Comparison with the Results of the Finite Difference
Calculations
Let us now compare the outcome of the calculations for the network of characteristics in ℜ 4 with the results of some numerical computations that were implemented
by using the finite difference equations of Sect. 4.5.3. These finite difference
equations incorporates the boundary condition at the end wall with the following
mathematical statement, u n, 100 0, to indicate that the velocity at all times t (¼nΔt)
goes to zero at x ¼ 10, noting that x ¼ jΔx, where Δx ¼ 0.1 and j ¼ 100. In addition,
the Eulerian position of a fluid element is denoted by x n, j which gives the position of
a specific element of the fluid (denoted by the index j) at time t (¼nΔt) whose initial
position is given by jΔx. For the numerical procedure the following parameters were
adopted; Δt ¼ 0.01, γ ¼ 1.4, while the ambient pressure and density were taken as
1.03 and 1, respectively, to comply with the speed of sound (c 0 ¼ 1.2) that was
assumed for calculating the network of characteristics in ℜ 4 .
Suppose we now consider some of the points in region ℜ 4 and let us take, for
example, point P 5 with coordinates, (9.019, 11.8). The Riemann invariants passing
through this point are R + ¼ 5.25 and R À ¼ À 5.5, hence, the particle velocity at this
point according to the equation, u ¼ (R + + R À )/2, is u ¼ À 0.125. Since the time at
this point is 11.8 and with Δt ¼ 0.01, this gives n ¼ 1180 to be used in the numerical
computations.
Table A.2 Shows the results
of the numerical calculation
for the coordinates of the network of points in region ℜ 4
Point in region ℜ 4
Coordinates (x, t) in ℜ 4
P 1
(10, 9.496)
P 2
(9.189, 10.173)
P 3
(10, 10.946)
P 4
(8.225, 10.916)
P 5
(9.019, 11.8)
P 6
(10, 12.784)
P 7
(7.154, 11.682)
P 8
(7.939, 12.587)
P 9
(8.931, 13.758)
P 10
(10, 14.887)
P 11
(5.731, 12.631)
P 12
(6.390, 13.736)
P 13
(7.334, 15.092)
P 14
(8.393, 16.421)
P 15
(10, 18.213)
P 16
(4.139, 13.626)
P 17
(4.705, 14.899)
P 18
(5.557, 16.462)
P 19
(6.561, 18.016)
P 20
(8.117, 20.101)
P 21
(10, 22.325)
Appendix A
327
Calculations
Let us now compare the outcome of the calculations for the network of characteristics in ℜ 4 with the results of some numerical computations that were implemented
by using the finite difference equations of Sect. 4.5.3. These finite difference
equations incorporates the boundary condition at the end wall with the following
mathematical statement, u n, 100 0, to indicate that the velocity at all times t (¼nΔt)
goes to zero at x ¼ 10, noting that x ¼ jΔx, where Δx ¼ 0.1 and j ¼ 100. In addition,
the Eulerian position of a fluid element is denoted by x n, j which gives the position of
a specific element of the fluid (denoted by the index j) at time t (¼nΔt) whose initial
position is given by jΔx. For the numerical procedure the following parameters were
adopted; Δt ¼ 0.01, γ ¼ 1.4, while the ambient pressure and density were taken as
1.03 and 1, respectively, to comply with the speed of sound (c 0 ¼ 1.2) that was
assumed for calculating the network of characteristics in ℜ 4 .
Suppose we now consider some of the points in region ℜ 4 and let us take, for
example, point P 5 with coordinates, (9.019, 11.8). The Riemann invariants passing
through this point are R + ¼ 5.25 and R À ¼ À 5.5, hence, the particle velocity at this
point according to the equation, u ¼ (R + + R À )/2, is u ¼ À 0.125. Since the time at
this point is 11.8 and with Δt ¼ 0.01, this gives n ¼ 1180 to be used in the numerical
computations.
Table A.2 Shows the results
of the numerical calculation
for the coordinates of the network of points in region ℜ 4
Point in region ℜ 4
Coordinates (x, t) in ℜ 4
P 1
(10, 9.496)
P 2
(9.189, 10.173)
P 3
(10, 10.946)
P 4
(8.225, 10.916)
P 5
(9.019, 11.8)
P 6
(10, 12.784)
P 7
(7.154, 11.682)
P 8
(7.939, 12.587)
P 9
(8.931, 13.758)
P 10
(10, 14.887)
P 11
(5.731, 12.631)
P 12
(6.390, 13.736)
P 13
(7.334, 15.092)
P 14
(8.393, 16.421)
P 15
(10, 18.213)
P 16
(4.139, 13.626)
P 17
(4.705, 14.899)
P 18
(5.557, 16.462)
P 19
(6.561, 18.016)
P 20
(8.117, 20.101)
P 21
(10, 22.325)
Appendix A
327