(b) Overtaking Shock Waves
Let us now consider the case of two plane shock waves following each other and
the overtaking of one shock by the other. The two shocks are generated by assuming
the piston has the following velocity;
u n,0 ¼ 0:8 if 0 n 500
¼ 0 if 500 < n 1250
¼ 1 if n > 1250
where n is the time-stepping index, hence, the numerical outputs for the piston’s
velocity and position are shown plotted in Fig. 4.63 where Δt ¼ 0.04 is the timestepping increment. Pressure plots of the two forward moving shock waves as a
function of position are shown plotted at four different times in Fig. 4.64 and the
markers in these plots show the position of the piston at these times.
After the second shock overtakes the first shock, a transmitted shock and a weak
reflected rarefaction wave result [17] as shown in Fig. 4.65 and the region between
them is divided by a contact surface. An expanded view of the numerical output for
this region (bounded by the marker at x ¼ 255 and forward front of the transmitted
shock) is shown in Fig. 4.66. One observes that this contact surface (indicated by the
marker at x ¼ 377) divides this region into two zones where the density and the
temperature (which is proportional to the product of the pressure and specific
volume) differ within the region while the pressure and particle velocity are the
same on both sides.
(c) Colliding Shock waves
The final example to be considered involves the collision of two shock waves
where one of the colliding shocks is produced by short-duration piston motion and
the shock’s subsequent reflection from an end wall. This returned shock then meets a
second outgoing shock wave which is produced by the further motion of the piston.
In the example to be considered here we will assume that the piston’s velocity is
given by,
u n,0 ¼ 0:4 if 0 n 500
¼ 0 if 500 < n 3000
¼ 0:6 if n > 3000
4.8 Numerical Examples of Plane Shocks
207
Let us now consider the case of two plane shock waves following each other and
the overtaking of one shock by the other. The two shocks are generated by assuming
the piston has the following velocity;
u n,0 ¼ 0:8 if 0 n 500
¼ 0 if 500 < n 1250
¼ 1 if n > 1250
where n is the time-stepping index, hence, the numerical outputs for the piston’s
velocity and position are shown plotted in Fig. 4.63 where Δt ¼ 0.04 is the timestepping increment. Pressure plots of the two forward moving shock waves as a
function of position are shown plotted at four different times in Fig. 4.64 and the
markers in these plots show the position of the piston at these times.
After the second shock overtakes the first shock, a transmitted shock and a weak
reflected rarefaction wave result [17] as shown in Fig. 4.65 and the region between
them is divided by a contact surface. An expanded view of the numerical output for
this region (bounded by the marker at x ¼ 255 and forward front of the transmitted
shock) is shown in Fig. 4.66. One observes that this contact surface (indicated by the
marker at x ¼ 377) divides this region into two zones where the density and the
temperature (which is proportional to the product of the pressure and specific
volume) differ within the region while the pressure and particle velocity are the
same on both sides.
(c) Colliding Shock waves
The final example to be considered involves the collision of two shock waves
where one of the colliding shocks is produced by short-duration piston motion and
the shock’s subsequent reflection from an end wall. This returned shock then meets a
second outgoing shock wave which is produced by the further motion of the piston.
In the example to be considered here we will assume that the piston’s velocity is
given by,
u n,0 ¼ 0:4 if 0 n 500
¼ 0 if 500 < n 3000
¼ 0:6 if n > 3000
4.8 Numerical Examples of Plane Shocks
207