Computational Fluid Dynamics for Incompressible Flows

1. Overview of CFD. 2. Governing Equations and Classification of PDE. 3. Finite Difference Method -- Fundamentals. 4. Finite Difference Methods -- Application. 5. Finite Volume Method. 6. Solution of Incompressible Navier-Stokes Equations. 7. Finite Volume Method for Complex Geometries. 8. Solution of Algebraic Equations. 9. Turbulence Modelling. 10. Grid Generation. 11. Best Practice Guidelines in CFD. Appendix 1. Area and Volume Calculation. Appendix 2. Transformation of Governing Equations to Generalized Curvilinear Coordinates. Appendix 3. Review of Vector Calculus. Appendix 4. Case Studies. References. Index. ;"This textbook covers fundamental and advanced concepts of computational fluid dynamics, a powerful and essential tool for fluid flow analysis. It discusses various governing equations used in computational fluid dynamics, their derivations, and the physical and mathematical significance of partial differential equations and the boundary conditions. It covers fundamental concepts of finite difference and finite volume methods for diffusion, convection-diffusion problems both for cartesian and non-orthogonal grids. The solution of algebraic equations arising due to finite difference and finite volume discretization are highlighted using direct and iterative methods. Pedagogical features including solved problems and unsolved exercises are interspersed throughout the text for better understanding. The textbook is primarily written for senior undergraduate and graduate students in the field of mechanical engineering and aerospace engineering, for a course on computational fluid dynamics and heat transfer. The textbook will be accompanied by teaching resources including solution manual for the instructors"-- Cover......Page 1 Half Title......Page 2 Title Page......Page 4 Copyright Page......Page 5 Dedication......Page 6 Table of Contents......Page 8 Preface......Page 20 Acknowledgements......Page 24 1.1 Introduction......Page 26 1.3 What Does a CFD Algorithm Do?......Page 27 1.4.1 Pre-Processor......Page 28 1.4.3 Post-Processor......Page 29 1.6.3 Finite Element Method......Page 30 1.7 Discretization Properties......Page 31 Questions:......Page 32 2.2.1 Mass (Continuity)......Page 34 2.2.1.1 Derivation of Continuity Equation......Page 35 2.2.2 Momentum Equations......Page 37 2.2.2.1 Derivation of X-Momentum Equation......Page 38 2.2.3 Energy Equation......Page 41 2.2.3.1 Derivation of Energy Equation......Page 43 2.2.4 General Scalar......Page 48 2.3.2 Compressible and Incompressible Flow......Page 49 2.5 Marching Problems......Page 50 2.6 Mathematical Classification......Page 51 2.7 Important Equations......Page 52 2.8 Boundary Conditions (BCs)......Page 53 2.8.3.2 Slip Wall......Page 54 Questions......Page 55 3.1 Introduction......Page 58 3.2 Taylor Series Expansion......Page 59 3.3 Unequal Grid Spacing......Page 63 3.4.1.1 Truncation Error......Page 64 3.4.2 Consistency......Page 65 3.4.3 Stability......Page 66 3.4.3.1 Von Neumann’s Method......Page 67 3.4.5 Lax’s Equivalence Theorem......Page 68 3.5 Examples......Page 69 Questions:......Page 71 4.2 One-Dimensional Diffusion Equations......Page 74 4.2.1.1 The Forward Time, Central Space......Page 75 4.2.1.2 The Richardson’s Method......Page 76 4.2.1.3 The DuFort-Frankel Method (D-FLeap-Frog Method)......Page 78 4.2.2.1 The Classical Implicit Method......Page 80 4.2.2.2 The Crank-Nicolson Method......Page 82 4.2.2.3 The Method of Weighted Averages......Page 85 4.3 One-Dimensional Transport Equations......Page 86 4.3.1.1 The FTCS Method......Page 87 4.3.1.2 Upwind Differencing......Page 88 4.3.1.3 The LaxMethod (Lax-FriedrichsMethod)......Page 91 4.3.1.4 The Lax-Wendroff Method......Page 92 4.3.1.6 The MacCormack Method......Page 93 4.3.1.7 The Beam-Warming Method......Page 94 4.3.1.8 The Implicit Method......Page 95 4.3.2.1 Central Difference......Page 96 4.3.2.3 The DuFort-Frankel Method......Page 97 4.4 Two-Dimensional Diffusion Equation......Page 98 4.4.1 The Explicit Method......Page 99 4.4.2.3 The Alternate Direction Implicit (ADI)Method......Page 100 4.4.2.5 Further Comments on Conservative vs.Non-Conservative Variables......Page 102 4.4.2.6 The Grid (Mesh) Independence Study......Page 103 4.5 Burgers’ Equation......Page 104 4.5.1.2 The Lax (Lax-Friedrichs) Method......Page 106 4.5.1.3 The Lax-Wendroff Method......Page 107 4.5.1.4 The MacCormack Method......Page 108 4.5.1.6 The Godunov Method......Page 109 4.5.1.7 The Roe Method......Page 114 4.5.2 The Viscid Burgers’ Equation......Page 117 4.5.2.2 The DuFort-Frankel Method......Page 118 4.5.2.4 The MacCormack Method......Page 119 4.7 Examples......Page 120 Questions......Page 132 5.1 Introduction......Page 136 5.2 The Diffusion Equation......Page 137 5.2.1 The Steady-State One-Dimensional Diffusion Equation......Page 138 5.2.2 Discretization of the Source Term......Page 140 5.2.3.2 Insulated Boundary......Page 141 5.2.4 Assembling the Algebraic Equations......Page 142 5.2.5 Extension to Two Dimensions......Page 143 5.2.7 Desirable Properties of a Discretization Scheme......Page 144 5.2.8 Further Comments on Interface DiffusionCoefficients......Page 145 5.3 The Convection-Diffusion Equation......Page 146 5.3.1.1 The Central Differencing Scheme......Page 147 5.3.1.2 The Upwind Differencing Scheme......Page 149 5.3.1.3 Exact Solution......Page 152 5.3.1.4 The Exponential Scheme......Page 153 5.3.1.5 The Hybrid Differencing Scheme......Page 154 5.3.1.6 The Second Order Upwind (SOU) Scheme......Page 155 5.3.1.7 The Quadratic Upstream Interpolation for Convective Kinetics (QUICK) Scheme......Page 157 5.3.1.8 The FROMM Scheme......Page 158 5.3.2 Deferred Correction Approach......Page 160 5.3.2.1 CDS......Page 161 5.3.2.2 SOU......Page 162 5.3.2.3 QUICK......Page 163 5.3.2.4 FROMM......Page 164 5.3.3 Extension to Two Dimension......Page 165 5.3.3.1 UDS......Page 166 5.3.3.2 CDS......Page 167 5.3.3.3 SOU......Page 168 5.3.3.4 QUICK......Page 169 5.3.3.5 FROMM......Page 170 5.3.4 Extension to Three Dimension......Page 171 5.3.4.3 SOU......Page 172 5.3.5 High Resolution and Bounded ConvectiveSchemes......Page 173 5.3.5.1 Normalized Variable Formulation......Page 174 5.3.5.3 High-Resolution Schemes......Page 176 5.3.5.4 The TVD Framework......Page 178 5.3.5.5 Implementation of Various ConvectiveSchemes in Code......Page 181 5.4 Time-Dependent Methods......Page 182 5.4.1.3 Central Differencing (Crank-Nicolson)......Page 183 5.5.2 Backward Differencing – Implicit Scheme......Page 184 5.5.3 Crank-Nicolson – Central Difference Scheme:......Page 185 5.8.1 Generalized Boundary Conditions......Page 186 5.8.2 Convective Boundary Conditions......Page 187 5.9 Examples......Page 188 5.10 Summary......Page 202 Questions......Page 204 6.1 Introduction......Page 206 6.2 Pressure-Velocity Coupling......Page 207 6.3 The Vorticity-Stream Function Method......Page 209 6.3.1 Boundary Conditions......Page 211 6.4.1 Co-located Storage of Variables......Page 212 6.4.2 Staggered Grid......Page 214 6.5.1 The Artificial Compressibility Method......Page 215 6.5.2 The Pressure Correction Approach......Page 216 6.5.2.2 The Fractional Step PressureProjection Method......Page 217 6.6.1 Derivation of Velocity Correction and Pressure Correction Equations......Page 218 6.6.1.1 Pressure and Velocity Corrections......Page 219 6.6.2 Pressure Correction Equation......Page 220 6.6.3 The SIMPLE Algorithm......Page 221 6.7 Variants of SIMPLE......Page 223 6.7.1 The SIMPLER Algorithm......Page 224 6.7.2 The SIMPLEC Algorithm......Page 226 6.7.3 The PISO (Pressure Implicit with Split Operator) Algorithm......Page 227 6.8 Summary......Page 229 Questions:......Page 230 7.2 Staggered Grid Algorithm......Page 232 7.3 The Co-located Grid Algorithm......Page 233 7.4 Discretization Methods for Non-Orthogonal Structured Grids......Page 235 7.4.2 The Transport Equation......Page 236 7.4.2.1 Discretization of Convective Flux......Page 237 7.4.2.2 Discretization of Diffusive Flux......Page 239 7.4.2.4 Implementation of the QUICK Scheme......Page 241 7.5.1 Derivation of Pressure Correction and Velocity Correction Equations......Page 242 7.5.2 Implementation of Momentum Interpolation......Page 244 7.6 Extension to Three Dimension......Page 247 7.6.2 Discretization of Convective Flux......Page 248 7.6.3 Discretization of Diffusive Flux......Page 249 7.6.5 Implementation of the QUICK Scheme......Page 252 7.6.6 Implementation of the SIMPLE Algorithm......Page 253 7.7.1 The Continuity Equation......Page 256 7.7.2.1 Discretization of Convective Flux......Page 258 7.7.2.2 Discretization of Diffusive Flux......Page 259 7.7.2.4 Implementation of the QUICK Scheme......Page 260 7.7.2.5 Derivation of Pressure Correctionand Velocity Correction Equation......Page 261 7.7.2.6 Implementation of the Momentum Interpolation......Page 263 7.8.2 The Transport Equation......Page 265 7.8.2.2 Discretization of Diffusive Flux......Page 267 7.9.1 Derivation of Pressure Correction and Velocity Correction Equations......Page 270 7.9.2 Implementation of the Momentum Interpolation......Page 272 7.9.3 Implementation of Higher-Order Schemes......Page 274 Questions......Page 275 8.2.1 Gauss Elimination......Page 278 8.2.2 LU Decomposition......Page 280 8.2.3 Tri-Diagonal Matrix Algorithm......Page 281 8.3.1 The Jacobi Method......Page 283 8.3.2 The Point Gauss-Seidel Method......Page 284 8.3.4 The Line Gauss-Seidel Method......Page 285 8.3.5 Convergence of the Iterative Methods......Page 286 8.4 Conjugate Gradient (CG) Methods......Page 287 8.4.2 The Pre-Conditioned CGS Method......Page 288 8.5.1 Introduction......Page 289 8.5.2 Pre-Conditioning by L-U Decomposition......Page 290 8.6 The Multigrid Method......Page 293 8.6.1 Coarsening Step......Page 294 8.6.3 Prolongation Step......Page 295 8.6.4 Cycling Strategy......Page 296 8.7 Examples......Page 297 8.8 Summary......Page 301 Questions:......Page 302 9.2.1 Characteristics of Turbulent Flows......Page 304 9.3 Direct Numerical Simulation (DNS)......Page 305 9.4.1 Reynolds-Averaged Navier-Stokes (RANS)Equations......Page 306 9.4.2 Eddy Viscosity Models Hypothesis......Page 309 9.5.1 Zero-Equation Models......Page 310 9.5.1.1 Structure of the Turbulent Boundary Layer......Page 311 9.5.2.1 The Cebaci-Smith Model......Page 312 9.5.3 The Transport Equation for Turbulent Kinetic Energy (One-Equation Model)......Page 313 9.5.4 Two-Equation Models......Page 314 9.5.4.2 The Wilcox k-ω Model......Page 315 9.5.4.3 The SST k- ω (Menter) TurbulenceModel......Page 316 9.5.4.4 Near-Wall Modifications forTwo-Equation Models......Page 317 9.6 Reynolds Stress Transport (Equation-based) Models (RSTMs)......Page 320 9.7 Large Eddy Simulation......Page 322 9.8 Summary......Page 323 Questions......Page 324 10.2 Geometry......Page 326 10.3 Grid Structure......Page 327 10.4 Classification of Grid Types......Page 328 10.5.1 Blocking Out Cells......Page 329 10.5.3 Body-Fitted Grids......Page 330 10.6 Mesh Quality......Page 331 10.7 Adaptive Grid......Page 332 10.8.1 Coordinate Transformation......Page 333 10.8.2.1 Algebraic Grid Generation......Page 335 10.8.2.2 Differential-Equation Based Techniques......Page 338 10.9 Unstructured Grid Generation......Page 339 10.9.1 Connectivity Information......Page 340 10.9.2 Triangular Grid Generation......Page 341 10.9.2.1 The Advancing-Front Technique......Page 342 10.9.2.2 The Delaunay-Based Method......Page 343 Questions......Page 345 11.2 Sources of Error......Page 348 11.2.3 Iteration or Convergence Errors......Page 349 11.3.1.1 Geometry Generation......Page 350 11.3.2.2 Time Discretization Errors......Page 351 11.3.4.1 Solution Algorithms......Page 352 11.3.4.3 Near-Wall Modeling......Page 353 11.3.6.1 Boundary Conditions......Page 354 11.4.3 Uncertainties......Page 356 11.5.1.2 Calculation Verification......Page 357 11.5.2 Validation......Page 358 Appendix 1: Area and Volume Calculation......Page 360 Appendix 2: Transformation of Governing Equations toGeneralized Curvilinear Coordinates......Page 368 Appendix 3: Review of Vector Calculus......Page 374 Appendix 4: Case Studies......Page 394 References......Page 406 Index......Page 412 This textbook covers fundamental and advanced concepts of computational fluid dynamics, a powerful and essential tool for fluid flow analysis. It discusses various governing equations used in the field, their derivations, and the physical and mathematical significance of partial differential equations and the boundary conditions. It covers fundamental concepts of finite difference and finite volume methods for diffusion, convection-diffusion problems both for cartesian and non-orthogonal grids. The solution of algebraic equations arising due to finite difference and finite volume discretization are highlighted using direct and iterative methods. Pedagogical features including solved problems and unsolved exercises are interspersed throughout the text for better understanding. The textbook is primarily written for senior undergraduate and graduate students in the field of mechanical engineering and aerospace engineering, for a course on computational fluid dynamics and heat transfer. The textbook will be accompanied by teaching resources including a solution manual for the instructors. Written clearly and with sufficient foundational background to strengthen fundamental knowledge of the topic. Offers a detailed discussion of both finite difference and finite volume methods. Discusses various higher-order bounded convective schemes, TVD discretisation schemes based on the flux limiter essential for a general purpose CFD computation. Discusses algorithms connected with pressure-linked equations for incompressible flow. Covers turbulence modelling like k-ε, k-ω, SST k-ω, Reynolds Stress Transport models. A separate chapter on best practice guidelines is included to help CFD practitioners. "This textbook covers fundamental and advanced concepts of computational fluid dynamics, a powerful and essential tool for fluid flow analysis. It discusses various governing equations used in computational fluid dynamics, their derivations, and the physical and mathematical significance of partial differential equations and the boundary conditions. It covers fundamental concepts of finite difference and finite volume methods for diffusion, convection-diffusion problems both for cartesian and non-orthogonal grids. The solution of algebraic equations arising due to finite difference and finite volume discretization are highlighted using direct and iterative methods. Pedagogical features including solved problems and unsolved exercises are interspersed throughout the text for better understanding. The textbook is primarily written for senior undergraduate and graduate students in the field of mechanical engineering and aerospace engineering, for a course on computational fluid dynamics and heat transfer. The textbook will be accompanied by teaching resources including solution manual for the instructors"-- Provided by publisher This textbook comprehensively discusses fundamental and advanced concepts of computational fluid dynamics and covers both finite difference and finite volume methods. It will serve as ideal study material for senior undergraduate and graduate students in the field of mechanical engineering and aerospace engineering.