Theory of shock waves and introduction to gas dynamics/Chapter 7
Chapter 7
Theory of Shock Waves. Introduction
In the preceding chapters we dealt with the cases where classic gas dynamics which operates with the concept of a continuous pressure distribution and uses differential equations to describe certain phenomena, but ignores viscosity and thermal conduction, runs into certain difficulties. Let us remind the reader of the nature of these difficulties.
In the chapter on sound propagation we established that the sound wave is subject to deformation as it propagates. The "wave crests", i.e., the places where the substance is compressed and moves in the direction of wave propagation, run ahead. Conversely, the "troughs", i.e., the expansion regions where the substance moves in a direction opposite to the propagation of sound, fall behind the wave as a whole. Thus, the sound wave, as it is deformed, lashes itself -- a phenomenon similar to the one observed when sea waves run on a shallow beach.
We have mentioned several times that the analogy between gas dynamics and phenomena occurring in liquids with a free surface has a very deep and far-reaching significance. In both cases there is a tendency towards a spontaneous increase in the gradients, toward a spontaneous formation of discontinuities during compression.
In the theory of outflow from a Laval nozzle we established that it is impossible to describe a number of intermediate regimes in a specific large region of counterpressure values by means of only the equations of continuous flow with constant entropy.
Finally, in the last problem investigated by us, namely, in the case of the motion of a gas caused by the sudden movement of a piston, this limitation of classical gas dynamics became particularly obvious. Thus we have seen that if the piston moves in the same direction of the gas, , and the differential equations of gas dynamics lead to absurd trivalent solutions, that is, solutions according to which in one and the same spot there must simultaneously exist three values for density, three values for temperature and three values for velocity.
All these cases indicate that there must be other forms of solution in gas dynamics which are not directly derived from the equations of ideal gases (ideal here refers to the absence of viscosity and thermal conduction). It can be expected that tor the conditions sought for a large value of gradients will be characteristic, so that in a given approximation they may be treated as the propagation of the discontinuity surfaces of velocity, pressure and density -- the so-called shock waves.
Before we go into the history of the problem of shock waves, we shall derive in an elementary form the equations of a shock wave, approximately in the same way as Hugoniot in his well-known book "On the Propagation of Discontinuities" [56]. We shall postulate the existence of a discontinuity (explosion), and shall not investigate how it was achieved, whether it is steady, and so on.