over time (t) . . . . . . 173 10.4 Distance along the domain (X) and Froude number (F r) for the no-slip and free-slip fronts in the lock-exchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 10.5 Time evolution of fraction of domain that contains fluid in three temperature classes. . 175 10.6 Diagnostic fields from the lid-driven cavity problem at steady state at 1/128 resolution. Left: the streamfunction. Right: the vorticity. The contour levels are taken from those given by Botella and Peyret [1998] in their tables 7 and 8. . . . . . . . . . . . . . . . . . 176 10.7 Convergence of the eight error metrics computed for the lid-driven cavity problem with mesh spacing. The eight metrics are described in the text. . . . . . . . . . . . . . . 177
LIST OF FIGURES xiii 10.8 Schematic of the domain for the three-dimensional flow past a backward facing step problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.9 Schematic of the domain for the two-dimensional flow past a backward facing step problem, showing adaptive mesh refinement. . . . . . . . . . . . . . . . . . . . . . . . . 178 10.10Snapshots of the velocity magnitude from the 2D run at times 5, 10 and 50 time units (top to bottom) into the simulation from the smaller benchmark run. The evolution of the dynamics to steady state can be seen, in particular the downstream movement of the streamline reattachment point (indicated by contours of U = 0). . . . . . . . . . . . 180 10.11Streamwise velocity profiles from the 2D run at x/h = 0, 2, 4 and 6 downstream of the step, where h = 1 is the step height. The converged solution is in red. The recirculation region is indicated by negative velocity in figure (b). It is clear that the reattachment point is between x/h = 2 and x/h = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.12From top to bottom: vertical plane cuts through the 3D domain showing the velocity magnitude at times 5, 25 and 50 time units. The evolution of the dynamics to steady state can be seen, in particular the downstream movement of the streamline reattachment point (indicated by contours of U = 0). . . . . . . . . . . . . . . . . . . . . . . . . 181 10.13Streamwise velocity profiles from the 3d run at x/h = 4, 6, 10 and 19 downstream of the step, where h = 1 is the step height. The converged solution is in red. The recirculation region is indicated by the negative velocity in figure (b). It is clear that the flow reattaches between x/h > 6 and 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10.14Streamlines and surface mesh in the flow past the sphere example. Top-left to bottomright show results from Reynolds numbers Re = 1, 10, 100, 1000. . . . . . . . . . . . . . 183 10.15Details of the mesh and flow at Re = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.16Comparison between the numerically calculated drag coefficients (CD, circles) and the correlation (10.2) (solid line) for Re = 1, 10, 100, 1000. . . . . . . . . . . . . . . . . . . . 184 10.17Velocity forcing term and analytic solutions for velocity and pressure for the rotating periodic channel test case. Note that each of these quantities is constant in the x direction.186 10.18Error in the pressure and velocity solutions for the rotating channel as a function of resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.19(a) Initial set-up of the water volume fraction, α1, and the velocity and pressure boundary conditions for the two-dimensional water column collapse validation problem [Zhou et al., 1999]. The presence of water is indicated as a grey region and the interface to air is delineated by contours of the volume fraction at 0.025, 0.5 and 0.975. The locations of the pressure (P2) and water depth gauges (H1, H2) are also indicated. (b) The adapted mesh used to represent the initial conditions. . . . . . . . . . . . . . . 189 10.20The evolution of the water volume fraction, α1, over several timesteps. The presence of water, α1 = 1, is indicated as a grey region and the interface to air, α1 = 0, is delineated by contours at α1 = 0.025, 0.5 and 0.975. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.21The evolution of the adaptive mesh over the same timesteps displayed in Figure 10.20. The mesh can be seen to closely follow the interface between the water and air. . . . . 191 10.22Comparison between the experimental (circles) and numerical water gauge data at H1 (i) and H2 (ii), x1 = 2.725m and 2.228m respectively. The letters along the top of the graphs indicate the times corresponding to Figure 10.20(a–h). Experimental data taken from Zhou et al. [1999] through Park et al. [2009]. . . . . . . . . . . . . . . . . . . . . . . 192 10.23Comparison between the experimental (circles) and numerical pressure gauge data at P2, x1 = 3.22m, x2 = 0.16m. The letters along the top of the graphs indicate the times corresponding to Figure 10.20(a–h). Experimental data taken from Zhou et al. [1999] through Park et al. [2009]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.24A vertical slice throught the domain showing the initial temperature stratification. The domain is a cylinder of radius 250 km and height 1 km. . . . . . . . . . . . . . . . . . . 194 10.25The temperature cross-section at a depth of 40m. . . . . . . . . . . . . . . . . . . . . . . 195 10.26Plots of the tidal harmonic amplitudes in the Mediterranean Sea from ICOM and the high resolution model of Tsimplis et al. [1995]. . . . . . . . . . . . . . . . . . . . . . . . 197
Page 1: Fluidity�Manual Applied Modelling
Page 5 and 6: Contents 1 Getting started 1 1.1 In
Page 7 and 8: CONTENTS vii 7.3 The Gmsh format .
Page 9 and 10: CONTENTS ix 9.4.3 Stat file diagnos
Page 11: List of Figures 2.1 Schematic of co
Page 15 and 16: Chapter 1 Getting started 1.1 Intro
Page 17 and 18: 1.2 Obtaining Fluidity 3 The third,
Page 19 and 20: 1.3 Building Fluidity 5 1.2.3.4 Oth
Page 21 and 22: 1.3 Building Fluidity 7 1.3.2.2 Spe
Page 23 and 24: 1.5 Running diamond 9 Signal handli
Page 25 and 26: Chapter 2 Model equations 2.1 How t
Page 27 and 28: 2.3 Fluid equations 13 2.2.2.2 Neum
Page 29 and 30: 2.3 Fluid equations 15 2.3.2.2 Comp
Page 31 and 32: 2.3 Fluid equations 17 2.3.4 Moment
Page 33 and 34: 2.4 Extensions, assumptions and der
Page 39 and 40: Chapter 3 Numerical discretisation
Page 41 and 42: 3.2 Spatial discretisation of the a
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3.3 The time loop 49 The coupled li
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3.4 Time discretisation of the adve
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3.6 Pressure equation for incompres
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3.7 Velocity and pressure element p
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3.8 Balance pressure 57 3.8 Balance
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3.10 Linear solvers 59 When reachin
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3.11 Algorithm for detectors (Lagra
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Chapter 4 Parameterisations Althoug
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4.1 Generic length scale turbulence
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4.3 Large-Eddy Simulation (LES) 67
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4.3 Large-Eddy Simulation (LES) 69
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Chapter 5 Embedded models The param
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5.1 Biology 73 where kN is the half
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Chapter 6 Adaptive remeshing 6.1 Mo
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6.4 Metric formation 77 By contrast
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6.6 Related topics 79 There are two
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Chapter 7 Mesh generation This chap
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7.2 The triangle format 83 1 Rema
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7.3 The Gmsh format 85 three parts:
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7.6 Decomposing meshes for parallel
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7.9 Non-Fluidity tools 89 7.7 Pseud
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7.9 Non-Fluidity tools 91 7.9.3 Imp
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Chapter 8 Configuring Fluidity 8.1
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8.3 The options tree 95 Figure 8.2:
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8.3 The options tree 97 8.3.4 IO Th
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8.3 The options tree 99 Checkpoint
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8.3 The options tree 101 8.3.5.3 Fi
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8.4 Meshes 103 if t < 4000.0: omega
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8.5 Material/Phase 105 8.4.2.4 Extr
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8.6 Fields 107 dent field). This is
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8.7 Advected quantities: momentum a
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8.7 Advected quantities: momentum a
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8.9 Solution of linear systems 113
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8.10 Equation of State (EoS) 115 8.
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8.11 Subgridscale Parameterisations
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8.12 Boundary conditions 119 8.12.3
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8.12 Boundary conditions 121 • ..
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8.16 Large scale low aspect ratio o
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8.16 Large scale low aspect ratio o
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8.17 Geophysical fluid dynamics pro
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8.18 Mesh adaptivity 129 An Element
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8.19 Multiple material/phase models
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8.19 Multiple material/phase models
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8.20 Compressible fluid model 135 t
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Chapter 9 Visualisation and Diagnos
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9.2 Online diagnostics 139 Gravitat
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9.2 Online diagnostics 141 Absolute
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9.2 Online diagnostics 143 Figure 9
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9.3 Offline diagnostics 145 Program
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9.3 Offline diagnostics 147 Additio
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9.3 Offline diagnostics 149 9.3.3.1
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9.3 Offline diagnostics 151 Script
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9.3 Offline diagnostics 153 TARGET
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9.3 Offline diagnostics 155 Figure
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9.4 The stat file 157 It is also po
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9.4 The stat file 159 9.4.3 Stat fi
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Temperature min fluid .../stat/incl
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9.4 The stat file 163 9.4.3.1 Surfa
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9.4 The stat file 165 ...
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Chapter 10 Examples 10.1 Introducti
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10.2 One dimensional advection 169
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10.3 The lock-exchange 171 10.2.4 E
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10.3 The lock-exchange 173 basic se
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10.4 Lid-driven cavity 175 domain f
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10.5 Backward facing step 177 Figur
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10.5 Backward facing step 179 No-no
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10.5 Backward facing step 181 Evolu
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10.6 Flow past a sphere: drag calcu
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10.7 Rotating periodic channel 185
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10.8 Water column collapse 187 Exam
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10.8 Water column collapse 189 (a)
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10.8 Water column collapse 191 (a)
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10.9 The restratification following
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10.10 Tides in the Mediterranean Se
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Bibliography M. T. Ainsworth and J.
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C. J. Cotter, D. A. Ham, and C. C.
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A. Birol Kara, Harley E. Hurlburt,
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J. R. Shewchuk. An introduction to
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Appendix A About this manual A.1 In
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A.3 Style guide 211 The options pro
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A.3 Style guide 213 A.3.8.3 Derivat
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Appendix B Mathematical notation Th
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Appendix C Useful numbers The table
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Appendix D Dimensional analysis D.1
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D.2 Dimensionless parameters 221 Na
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Appendix E The Fluidity Python stat
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E.6 Debugging with an interactive P
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Appendix F External libraries F.1 I
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F.4 Manual install of external libr
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Appendix G Troubleshooting G.1 Back
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Index absorption term . . . . . . .
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geostrophic balance . . . . . . . .
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