2.7.2 Piston Withdrawal at Constant Speed
Let us now apply the above considerations to the case where the piston is withdrawn
at a constant speed u 0 , hence, x p (t 0 ) ¼ À u 0 t 0 and u p (t 0 ) ¼ À u 0 so that Eq. (2.70)
becomes
x t
ð Þ ¼ Àu 0 t 0 þ c 0 À
γ þ 1
2
u 0
h
i
t À t 0
ð
Þ,
ð2:71Þ
which applies to all positive characteristics leaving the piston and all of these
characteristics have
u ¼ Àu 0 and c ¼ c 0 À
γ À 1
2
u 0
ð2:72Þ
with identical slopes of
c 0 À
γ þ 1
2
u 0 :
The first characteristic leaves the piston at t 0 ¼ 0 and Eq. (2.71) gives
x t
ð Þ ¼ c 0 À
γ þ 1
2
u 0
h
i
t
ð2:73Þ
and the region ℜ3 to the left of this characteristic as shown in Fig. 2.17 is a uniform
region according to Eq. (2.72) and the negative characteristics are also straight lines.
One can observe from Fig. 2.17 that there is a region ℜ2 with no positive
characteristics as Eq. (2.71) only applies to positive characteristics leaving the
piston; however, negative characteristics penetrate this region as Eq. (2.66) still
Fig. 2.17 Characteristics are sketched in the case where the piston is withdrawn at constant speed
(see text)
2.7 Application of Riemann Invariants to Simple Flow Problems
77