this plot at t ¼ 6.390 identifies that fluid element with an approximate initial position
of x ¼ 83Δx ¼ 8.3 and the approximate particle velocity (within the numerical
approximations) is given by u 1374, 83 , while the computed numerical output gives,
u 1374,83 ¼ À0:372,
which is in general agreement with the value of À0.375 based on the method of
characteristics. This particular particle velocity is shown plotted as a function of time
in Fig. A.12 and the marker at t ¼ 13.736 shows the particle velocity at this time as
À0.372.
By using the finite difference equations as presented in Sect. 4.5.3 some typical
particle paths were determined and they are shown in Fig. A.13 with the tube closed
at x ¼ 10. For comparison, the particle paths for an identical piston withdrawal
problem are shown in Fig. A.14 when the tube is not closed at the end.
Fig. A.10 Numerical
output showing the particle
velocity as a function of
time for the particle
originally located at
x ¼ 94Δx. The markers in
the plot show the particle
velocity of À0.128 at
t ¼ 11.8 (see text)
Fig. A.11 Particle position
at t ¼ 1374Δt as a function
of the initial position ( jΔx)
of the fluid elements
330
Appendix A