Introduction to Simple Shock Waves in Air

This book provides an elementary introduction to one-dimensional fluid flow problems involving shock waves in air. The differential equations of fluid flow are approximated by finite difference equations and these in turn are numerically integrated in a stepwise manner, with artificial viscosity introduced into the numerical calculations in order to deal with shocks. This treatment of the subject is focused on the finite-difference approach to solve the coupled differential equations of fluid flow and presents the results arising from the numerical solution using Mathcad programming. Both plane and spherical shock waves are discussed with particular emphasis on very strong explosive shocks in air. This expanded second edition features substantial new material on sound wave parameters, Riemann's method for numerical integration of the equations of motion, approximate analytical expressions for weak shock waves, short duration piston motion, numerical results for shockwave interactions, and new appendices on the piston withdrawal problem and numerical results for a closed shock tube. This text will appeal to students, researchers, and professionals in shock wave research and related fields. Students in particular will appreciate the benefits of numerical methods in fluid mechanics and the level of presentation.

Auteur
Dr. Seán Prunty is a former senior lecturer in electrical and electronic engineering at University College Cork Ireland. He has a primary degree and a Ph.D. degree, both in experimental physics, from the University of Dublin, Trinity College. He has thirty years of teaching experience and has carried out research in such areas as atomic physics and laser technology as well as in far-infrared polarimetry and electromagnetic scattering for plasma physics applications. He collaborated for many years on research in the fusion energy research area in Italy, England and Switzerland. Since his retirement in 2009 he has taken a particular interest in shock wave propagation.

Contenu

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

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

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 on Riemann's method

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

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 flow 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 for weak shocks

3.13 Conclusions

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 equation

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 &^0

4.8.4 Tube closed at end: a reflected shock 4.8.5 The numerical value of 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

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 Talyor'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

5.11.2 The pressure

5.11.3 The density

5.12 The density for small values of

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

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

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…