Appendix A
Further Consideration of the Piston Withdrawal Problem
In this appendix we will investigate in some detail the piston withdrawal problem
presented in Sect. 2.7.2 and sketched again here in Fig. A.1; we will do so by
considering a specific numerical example. We will consider a point on the head of
the expansion fan at a specific time t 0 which we will assume to be at t 0 ¼ 1, and we
will then follow the particle-path and the negative characteristic in the x-t plane that
passes through this point. We will go on to generate plots of the particle velocity, the
pressure and the density based on the method of characteristics and we will compare
the results with a numerical calculation using artificial viscosity that was similarly
implemented in Sect. 4.8.6 but having a much smaller grid interval. Finally, we will
consider this piston withdrawal problem in the case where the tube is closed at some
distance from the piston [1–3].
The Expansion Fan
We will assume that the air occupying the region x ! 0 at t ¼ 0 is at rest with
pressure, p 0 ¼ 1, density, ρ 0 ¼ 1 and with γ ¼ 1.4. The ambient speed of sound in
this region is given by the equation, c 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
γp 0 =ρ 0
p
, hence, c 0 ¼
ffiffi ffi
γ
p ¼ 1:183. The
piston is assumed to be suddenly withdrawn at a constant speed of u 0 ¼ 0.9, so that
its displacement is given by the equation, x(t) ¼ À u 0 t. The equation for the positive
characteristic at the head of the expansion fan is given by (see Sect. 2.7.1)
x ¼ c 0 t ¼ 1:183t
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7
313
Further Consideration of the Piston Withdrawal Problem
In this appendix we will investigate in some detail the piston withdrawal problem
presented in Sect. 2.7.2 and sketched again here in Fig. A.1; we will do so by
considering a specific numerical example. We will consider a point on the head of
the expansion fan at a specific time t 0 which we will assume to be at t 0 ¼ 1, and we
will then follow the particle-path and the negative characteristic in the x-t plane that
passes through this point. We will go on to generate plots of the particle velocity, the
pressure and the density based on the method of characteristics and we will compare
the results with a numerical calculation using artificial viscosity that was similarly
implemented in Sect. 4.8.6 but having a much smaller grid interval. Finally, we will
consider this piston withdrawal problem in the case where the tube is closed at some
distance from the piston [1–3].
The Expansion Fan
We will assume that the air occupying the region x ! 0 at t ¼ 0 is at rest with
pressure, p 0 ¼ 1, density, ρ 0 ¼ 1 and with γ ¼ 1.4. The ambient speed of sound in
this region is given by the equation, c 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
γp 0 =ρ 0
p
, hence, c 0 ¼
ffiffi ffi
γ
p ¼ 1:183. The
piston is assumed to be suddenly withdrawn at a constant speed of u 0 ¼ 0.9, so that
its displacement is given by the equation, x(t) ¼ À u 0 t. The equation for the positive
characteristic at the head of the expansion fan is given by (see Sect. 2.7.1)
x ¼ c 0 t ¼ 1:183t
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7
313