# How high could Concorde potentially reach by going fast and pulling up?

If Concorde was travelling at mach 2 and FL 600 then pulled up, it would enter a trajectory that went above its normal flight altitude before diving back down. I assume that it couldn't pull up to full control surface deflection in those conditions without the wings folding, so without damaging anything, how how high could Concorde go using this method? Is there a way to calculate/estimate this, or is it only provable by testing? I've searched google for answers but I can't find anything on this subject.

(An unrelated question that I couldn't find an answer to: what was the structural G limit of Concorde at cruising speed)

• Testing was done 4 this sort of thing in the 50s by Chuck Yeager using an F-104 with a booster rocket under the tail. It would do ballistic arcs from its service ceiling up to about 100000 ft+, helped along by the rocket. It needed nose attitude thrusters to control pitch attitude above about 90000 ft because the flight controls had insufficient dynamic pressure to work normally. In The Right Stuff, the bit with him spinning out and ejecting near the end of the movie was about that last one of those tests where it almost killed him when it stalled and spun on "reentry" because the att was off. – John K Aug 6 '20 at 12:10
• The part about structural limit can form an entire question by itself and could be asked separately. – Manu H Aug 6 '20 at 14:32
• you may be interested by the part about energy convertion of how it flies. You are looking for zoom climb and transformation of kinetic energy to potential energy – Manu H Aug 6 '20 at 14:36

## 2 Answers

There are a couple of ways to look at this question. The easiest is in first principals. Therefore imagine that the concord (flying at Mach 2 at FL 600) can suddenly and without energy loss pull up, such that it flies for example 10 degrees upwards. Also we assume the following:

• The engine flames out immediately, therefore no engine thrust.
• We neglect air resistance

How much higher would it go? The formula to calculate this is suprisingly simple: $$\Delta x = 1/2 \cdot \frac{(V_I \cdot sin(\gamma))^2}{g}$$ with $$V_I$$ being the inital velocity (590 m/s which is Mach 2), $$\gamma$$ being the climb angle and $$g$$ being the gravity of $$9.81 \frac{m}{s^2}$$.

I have calculated it for a couple of angles:

• $$\gamma = 10°$$: 535m
• $$\gamma = 30°$$: 4436m
• $$\gamma = 45°$$: 8871m
• $$\gamma = 60°$$: 13307m
• $$\gamma = 90°$$: 17742m

That gives us a first guess, but the question remains: Could it pull up at this high altitude? The answer is: Yes. Consider the formula to compute the dynamic pressure: $$q = 1/2 \cdot \rho \cdot V^2$$ The concorde takes-off at around $$113 \frac{m}{s}$$ at an air density of around $$1.225 \frac{kg}{m^3}$$. The air density at FL 600 is around 1/10th of the air density at sea level. However the crusing speed is roughly 5.2 times faster, which yields 2.7 times higher dynamic pressure at that altitude. Thus, the rudders definitly work at that altitude.

The real limiting factor here are the structural limits when you try to pull up at Mach 2. For example if you try to pull up your aircraft at Mach 2 with a constant acceleration of 1 G, you will follow an circle with a radius of ~50km. Even with 3 G acceleration, you will still follow an (upward) circle with a radius of roughly 16km. My guess is that the Concord can safely handle around 3 Gs (For example to be able to handle gusts at altitude). Additionally, you will loose a lot of your energy while pulling up, due to additional drag (remember, because of the increase angle of attack, the aircraft has more drag) and of course you are getting slower as you trade velocity versus height. Thus as some point you will not be able to pull-up any further because you lost too much speed. I tried to estimate how much higher you would have to fly in order to go down to stall speed1, and my result is 11.2 km of additional height. That would mean that you could reach a $$~70°$$ climb. However I neglected air-drag etc. and the fact that you will definitly not pull 3Gs at stall speed.

To get a definit answer, one would have to perform some computer simulations. Perhaps one of the popular flight simulations can help. But I would guesstimate, that at FL 600 going at Mach 2, you can probably (safely) pull the airplane up into a climb of around $$\gamma = 30°$$, which would yield around ~4.4km of additional height.

P.S: However the assumption that the engine will flame out at higher altitudes is probably true. My guess would be that the engine was designed such that it can not substantially exceed FL 600. But that is speculation.

1 I simply assumed that the stall speed is at the same dynamic pressure as at take-off condition. This yields a stall speed of $$357 \frac{m}{s}$$

• In a 3g pull-p the induced drag during the maneuver will be about 9 times higher than in straight flight. Even with engines running, the Concorde will drop to subsonic speed quickly. Also, as far as I remember, the g limit in cruise was lower than 3g. – Peter Kämpf Aug 6 '20 at 20:22
• Ah cool, how did you compute that? Perhaps We can calculate a more accurate answer? – U_flow Aug 6 '20 at 21:58
• Well, you could take into account that $g=9.75$ at $17742 m$ :-) – Breaking not so bad Aug 7 '20 at 18:47

At 60,0000’ I would expect the Concorde might be cruising at the edge of it’s performance limit.

If you try at “zoom climb” with any aircraft at the edge of it’s performance limit, the aircraft might be dangerously close to the stall speed.

• At 60,000 ft in standard atmosphere, Mach 2 is about 1147 KTAS and about 353 KIAS, which is lower than one might think, but not near stall speed yet. In fact, Concorde first slowed down at 60,000 ft (I forgot to what speed exactly) before actually starting to descent. – Bianfable Aug 6 '20 at 13:01
• To really answer the question, you should provide figures. Otherwise, I fear this is more a comment than an answer. E.g. you can add energy consideration (how much kinetic energy when cruising at 60000ft, and therefore how much height gained when transforming the kinetic energy to potential energy while slowing down to stall speed neglecting losses due to drag and gains due to thrust, therefore giving only a first approximation) – Manu H Aug 6 '20 at 14:31
• One would have to look at thrust available at 60,000 feet. At supersonic speeds, any maneuvering may produce huge amounts of drag, – Robert DiGiovanni Aug 6 '20 at 15:01
• This is called Coffin Corner, and doesn't sound very fun en.wikipedia.org/wiki/Coffin_corner_(aerodynamics) – Stuart Buckingham Aug 6 '20 at 17:51
• @StuartBuckingham "Coffin corner" does not apply to supersonic aircraft. There are analogous limitations, but they aren't the same. – pericynthion Aug 7 '20 at 1:03