Introduction to Simple Shock Waves in Air (eBook)

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.

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