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

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x CONTENTS A.3.2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 A.3.3 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 A.3.4 flml options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.3.5 Generating pdf and html output . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.3.6 Representing source code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.3.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.3.8 Mathematical notation conventions . . . . . . . . . . . . . . . . . . . . . . . . 212 B Mathematical notation 215 C Useful numbers 217 D Dimensional analysis 219 D.1 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 D.2 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 E The Fluidity Python state interface 223 E.1 System requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 E.2 Data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 E.2.1 Field objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 E.2.2 State objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 E.3 Predefined data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 E.4 Importing modules and accessing external data . . . . . . . . . . . . . . . . . . . . . . 225 E.5 The persistent dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 E.6 Debugging with an interactive Python session . . . . . . . . . . . . . . . . . . . . . . . 225 E.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 F External libraries 227 F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 F.2 List of external libraries and software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 F.3 Installing required libraries on Debian or Ubuntu . . . . . . . . . . . . . . . . . . . . . 228 F.4 Manual install of external libraries and software . . . . . . . . . . . . . . . . . . . . . . 228 F.4.1 Build environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 F.4.2 Compilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 F.4.3 Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 F.4.4 Numerical Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 F.4.5 VTK and supporting software . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 F.4.6 Supporting Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 G Troubleshooting 237 G.1 Backtraces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Index 239
List of Figures 2.1 Schematic of coordinates in a frame rotating with a sphere. The rotation is about a vector pointing from South to North pole. A point on the surface of the sphere x and its perpendicular distance from the axes of rotation x⊥ are shown. The latitude of x is given by ϕ and the unit vectors i, j and k represent a local coordinate axes at point x in the rotating frame: i points Eastwards, j points Northwards and k points in the radial outwards direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 One-dimensional (a, b) and two-dimensional (c, d) schematics of piecewise linear (a, c) and piecewise quadratic (b, d) continuous shape functions. The shape function has value 1 at node A descending to 0 at all surrounding nodes. The number of nodes per element, e, depends on the polynomial order while the support, s, extends to all the elements surrounding node A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Pure advection of a 1D top hat at CFL number 1/8 after 80 timesteps using a continuous Galerkin discretisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Pure advection of a 1D top hat at CFL number 1/8 after 80 timesteps using a continuous Galerkin discretisation with streamline-upwind stabilisation. . . . . . . . . . . . . 31 3.4 Pure advection of a 1D top hat at CFL number 1/8 after 80 timesteps using a continuous Galerkin discretisation with streamline-upwind Petrov-Galerkin stabilisation. . . 31 3.5 One-dimensional (a, b) and two-dimensional (c, d) schematics of piecewise linear (a, c) and piecewise quadratic (b, d) discontinuous shape functions. The shape function has value 1 at node A descending to 0 at all surrounding nodes. The number of nodes per element, e, depends on the polynomial order while the support, s, covers the same area as the element, e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.6 One-dimensional (a) and two-dimensional (b) schematics of piecewise constant, element centred shape functions. The shape function has value 1 at node A and across the element, e, descending to 0 at the element boundaries. As with other discontinuous shape functions, the support, s, coincides with the element, e. . . . . . . . . . . . . . . 40 3.7 Comparison between (a) a two-dimensional finite volume simplex mesh and (b) the equivalent control volume dual mesh (solid lines) constructed around a piecewise linear continuous finite element parent mesh (dashed lines). In the finite volume mesh the nodes (e.g. A) are element centred whereas in the control volume dual mesh the nodes are vertex based. In 2D the control volumes are constructed around A by connecting the centroids of the neighbouring triangles to the edge midpoints. See Figure 3.8 for the equivalent three-dimensional construction. . . . . . . . . . . . . . . . . . . . 40 3.8 The six dual control volume mesh faces within a piecewise linear tetrahedral parent mesh element. Each face is constructed by connecting the element centroid, the face centroids and the edge midpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.9 One-dimensional (a, b) and two-dimensional (c, d) schematics of piecewise constant control volume shape functions and dual meshes based on the parent (dashed lines) linear (a, c) and quadratic (b, d) finite element meshes. The shape function has value 1 at node A descending to 0 at the control volume boundaries. The support, s, coincides with the volume, v. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 xi
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: CONTENTS ix 9.4.3 Stat file diagnos
Page 13 and 14: LIST OF FIGURES xiii 10.8 Schematic
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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3.2 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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