Page 313 - Aerodynamics for Engineering Students

Compressible flow 295
-
u)

Direction of
T Values for successive
wave propagation particles in direction
of wave motion
- instant t
at
---- at instant t + 6t

Fig. 6.5

displacement, and hence velocity, pressure, etc., of an individual particle of gas is
changing continuously while it is under the influence of the passing impulse.
A more graphic way of expressing the gas conditions in the tube is to plot those of
successive particles in the direction of movement of the impulse, at a given instant of
time while the impulse is passing. Another curve of the particles’ velocities at an
instant later shows how individual particles behave.
Fig. 6.5 shows a typical set of curves for the passage of small pressure impulses,
and a matter of immediate interest is that an individual particle moves in the
direction of the wave propagation when its pressure is above the mean, and in the
reverse direction in the expansive phase.

6.3.1 The speed of sound (acoustic speed)
The changing conditions imposed on individual particles of gas as the pressure
pulse passes is now considered. As a first simple approach to defining the pulse
and its speed of propagation, consider the stream tube to have a velocity such that
the pulse is stationary, Fig. 6.6a. The flow upstream of the pulse has velocity u,
density p and pressure p, while the exit flow has these quantities changed by infini-
tesimal amounts to u + Su, p + Spy p + Sp.
The flow situation now to be considered is quasi-steady, assumed inviscid
and adiabatic (since the very small pressure changes take place too rapidly for
heat transfer to be significant), takes place in the absence of external forces, and is
one-dimensional, so that the differential equations of continuity and motion are
respectively

ap au
u-+p-=o (6.31)
ax ax