Planes are designed to fly level relative to gravity and the air pressure gradient, the plane does not care about what is going on at the surface. Just like the bottom of a lake does not affect how you swim. It is GRAVITY and AIR PRESSURE that determines the orientation of the plane. Let me explain.
So let's start with how planes fly and how we design them.
Planes are designed very specifically and deliberately around a concept called AERODYNAMIC STABILITY. A structure is considered aerodynamically stable if the forces from air pressure cause the structure to passively maintains its orientation relative to its velocity. In layman's terms, aerodynamic stability means the air will push the plane's nose in the direction the plane is moving.
A good visualization of aerodynamic stability is a dart. You'll notice how a dart has a heavy metal chassis near the front and wide fins near the back. You'll also notice that when you throw a dart it will fly in a ballistic arc with the nose pointed forward rather than flipping end over end, this is because the dart is aerodynamically stable.
There is one critical property of aerodynamic stability that explains how planes passively correct for the curvature of the earth:
The Center Of Mass Is In FRONT Of The Center Of Lift
Why do we do this? This is because designing the plane this way gives the plane two very desirable properties for flying vehicles. First, it means that the nose of the plane has the most inertia. That means that the nose of the plane is the most resistant to being deflected off course by turbulence forces, which makes it more energy efficient to keep the nose of the plane pointed in the direction you want it pointed. Secondly, if the plane starts to fall, the nose will fall first. This may sound counter intuitive, but it is a critical feature of aircrafts. Making the nose fall first means that if an engine failure occurs the nose will dip first allowing the pilot to maintain the orientation of the plane and attempt a gliding landing, which would save the crew. If this was not the case, the plane would flip uncontrollably end over end and the plane would be uncontrollable. The important take away here is that WHEN THE PLANE FALLS, IT FALLS NOSE FIRST!
So now let's talk about how lift works.
Lift is an upward force generated by the geometry of the planes wings. The amount of lift force you generate is directly dependent on the speed of air over the wing and the pressure of that air. When moving fast enough and with enough air pressure, a plane can generate enough lift to overcome the force of gravity, this is what makes the plane go UP.
What does F=MA tell us? An object will move in the direction of the force vector. Thus if gravity is greater than lift, you fall. If lift force is greater than gravity, you rise. If they are equal, you maintain altitude.
When lift force is greater than gravity, you move up. When lift force is less than gravity, you fall. When they are equal, this is an important critical point called Cruising Speed and Altitude. Cruising Speed and Altitude is the speed and altitude for a plane where the lift force is equal to the gravitational force, which means the plane will neither fall nor rise thus making it maintain altitude.
So, say the plane starts to accidentally climb in altitude because the earth is curving away.
What do we know about air pressure? Air pressure decreases as altitude increases. So if a plane is moving at a fixed speed and it climbs to a higher altitude, the air pressure drops, meaning the lift force drops, which means now gravity is a greater force than lift force, which means the plane will fall back down to its cruising altitude. As we discussed above, the plane falls nose first, thus as it drops in altitude and the air pressure increases again the nose will dip until the aerodynamic forces push the plane level again. If you wanted to increase in altitude, you would need to slam on the gas pedal of the plane and deliberately make it go WAY faster such that you can produce enough lift to maintain altitude, it CANNOT happen accidentally. Even so, jet engines also produce thrust proportional to air pressure, so there is a maximum altitude for every plane before the engines cannot move the plane fast enough to maintain altitude, and the plane will fall back down to a lower altitude.
The plane is designed such that gravity and the air pressure gradient will force the plane level to those two vectors.
So it would be impossible for planes to accidentally climb up to ridiculous altitudes on a spherical earth, since there is not enough air pressure at those altitudes for the plane to generate enough thrust or enough lift to climb that high. If they were to try to climb that high, the plane would fall back down to a cruising altitude nose first until the air pressure forces it level again.
On a flat earth, planes would not even be able to HAVE a cruising altitude. If we were in a sealed container with no gravitational force, the air pressure would be equal on all surfaces of the container. This means that air pressure would not drop with altitude as we have very clearly observed, this would mean that at a fixed speed planes would just climb forever until they hit the "dome". Obviously, we don't see this happening.
Not to mention, the amount of correction you would need for the earth's curvature is so small relative to other forces on the plane that it is generally ignored in the active control systems, as the plane corrects for this very minor error passively. Even if it wasn't, the active control system of the plane is set to maintain altitude and fly level to gravity, so there is no situation where the curvature of the earth will cause planes to fly off into space or turn upside down!
I hope this helps people understand. If you want to try this at home, you can demonstrate this with model propeller planes and an attached wifi camera. Fly the plane up until the engine and air isn't thick enough to maintain height. Move the center of mass around. Watch at it falls nose first. All of this is very demonstrable!
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