Optimal Modified Continuous Galerkin CFD (eBook)

A. J. Baker (Autor)

eBook Download: EPUB | PDF

2014
John Wiley & Sons (Verlag)
978-1-118-40308-2 (ISBN)

Infos

Covers the theory and applications of using weak form theory in incompressible fluid-thermal sciences

Giving you a solid foundation on the Galerkin finite-element method (FEM), this book promotes the use of optimal modified continuous Galerkin weak form theory to generate discrete approximate solutions to incompressible-thermal Navier-Stokes equations. The book covers the topic comprehensively by introducing formulations, theory and implementation of FEM and various flow formulations.

The author first introduces concepts, terminology and methodology related to the topic before covering topics including aerodynamics; the Navier-Stokes Equations; vector field theory implementations and large eddy simulation formulations.

• Introduces and addresses many different flow models (Navier-Stokes, full-potential, potential, compressible/incompressible) from a unified perspective
• Focuses on Galerkin methods for CFD beneficial for engineering graduate students and engineering professionals
• Accompanied by a website with sample applications of the algorithms and example problems and solutions

This approach is useful for graduate students in various engineering fields and as well as professional engineers.

Autor

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.

Inhalt

Optimal Modified Continuous Galerkin CFD 1
Contents 9
Preface 15
About the Author 19
Notations 21
1 Introduction 27
1.1 About This Book 27
1.2 The Navier–Stokes Conservation Principles System 28
1.3 Navier–Stokes PDE System Manipulations 31
1.4 Weak Form Overview 33
1.5 A Brief History of Finite Element CFD 35
1.6 A Brief Summary 37
References 38
2 Concepts, terminology, methodology 41
2.1 Overview 41
2.2 Steady DE Weak Form Completion 42
2.3 Steady DE GWSN Discrete FE Implementation 45
2.4 PDE Solutions, Classical Concepts 53
2.5 The Sturm–Liouville Equation, Orthogonality, Completeness 56
2.6 Classical Variational Calculus 59
2.7 Variational Calculus, Weak Form Duality 62
2.8 Quadratic Forms, Norms, Error Estimation 64
2.9 Theory Illustrations for Non-Smooth, Nonlinear Data 66
2.10 Matrix Algebra, Notation 70
2.11 Equation Solving, Linear Algebra 72
2.12 Krylov Sparse Matrix Solver Methodology 79
2.13 Summary 80
Exercises 80
References 82
3 Aerodynamics I: Potential flow, GWSh theory exposition, transonic flow mPDE shock capturing 85
3.1 Aerodynamics, Weak Interaction 85
3.2 Navier–Stokes Manipulations for Aerodynamics 86
3.3 Steady Potential Flow GWS 88
3.4 Accuracy, Convergence, Mathematical Preliminaries 92
3.5 Accuracy, Galerkin Weak Form Optimality 94
3.6 Accuracy, GWSh Error Bound 97
3.7 Accuracy, GWSh Asymptotic Convergence 99
3.8 GWSh Natural Coordinate FE Basis Matrices 102
3.9 GWSh Tensor Product FE Basis Matrices 108
3.10 GWSh Comparison with Laplacian FD and FV Stencils 113
3.11 Post-Processing Pressure Distributions 116
3.12 Transonic Potential Flow, Shock Capturing 118
3.13 Summary 122
Exercises 124
References 125
4 Aerodynamics II: boundary layers, turbulence closure modeling, parabolic Navier–Stokes 127
4.1 Aerodynamics, Weak Interaction Reprise 127
4.2 Navier–Stokes PDE System Reynolds Ordered 128
4.3 GWSh, n= 2 Laminar-Thermal Boundary Layer 130
4.4 GWSh + ?TS BL Matrix Iteration Algorithm 134
4.5 Accuracy, Convergence, Optimal Mesh Solutions 137
4.6 GWSh + ?TS Solution Optimality, Data Influence 141
4.7 Time Averaged NS, Turbulent BL Formulation 142
4.8 Turbulent BL GWSh + ?TS, Accuracy, Convergence 146
4.9 GWSh + ?TS BL Algorithm, TKE Closure Models 149
4.10 The Parabolic Navier–Stokes PDE System 155
4.11 GWSh + ?TS Algorithm for PNS PDE System 160
4.12 GWSh + ?TS k =1 NC Basis PNS Algorithm 163
4.13 Weak Interaction PNS Algorithm Validation 167
4.14 Square Duct PNS Algorithm Validation 173
4.15 Summary 174
Exercises 181
References 183
5 The Navier–Stokes Equations: theoretical fundamentals constraint, spectral analyses, mPDE theory, optimal Galerkin weak forms
5.1 The Incompressible Navier–Stokes PDE System 185
5.2 Continuity Constraint, Exact Enforcement 186
5.3 Continuity Constraint, Inexact Enforcement 190
5.4 The CCM Pressure Projection Algorithm 192
5.5 Convective Transport, Phase Velocity 194
5.6 Convection-Diffusion, Phase Speed Characterization 196
5.7 Theory for Optimal mGWSh + ?TS Phase Accuracy 203
5.8 Optimally Phase Accurate mGWSh + ?TS in n Dimensions 211
5.9 Theory for Optimal mGWSh Asymptotic Convergence 219
5.10 The Optimal mGWSh + ?TS k ˆ= 1 Basis NS Algorithm 227
5.11 Summary 229
Exercises 232
References 234
6 Vector Field Theory Implementations: vorticity-streamfunction, vorticity-velocity formulations 237
6.1 Vector Field Theory NS PDE Manipulations 237
6.2 Vorticity-Streamfunction PDE System, n = 2 239
6.3 Vorticity-Streamfunction mGWSh Algorithm 240
6.4 Weak Form Theory Verification, GWSh/mGWSh 245
6.5 Vorticity-Velocity mGWSh Algorithm, n = 3 254
6.6 Vorticity-Velocity GWSh+ ?TS Assessments, n = 3 259
6.7 Summary 269
Exercises 272
References 273
7 Classic State Variable Formulations: GWS/mGWSh + ?TS algorithms for Navier–Stokes accuracy, convergence, validation, BCs, radiation, ALE formulation
7.1 Classic State Variable Navier–Stokes PDE System 275
7.2 NS Classic State Variable mPDE System 277
7.3 NS Classic State Variable mGWSh + ?TS Algorithm 278
7.4 NS mGWSh + ?TS Algorithm Discrete Formation 280
7.5 mGWSh + ?TS Algorithm Completion 284
7.6 mGWSh + ?TS Algorithm Benchmarks, n = 2 286
7.7 mGWSh + ?TS Algorithm Validations, n = 3 294
7.8 Flow Bifurcation, Multiple Outflow Pressure BCs 308
7.9 Convection/Radiation BCs in GWSh + ?TS 309
7.10 Convection BCs Validation 314
7.11 Radiosity, GWSh Algorithm 321
7.12 Radiosity BC, Accuracy, Convergence, Validation 324
7.13 ALE Thermo-Solid-Fluid-Mass Transport Algorithm 328
7.14 ALE GWSh + ?TS Algorithm LISI Validation 330
7.15 Summary 336
Exercises 343
References 344
8 Time Averaged Navier–Stokes: mGWSh + ?TS algorithm for RaNS,Reynolds stress tensor closure models 345
8.1 Classic State Variable RaNS PDE System 345
8.2 RaNS PDE System Turbulence Closure 347
8.3 RaNS State Variable mPDE System 349
8.4 RaNS mGWSh + ?TS Algorithm Matrix Statement 351
8.5 RaNS mGWSh + ?TS Algorithm, Stability, Accuracy 357
8.6 RaNS Algorithm BCs for Conjugate Heat Transfer 363
8.7 RaNS Full Reynolds Stress Closure PDE System 367
8.8 RSM Closure mGWSh + ?TS Algorithm 371
8.9 RSM Closure Model Validation 373
8.10 Geologic Borehole Conjugate Heat Transfer 374
8.11 Summary 384
Exercises 389
References 390
9 Space Filtered Navier–Stokes: GWSh/mGWSh + ?TS for space filteredNavier–Stokes, modeled, analytical closure 391
9.1 Classic State Variable LES PDE System 391
9.2 Space Filtered NS PDE System 392
9.3 SGS Tensor Closure Modeling for LES 394
9.4 Rational LES Theory Predictions 397
9.5 RLES Unresolved Scale SFS Tensor Models 402
9.6 Analytical SFS Tensor/Vector Closures 407
9.7 Auxiliary Problem Resolution Via Perturbation Theory 409
9.8 LES Analytical Closure (arLES) Theory 412
9.9 arLES Theory mGWSh + ?TS Algorithm 413
9.10 arLES Theory mGWSh + ?TS Completion 417
9.11 arLES Theory Implementation Diagnostics 418
9.12 RLES Theory Turbulent BL Validation 429
9.13 Space Filtered NS PDE System on Bounded Domains 435
9.14 Space Filtered NS Bounded Domain BCs 436
9.15 ADBC Algorithm Validation, Space Filtered DE 438
9.16 arLES Theory Resolved Scale BCE Integrals 446
9.17 Turbulent Resolved Scale Velocity BC Optimal ?h-? 449
9.18 Resolved Scale Velocity DBC Validation A Re 456
9.19 arLES O(?2) State Variable Bounded Domain BCs 456
9.20 Well-Posed arLES Theory n = 3 Validation 459
9.21 Well-Posed arLES Theory n = 3 Diagnostics 467
9.22 Summary 472
Exercises 481
References 482
10 Summary-VVUQ: verification, validation, uncertainty quantification 485
10.1 Beyond Colorful Fluid Dynamics 485
10.2 Observations on Computational Reliability 486
10.3 Solving the Equations Right 487
10.4 Solving the Right Equations 490
10.5 Solving the Right Equations Without Modeling 492
10.6 Solving the Right Equations Well-Posed 494
10.7 Well-Posed Right Equations Optimal CFD 497
10.8 The Right Closing Caveat 499
References 500
Appendix A: Well-Posed arLES Theory PICMSS Template 501
Appendix B: Hypersonic Parabolic Navier–Stokes 509
B.1 High Speed External Aerodynamics 509
B.2 Compressible Navier–Stokes PDE System 510
B.3 Parabolic Compressible RaNS PDE System 514
B.4 Compressible PRaNS mPDE System Closure 516
B.5 Bow Shock Fitting, PRaNS State Variable IC 519
B.6 The PRaNS mGWSh + ?TS Algorithm 522
B.7 PRaNS mGWSh + ?TS Algorithm Completion 527
B.8 PRaNS Algorithm IC Generation 531
B.9 PRaNS mGWSh + ?TS Algorithm Validation 533
B.10 Hypersonic Blunt Body Shock Trajectory 541
B.11 Shock Trajectory Characteristics Algorithm 547
B.12 Blunt Body PRaNS Algorithm Validation 549
B.13 Summary 553
Exercises 558
References 559
Author Index 561
Subject Index 567

Textauszug

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;...

Produkt-Details

• Erscheint lt. Verlag: 10.3.2014
• Sprache: englisch
• Themenwelt: Informatik ► Weitere Themen ► CAD-Programme
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Allen J. Baker • compressible/incompressible aerodynamics • Control Process & Measurements • flow models • full-potential • large eddy simulation formulations • Maschinenbau • Maschinenbau - Entwurf • Materials Science • Materialwissenschaften • mechanical engineering • Mechanical Engineering - Design • Mess- u. Regeltechnik • Metalle u. Legierungen • Metals & Alloys • navier-stokes equations • Optimal Modified Continuous Galerkin CFD • Potential • vector field theory implementations
• ISBN-10: 1-118-40308-8 / 1118403088
• ISBN-13: 978-1-118-40308-2 / 9781118403082