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TABLE OF CONTENTS Copyright 1998 by Patrick J. Roache

Part I Overview

Chapter 1 Introduction

1.1 Sketch of Historical Development of CFD
1.2 The New Impetus Toward Higher Quality Solutions
1.3 A Personal Anecdote Illustrating the Importance of Systematic Code Verification
References for Chapter 1

Chapter 2 Semantics: Terminology, Taxonomies, and Definitions

2.1 Introduction
2.2 Semantics
2.3 Code Verification and Validation: Numerical vs. Conceptual Modeling
2.4 Code Confirmation
2.5 Benchmarks and Inter-Code Comparisons
2.6 Code Certification, Quality Assurance, and Accreditation
2.7 Verification of Calculations
2.8 Quantification of Uncertainty
2.9 Grid Convergence vs. Iterative Convergence
2.10 Error Taxonomies
2.11 Truncation Error vs. Discretization Error
2.12 Calibration and Tuning
2.13 Quality Assurance (QA) vs. Quality Work
2.14 Customer Illusions vs. Customer Care
2.15 Other Distinctions: Authors, Users, Modelers, Code and Software
2.16 Sensitivity, Uncertainty, and Risk
2.17 Etymology and Near Synonyms
2.18 Accuracy vs. Reliability
2.19 Additional Remarks on Verification
2.20 Conclusion: Limitations of Semantic Distinctions
References for Chapter 2

Part II Verification                                                                  backtotop2

Chapter 3 A Methodology for Accuracy Verification of Codes: the Method of Manufactured Solutions

3.1 Introduction
3.2 Warnings: the Division of Labor in Code Development and Use
3.3 Order of Convergence
3.4 A General Procedure for Generating an Analytical Solution for Code Accuracy Verification:
the Method of Manufactured Solutions
3.5 Example: Verification of a 3-D Poisson Equation Code in Nonorthogonal Grid
and a 3-D Grid Generation Code
3.6 Another Path to Manufactured Solutions
3.7 Code Verification Including Shock Waves
3.8 Need for a Theorem
3.9 Specific Analytical Solutions
3.10 Manufactured Solutions vs. Infinite Series Solutions
3.11 The Sensitivity of Grid Convergence Testing
3.12 Examples of Unanticipated Convergence Rates Determined by Systematic Grid Convergence Tests
3.13 Special Considerations for Problems with Multiple Scales: Turbulence Modeling
3.14 Warnings: What the Method Does Not “Verify”
3.15 Robustness and Confidence
References for Chapter 3

Chapter 4 Error Estimation for Quantification
of Uncertainty; Verification of Calculations

4.1 Introduction
4.2 Error Estimation for Grid Adaptation vs. Quantification of Uncertainty
4.3 Taxonomy for Additional Information for Error Estimates
4.4 Grid Refining and Coarsening
4.5 Levels of Simulation Use
4.6 Verification of Computer Round-off Errors
4.7 Effect of Differing formulations
References for Chapter 4

Chapter 5 Systematic Grid Convergence Studies                   backtotop2
 and the Grid Convergence Index (GCI)   

5.1 Introduction
5.2 Background on Grid Convergence Reporting
5.3 Richardson Extrapolation
5.4 A Generalization of Richardson Extrapolation
5.5 Richardson’s Extrapolation for
5.6 Grid Convergence Index for the Fine Grid Solution
5.7 Grid Convergence Index for the Coarse Grid Solution
5.8 Example GCI Calculation
5.9 Should the Coefficient Be “1” Or “3” Or “1.25”?
5.10 Additional Features of Grid Convergence Studies for Verification of Codes and Calculations
5.11 Conclusion
References for Chapter 5

Chapter 6 Applications of Systematic Grid Convergence       backtotop2
Studies and the Grid Convergence Index

6.1 Introduction
6.2 Two Further Examples of (Partial) Code Verification in Groundwater Flow
6.3 Issues in Calculation Verification
6.4 An Example of the Effective Grid Refinement Ratio
6.5 A Benchmark Problem for Driven Cavity Flow
6.6 A Benchmark Problem for Free Convection
6.7 Laminar Plane Jet Impinging on a Heated Flat Plate
6.8 A k-
e Model of a Free Shear Layer
6.9 Transonic Airfoil Calculations
6.10 Ordered Estimation of Far-Field Boundary Error
6.11 Artificial Dissipation Effects
6.12 Single and Dual Porosity Contaminant Transport: Source Term Location
6.13 Convergence Behaviors for Mixed-Order Methods
6.14 Grid Convergence of Zero Drag Coefficient
6.15 Anomalous Result Possibly Due to Grid Stretching
6.16 Non-Smooth Property Variation: Global Error Norms
6.17 Discrete Vortex Methods
6.18 Observed Convergence Rates for Euler Equations with Shocks
6.19 Completed Richardson Extrapolation
6.20 Truncation Error in Elliptic Grid Generation
6.21 One Dimensional Moving Adaptive Grid Problems
6.22 GCI Application in Solution Adaptive Grids with Non-Integer Grid Refinement
6.23 High Quality Grid Resolution Studies Leading to a Safety Factor of 1.25
6.24 Transport Code Verifications Using the GCI: Partitioning the Option Matrix
6.25 Turbulent Separated Flow: the Error Estimator of Celik and Karatekin
6.26 Level of Accuracy Estimates from Grid Convergence Studies
6.27 Other Examples of Careful Use of Richardson Extrapolation
6.28 Parameter Convergences of a Compressible Flow Code Near the Incompressible Limit
6.29 Justification of the Dupuit Approximation
6.30 Concluding Comment on Parameter Uncertainty vs. Numerical Uncertainty
References for Chapter 6

Chapter 7 Single Grid Error Estimators                                    backtotop2

7.1 Error Estimation from Higher or Lower Order Accuracy Solutions on the Same Grid (Category B)
7.2 Auxiliary PDE Solutions on the Same Grid (Category C)
7.3 Auxiliary Algebraic Evaluations on the Same Grid: Surrogate Estimators (Category D)
7.4 Time Accuracy Estimation
7.5 Concluding Remarks on Single Grid Error Estimators
References for Chapter 7

Chapter 8 Hard Stories

8.1 Factors Influencing Convergence Rates
8.2 Behavior of Quasi-Higher-Order Methods
8.3 Some Good News for Turbulence Modeling
8.4 The Myth of the Converged “Solution”
8.5 Esoteric Coding Mistakes
8.6 A False Verification Test of a Particle Tracker
8.8 Hard-Wired Data vs. User Input Data
8.9 Degraded Rate of Convergence Due to User Modeling Errors
8.10 Lessons from Nonlinear Dynamics
8.11 Adaptive and Local Time Stepping, and Steady State
8.12 Other Questions Related to the Steady State
References for Chapter 8
Part III Validation

Chapter 9 Difficulties With Experiments and Validation          backtotop2

9.1 Credulousness
9.2 Historical Methods of Validating Scientific Theories
9.3 The Theory Laden Experiment
9.4 Random and Systematic Errors in Experiments
9.5 Experimental Errors in Physical Properties
9.6 Boundary Conditions, Continuum and Numerical
9.7 Trends, Computational and Experimental
9.8 False Negatives and False Positives
9.9 “Nearby” Problems
9.10 Difficulty of the Option Tree
9.11 Data Sparsity and Lack of Synchronicity: Groundwater, Ocean/Lake, and Meteorology Modeling
9.12 The Effect of Parameter Resolution on Grid Convergence
9.13 Scale of Unsteadiness
9.14 Spatial Scales, Scaling Up, and Dimensionality
9.15 Assumptions of Periodicity
9.16 Other Difficulties of Validation in Aerospace
9.17 Universal Turbulence Models vs. Zonal Modeling
9.18 Strong and Weak Model Definitions and Model Validation
9.19 The Myth of the Totally Validated Code
References for Chapter 9

Chapter 10 Methodologies and Examples of                           backtotop2
Validations, Calibrations, and Certifications

10.1 Sources of Physical Modeling Errors in Aerodynamics CFD
10.2 Accuracy Level for Validation
10.3 Generic Models vs. Realistic Models for Validation and Calibration: Phases of Validation
10.4 CFD and Experimental Facility Corrections
10.5 Verification Must Be Independent of Validation: Airfoil Calculations
10.6 Synergism Between Computation and Validation Experiments
10.7 The Difficulty of Defining a “Nearby” Problem
10.8 Missing Experimental Details
10.9 Onset of 3 Dimensionality in Backstep Flow
10.10 Gray Area: “Validation” from a Calculated Benchmark
10.11 Gray Area: “Validation” of an Experimental Technique by a Computation
10.12 The MADE-2 Experience: Can Groundwater Flow Models Be Validated?
10.13 Dynamic Stall Wind Tunnel Data: Who Does the Tweaking?
10.14 Consortium Effort at CFD Code Certification
10.15 Simulation Team Responsibilities in Validation
10.16 Shifting Responsibilities and Gray Areas
10.17 WUA Benchmarks in 1994 and 1996
10.18 CFD Triathlons
10.19 Canadian CFD Society Test Case
10.20 Workshops
10.21 AGARD 1988 Validation of Computational Fluid Dynamics
10.22 A Case Study for CFD Code Validation Methodology
10.23 Joint Consideration of Experimental and Simulation Uncertainties
10.24 Dynamic Databases for Validation
References for Chapter 10

Part IV Broader Issues                                                             backtotop2

Chapter 11 Code Quality Assurance and Certification

11.1 Introduction
11.2 Quality Assurance (QA) vs. Quality Work
11.3 QA vs. Creativity
11.4 QA and Temperament Types
11.5 The Prevalence of Errors in Scientific Software: Use of Static Analyzers
11.6 Factors of Code Quality Assurance
11.7 Some Components of Project Code Certification
11.8 Engineering Teams and the Division of Labor
11.9 Personnel Roles, Code Levels, and Code Sources
11.10 Desirable (But Not Required) Code Characteristics
11.11 Code Documentation
11.12 Code Module Communication Structure
11.13 Code Updates
11.14 Built in Automatic Error Detection Tests
11.15 Designing for Code Maintenance
11.16 Commercial Codes and their Users
11.17 Code to Code Comparisons
11.18 General Software Certification and ISO-9000 Standards
11.19 QA for Large Public Policy Projects
11.20 QA of Analyses
11.21 QA / Certification of Users and Regulators
11.22 Assessment of Codes? Or Users?
11.23 QA Procedures
11.24 A Template for a QA System
11.25 Concluding Remarks On QA
References for Chapter 11

Chapter 12 Conclusions                                                           backtotop2

12.1 The Overall Process for Quantification of Uncertainty
12.2 Internet Archive
12.3 Fulfilling the Promise of Computational Power

Appendix A. Need for Control of Numerical Accuracy   

I. Introduction
II. Resistance and Objections
III. Difficulties in Applications
IV. Examples of What Can Be Done
V. Conclusions and Recommendations
Appendix: Editorial Policy Statement on the Control of Numerical Accuracy

Appendix B. Other Journal Policy Statements on Control of Numerical Accuracy

Journal of Heat Transfer Editorial Policy Statement on Numerical Accuracy
International Journal for Numerical Methods in Fluids-Editorial
AIAA Editorial Policy Statement on Numerical Accuracy and Experimental Uncertainty
Journal of Fluids Engineering-Editorial and Policy Statement
Policy Statement on the Control of Numerical Accuracy

Appendix C. Comment on Oreskes et al                                  backtotop2

Comment on “Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences”

Appendix D. CFD Parody: Will the Wind Tunnel Replace the Computer?


Appendix E. A Biographical Sketch of Lewis Fry Richardson



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