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

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24 Model equations at each point also requires that n� ϕi = 1. (2.52) i=1 In an n-material problem, the multi-material approach requires that n − 1 advection equations are solved, to describe the transport of the volume fraction of all but one of the materials. The volume fraction of the remaining material can be derived from the other volume fractions by n−1 ϕn = 1 − The transport of the i th volume fraction is given by where the volume fraction field at time zero must be specified. � ϕi. (2.53) i=1 ∂ϕi ∂t + u · ∇ϕi = 0, (2.54)
Chapter 3 Numerical discretisation 3.1 Introduction & some definitions This chapter covers the numerical discretisation of the model equations given in the previous chapter as they are assembled and solved in Fluidity. For a more general introduction into finite element theory we refer to a.o. Elman et al. [2005] and Gresho and Chan [1988]. In this chapter we define the domain we are solving over as Ω. The boundary to Ω is defined as ∂Ω and can be split into different sections, e.g. ∂Ω = ∂Ω N ∪ ∂Ω D where ∂Ω N is that part of the boundary over which Neumann conditions are applied and ∂Ω D is that part of the boundary over which Dirichlet conditions are applied. For clarity a subscript showing the field in question will be given on the N and D characters. The unit vector n is always assumed to be the outward facing normal vector to the domain. In the following this notation is used to describe both the normal vector at the boundary and between elements in the interior. 3.2 Spatial discretisation of the advection-diffusion equation The advection-diffusion equation for a scalar tracer c, is given in conservative form by ∂c + ∇ · (uc) − ∇ · µ · ∇c = 0. (3.1) ∂t We now define a partition of the boundary such that ∂Ω = ∂Ω N ∪ ∂Ω D , and impose boundary conditions on c: 3.2.1 Continuous Galerkin discretisation n · µ · ∇c = gNc on ∂Ω Nc , (3.2) c = gDc on ∂Ω Dc . (3.3) The Continuous Galerkin method (often abbreviated to CG, not to be confused with Conjugate Gradients) method, is a widely used finite element method in which the solution fields are constrained to be continuous between elements. The fields are only assumed to be C 0 continuous, that is to say there is no assumption that the gradient of a field is continuous over element boundaries. 25
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 and 12: List of Figures 2.1 Schematic of co
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 35 and 36: 2.4 Extensions, assumptions and der
Page 37: 2.4 Extensions, assumptions and der
Page 41 and 42: 3.2 Spatial discretisation of the a
Page 63 and 64: 3.3 The time loop 49 The coupled li
Page 65 and 66: 3.4 Time discretisation of the adve
Page 67 and 68: 3.6 Pressure equation for incompres
Page 69 and 70: 3.7 Velocity and pressure element p
Page 71 and 72: 3.8 Balance pressure 57 3.8 Balance
Page 73 and 74: 3.10 Linear solvers 59 When reachin
Page 75 and 76: 3.11 Algorithm for detectors (Lagra
Page 77 and 78: Chapter 4 Parameterisations Althoug
Page 79 and 80: 4.1 Generic length scale turbulence
Page 81 and 82: 4.3 Large-Eddy Simulation (LES) 67
Page 83 and 84: 4.3 Large-Eddy Simulation (LES) 69
Page 85 and 86: Chapter 5 Embedded models The param
Page 87 and 88: 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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