Figures 4.17, 4.18, 4.19 and 4.20 gives the value of the quantities plotted as a
function of the initial position of the fluid elements, however, we need these
quantities as a function of the position of the fluid elements at the times indicated
in order to determine, for example, the velocity of the reflected shock. Accordingly,
the discrete form of Eq. (4.3) is used in conjunction to those already presented in
Sect. 4.5.3 and this gives the position of the fluid elements as x 1000, j , x 1500, j and
x 1800, j at the times indicated where j identifies that fluid element whose initial
position was jΔx. When plotted, we obtain the following results for the particle
velocity, pressure and density as shown in Figs. 4.21, 4.22 and 4.23. We observe
from these plots that the marker at x ¼ 15 shows the position of the piston at
t ¼ 1000Δt: this position is consistent with what we expect as the piston moves at
a constant velocity of 0.3 (Arb. units), so that its new position at this time is
x ¼ 0.3 Â 1000Δt ¼ 15 from its initial position at x ¼ 0.
We can use these numerical results to make an estimate of the velocity of the
reflected shock wave and compare it with the theoretical predictions according to the
equations presented in Sect. 3.11. Specifically, Eq. (3.54) gives the Mach number M r
of the reflected shock in terms of the Mach number M i of the incident shock
according to
M r
M
2
r À 1
¼
M i
M
2
i À 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ
2 γ À 1
ð
Þ M
2
i À 1
À
Á γ þ
1
M
2
i
γ þ 1
ð
Þ
2
v
u
u
t
:
Fig. 4.21 Particle velocity as a function of the position of the fluid elements at the times indicated
for constant piston motion in a tube closed at the end. For the numerical procedure the following
parameters apply; γ ¼ 1.4, κ ¼ 1.5, Δx ¼ 0.4 and Δt ¼ 0.05 (see text)
4.8 Numerical Examples of Plane Shocks
165
function of the initial position of the fluid elements, however, we need these
quantities as a function of the position of the fluid elements at the times indicated
in order to determine, for example, the velocity of the reflected shock. Accordingly,
the discrete form of Eq. (4.3) is used in conjunction to those already presented in
Sect. 4.5.3 and this gives the position of the fluid elements as x 1000, j , x 1500, j and
x 1800, j at the times indicated where j identifies that fluid element whose initial
position was jΔx. When plotted, we obtain the following results for the particle
velocity, pressure and density as shown in Figs. 4.21, 4.22 and 4.23. We observe
from these plots that the marker at x ¼ 15 shows the position of the piston at
t ¼ 1000Δt: this position is consistent with what we expect as the piston moves at
a constant velocity of 0.3 (Arb. units), so that its new position at this time is
x ¼ 0.3 Â 1000Δt ¼ 15 from its initial position at x ¼ 0.
We can use these numerical results to make an estimate of the velocity of the
reflected shock wave and compare it with the theoretical predictions according to the
equations presented in Sect. 3.11. Specifically, Eq. (3.54) gives the Mach number M r
of the reflected shock in terms of the Mach number M i of the incident shock
according to
M r
M
2
r À 1
¼
M i
M
2
i À 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ
2 γ À 1
ð
Þ M
2
i À 1
À
Á γ þ
1
M
2
i
γ þ 1
ð
Þ
2
v
u
u
t
:
Fig. 4.21 Particle velocity as a function of the position of the fluid elements at the times indicated
for constant piston motion in a tube closed at the end. For the numerical procedure the following
parameters apply; γ ¼ 1.4, κ ¼ 1.5, Δx ¼ 0.4 and Δt ¼ 0.05 (see text)
4.8 Numerical Examples of Plane Shocks
165