Introduction to Simple Shock Waves in Air: With Numerical Solutions Using Artificial Viscosity

Preface to the Second Edition
Preface to the First Edition
Contents
Chapter 1: Brief Outline of the Equations of Fluid Flow
	1.1 Introduction
	1.2 Eulerian and Lagrangian Form of the Equations
	1.3 Some Elements of Thermodynamics
		1.3.1 Ideal Gas Equation
		1.3.2 The First Law of Thermodynamics
		1.3.3 Heat Capacity
		1.3.4 Isothermal Expansion or Compression of an Ideal Gas
		1.3.5 Reversible Adiabatic Process for an Ideal Gas
		1.3.6 Work Done by an Ideal Gas During an Adiabatic Expansion
		1.3.7 Alternate Form of the Equations for Specific Internal Energy and Enthalpy
		1.3.8 Ratio of the Specific Heats for Air
		1.3.9 The Second Law of Thermodynamics
	1.4 Conservation Equations in Plane Geometry
		1.4.1 Equation of Mass Conservation: The Continuity Equation
		1.4.2 Equation of Motion: The Momentum Equation
		1.4.3 Energy Balance Equation
	1.5 Constancy of the Entropy with Time for a Fluid Element
	1.6 Entropy Change for an Ideal Gas
	1.7 Spherical Geometry
		1.7.1 Continuity Equation
		1.7.2 Equation of Motion
		1.7.3 Equation of Energy Conservation
	1.8 Small Amplitude Disturbances: Sound Waves
	1.9 Typical Sound Wave Parameters
		1.9.1 Typical Sound Intensity in Normal Conversation
		1.9.2 Loud Sounds
	References
Chapter 2: Waves of Finite Amplitude
	2.1 Introduction
	2.2 Finite Amplitude Waves
	2.3 Change in Wave Profile
	2.4 Formation of a Normal Shock Wave
	2.5 Time and Place of Formation of Discontinuity
		2.5.1 Example: Piston Moving with Uniform Accelerated Velocity
		2.5.2 Example: Piston Moving with a Velocity u = atn, n> 0
	2.6 Another Form of the Equations: Riemann Invariants
		2.6.1 Solution of some First-Order Partial Differential Equations
		2.6.2 Nonlinear Equation
		2.6.3 An Example of Nonlinear Distortion
		2.6.4 The Breaking Time
	2.7 Application of Riemann Invariants to Simple Flow Problems
		2.7.1 Piston Withdrawal
		2.7.2 Piston Withdrawal at Constant Speed
		2.7.3 Piston Moving into a Tube
		2.7.4 Numerically Integrating the Equations of Motion Based Riemann´s Method
	References
Chapter 3: Conditions Across the Shock: The Rankine-Hugoniot Equations
	3.1 Introduction to Normal Shock Waves
	3.2 Conservation Equations
		3.2.1 Conservation of Mass
		3.2.2 Conservation of Momentum
		3.2.3 Conservation of Energy
	3.3 Thermodynamic Relations
	3.4 Alternative Notation for the Conservation Equations Across the Shock
	3.5 A Very Weak Shock
	3.6 Rankine-Hugoniot Equations
		3.6.1 Pressure and Density Changes for a Weak Shock
	3.7 Entropy Change of the Gas on Its Passage Through a Shock
	3.8 Other Useful Relationships in Terms of Mach Number
	3.9 Entropy Change Across the Shock in Terms of Mach Number
	3.10 Fluid Motion Behind the Shock in Terms of Shock Wave Parameters
	3.11 Reflection of a Plane Shock from a Rigid Plane Surface
	3.12 Approximate Analytical Expressions for Weak Shock Waves
		3.12.1 Shock Velocity for Weak Shocks
		3.12.2 Pressure Ratio for Weak Shocks
		3.12.3 Density Ratio for Weak Shocks
		3.12.4 Temperature Ratio for Weak Shocks
		3.12.5 Sound Speed Ratio for Weak Shocks
		3.12.6 Entropy Change for Weak Shocks
		3.12.7 Change in the Riemann Invariant R- for Weak Shocks
	3.13 Thickness of the Shock Wave Region
	3.14 Conclusions
	References
Chapter 4: Numerical Treatment of Plane Shocks
	4.1 Introduction
	4.2 The Need for Numerical Techniques
	4.3 Lagrangian Equations in Plane Geometry with Artificial Viscosity
		4.3.1 Continuity Equation
		4.3.2 Equation of Motion
		4.3.3 Equation of Energy Conservation
	4.4 Artificial Viscosity
		4.4.1 Equations for Plane-Wave Motion with Artificial Viscosity
		4.4.2 A Steady-State Plane Shock with Artificial Viscosity
		4.4.3 Variation in the Specific Volume Across the Shock
	4.5 The Numerical Procedure
		4.5.1 The Differential Equations for Plane Wave Motion: A Summary
		4.5.2 Finite Difference Expressions
		4.5.3 The Discrete Form of the Equations
	4.6 Stability of the Difference Equations
	4.7 Grid Spacing
	4.8 Numerical Examples of Plane Shocks
		4.8.1 Piston Generated Shock Wave
		4.8.2 Linear Ramp
		4.8.3 Piston Motion According to the Law u = atn; n > 0.
		4.8.4 Tube Closed at End: A Reflected Shock
		4.8.5 The Numerical Value of κ for the Artificial Viscosity
		4.8.6 Piston Withdrawal Generating an Expansion Wave
		4.8.7 The Shock Tube
		4.8.8 The Effect of Amplitude on Wave Propagation
		4.8.9 Short Duration Piston Motion: Shock Decay
		4.8.10 Some Numerical Results for Shock Wave Interactions
	4.9 Conclusions
	References
Chapter 5: Spherical Shock Waves: The Self-similar Solution
	5.1 Introduction
	5.2 Shock Wave from an Intense Explosion
	5.3 The Point Source Solution
	5.4 Taylor´s Analysis of Very Intense Shocks
		5.4.1 Momentum Equation
		5.4.2 Continuity Equation
		5.4.3 Energy Equation
	5.5 Derivatives at the Shock Front
	5.6 Numerical Integration of the Equations
	5.7 Energy of the Explosion
	5.8 The Pressure
	5.9 The Temperature
	5.10 The Pressure-Time Relationship for a Fixed Point
	5.11 Taylor´s Analytical Approximations for Velocity, Pressure and Density
		5.11.1 The Velocity phi
		5.11.2 The Pressure f
		5.11.3 The Density ψ
	5.12 The Density for Small Values of η
	5.13 The Temperature in the Central Region
	5.14 The Wasted Energy
	5.15 Taylor´s Second Paper
	5.16 Approximate Treatment of Strong Shocks
		5.16.1 Chernyi´s Approximation
		5.16.2 Bethe´s Approximation for Small Values of γ - 1
	5.17 Route to an Analytical Solution
	5.18 Analytical Solution Method
		5.18.1 The Analytical Expression for the Velocity
		5.18.2 The Analytical Expression for the Density
		5.18.3 The Analytical Expression for the Pressure
	References
Chapter 6: Numerical Treatment of Spherical Shock Waves
	6.1 Introduction
	6.2 Lagrangian Equations in Spherical Geometry
		6.2.1 Momentum Equation
		6.2.2 Continuity Equation
		6.2.3 Energy Equation
	6.3 Conservation Equations in Spherical Geometry: A Summary
	6.4 Difference Equations
	6.5 Numerical Solution of Spherical Shock Waves: The Point Source Solution
	6.6 Initial Conditions Using the Strong-Shock, Point-Source Solution
		6.6.1 The Pressure
		6.6.2 The Velocity
		6.6.3 The Density
	6.7 Specification of Initial Conditions
	6.8 Results of the Numerical Integration
	6.9 Shock Wave from a Sphere of High-Pressure, High-Temperature Gas
	6.10 Results of the Numerical Integration for the Expanding Sphere
		6.10.1 Pressure
		6.10.2 Density
		6.10.3 Velocity
	6.11 A Note on Grid Size
	6.12 Conclusions
	References
Appendix A
	Further Consideration of the Piston Withdrawal Problem
		The Expansion Fan
		Particle-Path
		Positive Characteristic Leaving the Piston´s Surface
		Parameter Variations According to the Method of Characteristics
		Parameter Variations According to the Numerical Calculations
		Tube Closed at End
		Boundary Condition
		Calculating the Coordinates of a Boundary Point
		Calculating the Coordinates of an Internal Point
		Comparison with the Results of the Finite Difference Calculations
	References
Appendix B
	Some Numerical Results for a Closed Shock Tube
Index