Optimal Modified Continuous Galerkin CFD
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A. J. Baker, The University of Tennessee, USA
Professor Baker is Professor Emeritus in both the Engineering Science Department and the Computational Mechanics Department at the University of Tennessee. Prior to this he was Director of UT CFD Laboratory at the University of Tennessee. He is a Fellow of International Association for Computational Mechanics (IACM) and the US Association for Computational Mechanics (USACM) as well as being an Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA). Professor Baker has devised many courses on FEA-related topics during his career and has also written multiple books and journal articles.
Content
About the Author xvii
Notations xix
1 Introduction 1
1.1 About This Book 1
1.2 The Navier-Stokes Conservation Principles System 2
1.3 Navier-Stokes PDE System Manipulations 5
1.4 Weak Form Overview 7
1.5 A Brief History of Finite Element CFD 9
1.6 A Brief Summary 11
References 12
2 Concepts, terminology, methodology 15
2.1 Overview 15
2.2 Steady DE Weak Form Completion 16
2.3 Steady DE GWSN Discrete FE Implementation 19
2.4 PDE Solutions, Classical Concepts 27
2.5 The Sturm-Liouville Equation, Orthogonality, Completeness 30
2.6 Classical Variational Calculus 33
2.7 Variational Calculus, Weak Form Duality 36
2.8 Quadratic Forms, Norms, Error Estimation 38
2.9 Theory Illustrations for Non-Smooth, Nonlinear Data 40
2.10 Matrix Algebra, Notation 44
2.11 Equation Solving, Linear Algebra 46
2.12 Krylov Sparse Matrix Solver Methodology 53
2.13 Summary 54
Exercises 54
References 56
3 Aerodynamics I: Potential flow, GWSh theory exposition, transonic flow mPDE shock capturing 59
3.1 Aerodynamics, Weak Interaction 59
3.2 Navier-Stokes Manipulations for Aerodynamics 60
3.3 Steady Potential Flow GWS 62
3.4 Accuracy, Convergence, Mathematical Preliminaries 66
3.5 Accuracy, Galerkin Weak Form Optimality 68
3.6 Accuracy, GWSh Error Bound 71
3.7 Accuracy, GWSh Asymptotic Convergence 73
3.8 GWSh Natural Coordinate FE Basis Matrices 76
3.9 GWSh Tensor Product FE Basis Matrices 82
3.10 GWSh Comparison with Laplacian FD and FV Stencils 87
3.11 Post-Processing Pressure Distributions 90
3.12 Transonic Potential Flow, Shock Capturing 92
3.13 Summary 96
Exercises 98
References 99
4 Aerodynamics II: boundary layers, turbulence closure modeling, parabolic Navier-Stokes 101
4.1 Aerodynamics, Weak Interaction Reprise 101
4.2 Navier-Stokes PDE System Reynolds Ordered 102
4.3 GWSh, n= 2 Laminar-Thermal Boundary Layer 104
4.4 GWSh + ¿TS BL Matrix Iteration Algorithm 108
4.5 Accuracy, Convergence, Optimal Mesh Solutions 111
4.6 GWSh +¿TS Solution Optimality, Data Influence 115
4.7 Time Averaged NS, Turbulent BL Formulation 116
4.8 Turbulent BL GWSh+ ¿TS, Accuracy, Convergence 120
4.9 GWSh+ ¿TS BL Algorithm, TKE Closure Models 123
4.10 The Parabolic Navier-Stokes PDE System 129
4.11 GWSh +¿TS Algorithm for PNS PDE System 134
4.12 GWSh +¿TS k=1 NC Basis PNS Algorithm 137
4.13 Weak Interaction PNS Algorithm Validation 141
4.14 Square Duct PNS Algorithm Validation 147
4.15 Summary 148
Exercises 155
References 157
5 The Navier-Stokes Equations: theoretical fundamentals; constraint, spectral analyses, mPDE theory, optimal Galerkin weak forms 159
5.1 The Incompressible Navier-Stokes PDE System 159
5.2 Continuity Constraint, Exact Enforcement 160
5.3 Continuity Constraint, Inexact Enforcement 164
5.4 The CCM Pressure Projection Algorithm 166
5.5 Convective Transport, Phase Velocity 168
5.6 Convection-Diffusion, Phase Speed Characterization 170
5.7 Theory for Optimal mGWSh+ ¿TS Phase Accuracy 177
5.8 Optimally Phase Accurate mGWSh + ¿TS in n Dimensions 185
5.9 Theory for Optimal mGWSh Asymptotic Convergence 193
5.10 The Optimal mGWSh ¿TS k 1 Basis NS Algorithm 201
5.11 Summary 203
Exercises 206
References 208
6 Vector Field Theory Implementations: vorticity-streamfunction, vorticity-velocity formulations 211
6.1 Vector Field Theory NS PDE Manipulations 211
6.2 Vorticity-Streamfunction PDE System, n= 2 213
6.3 Vorticity-Streamfunction mGWSh Algorithm 214
6.4 Weak Form Theory Verification, GWSh/mGWSh 219
6.5 Vorticity-Velocity mGWSh Algorithm, n= 3 228
6.6 Vorticity-Velocity GWSh+ ¿TS Assessments, n= 3 233
6.7 Summary 243
Exercises 246
References 247
7 Classic State Variable Formulations: GWS/mGWSh+¿TS algorithms for Navier-Stokes; accuracy, convergence, validation, BCs, radiation, ALE formulation 249
7.1 Classic State Variable Navier-Stokes PDE System 249
7.2 NS Classic State Variable mPDE System 251
7.3 NS Classic State Variable mGWSh+ ¿TS Algorithm 252
7.4 NS mGWSh +¿TS Algorithm Discrete Formation 254
7.5 mGWSh+ ¿TS Algorithm Completion 258
7.6 mGWSh+ ¿TS Algorithm Benchmarks, n=2 260
7.7 mGWSh+ ¿TS Algorithm Validations, n= 3 268
7.8 Flow Bifurcation, Multiple Outflow Pressure BCs 282
7.9 Convection/Radiation BCs in GWSh+ ¿TS 283
7.10 Convection BCs Validation 288
7.11 Radiosity, GWSh Algorithm 295
7.12 Radiosity BC, Accuracy, Convergence, Validation 298
7.13 ALE Thermo-Solid-Fluid-Mass Transport Algorithm 302
7.14 ALE GWSh +¿TS Algorithm LISI Validation 304
7.15 Summary 310
Exercises 317
References 318
8 Time Averaged Navier-Stokes: mGWSh+¿TS algorithm for RaNS, Reynolds stress tensor closure models 319
8.1 Classic State Variable RaNS PDE System 319
8.2 RaNS PDE System Turbulence Closure 321
8.3 RaNS State Variable mPDE System 323
8.4 RaNS mGWSh+¿TS Algorithm Matrix Statement 325
8.5 RaNS mGWSh + ¿TS Algorithm, Stability, Accuracy 331
8.6 RaNS Algorithm BCs for Conjugate Heat Transfer 337
8.7 RaNS Full Reynolds Stress Closure PDE System 341
8.8 RSM Closure mGWSh +¿TS Algorithm 345
8.9 RSM Closure Model Validation 347
8.10 Geologic Borehole Conjugate Heat Transfer 348
8.11 Summary 358
Exercises 363
References 364
9 Space Filtered Navier-Stokes: GWSh/mGWSh+¿TS for space filtered Navier-Stokes, modeled, analytical closure 365
9.1 Classic State Variable LES PDE System 365
9.2 Space Filtered NS PDE System 366
9.3 SGS Tensor Closure Modeling for LES 368
9.4 Rational LES Theory Predictions 371
9.5 RLES Unresolved Scale SFS Tensor Models 376
9.6 Analytical SFS Tensor/Vector Closures 381
9.7 Auxiliary Problem Resolution Via Perturbation Theory 383
9.8 LES Analytical Closure (arLES) Theory 386
9.9 arLES Theory mGWSh + ¿TS Algorithm 387
9.10 arLES Theory mGWSh + ¿TS Completion 391
9.11 arLES Theory Implementation Diagnostics 392
9.12 RLES Theory Turbulent BL Validation 403
9.13 Space Filtered NS PDE System on Bounded Domains 409
9.14 Space Filtered NS Bounded Domain BCs 410
9.15 ADBC Algorithm Validation, Space Filtered DE 412
9.16 arLES Theory Resolved Scale BCE Integrals 420
9.17 Turbulent Resolved Scale Velocity BC Optimal Oh-d 423
9.18 Resolved Scale Velocity DBC Validation 8 Re 430
9.19 arLES O(d2) State Variable Bounded Domain BCs 430
9.20 Well-Posed arLES Theory n = 3 Validation 433
9.21 Well-Posed arLES Theory n = 3 Diagnostics 441
9.22 Summary 446
Exercises 455
References 456
10 Summary-VVUQ: verification, validation, uncertainty quantification 459
10.1 Beyond Colorful Fluid Dynamics 459
10.2 Observations on Computational Reliability 460
10.3 Solving the Equations Right 461
10.4 Solving the Right Equations 464
10.5 Solving the Right Equations Without Modeling 466
10.6 Solving the Right Equations Well-Posed 468
10.7 Well-Posed Right Equations Optimal CFD 471
10.8 The Right Closing Caveat 473
References 474
Appendix A: Well-Posed arLES Theory PICMSS Template 475
Appendix B: Hypersonic Parabolic Navier-Stokes 483
B.1 High Speed External Aerodynamics 483
B.2 Compressible Navier-Stokes PDE System 484
B.3 Parabolic Compressible RaNS PDE System 488
B.4 Compressible PRaNS mPDE System Closure 490
B.5 Bow Shock Fitting, PRaNS State Variable IC 493
B.6 The PRaNS mGWSh+¿TS Algorithm 496
B.7 PRaNS mGWSh+¿TS Algorithm Completion 501
B.8 PRaNS Algorithm IC Generation 505
B.9 PRaNS mGWSh+¿TS Algorithm Validation 507
B.10 Hypersonic Blunt Body Shock Trajectory 515
B.11 Shock Trajectory Characteristics Algorithm 521
B.12 Blunt Body PRaNS Algorithm Validation 523
B.13 Summary 527
Exercises 532
References 533
Author Index 535
Subject Index 541
Notations
a expansion coefficient; speed of sound; characteristics coefficient A plane area; 1-D FE matrix prefix; coefficient AD approximate deconvolution ADBC approximate deconvolution boundary condition algorithm AF approximate factorization algorithm ALE arbitrary-lagrangian-eulerian algorithm [A] factored global matrix, RLES theory auxiliary problem matrix operator arLES essentially analytic LES closure theory b coefficient; boundary condition subscript {b} global data matrix B 2-D FE matrix prefix B(•) bilinear form B body force BC boundary condition BCE boundary commutation error integral BHE borehole heat exchanger BiSec bisected borehole heat exchanger BL boundary layer c coefficient; specific heat c phase velocity vector C 3-D matrix prefix; coefficient; chord; Courant number ≡ UΔt/Δx Cij cross stress tensor Cp aerodynamic pressure coefficient, = p/ρu2/2 CS Smagorinsky constant, its generalization CFD computational fluid dynamics CFL Courant number Cf skin friction coefficient CF/2 boundary layer skin friction coefficient CNFD Crank–Nicolson finite difference CS control surface CV control volume d(•) ordinary derivative, differential element d coefficient; FE matrix basis degree label, RSM distance; characteristics coefficient D binary diffusion coefficient; diagonal matrix D dimensionality, non-D diffusion coefficient ≡ Δt/Pah2 D(•) differential definition D(•) substantial derivative Dm(•) modified substantial derivative DES detached eddy simulation DE conservation of energy PDE DG discontinuous Galerkin weak form theory DM conservation of mass PDE DP conservation of momentum PDE DY conservation of species mass fraction PDE D(u, P) NS full stress tensor, diag[•] diagonal matrix [DIFF] laplacian diffusion matrix DNS direct numerical simulation e specific internal energy; element-dependent (subscript) e(·) error eN continuum approximation error eh discrete approximation error eijk alternating tensor eKL curl alternator on n = 2 EBV elliptic boundary value Ec Eckert number etaji coordinate transformation data E thermal energy; energy semi-norm (subscript) fj flux vector fn normal flux f(•) function of the argument f(vf, ) radiation view factor F(•) Fourier transform {F} weak form terminal algebraic statement F(k → i) Lambert's cosine law viewfactor FD finite difference FE finite element FV finite volume f efflux vector on ∂Ω F applied force g gravity magnitude; amplification factor; spatial filter function; characteristics enthalpy ratio g gravity Gr Grashoff number ≡ gβΔTL3/ν2 Gk→i Gebhart viewfactor GHP ground source heat pump GLS Galerkin least squares algorithm GWS Galerkin weak statement h mesh measure; discrete (superscript), heat transfer coefficient H boundary layer shape factor H Gauss quadrature weight; Hilbert space H.O.T. truncated Taylor series higher order terms i summation index; mesh node unit vector parallel to x I discrete matrix summation index, identity matrix iff if and only if I-EBV initial-elliptic boundary value j summation index, mesh node unit vector parallel to y J discrete matrix summation index [J] coordinate transformation jacobian [JAC] matrix statement jacobian k thermal conductivity; basis degree; index; diffusion coefficient element of the [DIFF] matrix average value of conductivity unit vector parallel to z K discrete matrix summation index element length; summation index differential operator on ∂Ω L reference length scale L discrete matrix summation index differential operator on Ω Lij Leonard stress tensor LES large eddy simulation, convolved Navier–Stokes PDEs m non-D wavenumber ≡ κh, integer [m] mass matrix mi point mass; discrete matrix summation index M particle system mass; domain matrix prefix; elements in Mi molecular mass [M] mPDE theory altered mass matrix Ma Mach number mGWS optimal modified Galerkin weak form mPDE modified partial differential equation mODE modified ordinary differential equation MLT mixing length theory n index; normal subscript; dimension of domain Ω;...
