Page 315 - Aerodynamics for Engineering Students

Compressible flow 297

If a disturbance of large amplitude, e.g. a rapid pressure rise, is set up there are
almost immediate physical limitations to its continuous propagation. The accelera-
tions of individual particles required for continuous propagation cannot be sustained
and a pressure front or discontinuity is built up. This pressure front is known as a
shock wave which travels through the gas at a speed, always in excess of the acoustic
speed, and together with the pressure jump, the density, temperature and entropy of
the gas increases suddenly while the normal velocity drops.
Useful and quite adequate expressions for the change of these flow properties
across the shock can be obtained by assuming that the shock front is of zero
thickness. In fact the shock wave is of finite thickness being a few molecular mean
free path lengths in magnitude, the number depending on the initial gas conditions
and the intensity of the shock.

6.4.1 One-dimensional properties of normal shock waves
Consider the flow model shown in Fig. 6.7a in which a plane shock advances
from right to left with velocity u1 into a region of still gas. Behind the shock the
velocity is suddenly increased to some value u in the direction of the wave. It is
convenient to superimpose on the system a velocity of u1 from left to right to bring
the shock stationary relative to the walls of the tube through which gas is flowing
undisturbed at u1 (Fig. 6.7b). The shock becomes a stationary discontinuity into
which gas flows with uniform conditions, p1, p1, u1, etc., and from which it flows with
uniform conditions, p2, p2, u2, etc. It is assumed that the gas is inviscid, and non-heat
conducting, so that the flow is adiabatic up to and beyond the discontinuity.
The equations of state and conservation for unit area of shock wave are:
State

(6.35)

Mass flow

Siationary
shock
(b)
Fig. 6.7