Aerothermodynamics 03/20/08

4.2.4.2 pg. 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. Doing so better displays how the outputs vary in the working regimes.

Mach number, variation in angle of attack, use of 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 also 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. (4.2.4.2.1) and (4.2.4.2.2) respectively. Once the launch vehicle reaches a speed of Mach 4.5, 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.

Fig. 4.2.4.2.1: Variation of bending moment with respect to Mach number at zero angle of attack. 1 Kg.

(Alex Woods, Jayme Zott)

Fig. 4.2.4.2.2: Variation of pitching moment with respect to Mach number at zero angle of attack. 1 Kg.

(Alex Woods, Jayme Zott)

The moments presented in Figs. (4.2.4.2.1) and (4.2.4.2.2) 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. (4.2.4.2.3), (4.2.4.2.4), and (4.2.4.2.5) respectively.

Fig. 4.2.4.2.3: Variation of normal force with respect to Mach number at zero angle of attack. 1 Kg.

(Alex Woods, Jayme Zott)

Fig. 4.2.4.2.4: Variation of axial force with respect to Mach number at zero angle of attack. 1 Kg.

(Alex Woods, Jayme Zott)

Fig. 4.2.4.2.5: Variation of shear force with respect to Mach number at zero angle of attack. 1 Kg.

(Alex Woods, Jayme Zott)

The variation of CD with Mach number at a constant zero angle of attack is shown in Fig. (4.2.4.2.6). Because the diameter of the 1 Kg launch vehicle is quite large, the coefficient of drag CD is also quite large.

Fig. 4.1.4.2.6: Impact of Mach number on CD at zero angle of attack. 1 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. 4.2.4.2.6 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. 4.2.4.2.7

Fig. 4.2.4.2.7: Impact of Mach number on CD at zero angle of attack based on historical data. 1 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. 4.2.4.2.8 shows the error caused by the using the CD trend based on historical geometries, rather than the CD determined directly from our own geometry.

Fig. 4.2.4.2.8: Comparison of CD based on historical data and CD based on dimensional analysis (linear perturbation theory). 1 Kg.

(Alex Woods, Jayme Zott)

Table 4.1.4.2.1 Summary of Maximum Aerodynamic Loading 1 Kg.
Aerodynamic Load / Subsonic / Supersonic
Bending Moment [Nm] / -773.0 / -388.7
Pitching Moment [Nm] / 357.8 / 179.9
Normal Force [N] / 94.2 / 47.3
Axial Force [N]
Shear Force [N]
Center of Pressure [% length]
Coefficient of Drag CD
Dynamic Pressure [Pa]
CD % error [%] / 335.7
-43.0
40.0
1.44
63 / 168.9
-21.6
40.0
0.81
240
21

Author: Jayme Zott