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What could the lowest altitude of Air France 447 have been to recover the (stalled) flight, where the co-pilot informed the captain that he was pulling up the whole time?

The co-pilot didn't inform the captain until 2000 feet, which obviously was too low.

Rodrigo de Azevedo
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Cloud
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    GXL888T stalled at 2,910 feet at level flight; and despite the captain actively trying to recover, it didn't. AAF447 was falling with high sink rate, so that suggests the minimum recovery altitude to be much higher. – kevin Jan 31 '18 at 12:48

1 Answers1

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For the stalled flight to recover, the nose needs to be pointed in the airstream, and then the aircraft pulled up with load factor below the ultimate load. From the accident report:

The recordings stopped at 2 h 14 min 28. The last recorded values were a vertical speed of -10,912 ft/min, a ground speed of 107 kt, pitch attitude of 16.2 degrees nose-up

Initial situation.

From earlier on in the report we can conclude that these values were typical for the complete stall from 36,000 ft to impact, except for the pitch attitude which was around zero for most of the stall. The flight state parameters were therefore:

  • Vertical speed was 10,000 feet/min = 51 m/s
  • Horizontal speed was 100 kts = 51 m/s, which constructs an airspeed of 72 m/s.
  • The engines were at TO thrust, creating a nose-up moment, compensated by the nose-down moment of the airstream hitting the horizontal stabiliser.
  • The automatic trim is retained in alternate law, so the stabiliser is trimmed full LE down = 14º.
  • Elevator is fully up = 30º on top of the trim - it is almost aligned with the stalled airstream.

enter image description here

Nose down manoeuvre.

  • At 72 m/s the aircraft is at about TO speed, and nose rotation speed should be comparable to TO rotation speed - a bit slower because one side of the elevator is stalled. It takes perhaps 5 - 10 seconds to rotate the nose down 45º (an estimated assumption from my part).
  • When at 45º nose down, gravity accelerates @ 0.7g and the engines at perhaps 0.25g. At this acceleration airspeed builds up with around 1 g = 9.8 m/s every second. If we take averages for nose down manoeuvre time and acceleration, airspeed builds up with 4.9 m/s for 7.5 seconds = 37 m/s. So when the nose is aligned the airspeed is about 110 m/s, but still accelerating fast.
  • During the 5 - 10 seconds nose down manoeuvre, the aircraft loses about 1,000 - 2,000 ft in altitude

Speed increase

Before the aircraft can be pulled from the dive, the true airspeed needs to be brought to the manoeuvre speed $V_a$. From the A330 FCOM:

The load alleviation is only available when :

  • The aircraft speed is above 250 knots.
  • The FLAPS lever is in the 0 position.
  • In normal or alternate law flight mode.

Above 250 knots the load alleviation system is active, in order to keep the maximum load factor at 2.5g. The manoeuvre speed (or cornering speed in military speak) seems to be set at 250 knots = 128 m/s. At 1g acceleration, it takes 2 seconds to increase speed from 110 to 128 m/s, during which the aircraft loses altitude of 500 ft:

$$\Delta h = sin(45) \cdot (V_0 \cdot t + \frac{1}{2} \cdot a \cdot t^2)$$

Pull-up manoeuvre.

Then during the pull-up manoeuvre, the load factor must stay under the limit load = 2.5g. 1g is taken up by gravity, and there is 1.5g available for the pull-up manoeuvre.

$$ m \cdot \Delta n \cdot g = m \cdot \frac {V^2}{R} \Rightarrow R = \frac {V^2}{\Delta n \cdot g}$$

With the values established above, we get R = 10,000 / (1.5 * 9.81) = 1,100 m = 3,300 ft. But that is at constant airspeed of 128 m/s, in reality the airspeed will still pick up at the beginning of the manoeuvre and the radius will be higher - let's say 4,000 ft. The velocity vector of the plane was 45º pointing downwards, so half the radius is used. This is from the moment the AoA is close to zero, which it needs to be in order to start the pull-up, which increases AoA again.

My estimate for the altitude required for a successful pull-up from the stalled situation, is therefore 2,000 + 500 + 2,000 = 4,500 ft if they end up skimming the tops of the waves. If they know exactly what to do, time things perfectly, and manage airspeed perfectly. The aircraft is protected against pulling too many g's, so once the nose is aligned the flight crew can pull the stick fully aft and let the aircraft manage the minimum pull-up radius. If for instance the pull-up is initiated 1 second later than the 2 secs estimated above, the airspeed increases to 138 m/s and the pull-up radius becomes 1,300 m = 4,000 ft, plus the aircraft loses an additional 300 ft during the 1 extra second diving down under full power - this 1 second delay in initiating the pull-up requires another 1,000 ft.

Recovering from a fully developed stall is now trained in Level D simulators. Picture below is from a company that makes the flight model and instructor station extension for any flight simulator to train the manoeuvre. Disclosure statement: I have done business with them in the past.

enter image description here

Koyovis
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    This is such a good answer, I will wait a day before accepting, but can't imagine a more thorough one. Thanks. I do wonder why this wasn't trained until recently though, it seems like a no-brainer, but I guess hindsight is 20-20. – Cloud Jan 31 '18 at 14:39
  • Yes indeed, hindsight. But a good thing that the aviation industry learns from previous accidents and trains for preventing them in the future. – Koyovis Jan 31 '18 at 14:47
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    Don’t you need time, with the AoA near zero, for airspeed to build up to something that will sustain your 3G +/- pull? Looks like you modeled pushing until zero AoA & then immediately pulling back again. Did I miss something else? Any estimate of airspeed attained during that push? Without sufficient airspeed, that 3G pull will put the aircraft right into a secondary stall. – Ralph J Jan 31 '18 at 15:23
  • @Koyovis Training could only begin after reliable post-stall aerodynamic models were developed which didn’t start for commercial airliners until a few years ago. Before, any post-stall simulator handling did incur risk of negative training due inaccurate post-stall aircraft handling behaviour. – Cpt Reynolds Jan 31 '18 at 16:01
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    The 30 seconds seems a bit... arbitrary. – Lnafziger Jan 31 '18 at 19:05
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    The report of GXL888T suggests that 1,000 ft would not be enough to pull up the aircraft. In that incident, the AoA recovered at 1,600 ft with nose down 54 degrees. At that moment the pilot input full aft stick and thrust idle, the aircraft impacted water 6 seconds later. It is also unlikely that the aircraft is capable of pulling and sustaining max G in that condition. Therefore I believe the 1,000 ft figure to be highly underestimated. – kevin Jan 31 '18 at 20:36
  • @kevin -- that brings up a great point... aft stick + idle thrust won't get you anything like 3 G's unless you have LOTS of airspeed already & are bleeding it off. So at some point, the thrust has to come up -- way up. Which increases the speed vector in the direction -- down -- that the nose is pointing. So rate of descent goes way up as well until pitch attitude nears level flight. The more guys examine this, the more I agree with your conclusion: *highly underestimated*. – Ralph J Jan 31 '18 at 22:17
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    @RalphJ The engines were at full throttle all the way down. – Koyovis Jan 31 '18 at 23:28
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    @cloud They trained stall prevention. The test program for that family of aircraft did not include full stall and data collection that would allow for a good enough simulator emulation of a stall. Practicing stalls in the aircraft is nowadays deemed to be too expensive for airliner class planes. Training stall prevention, if effective (in theory) should render the problem moot ... except when it doesn't. – KorvinStarmast Feb 01 '18 at 01:05
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    @CptReynolds Yes the flight simulator data was only valid within a limited region of AoA and sideslip. Outside this envelope, extrapolated data was used but that is totally unrealistic and invalid, as AA587 demonstrated. Actual data from crashes was used in developing full stall aerodynamic models, plus dynamic windtunnel data: the windtunnel model was measured while being rotated. Fully developed stall behaviour turns out to be determined by the aircraft config: low/high wing, T-tail, engines underwing etc. – Koyovis Feb 01 '18 at 01:37
  • +1 Nice calculations. When asked the same question the Airbus engineers didn't even want to hazard a guess with the data they had – TomMcW Feb 01 '18 at 02:05
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    @Lnafziger I've attempted to quantify the time required for the nose alignment a bit better. – Koyovis Feb 01 '18 at 04:05
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    I have not yet worked through your calculation in detail, but on first looks I am not sure whether your speed is indicated airspeed (for minimum flight speed purposes it should be) or true airspeed (for geometric calculations you need that). In any case, at high altitude the two differ a lot. Maybe you would like to add a note on that in your post? – Cpt Reynolds Feb 01 '18 at 18:49
  • 1g stalling speed from the AF447 report on the day in ambient conditions was around .65 Ma, which for 36000ft and ISA+15 is around 212kts IAS or 386kts TAS. – Cpt Reynolds Feb 01 '18 at 20:15
  • @CptReynolds It's true airspeed for the manoeuvres, I've added to the answer. – Koyovis Feb 02 '18 at 22:08
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    @Cloud: about the "no-brainer". Every pilot is trained from day one to fear and avoid stalls. The question in the case of AF447 is not about how the pilot dealt with the stall, but why he decided to not believe any of his instruments (they were all working fine) and keep full stick aft while falling from the sky, and not recognizing that the aircraft was in alternate law; together with poor CRM that allowed this to continue without the other two noticing. – Martin Argerami Feb 09 '18 at 00:50
  • You probably said this somewhere and I missed it, but where did you get the 100-kt horizontal speed from? – Vikki Mar 19 '23 at 15:59
  • @Koyovis: "The engines were at full throttle all the way down" - Full throttle, but were they at full thrust? At the extreme AoAs encountered during AF447's stalled descent, I'd expect there to be a considerable degree of engine-inlet-airflow-disruption-induced power loss, which the engines would take a nonzero amount of time to fully spool back up from even after lowering AoA to a sane value. – Vikki Mar 19 '23 at 16:05
  • One more note: once you've pushed over to θ=-45º, won't the velocity vector have steepened past -45º (because gravity) by the time you initiate the pull-up maneuver? – Vikki Mar 19 '23 at 16:11
  • Final (I think) question for now: "The velocity vector of the plane was 45º pointing downwards, so half the radius is used." Assuming that the plane's velocity vector is actually -45º (see previous comment), won't the pullup to level flight take 1-cos(45º)=0.293 of the radius, not half of it? – Vikki Mar 19 '23 at 16:20
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    @Vikki The 100 kts is a general representation, from the horizontal speed at crash and looking back at the general data before that. – Koyovis Mar 20 '23 at 09:37
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    @Vikki thrust & throttle - yes could be, the inlet flow could be partially stalled. Velocity vector pointing down: good point, less lift from a lower stalled wing flow. And yes for the cosine comment. Working flat out on a large project now, not sure if I have the time to update the answer (of 5 years ago ) – Koyovis Mar 20 '23 at 09:48