Aerothermodynamics
4.3.4.2
1
In our aerodynamic analysis, we use linear perturbation theory to determine the aerodynamic
loading on the launch vehicle. By integrating the change in pressure around the launch vehicle
we are able to solve for bending and pitching moments, drag coefficient, axial forces, normal
forces, shear forces, and the center of pressure location. All of these aerodynamic moments,
coefficients, and forces are based on the final geometry of the launch vehicle as well as the Mach
number, angle of attack, and time spent in the atmosphere. Linear perturbation theory is valid in
the subsonic and supersonic regimes, but falls apart in the transonic regime. For this reason, we
have truncated the aerodynamic outputs in the transonic regime. This better displays how the
outputs vary in the working regimes.
Mach number, variation in angle of attack, LITVC, stage separation, as well as wind gusts all
have a large impact on the aerodynamic loadings of the launch vehicle. As the launch vehicle
makes its way through the atmosphere, the change in density has a significant effect on the
impact of these forces and moments. The results for the variation of bending moment and
pitching moment with respect to Mach number at zero degree angle of attack can be found in
Figs. (4.3.4.2.1) and (4.3.4.2.2) respectively. Once the launch vehicle reaches a speed of Mach
4.4, it exits the atmosphere. At this point, the first stage has still not separated; therefore,
moments are shown as they act on the entire launch vehicle.
2000
Bending Moment (Nm)
0
0
1
2
3
4
5
-2000
-4000
-6000
-8000
-10000
Mach
Fig. 4.3.4.2.1: Variation of bending moment with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Aerothermodynamics
4.3.4.2
2
2000
Pitching Moment (Nm)
1500
1000
500
0
0
1
2
-500
3
4
5
Mach
Fig. 4.3.4.2.2: Variation of pitching moment with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
The moments presented in Figs. (4.3.4.2.1) and (4.3.4.2.2) correlate well with the magnitude of
moments expected for a launch vehicle of our size and shape.
The results for the variation of normal, axial, and shear forces with respect to Mach number at a
zero degree angle of attack can be found in Figs. (4.3.4.2.3), (4.3.4.2.4), and (4.3.4.2.5)
respectively.
350
300
Normal Force (N)
250
200
150
100
50
0
-50
0
1
2
3
Mach
Author: Jayme Zott
4
5
Aerothermodynamics
4.3.4.2
3
Fig. 4.3.4.2.3: Variation of normal force with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
1200
1000
Axial Force (N)
800
600
400
200
0
0
1
2
-200
3
4
5
Mach
Fig. 4.3.4.2.4: Variation of axial force with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
100
0
0
1
2
3
4
5
Shear Force (N)
-100
-200
-300
-400
-500
-600
Mach
Fig. 4.3.4.2.5: Variation of shear force with respect to Mach number at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Aerothermodynamics
4.3.4.2
4
The variation of CD with Mach number at a constant zero angle of attack is shown in Fig.
(4.3.4.2.6). Because the diameter of the 1 Kg launch vehicle is quite large, the coefficient of drag
CD is also quite large.
1.6
1.4
1.2
Cd
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Mach
Fig. 4.3.4.2.6: Impact of Mach number on CD at zero angle of attack. 5 Kg.
(Alex Woods, Jayme Zott)
The trajectory analysis requires the use of CD long before the final geometry is determined. Since
the linear perturbation theory requires complete knowledge of the launch vehicle geometry, the
CD variation shown in Fig. 4.3.4.2.6 is not used in the trajectory analysis. Instead, we use a CD
trend based on historical data. While the CD trend used in the trajectory analysis is not based on
our own geometry, it is based on successful launch vehicles with geometries similar to our final
design. The CD based on historical data at zero angle of attack is shown in the Fig. 4.3.4.2.7
Author: Jayme Zott
Aerothermodynamics
4.3.4.2
5
0.8
0.7
0.6
Cd
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Mach
Fig. 4.3.4.2.7: Impact of Mach number on CD at zero angle of attack based on historical data. 5 Kg.
(Jayme Zott)
Given additional time, we could complete a more complete trajectory analysis by including the
correct CD based on the linear perturbation theory into the trajectory code. Fig. 4.3.4.2.8 shows
the error caused by the using the CD trend based on historical geometries, rather than the CD
determined directly from our own geometry.
1.6
1.4
1.2
Cd
1
0.8
Cd (dimensional)
0.6
Cd (historical)
0.4
0.2
0
0
1
2
Mach
3
4
5
Fig. 4.3.4.2.8: Comparison of CD based on historical data and CD based on dimensional analysis (linear perturbation
theory). 5 Kg.
(Alex Woods, Jayme Zott)
Author: Jayme Zott
Aerothermodynamics
4.3.4.2
6
Table 4.3.4.2.1 Summary of Maximum Aerodynamic Loading 5 Kg.
Aerodynamic Load
Subsonic
Bending Moment [Nm]
-6505.0
Pitching Moment [Nm]
1246.0
Normal Force [N]
215.9
Axial Force [N]
760.0
Shear Force [N]
-407.4
Center of Pressure [% length]
38.2
Coefficient of Drag CD
1.31
Dynamic Pressure [Pa]
CD % error [%]
62.5
Author: Jayme Zott
Supersonic
-3346.0
640.9
111.0
390.9
-209.6
38.2
0.78
225
19.5