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Computational Fluid Dynamics
Fourth Edition
Volume 2

2000, 469 PP
   
Hoffmann, Chiang     

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The fundamental concepts of computational schemes established in the first volume are extended to the solution of Euler equations, Parabolized Navier-Stokes equations, and Navier-Stokes equations, along with treatment of boundary conditions.  In addition, chemically reacting flows, unstructured grids, finite volume schemes, and finite element method at the introductory level are included.

This volume begins with a review of the basic concepts which is presented in Chapter 10.  Subsequently, the transformation of the equations of fluid motion from physical space to computational space is provided in Chapter 11. This chapter also includes the linearization of the equations as well as the derivation of the Jacobian matrices.  Chapter 12 presents numerical schemes for the solution of the Euler equations for inviscid flow fields.  Specifications of the boundary conditions, along with illustrated examples, are provided in this chapter.  Chapter 13 presents Parabolized Navier-Stokes (PNS) equations and a numerical algorithm for solution.  The shock fitting procedure is discussed in this chapter as well.  The Navier-Stokes equations and various numerical schemes for solutions are discussed in Chapter 14.  Specification of boundary conditions, derivation of governing equations, and comparison of several types of boundary conditions are provided in Chapter 15.  An extension of the governing equations to include the effect of chemistry for hypersonic flowfield computations is included in Chapter 16.  To familiarize the reader with unstructured grids which are used in conjunction with finite volume and finite element schemes, they are introduced in Chapter 17.  It develops some fundamental concepts and explores two techniques for generation of unstructured grids in two dimensions.  Finally, finite volume schemes and finite element method are developed at the introductory level in Chapters 18 and 19, respectively.      

Several computer codes are developed based on the materials presented in this text.  These codes, manuals, and additional examples are presented in the test, Student Guide to CFD- Volume II.

Table of Contents   

    

Chapter:

    
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10 A Review
    
11 Transformation of the Equations of Fluid Motion from Physical Space to Computational Space
    
12
  
Euler Equations  
13
  
Parabolized Navier-Stokes Equations  
14
  
Navier-Stokes Equations  
15
  
Boundary Conditions  
16
  
An Introduction to High Temperature Gases
17
  
Grid Generation - Unstructured Grids
18
  
Finite Volume Method
19
  
Finite Element Method
  
  

    
Appendices
    

G An Introduction to Theory of Characteristics: 
Euler Equations
   
H Computation of Pressure at the Body Surface
  
I Rate of Formation of Species  
References
   
Index
   

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Preface
Introduction
   

Chapter Ten:
A Review
    

Introductory Remarks, Classification of Partial Differential Equations, Linear and Nonlinear PDEs, Classification Based on Characteristics, Boundary Conditions,  Finite Difference Equations, Parabolic Equations, One-Space Dimension, Multi-Space Dimensions Elliptic Equations, Hyperbolic Equations, Linear Equations, Nonlinear Equations, Stability Analysis,  Error Analysis,  Grid Generation – Structured Grids,  Transformation of the Equations from the Physical Space to Computational Space.

    


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Chapter Eleven:
Transformation of the Equations of Fluid Motion from Physical Space to Computational Space
    

Introductory Remarks,  Generalized Coordinate Transformation,  Equations for the Metrics,  Nondimensionalization of the Equations of Fluid Motion,  Navier-Stokes Equations,  Linearization,  Inviscid and Viscous Jacobian Matrices,  Thin-Layer Approximation,  Parabolized Navier-Stokes Equations,  Two-Dimensional Planar or Axisymmetric Formulation,  Incompressible Navier-Stokes Equations,  Inviscid and Viscous Jacobian Matrices,  Two-Dimensional Incompressible Navier-Stokes Equations,  Problems.

    


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Chapter Twelve:
Euler Equations
   

Introductory Remarks,  Euler Equations,  Quasi One-Dimensional Euler Equations,  Numerical Issues,  Explicit Formulations,  Steger and Warming Flux Vector Splitting,  Van Leer Flux Vector Splitting,  Modified Runge-Kutta Formulation,  Second-Order TVD Formulation,  Harten-Yee Upwind TVD,  Roe-Sweby Upwind TVD,  David-Yee Symmetric TVD,  Modified Runge-Kutta Scheme with TVD,  Implicit Formulations,  Steger and Warming Flux Vector Splitting,  Boundary Conditions,  Application 1: Diverging Nozzle Configuration,  Supersonic Inflow, Supersonic Outflow,  Analytical Solution,  Numerical Solutions,  Supersonic Inflow, Subsonic Outflow,  Analytical Solution,  Numerical Solutions,  Grid Clustering,  Global Time Step and Local Time Step,  Application 2: Shock Tube or Riemann Problem,  Problem Description,  Analytical Solution,  Numerical Solution,  Two-Dimensional Planar and Axisymmetric Euler Equations,  Numerical Considerations,  Explicit Formulations,  Steger and Warming Flux Vector Splitting,  Matrix Manipulations,  Existence of Zero Metrics Within the Domain,  Eigenvector Matrices,  Van Leer Flux Vector Splitting,  Modified Runge-Kutta Formulation,  Second-Order TVD Formulation,  Harten-Yee Upwind TVD,  Roe Sweby Upwind TVD,  David-Yee Symmetric TVD,  Modified Runge-Kutta Scheme with TVD,  Boundary Conditions,  Body Surface,  Symmetry,  Inflow,  Outflow,  Boundary Conditions Based on Characteristics,  Inflow Boundary,  Outflow Boundary,  Determination of Flow Variables,  Implicit Formulations,  Steger and Warming Flux Vector Splitting,  Computation of the Jacobian Matrices,  Boundary Conditions,  Application:  Axisymmetric/Two-Dimensional Problems,  Supersonic Channel Flow,  Grid Generation,  Numerical Scheme,  Analytical Solution,  The Physical Domain and Flow Conditions,  Initial Conditions and Time Step,  Results,  Axisymmetric Blunt Body,  Concluding Remarks,  Problems.

    


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Chapter Thirteen:
Parabolized Navier-Stokes Equations   

Introductory Remarks,  Governing Equations of Motion,  Streamwise Pressure Gradient,  Numerical Algorithm,  Boundary Conditions,  Extension to Three Dimensions,  Numerical Algorithm,  Numerical Damping Terms,  Shock Fitting Procedure,  Extension to Three-Dimensions,  Application,  Summary Objectives,  Problems.


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Chapter Fourteen:
Navier-Stokes Equations
   

Introductory Remarks,  Navier-Stokes Equations ,Thin-Layer Navier-Stokes Equations,  Numerical Algorithms,  Explicit Formulations,  MacCormack Explicit Formulation,  Flux Vector Splitting,  Modified Runge-Kutta Scheme,  Boundary Conditions,  Implicit Formulations,  Flux Vector Splitting,  Higher-Order Flux-Vector Splitting,  Second-Order Accuracy in Time,  LU Decomposition,  Extension to three-Dimensions,  Explicit Flux Vector Splitting Scheme,  Implicit Formulation,  Concluding Remarks,  Problems.

   


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Chapter Fifteen:
Boundary Conditions
   

Introductory Remarks,  Classification of Schemes for Specification of Boundary Conditions,  Category Two Boundary Conditions: Characteristics Based Boundary Conditions,  Mathematical Developments,  Slip Wall Boundary Condition,  Nonconservative (Primitive) Variables,  Conservative Variables,  No-Slip Wall Boundary Condition,  Nonconservative (Primitive) Variables,  Conservative Variables,  Inflow/Outflow Boundary Conditions,  Category Three Boundary Conditions: Addition of Buffer Layer,  Applications,  Application 1: Moving Shock Wave,  Application 2: Flow Over a Compression Corner,  Concluding Remarks.

   


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Chapter Sixteen:
An Introduction to High Temperature Gases
   

Introductory Remarks,  Fundamental Concepts,  Real Gas and Perfect Gas,  Partial Pressure,  Frozen Flow,  Equilibrium Flow,  Nonequilibrium Flow,  Various Modes of Energy,  Reaction Rates,  Five-Species Model,  Quasi One-Dimensional Flow/Equilibrium Chemistry,  Quasi One-Dimensional Flow/Nonequilibrium Chemistry,  Species Continuity Equation,  Coupling Schemes,  Numerical Procedure for the Loosely Coupled Scheme,  Applications,  Quasi One-Dimensional Flow,  Two-Dimensional Axisymmetric Flow,  Concluding Remarks.

   


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Chapter Seventeen:
Grid Generation - Unstructured Grids
    

Introductory Remarks,  Domain Nodalization,  Domain Triangulation,  The Advancing Front Method,  Simply-Connected Domain,  Multiply-Connected Domain,  The Delaunay Method,  Geometrical Description,  Outline of the Algorithm,  An Illustrative Example,  Concluding Remarks,  Problems.

    


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Chapter Eighteen:
Finite Volume Method
  

Introductory Remarks,  General Description of the Finite Volume Method,  Cell Centered Scheme,  Nodal Point Scheme,  Two-Dimensional Heat Conduction Equation,  Interior Triangles,  Boundary Triangles,  Dirichlet Type Boundary Condition,  Neumann Type Boundary Condition,  Flux Vector Splitting Scheme,  Interior Triangles,  Boundary Triangles,  Concluding Remarks,  Problems.

   


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Chapter Nineteen:
Finite Element Method
  

Introductory Remarks,  Optimization Techniques,  General Description and Development of the Finite Element Method,  Two-Dimensional Heat Conduction Equation,  Construction of the Global Matrix,  Boundary Conditions,  Reduction of the Half-Bandwidth of the Global Matrix.

   


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