within the range; 0.103 x 1.183, and these are shown plotted in Fig. A.3. With
γ ¼ 1.4, the exponents for the pressure and density are 7 and 5, respectively, which
illustrates that the decrease in these quantities within the expansion fan is quite rapid.
Parameter Variations According to the Numerical Calculations
Let us now compare the above results with those generated numerically using
artificial viscosity that are similar to those already presented in Sect. 4.8.6. We
note, however, in this case that the piston is withdrawn at a greater velocity of
0.9 (Arb. units) in comparison to the example considered in Sect. 4.8.6, and to ensure
greater accuracy, the grid increments Δx and Δt are reduced by an order of magnitude
so that Δx ¼ 0.01 and Δt ¼ 0.001. Other parameters are γ ¼ 1.4, p 0 ¼ 1, ρ 0 ¼ 1 and
κ ¼ 1.2. Figure A.4 shows the results of the numerical calculations for the particle
velocity, the pressure and the density (indicated by the solid lines) at t ¼ 1 (hence,
1000 time steps) and for comparison the variation of these quantities within the
expansion fan, according to the method of characteristics (as shown in Fig. A.3) is
included in the plots and displayed as broken lines.
Tube Closed at End
Let us now consider the slightly more complicated piston withdrawal problem where
the tube is closed at some position L as illustrated in Fig. A.5 and the positive
characteristics in the expansion fan reflect off the end wall. We must now deal with
the network of characteristics, both positive and negative, in the region agh and
designated as region ℜ 4 . This region has two families of curved characteristics and
we will utilize the numerical method of characteristics that was briefly discussed in
Sect. 2.7.4 in order to obtain a solution for this region. The solution involves the
calculation of the coordinates of the network of points for the characteristics in ℜ 4 .
To simplify the initial numerical values, we will assume that the piston is
withdrawn at a constant speed u p of 0.75 (Arb. units) and that the speed of sound
c 0 in the undisturbed air is 1.2 (Arb. units) and, as in the previous example, γ is taken
as 1.4. The ambient density ρ 0 and pressure p 0 within the tube are 1 and 1.03,
respectively; where the slightly higher pressure of 1.03 is in accordance with the
adopted speed of sound according to the equation, c 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
γp 0 =ρ 0
p
.
In general, the positive and negative characteristics are
dx
dt
C þ
¼ u þ c and
dx
dt
C À
¼ u À c,
respectively, and along C + we have the Riemann invariant
Appendix A
319
γ ¼ 1.4, the exponents for the pressure and density are 7 and 5, respectively, which
illustrates that the decrease in these quantities within the expansion fan is quite rapid.
Parameter Variations According to the Numerical Calculations
Let us now compare the above results with those generated numerically using
artificial viscosity that are similar to those already presented in Sect. 4.8.6. We
note, however, in this case that the piston is withdrawn at a greater velocity of
0.9 (Arb. units) in comparison to the example considered in Sect. 4.8.6, and to ensure
greater accuracy, the grid increments Δx and Δt are reduced by an order of magnitude
so that Δx ¼ 0.01 and Δt ¼ 0.001. Other parameters are γ ¼ 1.4, p 0 ¼ 1, ρ 0 ¼ 1 and
κ ¼ 1.2. Figure A.4 shows the results of the numerical calculations for the particle
velocity, the pressure and the density (indicated by the solid lines) at t ¼ 1 (hence,
1000 time steps) and for comparison the variation of these quantities within the
expansion fan, according to the method of characteristics (as shown in Fig. A.3) is
included in the plots and displayed as broken lines.
Tube Closed at End
Let us now consider the slightly more complicated piston withdrawal problem where
the tube is closed at some position L as illustrated in Fig. A.5 and the positive
characteristics in the expansion fan reflect off the end wall. We must now deal with
the network of characteristics, both positive and negative, in the region agh and
designated as region ℜ 4 . This region has two families of curved characteristics and
we will utilize the numerical method of characteristics that was briefly discussed in
Sect. 2.7.4 in order to obtain a solution for this region. The solution involves the
calculation of the coordinates of the network of points for the characteristics in ℜ 4 .
To simplify the initial numerical values, we will assume that the piston is
withdrawn at a constant speed u p of 0.75 (Arb. units) and that the speed of sound
c 0 in the undisturbed air is 1.2 (Arb. units) and, as in the previous example, γ is taken
as 1.4. The ambient density ρ 0 and pressure p 0 within the tube are 1 and 1.03,
respectively; where the slightly higher pressure of 1.03 is in accordance with the
adopted speed of sound according to the equation, c 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
γp 0 =ρ 0
p
.
In general, the positive and negative characteristics are
dx
dt
C þ
¼ u þ c and
dx
dt
C À
¼ u À c,
respectively, and along C + we have the Riemann invariant
Appendix A
319