35 years before I became an aircraft mechanic, I asked myself this same question. My friends and I finally decided that some serious magic is involved. Today, watching a 747 rise off the runway is still a magical sight. Physics provides a more accurate explanation though.
Air traveling over a wing is affected by Bernoulli's principle just as it is by passing through a constriction in a tube. The constriction forces the air to speed up, causing a drop in pressure.
Air traveling above and below a wing must reach the trailing edge in precisely the same amount of time. The curved shape of the upper wing creates a longer route for the air to travel. It's forced to accelerate in order to catch up with the bottom, creating the low pressure area. What we call "lift" is the resulting difference in pressure.
This interpretation of Bernoulli's principle has been taught to students and pilots since world war two. Unfortunately, the explanation is not completely accurate. As aircraft speed increases, the pressure differential does increase. Wind tunnel testing shows that airflow does speed up as it passes over the wing, but it actually reaches the trailing edge sooner than the lower airflow does. The "equal transit time" theory is quite incorrect for most situations.
When combined with "circulation theory", lift is described more accurately. Aeronautical engineers calculate lift through the interactions of airflow and vortexes. The mathematics are so horrific that it is largely ignored by most others.
According to my FAA Airframe Handbook, 75% of lift occurs over the top of the wing. The remaining 25% actually pushes up from beneath. A "high lift" wing has a larger upper curve (positive camber) than an ordinary wing. Engineers must strike a balance between lift, drag, and efficiency, depending on the requirements of the aircraft.
None of this explains why an aircraft can fly inverted, which is commonly done in many types. If the Bernoulli principle were fully accurate, inverted flight would be suicidal. The "angle of attack" of the wing allows this to be possible. In very simple terms, this refers to the nose-up attitude of the aircraft. A pilot can actually use the fuselage instead of the wing to create lift by flying sideways, with an appropriate angle of attack. Even a sheet of plywood can generate lift.
This illustrates a common misconception about the required shape of a wing. Because speed and angle of attack are the major controlling factors in lift, it's not essential
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