This book on computational techniques for thermal and fluid-dynamic problems arose from seminars given by the author at the Institute of Nuclear Energy Technology of Tsinghua University in Beijing, China. The book is composed of eight chapters-- some of which are characterized by a scholastic approach, others are devoted to numerical solution of ordinary differential equations of first order, and of partial differential equations of first and second order, respectively. In Chapter IV, basic concepts of consistency, stability and convergence of discretization algorithms are covered in some detail. Other parts of the book follow a less conventional approach, mainly informed by the author’s experience in teaching and development of computer programs. Among these is Chapter III, where the residual method of Orthogonal Collocations is presented in several variants, ranging from the classical Galerkin method to Point and Domain Collocations, applied to numerical solution of partial differential equations of first order. In most cases solutions of fluid dynamic problems are led through the discretization process, to the numerical solutions of large linear systems. Intended to impart a basic understanding of numerical techniques that would enable readers to deal with problems of Computational Fluid Dynamics at research level, the book is ideal as a reference for graduate students, researchers, and practitioners.
Author(s): Maurizio Bottoni
Series: Mechanical Engineering Series
Publisher: Springer
Year: 2021
Language: English
Pages: 551
City: Cham
Preface
Acknowledgements
Contents
List of Figures
List of Tables
List of Acronyms
About the Author
Chapter 1: Ordinary Differential Equations
1.1 Ordinary Differential Equations of First Order
1.1.1 Introduction
1.1.2 Overview of Analytical Solution Methods
1.1.2.1 Exact Differential Equations
1.1.2.2 Differential Equations with Separate Variables
1.1.2.3 Differential Equations with Separable Variables
1.1.2.4 Equations of the Type y' = f(ax + by)
1.1.2.5 Homogeneous Equations
1.1.2.6 Equations of the Type
First Case: The Lines Intersect in a Point (α,β)
Second Case: The Lines Are Parallel
1.1.2.7 Linear Differential Equations
1.1.2.8 Bernoulli Equation
1.1.2.9 Integrating Factor
1.1.2.10 Riccati Equation
1.1.2.11 Equations of the Form x = f(y') or y = f(y')
1.1.2.12 Clairaut Equation
1.1.2.13 D´Alembert - Lagrange Equation
1.1.3 Peano-Picard Solution Method
1.1.4 Euler, Euler-Cauchy and Heun Methods
1.1.5 Runge Kutta Methods
1.1.5.1 Theoretical Background
1.1.5.2 Example of Application to a Fluid-Dynamic Problem
1.1.6 Predictor-Corrector Methods
1.1.7 Systems of Ordinary Differential equations of First Order
1.1.7.1 Theoretical Background
1.1.7.2 Stiff Systems
1.2 Ordinary Differential equations of Second and Higher Order
1.2.1 Ordinary Differential Equations of Second Order
1.2.1.1 Reduction to a System of Ordinary Differential Equations of First Order
1.2.1.2 Finite Difference Method
1.2.2 Ordinary Differential Equations of Higher Order
1.3 The Thomas Algorithm
References
Chapter 2: Partial Differential Equations and the Method of Characteristics
2.1 Partial Differential Equations of First Order
2.1.1 General Definitions
2.1.2 Equations of Characteristics and Formal Solution
2.1.3 Solutions of Linear Damped Wave Equation
2.1.4 Quasi-Linear Undamped Wave
2.2 Partial Differential Equations of Second Order
2.2.1 General Definitions
2.2.2 Equation of Characteristics
2.2.3 The Method of Characteristics (Numerical Application)
2.2.4 Numerical Solution of the Laplace Equation
2.2.4.1 Numerical Solution with Monte Carlo Technique
2.2.4.2 Numerical Solution with the 5-Point Formula
2.2.4.3 Numerical Solution with Under-Relaxation
References
Chapter 3: Methods of Orthogonal Collocations (OC)
3.1 Quasi-linear Damped Wave Model Equation
3.2 Solution of Model Equation with the Subdomain Collocations Method
3.3 Method of Point Collocations
3.4 Galerkin Method of Orthogonal Collocations
3.5 Numerical Examples
3.6 Orthogonal Collocations Applied to One-Dimensional Equations of Sodium Vapor Flow
3.6.1 Subdomain Collocation Method
3.6.2 Point Collocation Method
3.7 Synopsis of the BLOW-3A Code
References
Chapter 4: Numerical Methods for the Solution of the Convection-Diffusion Equation and QUICK Algorithm
4.1 A Statistical Model of Brownian Motion Leading to the Convection-Diffusion Equation (One- and Two-Dimensional Random Walks)
4.2 Model Equation and General Definitions
4.3 Analysis of Convergence and Truncation Error for the Diffusion Equation
4.4 Stability Analysis for Diffusion Equation
4.4.1 The Matrix Method (Eigenvalue Criterion)
4.4.1.1 General Considerations
4.4.1.2 Leverrier Method for the Determination of the Eigenvalue of Maximum Modulus
4.4.1.3 Matrix Method for the Diffusion Equation
4.4.2 Fourier Stability Criterion After von Neumann
4.4.2.1 Explicit and Semi-Implicit Schemes
4.4.2.2 Leapfrog Scheme (Richardson´s Method)
4.4.2.3 Dufort-Frankel Scheme
4.4.2.4 Implicit Scheme
4.4.3 Hirt´s Stability Criterion (For Diffusion Equation)
4.5 The Convection Equation
4.5.1 Central Differences for the Convective Term
4.5.2 Upwind Differences for the Convective Term
4.5.3 The Problem of Numerical Diffusion
4.6 The Convection-Diffusion Equation
4.6.1 Central Differences for the Convective Term
4.6.2 Upwind Differences for the Convective Term
4.6.3 Hirt´s Stability Criterion (Central Differences)
4.7 Numerical Treatment of Conservation Equations with the QUICK Scheme
4.7.1 Introduction
4.7.2 Basic Equations of the Slip Model: Homogeneous Equilibrium Model [SM(HEM)]
4.7.3 Basic Equations of the Separated Phases Model (SPM)
4.7.4 Implementation of the QUICK Method in the Enthalpy Equation of the Slip Model
4.7.5 FRAM-Correction of the QUICK solution
4.7.6 Implementation of the QUICK Method in the Momentum Equation of the Slip Model
References
Chapter 5: Numerical Solution of Large Linear Systems
5.1 Basic Concepts and Definitions
5.2 The Young-Frankel Theory of Successive Over-Relaxation (SOR)
5.2.1 The Numerical Schemes
5.2.2 Relationship Between SOR, Gauss-Seidel, and Jacobi Methods
5.3 Relaxation Methods by Minimization of a Functional (Variational Methods)
5.3.1 N-Dimensional Case
5.3.2 Geometrical Representation of the Two-Dimensional Case
5.4 Gradient Methods as Subclass of Variational Methods
5.4.1 Gradient Methods Proper (Methods of Fastest Descent)
References
Chapter 6: Numerical Solution of Poisson Equation
6.1 Model Equation
6.2 5-Point Formula
6.3 9-Point Formula
6.4 Solution of Model Equation with the Classical Ritz Method
6.5 Alternative Derivation of the Element Equation Based on the Concept of Hilbert Space
6.6 Residual Method of Orthogonal Collocations (RMOC): Galerkin Variant
6.7 Residual Method of Orthogonal Collocations (RMOC): Subdomain Collocations
References
Chapter 7: Derivation and Numerical Solutions of Poisson-Like Equations
7.1 The Pressure Method: Poisson Equation for a Divergence-Free Velocity Field
7.2 Schematic Derivation of the Discrete Poisson Equation
7.2.1 Stability Analysis of Poisson Equation with the Fourier Method
7.3 Derivation of Poisson-Like Equations for Pressure and Enthalpy with Fully Implicit Treatment
7.3.1 Governing Equations
7.3.2 Calculation of Liquid and Vapor Velocity Components from Given Slip Velocity or Slip Ratio
7.3.3 Finite Difference Form of the Continuity Equation
7.3.4 Fully Implicit Treatment of Momentum Conservation Equation
7.3.5 Poisson Equation Describing the Pressure Distribution
7.3.6 Poisson Equation for Enthalpy Distribution
7.4 Solution of the Matrix Equation AX = B with the Method of Doolittle
7.5 Numerical Solution of the Poisson Equation with the Alternating Direction Implicit (ADI) Method
7.5.1 The Classical Variant
7.5.2 Advanced Variants of the ADI Method
7.5.3 Two-Dimensional Model Problem
7.5.4 Three-Dimensional Problem from Nuclear Reactor Safety Analysis
7.5.5 ADI Algorithm for Two-Phase Flow Calculations
7.6 Historical Note About the Computer Codes BACCHUS-3D/TP and COMMIX-2
References
Chapter 8: Numerical Treatment of the Transport Equations of Turbulence
8.1 Introduction to Turbulence
8.1.1 Kolmogorov Scales of Turbulence
8.1.2 Definitions of Main Turbulence Parameters
8.1.3 Analytical Expressions of the Energy Spectrum
8.1.4 Reynolds Stress Tensor and Anisotropy Tensor
8.1.5 Taylor´s Hypothesis and the Mixing Length Model
8.1.6 Dimensionless Quantities
8.1.7 Overview of Turbulence Models
8.1.7.1 The K-ε Turbulence Model
8.1.7.2 The Algebraic Stress Model (ASM)
8.1.7.3 The Large Eddies Simulation (LES) Model
8.1.7.4 Historical Development of Turbulence Models at the Argonne National Laboratory, USA
8.2 Transport Equations for Scalar Fluxes
8.3 Transport Equation for Variance of Temperature Fluctuations [g Equation]
8.4 Transport Equations for Reynolds Stress Model (RSM)
8.5 Transport Equation for Dissipation of Turbulent Kinetic Energy [ε Equation]
8.6 Synopsis of Governing Equations, Constants, Transport Quantities and Source Terms
8.7 Reynolds Equations for Homogeneous Turbulence
References
Appendix A: List of Computer Programs
Appendix A.I
Appendix A.II
Appendix A.III
Appendix A.IV
Appendix A.V
Appendix A.VI
Appendix A.VIII
Formula Synopsis: Concepts and Definitions, Matrix Algebra, Determinants, and Vector and Tensor Operations
Concepts and Definitions
Matrix Algebra
Vector and Tensor Operations
A Few Remarks on the Classification of Differential Equations
Partial Differential Equations of First Order
Systems of Partial Differential Equations of First Order
Partial Differential Equations of Second Order
Partial Differential Equations of Higher Order (n > 2)
Nomenclature
Greek Symbols
Indices
Special Symbols
Historical Notes
References
Literature (Relevant for the Topics Treated but Not Quoted in the Text)
Index
