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Fundamental Equations
of Fluid Mechanics

1996, 509 PP
     Hoffmann, Chiang, Siddiqui, Papadakis
     

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This text has been prepared with the primary goal of assembling the broad range of fundamental equations of fluid mechanics and auxiliary relations in fluids.  The text has been written as a reference text to be used by practicing engineers, scientists, students and instructors.  It is presumed that the reader has had reasonable exposure to fluid mechanics. 

Formulations are presented with the necessary explanations, and, when appropriate, they are expressed in different coordinate systems.  Furthermore, different forms of the equations, based on the imposed assumptions, are provided.  The governing equations of fluid mechanics are presented in four chapters, whereas computational fluid mechanics is presented in Chapters Five and Six, and auxiliary relations and fluid properties are presented in Chapter Seven.  An extensive set of conversion factors is presented in Chapter Eight, and some selected mathematical relations are presented in Chapter Nine.  


Table of Contents   

    

Chapter:

    
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the Table of Contents for that Chapter
       

1

Equations of Fluid Motion  -  Navier-Stokes Equations
    
2
  
Equations of Fluid Motion for Inviscid Flows  
3
  
Boundary Layer Equations  
4
  
Hypersonic / Chemistry Equations  
5 Transformation of the Equations of Fluid Motion from Physical Space to Computational Space
  
6
  
Computational Fluid Dynamics   
7
  
Auxilary Relations and  Fluid Properties
8
  
Conversion Factors
9
  
Mathematical Relations

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Preface
Introduction

   

Chapter One:
Equations of Fluid Motion - Navier-Stokes Equations
    

Introductory Remarks, Coordinate Systems; Generalized Orthogonal Coordinate Systems, Commonly Used Coordinate Systems, Integral Formulations, Differential Formulations (Cartesian, Cylindrical, Spherical, and General Orthogonal Coordinates); Conservation of Mass, Conservation of Linear Momentum, Conservation of Energy, Flux Vector Formulations, Two-Dimensional Planar and Axisymmetric Formulations, Nondimensionalization of the Equations of Fluid Motion, Incompressible Navier-Stokes Equations; Primitive Variable Formulations, Vorticity-Stream Function Formulations, Poisson Equation for Pressure, Irrotational, Incompressible Flow, Selected Solutions of the Navier-Stokes Equations for Internal Flows; Fully Developed Laminar Flow, Fully Developed Flow in a Duct of Circular Cross Section; Skin Friction—Laminar Flow, Skin Friction—Turbulent Flow, Heat Transfer for Fully Developed Laminar Flow with Constant Surface Temperature, Heat Transfer for Fully Developed Flows in Ducts of Circular Cross Section—Laminar Flow, Turbulent Flow—Constant Heat Flux Boundary—Constant Surface Temperature Boundary, Fully Developed Flow for Noncircular Cross Sections, Fully Developed Flow in Circular Tube Annulus; Skin Friction, Heat Transfer—Laminar Flow, Turbulent Flow, One-Dimensional Flow with Friction.

    


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Chapter Two:
Equations of Fluid Motion for Inviscid Flows    

Introductory Remarks, Euler’s Equations, Bernoulli’s Equation, Velocity Potential Function, Velocity Potential Equation, Velocity Potential Equation for Small Perturbations, Thin Airfoil Theory; Circulation, Vorticity, Irrotational Flow, Stokes’ Theorem, Vortex Sheet, Kutta-Joukowski Theorem, Kutta Condition, Symmetric Airfoil, Cambered Airfoil, Thin Airfoil with Flap/Slat, Finite Wing Theory, Downwash, Effective Angle of Attack, Biot-Savart Law, Helmholtz Vortex Theorem, Elliptic Spanwise Circulation Distribution, General Spanwise Circulation Distribution, Isentropic Relations, Stagnation Properties, Characteristic Mach Number, Crocco’s Theorem, Entropy Equation, Pressure Coefficient, One-Dimensional Flow, One-Dimensional Flow with Heat Transfer, Shock Waves, Normal Shock Equations, Shock Thickness, Rankine-Hugoniot Relations, Oblique Shock Equation, Conical Flow, Prandtl-Meyer Expansion Waves, Quasi One-Dimensional Flow, Lift Coefficient, Drag Coefficient.

    


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Chapter Three:
Boundary Layer Equations

   

Introductory Remarks, Laminar Boundary-Layer Equations; Axisymmetric Flows, Integral Equation of the Boundary Layer, Turbulent Boundary-Layer Equations, Turbulence Models; General Remarks, Turbulent Shear Stress and Heat Flux, Zero-Equation Turbulence Models; Inner/Outer Region Model, Outer Law or the Law of the Wake, Cebeci/Smith Model, Baldwin/Lomax Model, One-Equation Models; Baldwin-Barth One-Equation Model, Spalart-Allmaras One-Equation Model, Two-Equation Models;  Two-Equation Model, Analytical Solutions of the Boundary-Layer Equations; Laminar Incompressible Flows Past Flat Surfaces, Turbulent Incompressible Flows Past Flat Surfaces, Simple Formulas for Forced Convection, Flow Over the Tube Bundles, Hydrodynamic Entrance Length, Thermal Entrance Length-Laminar Flow; Constant Heat Flux, Thermal Entry Length for Noncircular Tubes and Ducts in Laminar Flow, Flow Between Parallel Planes, Rectangular Ducts, Turbulent Flow Inside Tubes; Universal Velocity Distribution Laws, Friction Coefficient in Fully Developed Turbulent Flow, Turbulent Flow in Rough Pipes, Turbulent Flow Inside Tubes-Heat Transfer, Thermal Entry Length for Turbulent Flow in Circular Tubes, Free Convection Boundary Layers.

    


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Chapter Four:
Hypersonic / Chemistry Equations   

Introductory Remarks, Fundamental Concepts and Definitions; Real Gas, Perfect Gas, Partial Pressure, Various Modes of Energy, Compressibility Factor, Thermodynamic Properties of Mixture, Thermally Perfect Gas, Calorically Perfect Gas, Equation of State and the Speed of Sound for Mixture, Equilibrium, Nonequilibrium, Frozen Flows, Chemical Equilibrium, Thermal Equilibrium, Damkohler Number, Knudson Number, Navier-Stokes Equations for Nonequalibrium Flows, Temperature Models, Transport Properties; Diffusion Velocity of Species, Viscosity, Thermal Conductivity, Vibrational Heat Conduction Flux, T-V Energy-Exchange Process, Chemical Reactions; Rate Equations, Determination of the Rate Constants, Determination of Mass Production Rate, Thermodynamic Properties of Air, Curve Fits for the Thermodynamic Properties of Equilibrium Air Suitable for Numerical Computations, Newtonian Flow, Heat Transfer Rates; Stagnation Point Heating, Surface Heat Flux Distributions.


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

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

   


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Chapter Six:
Computational Fluid Dynamics
    

Introductory Remarks, Classification of Partial Differential Equations, Linear and Nonlinear PDE’s, Classification Based on Characteristics, Boundary Conditions, Finite Difference Expressions, Finite Difference Equations; Explicit and Implicit Formulations, Modified Equations, Parabolic Equations, Elliptic Equations, Hyperbolic Equations, Stability Analysis, Error Analysis, Grid Generation-Structured Grids, Incompressible Navier-Stokes Equations, Vorticity Transport Equation, Stream Function Equation, Primitive Variable Formulations, Finite Difference Equations, Navier-Stokes Equations, MacCormack Explicit, Flux Vector Splitting, Runge Kutta Explicit, Beta Formulation.

   


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Chapter Seven:
Auxiliary Relations and Fluid Properties
    

Introductory Remarks, Viscosity, Viscosity from the Point of View of Kinetic Theory, Pressure and Temperature Dependency of Viscosity for Air, Viscosity of Liquids, Kinematic Viscosity, Bulk Viscosity, Thermal Conductivity, Pressure and Temperature Dependency of Thermal Conductivity for Air, Thermal Conductivity of Liquids, Specific Heats, Equation of State, Speed of Sound, Prandtl Number, Turbulent Prandtl Number, U.S. Standard Atmosphere. 

   


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Chapter Eight:
Conversion Factors    
    


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Chapter Nine:
Mathematical Relations
   

Algebra; Binomial Expansions, Algebraic Equations, Series Expansions; Common Functions, Binomial Series, Taylor Series, Trigonometric Relations, Hyperbolic Functions, Functions of Complex Variables, Vector Algebra; Cartesian Coordinate System, Cylindrical Coordinate System, Spherical Coordinate System, General Orthogonal Coordinate System.

   


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