Computational Fluid
Dynamics
Fourth Edition
Volume 1
2000, 486 PP
Hoffmann, Chiang
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Table of Contents
This
threevolume 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 NavierStokes, and NavierStokes
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
NavierStokes 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 NavierStokes equations for incompressible flows in
primitive variables, as well as vorticitystream 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: 
Click on Chapter Title to View
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 NavierStokes Equations 
8

Incompressible
NavierStokes 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 
BlockTridiagonal
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, SecondOrder
Partial Differential Equations, Elliptic Equations, Parabolic Equations, Hyperbolic
Equations, Model Equations, System of FirstOrder Partial Differential Equations,
System of SecondOrder 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 DuFortFrankel Method, Implicit
Methods, The Laasonen Method, The CrankNicolson Method, The Beta Formulation, Applications,
Analysis, Parabolic Equations in TwoSpace Dimensions, Approximate Factorization, Fractional
Step Methods, Extension to ThreeSpace 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 GaussSeidel Iteration Method, The Line GaussSeidel Iteration Method,
Point Successive OverRelaxation Method (PSOR), Line Successive OverRelaxation 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 LaxWendroff Method, Implicit Formulations, Euler's BTCS Method,
Implicit First Upwind Differencing Method, CrankNicolson Method, Splitting Methods, MultiStep
Methods, Richtmyer/LaxWendroff MultiStep Method,The MacCormack Method,
Applications to a Linear Problem, Nonlinear Problem, The Lax Method, The LaxWendroff Method,
The MacCormack Method, The Beam and Warming Implicit Method, Explicit FirstOrder Upwind Scheme,
Implicit FirstOrder Upwind Scheme, RungeKutta Method, Modified RungeKutta Method, Linear
Damping Application, Flux Corrected Transport, Application, Classification
of Numerical Schemes, Monotone Schemes, Total Variation Diminishing Schemes,
Essentially NonOscillatory Schemes, TVD Formulations, First Order TVD Schemes, Entropy Condition,
Application, SecondOrder TVD Schemes, HartenYee Upwind TVD Limiters,
RoeSweby Upwind TVD Limiters, DavisYee Symmetric TVD Limiters, Modified
RungeKutta Method with TVD, Summary Objectives, Problems.


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Chapter Seven:
Scalar Representation of the NavierStokes Equations


Introductory Remarks, Model Equation, Equations of Fluid Motion, Numerical Algorithms,
FTCS Explicit, FTBCS Explicit, DuFortFrankel Explicit, MacCormack Explicit,
MacCormack Implicit, BTCS Implicit, BTBCS Implicit, Applications: Nonlinear Problem, FTCS Explicit,
FTBCS Explicit, DuFortFrankel Explicit, MacCormick Explicit, MacCormick Implicit, BTCS Implicit,
BTBCS Implicit, Modified RungeKutta, SecondOrder TVD Schemes, Summary Objectives, Problems.


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Chapter Eight:
Incompressible NavierStokes Equations


Introductory Remarks, Incompressible NavierStokes Equations, Primitive Variable Formulations,
VorticityStream Function Formulations, Comments on Formulations, Poisson Equation for Pressure:
Primitive Variables, Poisson Equation for Pressure: VorticityStream Function Formulation,
Numerical Algorithms: Primitive Variables, Steady Flows, Artificial Compressibility, Solution on a
Regular Grid, CrankNicolson Implicit, Boundary Conditions, Body Surface, FarField,
Symmetry, Inflow, Outflow, An Example, Staggered Grid, Marker and Cell Method, Implementation of the
Boundary Conditions, DuFortFrankel Scheme, Use of the Poisson Equation for Pressure,
Unsteady Incompressible NavierStokes Equations, Numercial Alogithms: VorticityStream Functions
Formulations, Vorticity Transport Equation, Steam Function Equation, Boundary Conditions,
Body Surface, FarField, 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, SimplyConnected Domain, DoublyConnected Domain, MultiplyConnected Domain,
Coordinate System Control, Grid Point Clustering, Orthogonality at the Surface, Hyperbolic
Grid Generation Techniques, Parabolic Grid Generators, Problems.


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