let us now investigate this using the numerical procedure and in order to do so we will
consider the case with n ¼ 2. Let us assume the piston moves according to;
u t
ð Þ ¼ at
2 ; 10 ! t ! 0
¼ 1; t > 10,
where a ¼ 0.01. As in the case of the linear ramp, we will assume that γ ¼ 1.4 and
c 0 ¼
ffiffi ffi
γ
p as the pressure and density in the undisturbed medium have unity values.
Using these parameters in the above equations one finds that t shock ¼ 14.62 and
x shock ¼ 14.63 (both in arbitrary units). For the numerical procedure the following
spatial and time increments were assumed; Δx ¼ 0.3 and Δt ¼ 0.05. Figure 4.15
shows the particle velocity as a function of position at five different times. One can
verify from the plots that the forward front of the wave advances at the sonic speed c 0
while the velocity profile begins to exhibit an almost vertical profile with increasing
time and that the onset of this vertical profile occurs at an intermediate point and not
at the forward front of the wave. The broken line in Fig. 4.15 corresponds to t ¼ 14.6
and represents the approximate time taken for the shock to form according to the
above equation for t shock while its position is in conformity with the equation above
for x shock . Figure 4.16 shows a plot of the artificial viscosity q as a function of
position at times similar to those chosen for Fig. 4.15 (the artificial viscosity is
negligible at t ¼ 7.5 and at t ¼ 10 in comparison to the viscosity at larger values of
t and, accordingly, they are not visible in the scale adopted for this figure). Since the
artificial viscosity only becomes dominant at the shock front we not only see a
significant increase in the value of q at t ¼ 14.6, but also a distinctive narrowing of
the q-profile that one expects to observe in the vicinity of the shock front. Accordingly, it can be seen that the numerical results obtained here are in accord with the
analytical predictions.
4.8.4 Tube Closed at End: A Reflected Shock
Let us now consider again our original example of piston motion in a tube but, in this
instance, we will assume that the tube is closed at some position L on the right-hand
side so that the following boundary condition applies, namely, u(t, L) ¼ 0. Here we
take L ¼ 80 units, and with Δx ¼ 0.4 in this particular case, we will divide the space
into 200 evenly spaced points separated Δx units apart. The following parameters are
assumed: γ ¼ 1.4, κ ¼ 1.5, Δx ¼ 0.4, Δt ¼ 0.05, ρ 0 ¼ 1. Similar to the example in
Sect. 4.8.1, the piston motion is given by u p (t, 0) ¼ 0.3 and the boundary condition at
the end of the tube is also included in the numerical procedure.
Plots showing the particle velocity, pressure and density are shown in Figs. 4.17,
4.18 and 4.19 for three different times. In addition, a plot of the artificial viscosity at
t ¼ 50 is included as shown in Fig. 4.20 which enables one to determine position of
the shock front with some accuracy from the maximum value of q. The plots
160
4 Numerical Treatment of Plane Shocks
consider the case with n ¼ 2. Let us assume the piston moves according to;
u t
ð Þ ¼ at
2 ; 10 ! t ! 0
¼ 1; t > 10,
where a ¼ 0.01. As in the case of the linear ramp, we will assume that γ ¼ 1.4 and
c 0 ¼
ffiffi ffi
γ
p as the pressure and density in the undisturbed medium have unity values.
Using these parameters in the above equations one finds that t shock ¼ 14.62 and
x shock ¼ 14.63 (both in arbitrary units). For the numerical procedure the following
spatial and time increments were assumed; Δx ¼ 0.3 and Δt ¼ 0.05. Figure 4.15
shows the particle velocity as a function of position at five different times. One can
verify from the plots that the forward front of the wave advances at the sonic speed c 0
while the velocity profile begins to exhibit an almost vertical profile with increasing
time and that the onset of this vertical profile occurs at an intermediate point and not
at the forward front of the wave. The broken line in Fig. 4.15 corresponds to t ¼ 14.6
and represents the approximate time taken for the shock to form according to the
above equation for t shock while its position is in conformity with the equation above
for x shock . Figure 4.16 shows a plot of the artificial viscosity q as a function of
position at times similar to those chosen for Fig. 4.15 (the artificial viscosity is
negligible at t ¼ 7.5 and at t ¼ 10 in comparison to the viscosity at larger values of
t and, accordingly, they are not visible in the scale adopted for this figure). Since the
artificial viscosity only becomes dominant at the shock front we not only see a
significant increase in the value of q at t ¼ 14.6, but also a distinctive narrowing of
the q-profile that one expects to observe in the vicinity of the shock front. Accordingly, it can be seen that the numerical results obtained here are in accord with the
analytical predictions.
4.8.4 Tube Closed at End: A Reflected Shock
Let us now consider again our original example of piston motion in a tube but, in this
instance, we will assume that the tube is closed at some position L on the right-hand
side so that the following boundary condition applies, namely, u(t, L) ¼ 0. Here we
take L ¼ 80 units, and with Δx ¼ 0.4 in this particular case, we will divide the space
into 200 evenly spaced points separated Δx units apart. The following parameters are
assumed: γ ¼ 1.4, κ ¼ 1.5, Δx ¼ 0.4, Δt ¼ 0.05, ρ 0 ¼ 1. Similar to the example in
Sect. 4.8.1, the piston motion is given by u p (t, 0) ¼ 0.3 and the boundary condition at
the end of the tube is also included in the numerical procedure.
Plots showing the particle velocity, pressure and density are shown in Figs. 4.17,
4.18 and 4.19 for three different times. In addition, a plot of the artificial viscosity at
t ¼ 50 is included as shown in Fig. 4.20 which enables one to determine position of
the shock front with some accuracy from the maximum value of q. The plots
160
4 Numerical Treatment of Plane Shocks