Page 5 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONTENTS
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7.6.2
Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . 211
7.6.3
7.6.4 Time-step calculation .. ... .. .. .. .. .. .. ... .. .. .. 210
Steady and transient solution methods . . . . . . . . . . . . . . . . 212
7.7 Artificial Compressibility Scheme . . . . . . . . . . . . . . . . . . . . . . . 213
7.8 Nusselt Number, Drag and Stream Function . . . . . . . . . . . . . . . . . 213
7.8.1 Nusselt number . .. .. ... .. .. .. .. .. .. ... .. .. .. 214
7.8.2 Drag calculation .. .. ... .. .. .. .. .. .. ... .. .. .. 215
7.8.3 Stream function . .. .. ... .. .. .. .. .. .. ... .. .. .. 216
7.9 Mesh Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.10 Laminar Isothermal Flow .. .. ... .. .. .. .. .. .. ... .. .. .. 218
7.10.1 Geometry, boundary and initial conditions . . . . . . . . . . . . . . 218
7.10.2 Solution . .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 219
7.11 Laminar Non-isothermal Flow . ... .. .. .. .. .. .. ... .. .. .. 220
7.11.1 Forced convection heat transfer . . . . . . . . . . . . . . . . . . . . 220
7.11.2 Buoyancy-driven convection heat transfer . . . . . . . . . . . . . . 223
7.11.3 Mixed convection heat transfer . . . . . . . . . . . . . . . . . . . . 227
7.12 Introduction to Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.12.1 Solution procedure and result .. .. .. .. .. .. ... .. .. .. 233
7.13 Extension to Axisymmetric Problems . . . . . . . . . . . . . . . . . . . . . 234
7.14 Summary .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 235
7.15 Exercise . .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 236
Bibliography . .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 236

8 Convection in Porous Media 240
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.2 Generalized Porous Medium Flow Approach . . . . . . . . . . . . . . . . . 243
8.2.1 Non-dimensional scales ... .. .. .. .. .. .. ... .. .. .. 245
8.2.2 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.3 Discretization Procedure .. .. ... .. .. .. .. .. .. ... .. .. .. 247
8.3.1 Temporal discretization ... .. .. .. .. .. .. ... .. .. .. 247
8.3.2 Spatial discretization .. ... .. .. .. .. .. .. ... .. .. .. 249
8.3.3 Semi- and quasi-implicit forms . .. .. .. .. .. ... .. .. .. 252
8.4 Non-isothermal Flows . .. .. ... .. .. .. .. .. .. ... .. .. .. 254
8.5 Forced Convection . .. .. .. ... .. .. .. .. .. .. ... .. .. .. 255
8.6 Natural Convection. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 256
8.6.1 Constant porosity medium .. .. .. .. .. .. .. ... .. .. .. 258
8.7 Summary .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 262
8.8 Exercise. .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 262
Bibliography . .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 262

9 Some Examples of Fluid Flow and Heat Transfer Problems 265
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.2 Isothermal Flow Problems . .. ... .. .. .. .. .. .. ... .. .. .. 265
9.2.1 Steady state problems . ... .. .. .. .. .. .. ... .. .. .. 265
9.2.2 Transient flow . .. .. ... .. .. .. .. .. .. ... .. .. .. 277