Computational Modeling for Fluid Flow and Interfacial Transport

by Shyy, Wei

• ISBN13: 9780486453033
• ISBN10: 0486453030
• Format: Paperback
• Copyright: 2006-10-06
• Publisher: Dover Publications

Summary

Practical applications and examples highlight this treatment of computational modeling for handling complex flowfields. The author draws upon personal research to address both macroscopic and microscopic features. A reference for researchers and graduate students of many backgrounds, it also functions as a text for learning essential computation elements. 1994 edition.

Table of Contents

PREFACE	vii
PART I. BASIC CONCEPTS OF FINITE DIFFERENCE METHODS
CHAPTER I. INTRODUCTION TO FINITE DIFFERENCE METHODS
1. Basic Concepts of Finite Difference Schemes	3	(4)
1.1 Finite Difference Operators
1.2 Order of Accuracy
2. Solution of Finite Difference Equations	7	(2)
3. Order of Accuracy of a Difference Scheme: Global and Local	9	(5)
3.1 Scheme I. Δ—operator
3.2 Scheme II. δ—operator
3.3 Boundary Treatment
4. Stability of Difference Schemes	14	(9)
4.1 Illustration of the Instability of a Finite Difference Scheme
4.2 Stability
4.3 Forward Difference (Explicit Euler Method)
4.4 Backward Difference (Implicit Euler Method)
4.5 Central Difference
4.6 Trapezoid Difference
4.7 Some Relevant Facts
4.8 The Issue of Accuracy Versus Stability
4.9 Systems of Equations
5. Discretization of PDE's and Error Norms	23	(2)
5.1 Discretization of PDEs: an Example
5.2 Local Error and its Relationship to Global Error
5.3 Error Norms
6. Further Reading	25	(2)
CHAPTER II. PARABOLIC EQUATIONS
1. Background on PDEs	27	(2)
1.1 Three Operators and Classes of Equations
1.2 Classification of Equations: Second—Order PDEs
2. Analytical Background for Parabolic PDEs	29	(1)
2.1 Fourier Analysis
2.2 The Min-Max Principle
2.3 Energy Identity
2.4 Characteristics of Solutions of Parabolic PDEs
3. Explicit Numerical Schemes for Parabolic PDEs	30	(10)
3.1 An Example
3.2 Methods Employed to Analyze Stability of Difference Schemes
3.3 Consistency, Stability and Convergence
3.4 Lax's Equivalence Theorem
3.5 DuFort-Frankel Explicit Scheme
4. General Two-Step Schemes for Parabolic PDEs	40	(4)
4.1 Stability
4.2 Accuracy Improvement Via Manipulation of the Degree of Implicitness
5. Keller's Box Method	44	(2)
6. Leap-Frog Scheme	46	(2)
6.1 Heat Conduction Equation
6.2 First-Order ODE
6.3 The Wave Equation
7. Multi-Dimensional Problems	48	(1)
8. Choice of Method: Explicit or Implicit?	49	(1)
9. Solution Methods for Implicit Scheme: the ADI Method	50	(1)
10. Nonuniform Meshes	51	(1)
11. Treatment of Boundary Conditions	52	(2)
11.1 Central Difference Treatment of the Boundary Condition
11.2 One-Sided Treatment of the Boundary Condition
12. Equations Containing Variable Coefficients	54	(3)
12.1 Equations of Variable Diffusivity
12.2 The 1-D Unsteady Conduction Equation in Different Coordinates
13. Summary of Schemes for the Heat Equation Ut = Uxx	57	(2)
CHAPTER III. ELLIPTIC EQUATIONS
1. Introduction	59	(1)
2. Basic Analysis and Analogy between Difference and Differential Equations	59	(4)
2.1 Positivity Property
2.2 Boundedness Property
2.3 Stability and Convergence
3. Laplace Equation in a Square	63	(3)
3.1 Dirichlet BC
3.2 Neumann BC
4. Solution of Linear Equations: Classical Iterative Methods	66	(14)
4.1 Basics
4.2 Convergence of Iterative Methods
4.3 Asymptotic Rate of Convergence
4.4 A Sufficient Condition for Convergence: Diagonal Dominance
4.5 Analogy Between Iterative Methods for Elliptic Equations and Time Marching Methods for Parabolic Equations
4.6 Convergence Rate of Classical Iterative Methods
4.7 A Simple Acceleration Technique
4.8 Line or Block Iterative Methods
4.9 Effect of Grid Aspect Ratio on Convergence Rate
4.10 First-Order Derivatives
5. Stone's Strongly Implicit Procedure (SIP)	80	(9)
6. Multigrid Method	81	(8)
6.1 Heuristic Analysis
6.2 Spectral Analysis
6.3 Restriction, Prolongation and Scheduling
CHAPTER IV. HYPERBOLIC EQUATIONS
1. Introduction and Analytical Background	89	(4)
1.1 First-Order Hyperbolic Equation
1.2 Second-Order Hyperbolic Equation
1.3 Conservation Laws and Weak Form Solutions
2. Naive Schemes	93	(4)
2.1 Scheme I: Forward in Time and Space
2.2 Scheme II: Forward in Time and Central in Space
2.3 Scheme III: Forward in Time and Backward in Space
3. More Complicated Schemes	97	(6)
3.1 Courant-Isaacson-Rees (CIR) Scheme
3.2 Lax-Friedrichs Scheme
3.3 Lax-Wendroff Scheme
3.4 Unified Expression: q-Schemes
3.5 Leapfrog Scheme
3.6 Crank-Nicolson Scheme for Ut + aUx = 0
3.7 Box Scheme for Ut + aU, = 0
4. Numerical Dissipation and Dispersion	103	(6)
4.1 Background
4.2 Monotone Schemes
4.3 Godunov's Theorem
4.4 High Resolution Schemes
4.5 Relation of Cell Peclet Number with Numerical Dispersion and Dissipation
5. System of Equations	109	(4)
5.1 Introduction
5.2 Application of the CIR Scheme to the System of Equations
PART II PRESSURE-BASED ALGORITHMS AND THEIR APPLICATIONS
CHAPTER V. PRESSURE-BASED ALGORITHMS
1. Introduction	113	(3)
2. Pressure-Based Formulation for All Flow Speeds	116	(12)
2.1 Overview of a Popular Pressure-Based Method
2.2 Governing Equations
2.3 Formulation of the Pressure Correction Equation
2.4 Compressibility Effect
3. Choice of Velocity Variables	128	(5)
3.1 Basic Notions
3.2 Physical and Geometric Conservation Laws
4. Grid	133	(10)
4.1 Staggered Grid Layout
4.2 Adaptive Grid
5. Open Boundary Treatment	143	(7)
6. Convection Treatment	150	(34)
6.1 Background
6.2 Non-Linear Filtering Method
6.3 Some Currently Popular Schemes for Recirculating Flows
6.4 Controlled Variation Scheme
7. Convergence and Matrix Solver	184	(19)
7.1 General Characteristics
7.2 Multigrid Methodology
7.3 Examples
7.4 Turbulent Flow Treatment
8. Composite Grid Method	203	(35)
8.1 Introduction
8.2 Organization of Composite Grids
8.3 Global Conservation Conditions for the Staggered Grid
8.4 Conservation Strategy for Pressure-Based Method
8.5 Explicit Conservation Procedure
8.6 Conditions for Local and Global Conservation
8.7 Composite Grid Calculation
8.8 Composite Multigrid
9. Concluding Remarks	238	(1)
CHAPTER VI. PRACTICAL APPLICATIONS
1. Three-Dimensional Combustor Flow Simulation	239	(5)
1.1 Background
1.2 Combustion Model
1.3 Numerical Algorithm
1.4 Case Studies
2. High Pressure Discharge Lamp	244	(14)
2.1 Background
2.2 Case Studies
3. Two-Phase Thermocapillary Flow under Normal and Microgravity Conditions	258	(5)
3.1 Background
3.2 Case Studies
4. Flow in a 360° Passage with Multiple Airfoils	263	(12)
4.1 Background and Problem Formulation
4.2 Results and Discussion
5. Concluding Remarks	275	(4)
PART III. INTERFACIAL TRANSPORT
CHAPTER VII. BASIC CONCEPTS OF THERMODYNAMICS
1. Basic Concepts	279	(4)
1.1 Thermodynamic Equilibrium and Basic Laws
1.2 The Gibbs Equation
1.3 The Gibbs - Duhem Equation
1.4 Interpretation of Free Energy and Chemical Potential under Phase Equilibrium
2. Thermodynamic Potentials	283	(2)
2.1 Gibbs Free Energy
2.2 Condition for Equilibrium
2.3 Implications of Equilibrium Conditions during Phase Change
3. Thermodynamics of Interfaces	285	(3)
3.1 A Heuristic Account of Interface Formation
3.2 Origins of the Forces
3.3 Latent Heat
3.4 Latent Heat of Vaporization: ΔHl-g
3.5 Solidification and Melting
3.6 Enthalpy as a Function of Temperature
4. Phase Equilibria	288	(5)
4.1 The Clausius-Clapeyron Equation
4.2 Interpretation of the Clausius-Clapeyron Equation
5. Surface Tension and the Gibbs-Thomson Effect	293	(5)
5.1 A Simple Kinetic Estimation
5.2 The Gibbs-Thomson Effect
5.3 The Young-Laplace Equation
5.4 The Temperature Form of the Gibbs-Thomson Equation
6. Further Reading	298	(1)
CHAPTER VIII. THERMOFLUID PHENOMENA INVOLVING CAPILLARITY AND GRAVITY
1. Meniscus Formation and Contact Angle	299	(25)
1.1 Edge-Defined Fibre Growth Process
1.2 Boundary Condition: the Contact Angle
1.3 Basic Meniscus Behavior with a Static Contact Angle
1.4 Quasi-equilibrium Motion with Hysteresis
1.5 Surface Tension and Capillary Flow
2. Jet Breakup and Drop Formation	324	(4)
2.1 Basic Notions
2.2 Cylindrical Jets
3. Benard Convection	328	(14)
3.1 Surface Tension Induced Convection
3.2 Linear Stability Analysis
3.3 Normal Mode Analysis
3.4 Buoyancy with Surface Tension
3.5 Scaling Analysis
4. Case Studies	342	(10)
4.1 Buoyancy Induced Convection
4.2 Surface Tension Driven Convection
4.3 Double-Diffusive Convection in a Square Cavity
5. Further Reading	352	(1)
CHAPTER IX. PHYSICAL AND COMPUTATIONAL ISSUES IN PHASE-CHANGE DYNAMICS
1. The Driving Force for Solidification	353	(1)
2. Nucleation in Pure Materials	354	(12)
2.1 Homogeneous Nucleation
2.2 Heterogeneous Nucleation
2.3 Heat Conduction with Phase Change
2.4 Heat Flow and Interface Stability for Pure Materials
2.5 Steady State Solution of the Planar Phase Change Problem
2.6 Ivantsov Solution of the Phase Change Problem
2.7 Inclusion of the Gibbs-Thomson Effect
3. Alloy Solidification	366	(6)
3.1 Linear Stability Analysis
3.2 Constitutional Supercooling
3.3 Linear Stability Analysis of Mullins and Sekerka
3.4 Effect of Convection on Interface Morphology
4. General Formulation for Interface Tracking	372	(4)
4.1 Literature Survey
4.2 Governing Laws
5. Modelling and Related Issues	376	(20)
5.1 Issues of Scaling
5.2 A Computational Procedure for Tracking Phase Fronts
5.3 Motion of Curved Fronts
5.4 A Scaling Procedure for the Conduction Driven Phase Change Problem
5.5 Morphological Model with Convection Effect
5.6 Examples
6. Tracking of Highly Distorted Fronts	396	(28)
6.1 Background
6.2 Basic Methodology
6.3 Mergers of Interfaces
6.4 Summary of Merger Procedure
6.5 Results for Non—Merging Interfaces
6.6 Results for Merging Interfaces
7. Enthalpy Formulation for Phase Change Problem	424	(8)
7.1 Basic Concepts and Implementation
7.2 Convection Effects
8. Case Studies	432	(31)
8.1 Effect of Convection on Phase Change
8.2 Phase Change with Natural Convection Only
8.3. Interaction of Thermocapillary and Natural Convection Flows During Phase Change
8.4 Effect of Gravity Jitter
8.5 Effect of Geometry on Transport Characteristics
8.6 Ingot Casting
8.7 Low Pressure Conditions
9. Further Reading	463	(2)
REFERENCES	465	(34)
INDEX	499