# Modelling SpaceX's lift and drag versus angle of attack and Mach number

I've been trying to find data in the literature that would provide analytical expressions for the relationship between the lift and drag of an object similar in shape and size to the Falcon 9 first-stage (i.e. a large, high-speed cylinder), under different angle of attack and Mach number values (that is, $C_{L}=C_{L}(\alpha,M)$ and $C_{D}=C_{D}(\alpha,M)$). One paper I found had a fairly simple relationship given by

$$C_{L}=-0.041065+0.016292\alpha+0.0002602\alpha^{2}$$ $$C_{D}=0.080505-0.03026C_{L}+0.86495C_{L}^{2}$$

Where the angle of attack, $\alpha$, was scheduled with respect to velocity (i.e. acting as a passive control input) by $$\alpha = 40$$ for $V>4570 m/s$ $$\alpha = 40-0.20705(V-4570)^{2}/340^{2}$$ for $V\leq4570 m/s$

However, since it seems like SpaceX use angle-of-attack as an active aerodynamic control input, I wanted to try and incorporate this into an optimal control simulation that already takes thrust control into account, in order to give the simulation a bit more fidelity. Are there any possible models that were developed for, say, ICBMs re-entering the Earth's atmosphere that could be used as rough models for the Falcon 9's lift and drag with respect to angle-of-attack and Mach number?

• Nuclear weapon re-entry aero models? I can see no problem with obtaining that information! NSA - that was sarcasm! – Organic Marble Sep 4 '17 at 16:40
• That would be nice, but I was thinking more along the lines of open research in the literature – InquisitiveInquirer Sep 4 '17 at 20:43
• How can $C_L$ be non-zero at $\alpha=0$? If the rocket body is oriented parallel to the direction it is moving, shouldn't there be zero lift? Could you cite the paper this is from? – uhoh Sep 7 '17 at 12:56

## 1 Answer

Unfortunately you'll need to eyeball the graphs yourself to try and create any analytical functions - but Rocket Propulsion Elements (7th ed.) by Sutton and Biblarz has some graphs of $C_{L}(\alpha,M)$ and $C_{D}(\alpha,M)$ for the German V-2 in Section 4.2.

These plots assume no interactions with the exhaust plume, and the V-2 isn't quite cylindrical, but it should get you started.

If you're looking to model drag during supersonic retropropulsion as well, there's a fantastic thread on r/SpaceX discussing that, specifically with the Falcon 9 in mind.