Introduction to Simple Shock Waves in Air

This chapter provides a brief review of the equations of fluid flow. The one-dimensional forms of the equations that apply to non-viscous flow are presented in both plane and spherical geometry. The Eulerian and Lagrangian descriptions of fluid motion are briefly outlined. Since this book deals with shock waves in air, which is considered to behave as a perfect gas, those elements of thermodynamics that apply to perfect gases and that have relevance to fluid motion are also presented. The chapter concludes with a brief look at small amplitude disturbances and the speed at which they are propagated in air. The equations presented in this chapter are important for the subsequent chapters.

The propagation of waves of finite amplitude and the change in wave profile leading to the formation of shock waves is the subject matter of this chapter. Simple examples of piston motion are presented to illustrate the formation of a normal shock wave and the time and place for its formation are also discussed. A brief introduction to Riemann invariants and the method of characteristics is presented and some examples to illustrate the method of characteristics in solving partial differential equations are outlined. The solution of nonlinear equations that result in the characteristics intersecting and the formation of shock waves are illustrated. The chapter concludes with some examples to illustrate the application of the method of characteristics and Riemann invariants to simple flow problems involving piston motion.

This chapter deals with the Rankine-Hugoniot relations connecting the states on both sides of a shock wave. These relationships are shown to produce important formulae connecting the pressure, density and temperature ratios on either side of the shock surface. In addition, other useful relationships are derived and presented in terms of the Mach number and the limiting form of these relationships in the case of weak and very strong shocks is discussed. The increase in the entropy of a gas on its passage through a shock is also considered. Finally, the reflection of a plane shock from a rigid plane surface is discussed and a relationship connecting the Mach numbers of the incident and reflected shocks is presented.

The numerical solution of several examples of plane shock waves using artificial viscosity and their comparison with theoretical predictions is the dominant feature of this chapter. The Lagrangian form of the equations in plane geometry is derived and after a short introduction to finite difference representations of differential equations, the discrete form of the equations is presented. Numerical solutions involving plane shocks arising from piston motion are presented, discussed and compared with the predictions of the Rankine-Hugoniot equations of Chap. 3. Reflected shocks are also considered. Piston withdrawal from a tube that generates an expansion wave is also discussed and the numerical results are compared with the predictions based on the method of characteristics presented in Chap. 2. Finally, some numerical results arising from an analysis of the shock tube are presented and discussed.

This particular chapter has an independent character and deals almost exclusively with Taylor’s analysis of very strong spherical shocks. The presentation follows Taylor’s analysis and notation in relation to the similarity solution for the point source explosion in air. Following his analysis, the partial differential equations in Eulerian form are reduced to a set of coupled ordinary differential equations which are numerically integrated. Taylor’s analytical approximations for the pressure, density and velocity are presented and these turn out to be remarkably accurate when compared to the numerical solutions. His analysis of the energy left in the atmosphere after the blast wave has propagated away has also been presented and discussed. The chapter concludes with an approximate treatment of very strong shock, which is based on the particular nature of the point source solution where most of the material is piled up at the shock front.

The numerical treatment of very strong spherical shock waves is the subject matter of this chapter. The Lagrangian equations in normalized form with artificial viscosity included are presented for spherical symmetric flow while radial distances are normalized with respect to a length based on the total blast energy and ambient air pressure. Two specific numerical procedures are presented; one in relation to the point source explosion and the second in relation to the expansion of very hot high-pressure isothermal sphere into the surrounding atmosphere. In relation to the point source explosion, Taylor’s strong shock solution was taken as initial conditions with an initial pressure of 1000 atmospheres and the equations are numerically integrated over a time interval where the pressure at the shock front drops to just a few atmospheres. The isothermal sphere had a starting pressure of 1000 atmospheres and the numerical procedure was run over a time interval where the overpressure at the shock front was down to less than 0.2 atmospheres. Several plots of pressure, density and particle velocity are presented.