dx
dt
¼ c 0 and
dx
dt
¼ Àc 0 , respectively:
These characteristics are straight lines, hence,
x ¼ c 0 t þ x0 þ and x ¼ Àc 0 t þ x0 À
where x0 + and x0 À are the intercepts on the x-axis for the C + and C À characteristics,
respectively. The region ℜ1 traversed by these characteristics (shown as dashed
lines) is classified as a uniform region where u ¼ 0 and c ¼ c 0 , and the region is
bounded by the characteristic x ¼ c 0 t as shown in Fig. 2.15.
Let us now consider region ℜ2. In this region the positive characteristics start at
the piston surface while the negative characteristics that start at t ¼ 0 penetrate into
this region, hence, Eq. (2.66) still applies to this region, that is,
u À
2c
γ À 1
¼ À
2c 0
γ À 1
and solving this equation for c gives
c ¼ c 0 þ
γ À 1
2
u:
ð2:67Þ
However, u + 2c/(γ À 1) is a constant on the positive characteristic and the
constant it “picks up” is dependent on where the characteristic intersects the piston
surface; call this constant K s , then on C + we have
u þ
2c
γ À 1
¼ K s
and using Eq. (2.67) in this latter equation gives
Fig. 2.15 Piston withdrawal showing the characteristics (broken lines) in the uniform region (see
text)
2.7 Application of Riemann Invariants to Simple Flow Problems
75