155
Using Equations (8.18) and (8.19) we can nd the position and velocity at the end of each interval in terms of their
values at the beginning. A general algorithm of the simulations is presented in Figure 8.7.
Identify the parameters, m, A, C
D
, C
L
, and r
Choose the time interval Δt and the
initial values of x, y, v
x
, v
y
, and t
Choose the maximum number of intervals
N. The maximum time will be t
max
= NΔt
Calculate the
acceleration
components
a
x
and a
y
Plot x, y, v
x
, v
y
, a
x
, and a
y
Calculate the new velocity
components v
x
and v
y
using Equation 13
Calculate the new coordinates
x and y using Equation 14
Increment the time by Δt
Stop
Iterate these steps
while n < N or t < t
max
Find flow function for
arbitrary cross section area
Transform function to
Laplace equation with the boundary
condition on the stream contours
Using a Schwarz's integral
calculate function
Find u(x, y) by Lavrentyev-
Shabat method
Find equivalent radius
by Polya-Szego method
Describe turbulent flow of air
by 2-layered Prandtle -Taylor
structure model
Figure 8.7. General algorithm of numerical analysis.
Projectile trajectories were numerically simulated according to the algorithm provided in Figure 8.7 with time inter-
val ∆t = 1 s for the fragment with mass m = 11g, face area A = 2.21 cm
2
, detonation velocity v
0
= 1,000 m/s, and initial
launch angle θ = 45°. A variation of numerically simulated projectile trajectories with drag coecient, detonation
velocity, initial launch angle, and mass of the fragment are shown in Figures 8.8–8.11, respectively.
8.2 SAMPLE PROBLEM
156 8. TOPIC AK-8
0
50
100
150
200
250
300
350
0 50 100 150 200 250 300 350 400 450 500
y, m
x, m
Numerical Simulations
m=11 g; A=2.21 cm
2
; V
0
=1,000 m/s; q=45
°
Cd = 0.5; k=39 m/s
Cd = 0.7; k=33 m/s
Cd = 0.9; k=29 m/s
Figure 8.8. Variation of numerically simulated projectile trajectories with drag coecient.
157
0
50
100
150
200
250
0 50 100 150 200 250 300 350
y, m
x, m
Numerical Simulations
m=11 g; A=2.21 cm
2
; C
D
=0.7; k=33 m/s; q=45
°
600 m/s
800 m/s
1000 m/s
Figure 8.9. Variation of numerically simulated projectile trajectories with initial launch velocities.
8.2 SAMPLE PROBLEM
158 8. TOPIC AK-8
0
50
100
150
200
250
0 50 100 150 200 250 300 350 400
y, m
x, m
Numerical Simulations
m=11 g; A=2.21 cm
2
; v
0
=1,000 m/s; C
D
=0.7; k=33 m/s
15
30
45
Figure 8.10. Variation of numerically simulated projectile trajectories with initial launch angles.
15°
30°
45°
159
0
50
100
150
200
250
0 50 100 150 200 250 300 350
y, m
x, m
Numerical Simulations
q=45
°
; A=2.21 cm
2
; v
0
=1,000 m/s; C
D
=0.7; k=33 m/s
m=11 g
m=50 g
m=100 g
Figure 8.11. Variation of numerically simulated projectile trajectories with fragment mass.
REFERENCES
Cooper, P. W. (1996). Explosives Engineering. New York:Wiley-VCH.
Küchemann, D. (2012). e Aerodynamic Design of Aircraft. AIAA Education Series. Reston, VA: American Institute
of Aeronautics & Astronautics.
REFERENCES
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