When a non-pilot hears the word stall, it brings to mind what happens when a car stalls - the engine quits. It seems like that would be a dangerous scenario in an airplane.
From a non-pilot perspective, what happens when an airplane stalls and why is it important for pilots to practice it?
Stall was an unfortunate choice of words for an engine that suddenly quits since the aerodynamic stall in aviation means something very different and isn't related to the aircraft engine at all1.
To a non-pilot, an aerodynamic stall can best be described as the situation where there is not enough air flowing over the wings to create the amount of lift needed to hold up the airplane.
The main reason that student pilots practice stalls is to learn the telltale signs that occur just before it happens and to make the recovery procedure automatic. If pilots can recognize an impending stall, they can take corrective action to either avoid the stall altogether or to recover as quickly as possible.
Outside of training, inadvertent stalls typically only occur shortly before landing and after takeoff, when the pilot gets distracted while already at a slow speed. In both of these situations the airplane is very close to the ground, immediately requiring the correct action from the pilot in order to avoid a crash. This needs to be instinctive and corrected using muscle memory so that it is accomplished as rapidly as possible.
The next logical question is usually: How does a pilot fix an airplane that has stalled?
Fortunately, airplanes are designed so that even during a stall the tail is still effective2 and the pilot is able to use it to force the nose down. This makes the airplane go faster, since it is pointed down towards the ground, and gets more air moving over the wing which allows it to create enough lift for the airplane to start flying again. During practice it is usually pretty uneventful, but when it happens at a low altitude there may not be enough time to regain flying speed before the airplane crashes.
For more information, AOPA has a great Safety Publication targeting flight instructors called Why we teach slow flight and stalls which is available on their website.
1 However, the sail on a sailboat can also "stall" when there isn't enough wind and since they have been around since 3,000 BC I guess that technically this usage of the word applies to both situations.
2 There are some stalls in particular airplane designs known as deep stalls that can be unrecoverable. I don't think that this is important when describing it to a layperson though.
An engine stall and an aerodynamic stall are completely different. In aviation, an engine stall is referred to as an engine failure, and an aerodynamic stall is simply referred to as a stall.
An aerodynamic stall happens when the wing stops producing lift because the Angle of Attack is too high. This is usually, but not always, caused by pulling back on the stick without adjusting power appropriately. Various factors including the weight of the aircraft, flaps, and icing can change the angle of attack the aircraft stalls at.
A stall happens when the wing no longer creates lift. This happens when the speed of the air going over the wing decreases too much. Basically, this is why the advice one mother gave her son during World War I, 'Fly low and slow, I don't want you to get hurt!' makes very little sense aerodynamically speaking.
Pilots practice stalls to learn the warning signs that they are entering a stall, and to practice recovering from a stall should they ever end up in one.
Also, as brinky stated in a comment to this answer:
While in flight training or when moving to a different airframe, pilots practice putting the aircraft in a stall and then recovering from it to also develop "muscle-type" memory, because stalls are probably the single most unsettling and unpredictable behavior that all aircraft share, and different airframes may react very differently to stalls. Stalls are unforgiving and must be rectified promptly, due to rapid altitude loss.
The stalls that pilots practice are aerodynamic stalls not engine stalls. It happens when the critical angle of attack is exceeded. [Typically the nose is pitched up too much is what it means]. It results in airflow separation that means that the wing no longer is generating any [significant] lift. As soon as the pilot recovers by letting the nose down and regaining some airspeed they are flying again. We practice these for the purpose of being able to recognize the onset of a stall and being able to recover from an inadvertent stall. This is important because while it is actually kind of fun at altitude, near the ground (like while on approach or departure for example) it can be deadly.
When the airplane is at a "regular" angle of attack (angle with the direction of the wind), with the nose more or less forward, the wing works as designed and produces lift.
If the airplane turns the nose straight up while continuing to go forward, it's intuitive that the wings stop producing lift, as they are just vertical walls against the wind at this point. (The plane can still fly in this condition if the engines alone can pull all the weight, but that's only common in fighter and acrobat planes).
It turns out that the transition between these two is relatively sudden. As the angle of attack increases, wing lift goes up and up and up, then suddenly drops sharply as the smooth air flow detaches from the back of the wing. That's the stall. It can also happen when lowering speed while keeping the angle constant.
As to why pilots practice it, nothing to add to what flyingfisch said.
Stall has to do with the attachment of the boundary layer on the upper surface of the wing. When the airflow separates (video) from the surface it stops generating lift and the wing stalls. Lift is the aerodynamic force that keeps the aircraft airborne. It's the reaction of the air to the mass of the airplane.
Now, lightheartedly, to make a super simple analogy, let's say that the equivalent of lift when you walk is the upward reaction of the floor to the mass of your body (which is actually true); imagine also that you are walking and suddenly a trapdoor opens under your feet (airflow separation). As soon as you feel that you're falling you instinctively open your arms trying to grab something while your eyes are looking around for something to hold on, say a handle or the edges of the trapdoor. This is the situation that one could say that you have stalled. Your normal walking was interrupted and now you're falling.
Pilots train in order to recognize the symptoms and learn where the "handles" are -i.e. necessary actions to take- when this situation occurs.
"Stall" in aviation generally means an aerodynamic stall. An "engine stall" would be called just that.
There is an excellent understandable writeup of an Airbus crash from a stall, Air France flight 447, on the UK's Daily Telegraph.
A wing provides lift to a plane as a direct result of the air flowing over the surface. A stationary plane falls - without airspeed, the wings cannot provide lift. A stall occurs when the airspeed falls too low, and the lift provided by the wings cannot maintain flight.
Airspeed can fall too low for several reasons; for AF447 it was the most common case, that the plane's angle of attack was raised too high, and the engines could not provide enough thrust to keep the plane flying above its stall speed. In this scenario, memory of stall practices should kick in and the pilot should point the nose down to regain airspeed, allowing the wings to provide more lift.
Interestingly, the article suggests that the Airbus controls take commands rather than being a modifiable reflection of the current state of the aircraft, so once the stick was released it wasn't apparent that the plane was trying to climb to the other pilot. Eventually airspeed dropped so low that the airspeed sensors themselves switched off, silencing the stall warning. When the pilots realised the mistake and pointed the nose back down, the increasing airspeed re-enabled the airspeed sensors, which restarted the stall warning, perhaps confusing the crew further. The captain realised just before impact: "10 degrees of pitch..."
Practising of scenarios is to prepare crew so that in the event of some danger or failure, they do not panic but respond appropriately and correct the problem.
A stall can be observed visually on an aircraft with a flexible cloth wing, such as a hang glider or micro-light. To land a hang glider, one deliberately triggers a stall which spreads out across the wing surface, and you can see the fabric going limp.
In paragliding, the pilot stalls the wing on purpose sometimes in order to recover the glider from some situation - when the lines are messed up or there's something else.
It is similar to rebooting your computer.
A stall is when the plane exceeds the linear and predictable. If it hits the air too steep, the flow over the wings collapses into chaos and the behaviour is not always predictable.
Pilots practice to develop the skills to recover into the linear range of flight
See also spins and look up the number of people who have died after stalling a "laminar flow" wing (the Gee-Bee racer comes to mind here).
A stall is when there is not enough air flowing over the wings to achieve the amount of lift necessary to keep the airplane flying. The telltale sign of an impending stall is when the airplane begins to buffet. The recovery procedure includes unloading the airframe by pushing forward on the yoke(the airplane equivalent of a steering wheel) and adding engine power. In a stall where the airplane's weight and balance is well configured, the nose will be heavier and will come down once the wings have stalled, making it easier for the pilot to recover even if the tailplane is not super effective after the stall. Unrecoverable stalls result when there is an aft Center of Gravity, and there is not enough authority in the elevator to force the nose down.
why pilots practice it?
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