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

by Seán Prunty

Overview

This book provides an elementary introduction to some 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. Artificial viscosity is introduced into the numerical calculations in order to deal with shocks. The presentation is restricted to the finite-difference approach to solve the coupled differential equations of fluid flow as distinct from finite-volume or finite-element methods. This text 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 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.

Product Details

ISBN-13:	9783030636050
Publisher:	Springer International Publishing
Publication date:	01/23/2021
Series:	Shock Wave and High Pressure Phenomena
Edition description:	2nd ed. 2021
Pages:	344
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

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.

Table of Contents

1 Brief outline of the equations of fluid flow

1.1 Introduction

1.2 Eulerian and Lagrangian form of the equations

1.3 Conservation equations in plane geometry

1.3.1 Equation of mass conservation: the continuity equation

1.3.2 Equation of motion: the momentum equation

1.3.3 Energy balance equation

1.4 Constancy of the entropy with time for a fluid element

1.5 Entropy change for an ideal gas

1.6 Spherical geometry

1.6.1 Continuity equation

1.6.2 Equation of motion

1.6.3 Equation of energy conservation

1.7 Small amplitude disturbances: sound waves

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 >0

2.6 Another forms of the equations: Riemann invariants

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 Rankine-Hugoniot equations

3.6 Other useful relationships in terms of Mach number

3.7 Fluid flow behind the shock in terms of shock wave parameters

3.8 Reflection of a plane shock from a rigid plane surface

3.9 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 The differential equations for plane wave motion: a summary

4.5 Difference 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 The shock tube

4.8.4 Tube closed at end

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

5.17 Conclusions

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 Results of the numerical integration

6.8 Shock wave from a sphere of high pressure, high temperature gas

6.9 Results of the numerical integration for the expanding sphere

6.9.1 The pressure

6.9.2 The density

6.9.3 The velocity

6.10 A final note