### Theoretical and Computational Approaches, Third Edition

### Theoretical and Computational Approaches, Third Edition

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Many introductions to fluid dynamics offer an illustrative approach that demonstrates some aspects of fluid behavior, but often leave you without the tools necessary to confront new problems. For more than a decade, **Fluid Dynamics: Theoretical and Computational Approaches** has supplied these missing tools with a constructive approach that made the book a bestseller. Now in its third edition, it supplies even more computational skills in addition to a solid foundation in theory.

After laying the groundwork in theoretical fluid dynamics, independent of any particular coordinate system in order to allow coordinate transformation of the equations, the author turns to the technique of writing Navier–Stokes and Euler’s equations, flow of inviscid fluids, laminar viscous flow, and turbulent flow. He also includes requisite mathematics in several “Mathematical Expositions” at the end of the book and provides abundant end-of-chapter problems.

**What’s New in the Third Edition?**

By integrating a strong theoretical foundation with practical computational tools, **Fluid Dynamics: Theoretical and Computational Approaches, Third Edition** is an indispensable guide to the methods needed to solve new and unfamiliar problems in fluid dynamics.

**Important Nomenclature**

**Kinematics of Fluid Motion**

Introduction to Continuum Motion

Fluid Particles

Inertial Coordinate Frames

Motion of a Continuum

The Time Derivatives

Velocity and Acceleration

Steady and Nonsteady Flow

Trajectories of Fluid Particles and Streamlines

Material Volume and Surface

Relation between Elemental Volumes

Kinematic Formulas of Euler and Reynolds

Control Volume and Surface

Kinematics of Deformation

Kinematics of Vorticity and Circulation

References

Problems

**The Conservation Laws and the Kinetics of Flow**

Fluid Density and the Conservation of Mass

Principle of Mass Conservation

Mass Conservation Using a Control Volume

Kinetics of Fluid Flow

Conservation of Linear and Angular Momentum

Equations of Linear and Angular Momentum

Momentum Conservation Using a Control Volume

Conservation of Energy

Energy Conservation Using a Control Volume

General Conservation Principle

The Closure Problem

Stokes’ Law of Friction

Interpretation of Pressure

The Dissipation Function

Constitutive Equation for Non-Newtonian Fluids

Thermodynamic Aspects of Pressure and Viscosity

Equations of Motion in Lagrangian Coordinates

References

Problems

**The Navier–Stokes Equations**

Formulation of the Problem

Viscous Compressible Flow Equations

Viscous Incompressible Flow Equations

Equations of Inviscid Flow (Euler’s Equations)

Initial and Boundary Conditions

Mathematical Nature of the Equations

Vorticity and Circulation

Some Results Based on the Equations of Motion

Nondimensional Parameters in Fluid Motion

Coordinate Transformation

Streamlines and Stream Surfaces

Navier–Stokes Equations in Stream Function Form

References

Problems

**Flow of Inviscid Fluids**

Introduction

Part I: Inviscid Incompressible Flow

The Bernoulli Constant

Method of Conformal Mapping in Inviscid Flows

Sources, Sinks, and Doublets in Three Dimensions

Part II: Inviscid Compressible Flow

Basic Thermodynamics

Subsonic and Supersonic Flow

Critical and Stagnation Quantities

Isentropic Ideal Gas Relations

Unsteady Inviscid Compressible Flow in One-dimension

Steady Plane Flow of Inviscid Gases

Theory of Shock Waves

References

Problems

**Laminar Viscous Flow**

Part I: Exact Solutions

Introduction

Exact Solutions

Exact Solutions for Slow Motion

Part II: Boundary Layers

Introduction

Formulation of the Boundary Layer Problem

Boundary Layer on 2-D Curved Surfaces

Separation of the 2-D Steady Boundary Layers

Transformed Boundary Layer Equations

Momentum Integral Equation

Free Boundary Layers

Numerical Solution of the Boundary Layer Equation

Three-Dimensional Boundary Layers

Momentum Integral Equations in Three Dimensions

Separation and Attachment in Three Dimensions

Boundary Layers on Bodies of Revolution and Yawed Cylinders

Three-Dimensional Stagnation Point Flow

Boundary Layer On Rotating Blades

Numerical Solution of 3-D Boundary Layer Equations

Unsteady Boundary Layers

Second-Order Boundary Layer Theory

Inverse Problems in Boundary Layers

Formulation of the Compressible Boundary Layer Problem

Part III: Navier–Stokes Formulation

Incompressible Flow

Compressible Flow

Hyperbolic Equations and Conservation Laws

Numerical Transformation and Grid Generation

Numerical Algorithms for Viscous Compressible Flows

Thin-Layer Navier–Stokes Equations (TLNS)

References

Problems

**Turbulent Flow**

Part I: Stability Theory and the Statistical Description of Turbulence

Introduction

Stability of Laminar Flows

Formulation for Plane-Parallel Laminar Flows

Temporal Stability at Inﬁnite Reynolds Number

Numerical Algorithm for the Orr–Sommerfeld Equation

Transition to Turbulence

Statistical Methods in Turbulent Continuum Mechanics

Statistical Concepts

Internal Structure in Physical Space

Internal Structure in the Wave-Number Space

Theory of Universal Equilibrium

Part II: Development of Averaged Equations

Introduction

Averaged Equations for Incompressible Flow

Averaged Equations for Compressible Flow

Turbulent Boundary Layer Equations

Part III: Basic Empirical and Boundary Layer Results in Turbulence

The Closure Problem

Prandtl’s Mixing-Length Hypothesis

Wall-Bound Turbulent Flows

Analysis of Turbulent Boundary Layer Velocity Proﬁles

Momentum Integral Methods in Boundary Layers

Differential Equation Methods in 2-D Boundary Layers

Part IV: Turbulence Modeling

Generalization of Boussinesq’s Hypothesis

Zero-Equation Modeling in Shear Layers

One-Equation Modeling

Two-Equation (*K*-Î) Modeling

Reynolds’ Stress Equation Modeling

Application to 2-D Thin Shear Layers

Algebraic Reynolds’ Stress Closure

Development of A Nonlinear Constitutive Equation

Current Approaches to Nonlinear Modeling

Heuristic Modeling

Modeling for Compressible Flow

Three-Dimensional Boundary Layers

Illustrative Analysis of Instability

Basic Formulation of Large Eddy Simulation

References

Problems

**Mathematical Exposition 1: Base Vectors and Various Representations**

Introduction

Representations in Rectangular Cartesian Systems

Scalars, Vectors, and Tensors

Differential Operations On Tensors

Multiplication of A Tensor and A Vector

Scalar Multiplication of Two Tensors

A Collection of Usable Formulas

Taylor Expansion in Vector Form

Principal Axes of a Tensor

Transformation of T to the Principal Axes

Quadratic Form and the Eigenvalue Problem

Representation in Curvilinear Coordinates

Christoffel Symbols in Three Dimensions

Some Derivative Relations

Scalar and Double Dot Products of Two Tensors

**Mathematical Exposition 2: Theorems of Gauss, Green, and Stokes**

Gauss’ Theorem

Green’s Theorem

Stokes’ Theorem

**Mathematical Exposition 3: Geometry of Space and Plane Curves**

Basic Theory of Curves

**Mathematical Exposition 4: Formulas for Coordinate Transformation**

Introduction

Transformation Law for Scalars

Transformation Laws for Vectors

Transformation Laws for Tensors

Transformation Laws for the Christoffel Symbols

Some Formulas in Cartesian and Curvilinear Coordinates

**Mathematical Exposition 5: Potential Theory**

Introduction

Formulas of Green

Potential Theory

General Representation of a Vector

An Application of Green’s First Formula

**Mathematical Exposition 6: Singularities of the First-Order ODEs**

Introduction

Singularities and Their Classiﬁcation

**Mathematical Exposition 7: Geometry of Surfaces**

Basic Deﬁnitions

Formulas of Gauss

Formulas of Weingarten

Equations of Gauss

Normal and Geodesic Curvatures

Grid Generation in Surfaces

**Mathematical Exposition 8: Finite Difference Approximation Applied to PDEs**

Introduction

Calculus of Finite Differences

Iterative Root Finding

Numerical Integration

Finite Difference Approximations of Partial Derivatives

Finite Difference Approximation of Parabolic PDEs

Finite Difference Approximation of Elliptic Equations

**Mathematical Exposition 9: Frame Invariancy**

Introduction

Orthogonal Tensor

Arbitrary Rectangular Frames of Reference

Check for Frame Invariancy

Use of Q

References for the Mathematical Expositions

Index

Many introductions to fluid dynamics offer an illustrative approach that demonstrates some aspects of fluid behavior, but often leave you without the tools necessary to confront new problems. For more than a decade, **Fluid Dynamics: Theoretical and Computational Approaches** has supplied these missing tools with a constructive approach that made the book a bestseller. Now in its third edition, it supplies even more computational skills in addition to a solid foundation in theory.

After laying the groundwork in theoretical fluid dynamics, independent of any particular coordinate system in order to allow coordinate transformation of the equations, the author turns to the technique of writing Navier–Stokes and Euler’s equations, flow of inviscid fluids, laminar viscous flow, and turbulent flow. He also includes requisite mathematics in several “Mathematical Expositions” at the end of the book and provides abundant end-of-chapter problems.

**What’s New in the Third Edition?**

By integrating a strong theoretical foundation with practical computational tools, **Fluid Dynamics: Theoretical and Computational Approaches, Third Edition** is an indispensable guide to the methods needed to solve new and unfamiliar problems in fluid dynamics.

**Important Nomenclature**

**Kinematics of Fluid Motion**

Introduction to Continuum Motion

Fluid Particles

Inertial Coordinate Frames

Motion of a Continuum

The Time Derivatives

Velocity and Acceleration

Steady and Nonsteady Flow

Trajectories of Fluid Particles and Streamlines

Material Volume and Surface

Relation between Elemental Volumes

Kinematic Formulas of Euler and Reynolds

Control Volume and Surface

Kinematics of Deformation

Kinematics of Vorticity and Circulation

References

Problems

**The Conservation Laws and the Kinetics of Flow**

Fluid Density and the Conservation of Mass

Principle of Mass Conservation

Mass Conservation Using a Control Volume

Kinetics of Fluid Flow

Conservation of Linear and Angular Momentum

Equations of Linear and Angular Momentum

Momentum Conservation Using a Control Volume

Conservation of Energy

Energy Conservation Using a Control Volume

General Conservation Principle

The Closure Problem

Stokes’ Law of Friction

Interpretation of Pressure

The Dissipation Function

Constitutive Equation for Non-Newtonian Fluids

Thermodynamic Aspects of Pressure and Viscosity

Equations of Motion in Lagrangian Coordinates

References

Problems

**The Navier–Stokes Equations**

Formulation of the Problem

Viscous Compressible Flow Equations

Viscous Incompressible Flow Equations

Equations of Inviscid Flow (Euler’s Equations)

Initial and Boundary Conditions

Mathematical Nature of the Equations

Vorticity and Circulation

Some Results Based on the Equations of Motion

Nondimensional Parameters in Fluid Motion

Coordinate Transformation

Streamlines and Stream Surfaces

Navier–Stokes Equations in Stream Function Form

References

Problems

**Flow of Inviscid Fluids**

Introduction

Part I: Inviscid Incompressible Flow

The Bernoulli Constant

Method of Conformal Mapping in Inviscid Flows

Sources, Sinks, and Doublets in Three Dimensions

Part II: Inviscid Compressible Flow

Basic Thermodynamics

Subsonic and Supersonic Flow

Critical and Stagnation Quantities

Isentropic Ideal Gas Relations

Unsteady Inviscid Compressible Flow in One-dimension

Steady Plane Flow of Inviscid Gases

Theory of Shock Waves

References

Problems

**Laminar Viscous Flow**

Part I: Exact Solutions

Introduction

Exact Solutions

Exact Solutions for Slow Motion

Part II: Boundary Layers

Introduction

Formulation of the Boundary Layer Problem

Boundary Layer on 2-D Curved Surfaces

Separation of the 2-D Steady Boundary Layers

Transformed Boundary Layer Equations

Momentum Integral Equation

Free Boundary Layers

Numerical Solution of the Boundary Layer Equation

Three-Dimensional Boundary Layers

Momentum Integral Equations in Three Dimensions

Separation and Attachment in Three Dimensions

Boundary Layers on Bodies of Revolution and Yawed Cylinders

Three-Dimensional Stagnation Point Flow

Boundary Layer On Rotating Blades

Numerical Solution of 3-D Boundary Layer Equations

Unsteady Boundary Layers

Second-Order Boundary Layer Theory

Inverse Problems in Boundary Layers

Formulation of the Compressible Boundary Layer Problem

Part III: Navier–Stokes Formulation

Incompressible Flow

Compressible Flow

Hyperbolic Equations and Conservation Laws

Numerical Transformation and Grid Generation

Numerical Algorithms for Viscous Compressible Flows

Thin-Layer Navier–Stokes Equations (TLNS)

References

Problems

**Turbulent Flow**

Part I: Stability Theory and the Statistical Description of Turbulence

Introduction

Stability of Laminar Flows

Formulation for Plane-Parallel Laminar Flows

Temporal Stability at Inﬁnite Reynolds Number

Numerical Algorithm for the Orr–Sommerfeld Equation

Transition to Turbulence

Statistical Methods in Turbulent Continuum Mechanics

Statistical Concepts

Internal Structure in Physical Space

Internal Structure in the Wave-Number Space

Theory of Universal Equilibrium

Part II: Development of Averaged Equations

Introduction

Averaged Equations for Incompressible Flow

Averaged Equations for Compressible Flow

Turbulent Boundary Layer Equations

Part III: Basic Empirical and Boundary Layer Results in Turbulence

The Closure Problem

Prandtl’s Mixing-Length Hypothesis

Wall-Bound Turbulent Flows

Analysis of Turbulent Boundary Layer Velocity Proﬁles

Momentum Integral Methods in Boundary Layers

Differential Equation Methods in 2-D Boundary Layers

Part IV: Turbulence Modeling

Generalization of Boussinesq’s Hypothesis

Zero-Equation Modeling in Shear Layers

One-Equation Modeling

Two-Equation (*K*-Î) Modeling

Reynolds’ Stress Equation Modeling

Application to 2-D Thin Shear Layers

Algebraic Reynolds’ Stress Closure

Development of A Nonlinear Constitutive Equation

Current Approaches to Nonlinear Modeling

Heuristic Modeling

Modeling for Compressible Flow

Three-Dimensional Boundary Layers

Illustrative Analysis of Instability

Basic Formulation of Large Eddy Simulation

References

Problems

**Mathematical Exposition 1: Base Vectors and Various Representations**

Introduction

Representations in Rectangular Cartesian Systems

Scalars, Vectors, and Tensors

Differential Operations On Tensors

Multiplication of A Tensor and A Vector

Scalar Multiplication of Two Tensors

A Collection of Usable Formulas

Taylor Expansion in Vector Form

Principal Axes of a Tensor

Transformation of T to the Principal Axes

Quadratic Form and the Eigenvalue Problem

Representation in Curvilinear Coordinates

Christoffel Symbols in Three Dimensions

Some Derivative Relations

Scalar and Double Dot Products of Two Tensors

**Mathematical Exposition 2: Theorems of Gauss, Green, and Stokes**

Gauss’ Theorem

Green’s Theorem

Stokes’ Theorem

**Mathematical Exposition 3: Geometry of Space and Plane Curves**

Basic Theory of Curves

**Mathematical Exposition 4: Formulas for Coordinate Transformation**

Introduction

Transformation Law for Scalars

Transformation Laws for Vectors

Transformation Laws for Tensors

Transformation Laws for the Christoffel Symbols

Some Formulas in Cartesian and Curvilinear Coordinates

**Mathematical Exposition 5: Potential Theory**

Introduction

Formulas of Green

Potential Theory

General Representation of a Vector

An Application of Green’s First Formula

**Mathematical Exposition 6: Singularities of the First-Order ODEs**

Introduction

Singularities and Their Classiﬁcation

**Mathematical Exposition 7: Geometry of Surfaces**

Basic Deﬁnitions

Formulas of Gauss

Formulas of Weingarten

Equations of Gauss

Normal and Geodesic Curvatures

Grid Generation in Surfaces

**Mathematical Exposition 8: Finite Difference Approximation Applied to PDEs**

Introduction

Calculus of Finite Differences

Iterative Root Finding

Numerical Integration

Finite Difference Approximations of Partial Derivatives

Finite Difference Approximation of Parabolic PDEs

Finite Difference Approximation of Elliptic Equations

**Mathematical Exposition 9: Frame Invariancy**

Introduction

Orthogonal Tensor

Arbitrary Rectangular Frames of Reference

Check for Frame Invariancy

Use of Q

References for the Mathematical Expositions

Index

Many introductions to fluid dynamics offer an illustrative approach that demonstrates some aspects of fluid behavior, but often leave you without the tools necessary to confront new problems. For more than a decade, **Fluid Dynamics: Theoretical and Computational Approaches** has supplied these missing tools with a constructive approach that made the book a bestseller. Now in its third edition, it supplies even more computational skills in addition to a solid foundation in theory.

After laying the groundwork in theoretical fluid dynamics, independent of any particular coordinate system in order to allow coordinate transformation of the equations, the author turns to the technique of writing Navier–Stokes and Euler’s equations, flow of inviscid fluids, laminar viscous flow, and turbulent flow. He also includes requisite mathematics in several “Mathematical Expositions” at the end of the book and provides abundant end-of-chapter problems.

**What’s New in the Third Edition?**

By integrating a strong theoretical foundation with practical computational tools, **Fluid Dynamics: Theoretical and Computational Approaches, Third Edition** is an indispensable guide to the methods needed to solve new and unfamiliar problems in fluid dynamics.

**Important Nomenclature**

**Kinematics of Fluid Motion**

Introduction to Continuum Motion

Fluid Particles

Inertial Coordinate Frames

Motion of a Continuum

The Time Derivatives

Velocity and Acceleration

Steady and Nonsteady Flow

Trajectories of Fluid Particles and Streamlines

Material Volume and Surface

Relation between Elemental Volumes

Kinematic Formulas of Euler and Reynolds

Control Volume and Surface

Kinematics of Deformation

Kinematics of Vorticity and Circulation

References

Problems

**The Conservation Laws and the Kinetics of Flow**

Fluid Density and the Conservation of Mass

Principle of Mass Conservation

Mass Conservation Using a Control Volume

Kinetics of Fluid Flow

Conservation of Linear and Angular Momentum

Equations of Linear and Angular Momentum

Momentum Conservation Using a Control Volume

Conservation of Energy

Energy Conservation Using a Control Volume

General Conservation Principle

The Closure Problem

Stokes’ Law of Friction

Interpretation of Pressure

The Dissipation Function

Constitutive Equation for Non-Newtonian Fluids

Thermodynamic Aspects of Pressure and Viscosity

Equations of Motion in Lagrangian Coordinates

References

Problems

**The Navier–Stokes Equations**

Formulation of the Problem

Viscous Compressible Flow Equations

Viscous Incompressible Flow Equations

Equations of Inviscid Flow (Euler’s Equations)

Initial and Boundary Conditions

Mathematical Nature of the Equations

Vorticity and Circulation

Some Results Based on the Equations of Motion

Nondimensional Parameters in Fluid Motion

Coordinate Transformation

Streamlines and Stream Surfaces

Navier–Stokes Equations in Stream Function Form

References

Problems

**Flow of Inviscid Fluids**

Introduction

Part I: Inviscid Incompressible Flow

The Bernoulli Constant

Method of Conformal Mapping in Inviscid Flows

Sources, Sinks, and Doublets in Three Dimensions

Part II: Inviscid Compressible Flow

Basic Thermodynamics

Subsonic and Supersonic Flow

Critical and Stagnation Quantities

Isentropic Ideal Gas Relations

Unsteady Inviscid Compressible Flow in One-dimension

Steady Plane Flow of Inviscid Gases

Theory of Shock Waves

References

Problems

**Laminar Viscous Flow**

Part I: Exact Solutions

Introduction

Exact Solutions

Exact Solutions for Slow Motion

Part II: Boundary Layers

Introduction

Formulation of the Boundary Layer Problem

Boundary Layer on 2-D Curved Surfaces

Separation of the 2-D Steady Boundary Layers

Transformed Boundary Layer Equations

Momentum Integral Equation

Free Boundary Layers

Numerical Solution of the Boundary Layer Equation

Three-Dimensional Boundary Layers

Momentum Integral Equations in Three Dimensions

Separation and Attachment in Three Dimensions

Boundary Layers on Bodies of Revolution and Yawed Cylinders

Three-Dimensional Stagnation Point Flow

Boundary Layer On Rotating Blades

Numerical Solution of 3-D Boundary Layer Equations

Unsteady Boundary Layers

Second-Order Boundary Layer Theory

Inverse Problems in Boundary Layers

Formulation of the Compressible Boundary Layer Problem

Part III: Navier–Stokes Formulation

Incompressible Flow

Compressible Flow

Hyperbolic Equations and Conservation Laws

Numerical Transformation and Grid Generation

Numerical Algorithms for Viscous Compressible Flows

Thin-Layer Navier–Stokes Equations (TLNS)

References

Problems

**Turbulent Flow**

Part I: Stability Theory and the Statistical Description of Turbulence

Introduction

Stability of Laminar Flows

Formulation for Plane-Parallel Laminar Flows

Temporal Stability at Inﬁnite Reynolds Number

Numerical Algorithm for the Orr–Sommerfeld Equation

Transition to Turbulence

Statistical Methods in Turbulent Continuum Mechanics

Statistical Concepts

Internal Structure in Physical Space

Internal Structure in the Wave-Number Space

Theory of Universal Equilibrium

Part II: Development of Averaged Equations

Introduction

Averaged Equations for Incompressible Flow

Averaged Equations for Compressible Flow

Turbulent Boundary Layer Equations

Part III: Basic Empirical and Boundary Layer Results in Turbulence

The Closure Problem

Prandtl’s Mixing-Length Hypothesis

Wall-Bound Turbulent Flows

Analysis of Turbulent Boundary Layer Velocity Proﬁles

Momentum Integral Methods in Boundary Layers

Differential Equation Methods in 2-D Boundary Layers

Part IV: Turbulence Modeling

Generalization of Boussinesq’s Hypothesis

Zero-Equation Modeling in Shear Layers

One-Equation Modeling

Two-Equation (*K*-Î) Modeling

Reynolds’ Stress Equation Modeling

Application to 2-D Thin Shear Layers

Algebraic Reynolds’ Stress Closure

Development of A Nonlinear Constitutive Equation

Current Approaches to Nonlinear Modeling

Heuristic Modeling

Modeling for Compressible Flow

Three-Dimensional Boundary Layers

Illustrative Analysis of Instability

Basic Formulation of Large Eddy Simulation

References

Problems

**Mathematical Exposition 1: Base Vectors and Various Representations**

Introduction

Representations in Rectangular Cartesian Systems

Scalars, Vectors, and Tensors

Differential Operations On Tensors

Multiplication of A Tensor and A Vector

Scalar Multiplication of Two Tensors

A Collection of Usable Formulas

Taylor Expansion in Vector Form

Principal Axes of a Tensor

Transformation of T to the Principal Axes

Quadratic Form and the Eigenvalue Problem

Representation in Curvilinear Coordinates

Christoffel Symbols in Three Dimensions

Some Derivative Relations

Scalar and Double Dot Products of Two Tensors

**Mathematical Exposition 2: Theorems of Gauss, Green, and Stokes**

Gauss’ Theorem

Green’s Theorem

Stokes’ Theorem

**Mathematical Exposition 3: Geometry of Space and Plane Curves**

Basic Theory of Curves

**Mathematical Exposition 4: Formulas for Coordinate Transformation**

Introduction

Transformation Law for Scalars

Transformation Laws for Vectors

Transformation Laws for Tensors

Transformation Laws for the Christoffel Symbols

Some Formulas in Cartesian and Curvilinear Coordinates

**Mathematical Exposition 5: Potential Theory**

Introduction

Formulas of Green

Potential Theory

General Representation of a Vector

An Application of Green’s First Formula

**Mathematical Exposition 6: Singularities of the First-Order ODEs**

Introduction

Singularities and Their Classiﬁcation

**Mathematical Exposition 7: Geometry of Surfaces**

Basic Deﬁnitions

Formulas of Gauss

Formulas of Weingarten

Equations of Gauss

Normal and Geodesic Curvatures

Grid Generation in Surfaces

**Mathematical Exposition 8: Finite Difference Approximation Applied to PDEs**

Introduction

Calculus of Finite Differences

Iterative Root Finding

Numerical Integration

Finite Difference Approximations of Partial Derivatives

Finite Difference Approximation of Parabolic PDEs

Finite Difference Approximation of Elliptic Equations

**Mathematical Exposition 9: Frame Invariancy**

Introduction

Orthogonal Tensor

Arbitrary Rectangular Frames of Reference

Check for Frame Invariancy

Use of Q

References for the Mathematical Expositions

Index

**Fluid Dynamics: Theoretical and Computational Approaches** has supplied these missing tools with a constructive approach that made the book a bestseller. Now in its third edition, it supplies even more computational skills in addition to a solid foundation in theory.

**What’s New in the Third Edition?**

**Fluid Dynamics: Theoretical and Computational Approaches, Third Edition** is an indispensable guide to the methods needed to solve new and unfamiliar problems in fluid dynamics.

**Important Nomenclature**

**Kinematics of Fluid Motion**

Introduction to Continuum Motion

Fluid Particles

Inertial Coordinate Frames

Motion of a Continuum

The Time Derivatives

Velocity and Acceleration

Steady and Nonsteady Flow

Trajectories of Fluid Particles and Streamlines

Material Volume and Surface

Relation between Elemental Volumes

Kinematic Formulas of Euler and Reynolds

Control Volume and Surface

Kinematics of Deformation

Kinematics of Vorticity and Circulation

References

Problems

**The Conservation Laws and the Kinetics of Flow**

Fluid Density and the Conservation of Mass

Principle of Mass Conservation

Mass Conservation Using a Control Volume

Kinetics of Fluid Flow

Conservation of Linear and Angular Momentum

Equations of Linear and Angular Momentum

Momentum Conservation Using a Control Volume

Conservation of Energy

Energy Conservation Using a Control Volume

General Conservation Principle

The Closure Problem

Stokes’ Law of Friction

Interpretation of Pressure

The Dissipation Function

Constitutive Equation for Non-Newtonian Fluids

Thermodynamic Aspects of Pressure and Viscosity

Equations of Motion in Lagrangian Coordinates

References

Problems

**The Navier–Stokes Equations**

Formulation of the Problem

Viscous Compressible Flow Equations

Viscous Incompressible Flow Equations

Equations of Inviscid Flow (Euler’s Equations)

Initial and Boundary Conditions

Mathematical Nature of the Equations

Vorticity and Circulation

Some Results Based on the Equations of Motion

Nondimensional Parameters in Fluid Motion

Coordinate Transformation

Streamlines and Stream Surfaces

Navier–Stokes Equations in Stream Function Form

References

Problems

**Flow of Inviscid Fluids**

Introduction

Part I: Inviscid Incompressible Flow

The Bernoulli Constant

Method of Conformal Mapping in Inviscid Flows

Sources, Sinks, and Doublets in Three Dimensions

Part II: Inviscid Compressible Flow

Basic Thermodynamics

Subsonic and Supersonic Flow

Critical and Stagnation Quantities

Isentropic Ideal Gas Relations

Unsteady Inviscid Compressible Flow in One-dimension

Steady Plane Flow of Inviscid Gases

Theory of Shock Waves

References

Problems

**Laminar Viscous Flow**

Part I: Exact Solutions

Introduction

Exact Solutions

Exact Solutions for Slow Motion

Part II: Boundary Layers

Introduction

Formulation of the Boundary Layer Problem

Boundary Layer on 2-D Curved Surfaces

Separation of the 2-D Steady Boundary Layers

Transformed Boundary Layer Equations

Momentum Integral Equation

Free Boundary Layers

Numerical Solution of the Boundary Layer Equation

Three-Dimensional Boundary Layers

Momentum Integral Equations in Three Dimensions

Separation and Attachment in Three Dimensions

Boundary Layers on Bodies of Revolution and Yawed Cylinders

Three-Dimensional Stagnation Point Flow

Boundary Layer On Rotating Blades

Numerical Solution of 3-D Boundary Layer Equations

Unsteady Boundary Layers

Second-Order Boundary Layer Theory

Inverse Problems in Boundary Layers

Formulation of the Compressible Boundary Layer Problem

Part III: Navier–Stokes Formulation

Incompressible Flow

Compressible Flow

Hyperbolic Equations and Conservation Laws

Numerical Transformation and Grid Generation

Numerical Algorithms for Viscous Compressible Flows

Thin-Layer Navier–Stokes Equations (TLNS)

References

Problems

**Turbulent Flow**

Part I: Stability Theory and the Statistical Description of Turbulence

Introduction

Stability of Laminar Flows

Formulation for Plane-Parallel Laminar Flows

Temporal Stability at Inﬁnite Reynolds Number

Numerical Algorithm for the Orr–Sommerfeld Equation

Transition to Turbulence

Statistical Methods in Turbulent Continuum Mechanics

Statistical Concepts

Internal Structure in Physical Space

Internal Structure in the Wave-Number Space

Theory of Universal Equilibrium

Part II: Development of Averaged Equations

Introduction

Averaged Equations for Incompressible Flow

Averaged Equations for Compressible Flow

Turbulent Boundary Layer Equations

Part III: Basic Empirical and Boundary Layer Results in Turbulence

The Closure Problem

Prandtl’s Mixing-Length Hypothesis

Wall-Bound Turbulent Flows

Analysis of Turbulent Boundary Layer Velocity Proﬁles

Momentum Integral Methods in Boundary Layers

Differential Equation Methods in 2-D Boundary Layers

Part IV: Turbulence Modeling

Generalization of Boussinesq’s Hypothesis

Zero-Equation Modeling in Shear Layers

One-Equation Modeling

Two-Equation (*K*-Î) Modeling

Reynolds’ Stress Equation Modeling

Application to 2-D Thin Shear Layers

Algebraic Reynolds’ Stress Closure

Development of A Nonlinear Constitutive Equation

Current Approaches to Nonlinear Modeling

Heuristic Modeling

Modeling for Compressible Flow

Three-Dimensional Boundary Layers

Illustrative Analysis of Instability

Basic Formulation of Large Eddy Simulation

References

Problems

**Mathematical Exposition 1: Base Vectors and Various Representations**

Introduction

Representations in Rectangular Cartesian Systems

Scalars, Vectors, and Tensors

Differential Operations On Tensors

Multiplication of A Tensor and A Vector

Scalar Multiplication of Two Tensors

A Collection of Usable Formulas

Taylor Expansion in Vector Form

Principal Axes of a Tensor

Transformation of T to the Principal Axes

Quadratic Form and the Eigenvalue Problem

Representation in Curvilinear Coordinates

Christoffel Symbols in Three Dimensions

Some Derivative Relations

Scalar and Double Dot Products of Two Tensors

**Mathematical Exposition 2: Theorems of Gauss, Green, and Stokes**

Gauss’ Theorem

Green’s Theorem

Stokes’ Theorem

**Mathematical Exposition 3: Geometry of Space and Plane Curves**

Basic Theory of Curves

**Mathematical Exposition 4: Formulas for Coordinate Transformation**

Introduction

Transformation Law for Scalars

Transformation Laws for Vectors

Transformation Laws for Tensors

Transformation Laws for the Christoffel Symbols

Some Formulas in Cartesian and Curvilinear Coordinates

**Mathematical Exposition 5: Potential Theory**

Introduction

Formulas of Green

Potential Theory

General Representation of a Vector

An Application of Green’s First Formula

**Mathematical Exposition 6: Singularities of the First-Order ODEs**

Introduction

Singularities and Their Classiﬁcation

**Mathematical Exposition 7: Geometry of Surfaces**

Basic Deﬁnitions

Formulas of Gauss

Formulas of Weingarten

Equations of Gauss

Normal and Geodesic Curvatures

Grid Generation in Surfaces

**Mathematical Exposition 8: Finite Difference Approximation Applied to PDEs**

Introduction

Calculus of Finite Differences

Iterative Root Finding

Numerical Integration

Finite Difference Approximations of Partial Derivatives

Finite Difference Approximation of Parabolic PDEs

Finite Difference Approximation of Elliptic Equations

**Mathematical Exposition 9: Frame Invariancy**

Introduction

Orthogonal Tensor

Arbitrary Rectangular Frames of Reference

Check for Frame Invariancy

Use of Q

References for the Mathematical Expositions

Index

**Fluid Dynamics: Theoretical and Computational Approaches** has supplied these missing tools with a constructive approach that made the book a bestseller. Now in its third edition, it supplies even more computational skills in addition to a solid foundation in theory.

**What’s New in the Third Edition?**

**Fluid Dynamics: Theoretical and Computational Approaches, Third Edition** is an indispensable guide to the methods needed to solve new and unfamiliar problems in fluid dynamics.

**Important Nomenclature**

**Kinematics of Fluid Motion**

Introduction to Continuum Motion

Fluid Particles

Inertial Coordinate Frames

Motion of a Continuum

The Time Derivatives

Velocity and Acceleration

Steady and Nonsteady Flow

Trajectories of Fluid Particles and Streamlines

Material Volume and Surface

Relation between Elemental Volumes

Kinematic Formulas of Euler and Reynolds

Control Volume and Surface

Kinematics of Deformation

Kinematics of Vorticity and Circulation

References

Problems

**The Conservation Laws and the Kinetics of Flow**

Fluid Density and the Conservation of Mass

Principle of Mass Conservation

Mass Conservation Using a Control Volume

Kinetics of Fluid Flow

Conservation of Linear and Angular Momentum

Equations of Linear and Angular Momentum

Momentum Conservation Using a Control Volume

Conservation of Energy

Energy Conservation Using a Control Volume

General Conservation Principle

The Closure Problem

Stokes’ Law of Friction

Interpretation of Pressure

The Dissipation Function

Constitutive Equation for Non-Newtonian Fluids

Thermodynamic Aspects of Pressure and Viscosity

Equations of Motion in Lagrangian Coordinates

References

Problems

**The Navier–Stokes Equations**

Formulation of the Problem

Viscous Compressible Flow Equations

Viscous Incompressible Flow Equations

Equations of Inviscid Flow (Euler’s Equations)

Initial and Boundary Conditions

Mathematical Nature of the Equations

Vorticity and Circulation

Some Results Based on the Equations of Motion

Nondimensional Parameters in Fluid Motion

Coordinate Transformation

Streamlines and Stream Surfaces

Navier–Stokes Equations in Stream Function Form

References

Problems

**Flow of Inviscid Fluids**

Introduction

Part I: Inviscid Incompressible Flow

The Bernoulli Constant

Method of Conformal Mapping in Inviscid Flows

Sources, Sinks, and Doublets in Three Dimensions

Part II: Inviscid Compressible Flow

Basic Thermodynamics

Subsonic and Supersonic Flow

Critical and Stagnation Quantities

Isentropic Ideal Gas Relations

Unsteady Inviscid Compressible Flow in One-dimension

Steady Plane Flow of Inviscid Gases

Theory of Shock Waves

References

Problems

**Laminar Viscous Flow**

Part I: Exact Solutions

Introduction

Exact Solutions

Exact Solutions for Slow Motion

Part II: Boundary Layers

Introduction

Formulation of the Boundary Layer Problem

Boundary Layer on 2-D Curved Surfaces

Separation of the 2-D Steady Boundary Layers

Transformed Boundary Layer Equations

Momentum Integral Equation

Free Boundary Layers

Numerical Solution of the Boundary Layer Equation

Three-Dimensional Boundary Layers

Momentum Integral Equations in Three Dimensions

Separation and Attachment in Three Dimensions

Boundary Layers on Bodies of Revolution and Yawed Cylinders

Three-Dimensional Stagnation Point Flow

Boundary Layer On Rotating Blades

Numerical Solution of 3-D Boundary Layer Equations

Unsteady Boundary Layers

Second-Order Boundary Layer Theory

Inverse Problems in Boundary Layers

Formulation of the Compressible Boundary Layer Problem

Part III: Navier–Stokes Formulation

Incompressible Flow

Compressible Flow

Hyperbolic Equations and Conservation Laws

Numerical Transformation and Grid Generation

Numerical Algorithms for Viscous Compressible Flows

Thin-Layer Navier–Stokes Equations (TLNS)

References

Problems

**Turbulent Flow**

Part I: Stability Theory and the Statistical Description of Turbulence

Introduction

Stability of Laminar Flows

Formulation for Plane-Parallel Laminar Flows

Temporal Stability at Inﬁnite Reynolds Number

Numerical Algorithm for the Orr–Sommerfeld Equation

Transition to Turbulence

Statistical Methods in Turbulent Continuum Mechanics

Statistical Concepts

Internal Structure in Physical Space

Internal Structure in the Wave-Number Space

Theory of Universal Equilibrium

Part II: Development of Averaged Equations

Introduction

Averaged Equations for Incompressible Flow

Averaged Equations for Compressible Flow

Turbulent Boundary Layer Equations

Part III: Basic Empirical and Boundary Layer Results in Turbulence

The Closure Problem

Prandtl’s Mixing-Length Hypothesis

Wall-Bound Turbulent Flows

Analysis of Turbulent Boundary Layer Velocity Proﬁles

Momentum Integral Methods in Boundary Layers

Differential Equation Methods in 2-D Boundary Layers

Part IV: Turbulence Modeling

Generalization of Boussinesq’s Hypothesis

Zero-Equation Modeling in Shear Layers

One-Equation Modeling

Two-Equation (*K*-Î) Modeling

Reynolds’ Stress Equation Modeling

Application to 2-D Thin Shear Layers

Algebraic Reynolds’ Stress Closure

Development of A Nonlinear Constitutive Equation

Current Approaches to Nonlinear Modeling

Heuristic Modeling

Modeling for Compressible Flow

Three-Dimensional Boundary Layers

Illustrative Analysis of Instability

Basic Formulation of Large Eddy Simulation

References

Problems

**Mathematical Exposition 1: Base Vectors and Various Representations**

Introduction

Representations in Rectangular Cartesian Systems

Scalars, Vectors, and Tensors

Differential Operations On Tensors

Multiplication of A Tensor and A Vector

Scalar Multiplication of Two Tensors

A Collection of Usable Formulas

Taylor Expansion in Vector Form

Principal Axes of a Tensor

Transformation of T to the Principal Axes

Quadratic Form and the Eigenvalue Problem

Representation in Curvilinear Coordinates

Christoffel Symbols in Three Dimensions

Some Derivative Relations

Scalar and Double Dot Products of Two Tensors

**Mathematical Exposition 2: Theorems of Gauss, Green, and Stokes**

Gauss’ Theorem

Green’s Theorem

Stokes’ Theorem

**Mathematical Exposition 3: Geometry of Space and Plane Curves**

Basic Theory of Curves

**Mathematical Exposition 4: Formulas for Coordinate Transformation**

Introduction

Transformation Law for Scalars

Transformation Laws for Vectors

Transformation Laws for Tensors

Transformation Laws for the Christoffel Symbols

Some Formulas in Cartesian and Curvilinear Coordinates

**Mathematical Exposition 5: Potential Theory**

Introduction

Formulas of Green

Potential Theory

General Representation of a Vector

An Application of Green’s First Formula

**Mathematical Exposition 6: Singularities of the First-Order ODEs**

Introduction

Singularities and Their Classiﬁcation

**Mathematical Exposition 7: Geometry of Surfaces**

Basic Deﬁnitions

Formulas of Gauss

Formulas of Weingarten

Equations of Gauss

Normal and Geodesic Curvatures

Grid Generation in Surfaces

**Mathematical Exposition 8: Finite Difference Approximation Applied to PDEs**

Introduction

Calculus of Finite Differences

Iterative Root Finding

Numerical Integration

Finite Difference Approximations of Partial Derivatives

Finite Difference Approximation of Parabolic PDEs

Finite Difference Approximation of Elliptic Equations

**Mathematical Exposition 9: Frame Invariancy**

Introduction

Orthogonal Tensor

Arbitrary Rectangular Frames of Reference

Check for Frame Invariancy

Use of Q

References for the Mathematical Expositions

Index

**Fluid Dynamics: Theoretical and Computational Approaches** has supplied these missing tools with a constructive approach that made the book a bestseller. Now in its third edition, it supplies even more computational skills in addition to a solid foundation in theory.

**What’s New in the Third Edition?**

**Fluid Dynamics: Theoretical and Computational Approaches, Third Edition** is an indispensable guide to the methods needed to solve new and unfamiliar problems in fluid dynamics.

**Important Nomenclature**

**Kinematics of Fluid Motion**

Introduction to Continuum Motion

Fluid Particles

Inertial Coordinate Frames

Motion of a Continuum

The Time Derivatives

Velocity and Acceleration

Steady and Nonsteady Flow

Trajectories of Fluid Particles and Streamlines

Material Volume and Surface

Relation between Elemental Volumes

Kinematic Formulas of Euler and Reynolds

Control Volume and Surface

Kinematics of Deformation

Kinematics of Vorticity and Circulation

References

Problems

**The Conservation Laws and the Kinetics of Flow**

Fluid Density and the Conservation of Mass

Principle of Mass Conservation

Mass Conservation Using a Control Volume

Kinetics of Fluid Flow

Conservation of Linear and Angular Momentum

Equations of Linear and Angular Momentum

Momentum Conservation Using a Control Volume

Conservation of Energy

Energy Conservation Using a Control Volume

General Conservation Principle

The Closure Problem

Stokes’ Law of Friction

Interpretation of Pressure

The Dissipation Function

Constitutive Equation for Non-Newtonian Fluids

Thermodynamic Aspects of Pressure and Viscosity

Equations of Motion in Lagrangian Coordinates

References

Problems

**The Navier–Stokes Equations**

Formulation of the Problem

Viscous Compressible Flow Equations

Viscous Incompressible Flow Equations

Equations of Inviscid Flow (Euler’s Equations)

Initial and Boundary Conditions

Mathematical Nature of the Equations

Vorticity and Circulation

Some Results Based on the Equations of Motion

Nondimensional Parameters in Fluid Motion

Coordinate Transformation

Streamlines and Stream Surfaces

Navier–Stokes Equations in Stream Function Form

References

Problems

**Flow of Inviscid Fluids**

Introduction

Part I: Inviscid Incompressible Flow

The Bernoulli Constant

Method of Conformal Mapping in Inviscid Flows

Sources, Sinks, and Doublets in Three Dimensions

Part II: Inviscid Compressible Flow

Basic Thermodynamics

Subsonic and Supersonic Flow

Critical and Stagnation Quantities

Isentropic Ideal Gas Relations

Unsteady Inviscid Compressible Flow in One-dimension

Steady Plane Flow of Inviscid Gases

Theory of Shock Waves

References

Problems

**Laminar Viscous Flow**

Part I: Exact Solutions

Introduction

Exact Solutions

Exact Solutions for Slow Motion

Part II: Boundary Layers

Introduction

Formulation of the Boundary Layer Problem

Boundary Layer on 2-D Curved Surfaces

Separation of the 2-D Steady Boundary Layers

Transformed Boundary Layer Equations

Momentum Integral Equation

Free Boundary Layers

Numerical Solution of the Boundary Layer Equation

Three-Dimensional Boundary Layers

Momentum Integral Equations in Three Dimensions

Separation and Attachment in Three Dimensions

Boundary Layers on Bodies of Revolution and Yawed Cylinders

Three-Dimensional Stagnation Point Flow

Boundary Layer On Rotating Blades

Numerical Solution of 3-D Boundary Layer Equations

Unsteady Boundary Layers

Second-Order Boundary Layer Theory

Inverse Problems in Boundary Layers

Formulation of the Compressible Boundary Layer Problem

Part III: Navier–Stokes Formulation

Incompressible Flow

Compressible Flow

Hyperbolic Equations and Conservation Laws

Numerical Transformation and Grid Generation

Numerical Algorithms for Viscous Compressible Flows

Thin-Layer Navier–Stokes Equations (TLNS)

References

Problems

**Turbulent Flow**

Part I: Stability Theory and the Statistical Description of Turbulence

Introduction

Stability of Laminar Flows

Formulation for Plane-Parallel Laminar Flows

Temporal Stability at Inﬁnite Reynolds Number

Numerical Algorithm for the Orr–Sommerfeld Equation

Transition to Turbulence

Statistical Methods in Turbulent Continuum Mechanics

Statistical Concepts

Internal Structure in Physical Space

Internal Structure in the Wave-Number Space

Theory of Universal Equilibrium

Part II: Development of Averaged Equations

Introduction

Averaged Equations for Incompressible Flow

Averaged Equations for Compressible Flow

Turbulent Boundary Layer Equations

Part III: Basic Empirical and Boundary Layer Results in Turbulence

The Closure Problem

Prandtl’s Mixing-Length Hypothesis

Wall-Bound Turbulent Flows

Analysis of Turbulent Boundary Layer Velocity Proﬁles

Momentum Integral Methods in Boundary Layers

Differential Equation Methods in 2-D Boundary Layers

Part IV: Turbulence Modeling

Generalization of Boussinesq’s Hypothesis

Zero-Equation Modeling in Shear Layers

One-Equation Modeling

Two-Equation (*K*-Î) Modeling

Reynolds’ Stress Equation Modeling

Application to 2-D Thin Shear Layers

Algebraic Reynolds’ Stress Closure

Development of A Nonlinear Constitutive Equation

Current Approaches to Nonlinear Modeling

Heuristic Modeling

Modeling for Compressible Flow

Three-Dimensional Boundary Layers

Illustrative Analysis of Instability

Basic Formulation of Large Eddy Simulation

References

Problems

**Mathematical Exposition 1: Base Vectors and Various Representations**

Introduction

Representations in Rectangular Cartesian Systems

Scalars, Vectors, and Tensors

Differential Operations On Tensors

Multiplication of A Tensor and A Vector

Scalar Multiplication of Two Tensors

A Collection of Usable Formulas

Taylor Expansion in Vector Form

Principal Axes of a Tensor

Transformation of T to the Principal Axes

Quadratic Form and the Eigenvalue Problem

Representation in Curvilinear Coordinates

Christoffel Symbols in Three Dimensions

Some Derivative Relations

Scalar and Double Dot Products of Two Tensors

**Mathematical Exposition 2: Theorems of Gauss, Green, and Stokes**

Gauss’ Theorem

Green’s Theorem

Stokes’ Theorem

**Mathematical Exposition 3: Geometry of Space and Plane Curves**

Basic Theory of Curves

**Mathematical Exposition 4: Formulas for Coordinate Transformation**

Introduction

Transformation Law for Scalars

Transformation Laws for Vectors

Transformation Laws for Tensors

Transformation Laws for the Christoffel Symbols

Some Formulas in Cartesian and Curvilinear Coordinates

**Mathematical Exposition 5: Potential Theory**

Introduction

Formulas of Green

Potential Theory

General Representation of a Vector

An Application of Green’s First Formula

**Mathematical Exposition 6: Singularities of the First-Order ODEs**

Introduction

Singularities and Their Classiﬁcation

**Mathematical Exposition 7: Geometry of Surfaces**

Basic Deﬁnitions

Formulas of Gauss

Formulas of Weingarten

Equations of Gauss

Normal and Geodesic Curvatures

Grid Generation in Surfaces

**Mathematical Exposition 8: Finite Difference Approximation Applied to PDEs**

Introduction

Calculus of Finite Differences

Iterative Root Finding

Numerical Integration

Finite Difference Approximations of Partial Derivatives

Finite Difference Approximation of Parabolic PDEs

Finite Difference Approximation of Elliptic Equations

**Mathematical Exposition 9: Frame Invariancy**

Introduction

Orthogonal Tensor

Arbitrary Rectangular Frames of Reference

Check for Frame Invariancy

Use of Q

References for the Mathematical Expositions

Index