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Computational Fluid Dynamics
Fourth Edition
Volume 1


2000, 486 PP

     Hoffmann, Chiang
     

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This three-volume text is designed for use in introductory, intermediate, and advanced courses in computational fluid dynamics (CFD) and computational fluid turbulence (CFT). The fundamentals of computational schemes are established in the first volume, presented in nine chapters. The first seven chapters include basic concepts and introductory topics, whereas Chapters 8 and 9 cover advanced topics. In the second volume, the fundamental concepts are extended for the solution of the Euler,  Parabolized Navier-Stokes, and Navier-Stokes equations. Finally, unstructured grid generation schemes, finite volume techniques, and finite element method are explored in the second volume. In the third volume, turbulent flows and several computational procedures for the solution of turbulent flows are addressed. The first two volumes are designed such that they can be easily adapted to two sequential courses in CFD. Students with an interest in fluid mechanics and heat transfer should have sufficient background to undertake these courses. In addition, fundamental  knowledge of programming and graphics is essential for the applications of methods presented throughout the text. Typically, the first course is offered at the undergraduate level, whereas the second course can be offered at the graduate level. The third volume of the text is designed for a course with the major emphasis on turbulent flows.          

The general approach and presentation of the material is intended to be brief, with emphasis on applications. A fundamental background is established in the first seven chapters, where various model equations are presented, and the procedures used for the numerical solutions are illustrated. For purposes of analysis, the numerical solutions of the sample problems are presented in tables. In many instances, the behavior of a solution can be easily analyzed by considering graphical presentations of the results; therefore, they are included in the text as well. Before attempting to solve the problems proposed at the end of each chapter, the student should try to generate numerical solutions of the sample problems, using codes developed individually or available codes modified for the particular application. The results should be verified by comparing them with the solutions presented in the text. If an analytical solution for the proposed problem is available, the numerical solution should be compared to the analytical solution.          

The emphasis in the first volume is on finite difference methods. Chapter 1 classifies the various partial differential equations, and presents some fundamental concepts and definitions. Chapter 2 describes how to achieve approximate representation of partial derivatives with finite difference equations. Chapter 3 discusses procedures for solving parabolic equations. Stability analysis is presented in Chapter 4. The order for Chapters 3 and 4 can be reversed. In fact, the results of stability analysis are required for the solution of parabolic equations in Chapter 3. The reason that the solution procedure of parabolic equations is developed first in Chapter 3 is to spread the computer code developments, since they require a substantial amount of time compared to other assignments. This will prevent the concentration of code development in the latter part of the course. Procedures for solving elliptic and hyperbolic partial differential equations are presented in Chapters 5 and 6, respectively. Chapter 7 presents a scalar model equation equivalent of the Navier-Stokes equations. In this chapter, numerical algorithms are investigated to solve a scalar model equation, which includes unsteady, convective, and diffusive terms.          

The solution schemes established in the first seven chapters are extended to the solution of a system of partial differential equations in Chapter 8.In particular, the Navier-Stokes equations for incompressible flows in primitive variables, as well as vorticity-stream function formulations, are reviewed. Subsequently, the numerical schemes and specification of appropriate boundary conditions are introduced. Finally, Chapter 9 is designed to introduce the structured grid generation techniques. Various schemes, along with applications, are illustrated in this chapter.          

In addition to this three volume text, Computational Fluid Dynamics, a three volume text, Student Guide to CFD, has been developed. The text, Student Guide to CFD, includes computer codes, description of input/output, and additional example problems. However, it is important to emphasize that computer code developments an important aspect of CFD, and that, in fact, one learns a great deal about the numerical schemes and their behavior as one develops, debugs, and validates his or her own computer code. Therefore, it is important to state here that the computer codes provided in the text Student Guide to CFD should not be used as an avenue to replace that aspect of CFD and that code development must be an important objective of the learning process. However, these codes can be used as a basis upon which one may develop other codes, or the codes can be modified for other applications.

Table of Contents   

    

Chapter:

    
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the Table of Contents for that Chapter
       

1

Classification of Partial Differential Equations
    
2
  
Finite Difference Formulations  
3
  
Parabolic Partial Differential Equations  
4
  
Stability Analysis  
5
  
Elliptic Equations  
6
  
Hyperbolic Equations  
7
  
Scalar Representation of the Navier-Stokes Equations
8
  
Incompressible Navier-Stokes Equations
9
  
Grid Generation - Structured Grids
  
  

    
Appendices
    

A An Introduction to Theory of Characteristics: 
Wave Propogation
   
B Tridiagonal System of Equations
  
C Derivation of Partial Derivatives for the Modified Equations
   
D Basic Equations of Fluid Mechanics
   
E Block-Tridiagonal System of Equations
   
F Derivatives in the Computational Domain
   
References
   
Index
   

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Preface
Introduction
   

Chapter One:
Classification of Partial Differential Equations

    

Introductory Remarks, Linear and Nonlinear Partial Differential Equations, Second-Order Partial Differential Equations, Elliptic Equations, Parabolic Equations, Hyperbolic Equations, Model Equations, System of First-Order Partial Differential Equations, System of Second-Order Partial Differential Equations, Initial and Boundary Conditions, Remarks and Definitions, Summary Objectives, Problems.

    


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Chapter Two:
Finite Difference Formulations
    

Introductory Remarks, Taylor Series Expansion, Finite Difference by Polynomials, Finite Difference Equations, Applications, Finite Difference Approximation of Mixed Partial Derivatives, Taylor Series Expansion, The Use of Partial Derivatives with Respect to One Independent Variable, Summary Objectives, Problems.

    


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Chapter Three:
Parabolic Partial Differential Equations

   

Introductory Remarks, Finite Difference Formulations, Explicit Methods, The Forward Time/Central Space Method, The Richardson Method, The DuFort-Frankel Method, Implicit Methods, The Laasonen Method, The Crank-Nicolson Method, The Beta Formulation, Applications, Analysis, Parabolic Equations in Two-Space Dimensions, Approximate Factorization, Fractional Step Methods, Extension to Three-Space Dimensions, Consistency Analysis of Finite Difference Equations, Linearization, Irregular Boundaries, Summary Objectives, Problems.

    


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Chapter Four:
Stability Analysis  
   

Introductory Remarks, Discrete Perturbation Stability Analysis, Von Neumann Stability Analysis, Multidimensional Problems, Error Analysis, Modified Equation, Artificial Viscosity, Summary Objectives, Problems.


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Chapter Five:
Elliptic Equations
   

Introductory Remarks, Finite Difference Formulations, Solution Algorithms, The Jacobi Iteration Method, The Point Gauss-Seidel Iteration Method, The Line Gauss-Seidel Iteration Method, Point Successive Over-Relaxation Method (PSOR), Line Successive Over-Relaxation Method (LSOR), The Alternating Direction Implicit Method (ADI), Applications, Summary Objectives, Problems.

   


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Chapter Six:
Hyperbolic Equations
    

Introductory Remarks, Finite Difference Formulations, Explicit Formulations, Euler's FTFS Method, Euler's FTCS Method, The First Upwind Differencing Method, The Lax Method, Midpoint Leapfrog Method, The Lax-Wendroff Method, Implicit Formulations, Euler's BTCS Method, Implicit First Upwind Differencing Method, Crank-Nicolson Method, Splitting Methods, Multi-Step Methods, Richtmyer/Lax-Wendroff Multi-Step Method,The MacCormack Method, Applications to a Linear Problem, Nonlinear Problem, The Lax Method, The Lax-Wendroff Method, The MacCormack Method, The Beam and Warming Implicit Method, Explicit First-Order Upwind Scheme, Implicit First-Order Upwind Scheme, Runge-Kutta Method, Modified Runge-Kutta Method, Linear Damping Application, Flux Corrected Transport, Application, Classification of Numerical Schemes, Monotone Schemes, Total Variation Diminishing Schemes, Essentially Non-Oscillatory Schemes, TVD Formulations, First Order TVD Schemes, Entropy Condition, Application, Second-Order TVD Schemes, Harten-Yee Upwind TVD Limiters, Roe-Sweby Upwind TVD Limiters, Davis-Yee Symmetric TVD Limiters, Modified Runge-Kutta Method with TVD, Summary Objectives, Problems.

   


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Chapter Seven:
Scalar Representation of the Navier-Stokes Equations
    

Introductory Remarks, Model Equation, Equations of Fluid Motion, Numerical Algorithms, FTCS Explicit, FTBCS Explicit, DuFort-Frankel Explicit, MacCormack Explicit, MacCormack Implicit, BTCS Implicit, BTBCS Implicit, Applications: Nonlinear Problem, FTCS Explicit, FTBCS Explicit, DuFort-Frankel Explicit, MacCormick Explicit, MacCormick Implicit, BTCS Implicit, BTBCS Implicit, Modified Runge-Kutta, Second-Order TVD Schemes, Summary Objectives, Problems.

   


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Chapter Eight:
Incompressible Navier-Stokes Equations
    

Introductory Remarks, Incompressible Navier-Stokes Equations, Primitive Variable Formulations, Vorticity-Stream Function Formulations, Comments on Formulations, Poisson Equation for Pressure: Primitive Variables, Poisson Equation for Pressure: Vorticity-Stream Function Formulation, Numerical Algorithms: Primitive Variables, Steady Flows, Artificial Compressibility, Solution on a Regular Grid, Crank-Nicolson Implicit, Boundary Conditions, Body Surface, Far-Field, Symmetry, Inflow, Outflow, An Example, Staggered Grid, Marker and Cell Method, Implementation of the Boundary Conditions, DuFort-Frankel Scheme, Use of the Poisson Equation for Pressure, Unsteady Incompressible Navier-Stokes Equations, Numercial Alogithms: Vorticity-Stream Functions Formulations, Vorticity Transport Equation, Steam Function Equation, Boundary Conditions, Body Surface, Far-Field, Symmetry, Inflow, Outflow, Application, Temperature Field, The Energy Equation, Numerical Schemes, Boundary Conditions, Problems.

    


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Chapter Nine:
Grid Generation - Structured Grids
   

Introductory Remarks, Transformation of the Governing Partial Differential, Metrics and the Jacobian of Transformation, Grid Generation Techniques, Algebraic Grid Generation Techniques, Partial Differential Equation Techniques, Elliptic Grid Generators, Simply-Connected Domain, Doubly-Connected Domain, Multiply-Connected Domain, Coordinate System Control, Grid Point Clustering, Orthogonality at the Surface, Hyperbolic Grid Generation Techniques, Parabolic Grid Generators, Problems.

   


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