2.7.3 Piston Moving into a Tube
Suppose, on the other hand, the piston moves into the tube with uniform acceleration
a, such that, its velocity is u p (t) ¼ at and its displacement at time t is given by
x p (t) ¼ at
2 /2 as shown in Fig. 2.19.
By assuming that the air is at rest in the tube at t ¼ 0 we can easy verify, as in the
case where we considered the piston being withdrawn, that we obtain a uniform
region bounded by the characteristic x ¼ c 0 t as shown in Fig. 2.20. The characteristics are shown as broken lines.
Taking a point A in the air adjacent to the piston as shown in Fig. 2.20 and let us
consider a negative characteristic from the uniform region to A; the Riemann
invariant for this negative characteristic is
u A À
2c A
γ À 1
¼ À
2c 0
γ À 1
and solving for c A gives
Fig. 2.19 Piston moving
into a tube with uniform
acceleration and the
associated plot of the piston
path on the xt-plane
Fig. 2.20 Characteristics
are sketched for the uniform
region when the piston
moves into the tube (see
text)
2.7 Application of Riemann Invariants to Simple Flow Problems
81