The fundamentals of aviation
Airplanes are supported in flight by air pressure which is reconfigured when air flows over the wing of the airplane.
Each object immersed in fluid (e.g. air or water) is subject to a pressure on all its surfaces, a force on each unit of area due to the weight of the air or water that is above (even if some surfaces are oriented downwards or laterally). In the absence of motion -- for example, when the airplane is stationary on the runway -- a wing is subject to equal pressure on both its upper and lower sides, and therefore does not tend to move up or down.
When the airplane is in flight, air flows over the wing, and the shape of the wing section -- curved at the top and flat or almost flat at the bottom -- reduces the pressure on the top, causing pressure to rise from below, which then exerts a holding force ("lift"). The lift increases if the front of the wing is slightly raised, so that the wing "bites" the moving air at a smaller angle ("angle of attack"), and, for a given lift, this type of wing encounters much less air resistance ("atmospheric friction") than a flat kite.
This section contains examples from aviation, relating to wing trim and variable pitch propellers, to illustrate reference systems, the sum of vectors and their decomposition.
The fundamentals of aviation
Airplanes are supported in flight by air pressure which is reconfigured when air flows over the wing of the airplane.
Each object immersed in fluid (e.g. air or water) is subject to a pressure on all its surfaces, a force on each unit of area due to the weight of the air or water that is above (even if some surfaces are oriented downwards or laterally). In the absence of motion -- for example, when the airplane is stationary on the runway -- a wing is subject to equal pressure on both its upper and lower sides, and therefore does not tend to move up or down.
When the airplane is in flight, air flows over the wing, and the shape of the wing section -- curved at the top and flat or almost flat at the bottom -- reduces the pressure on the top, causing pressure to rise from below, which then exerts a holding force ("lift"). The lift increases if the front of the wing is slightly raised, so that the wing "bites" the moving air at a smaller angle ("angle of attack"), and, for a given lift, this type of wing encounters much less air resistance ("atmospheric friction") than a flat kite.
Reference systems
But, wait a minute -- is the airplane moving or the air?
It depends on the reference system! In the air or ground reference system, the airplane actually moves. But you can also calculate everything with respect to the reference system integral with the airplane, where the air moves. As long as the plane flies in a straight line at a constant speed, the same laws apply.
(In the next sections it will be shown that you can also extend all this to a flight over a curved trajectory, as long as you include the centrifugal force and the force of Coriolis, "inertial" forces that only manifest in calculations in a reference system in motion).
Working in the aircraft's reference system, for example, it becomes easy to include the effects of wind, whose velocity is simply added (sum of vectors!) to the air velocity as perceived by the airplane.
To evaluate how a wing behaves in flight, instead of moving it in still air, it can be equally mounted in a laboratory and invested with a jet of air. The two physical processes are identical. This is the principle of the wind tunnel -- an environment with a large fan blowing a jet of air (or rather, sucking it in, so that the flow produced is more regular), and inside the wing sections can be assembled and tested.
The wind tunnel built by Orville and Wilbur Wright, inventors of the first working airplane, was not the first one -- there were already others at that time -- but it was the first to be used to actually design a flying machine.
The Wright brothers used small-scale replicas of their wings and measured their lift and atmospheric friction using delicate scales (there is a theory of small-scale replication behaviour).
A reconstruction of their original wind tunnel, as well as an exhibition of the scales with which they measured the lift and resistance of the air, are on public display at the Franklin Institute of the Philadelphia Museum. By clicking here you can visit a website describing that exhibition, with additional hyperlinks that can help you build your own wind tunnel.
Arrow Wings
The wings of small airplanes, which fly at low speeds, are generally orthogonal to the fuselage, a configuration that offers the best efficiency. On large commercial aircraft, or fast military jets, on the other hand, the wings are often configured as an arrow; some military jets can even vary the orientation of the wings -- orthogonal for best takeoff and landing efficiency, and oriented as an arrow for flight close to the speed of sound.
At the speed of sound, the air resistance ("atmospheric friction") increases dramatically, as the air cannot get out of the nose of the aircraft in time, so it is compressed and heated. Heat is a form of energy, and to produce it something else it has to give up its energy -- in this case, it is motion, which thus causes increasing friction; the "lift" of the wings therefore suffers. In fact, these problems begin long before the speed of sound is reached, because part of the airflow above the wings has a higher speed and can then reach the speed of sound before the airplane reaches it.
In this way, even if the air strikes the airplane with velocity v, the velocity vector can be broken down into two components perpendicular to each other -- a velocity of the flow v sin s directed along the wing, and a velocity of the flow v cos s directed perpendicularly to it. Both these components are less than v, since both (sin s) and (cos s) are always less than 1.
It can now be said that the flow of air along the wing does not cause any accumulation effect, and does not affect lift or atmospheric friction, and can therefore be ignored. Only the perpendicular flow v cos s has such effects, and, in a gross theory, the efficiency of the wings depends only on how close to the speed of sound is the speed of the perpendicular component. In this way, the arrow wings allow the plane to fly at a speed a little closer to that of sound, without suffering the negative effects. The Airbus 320, for example, has its wings bent back about 25 degrees. To visit a website with a detailed discussion about arrow wings, you can click here.
Propellers
The propellers of an airplane function as small rotating wings, the effect of which is to "pull" the plane forward (this tensile force is known as a thrust). Probably the greatest advantage obtained by the Wright brothers from their wind tunnel was not that of the design of the wings (a rough design, limited by the available technology), but of the design of the propellers, which were twice as efficient as the other propellers of their time.
Again, it is more convenient to consider the static propeller and the moving air instead. We can also ignore the fact that the propeller moves circularly, considering only a small segment of that circular motion, along which the motion takes place almost in a straight line.
(It should be noted, however, that each part of the propeller blade moves at a different speed. It is necessary to subdivide the propeller into sections, each at a different distance from the central axis, and then to study separately the forces on each section. Here we will concentrate on the terminal sections of the blades, whose speed v1 is the highest and which therefore generate the highest thrust).
What complicates the situation is the fact that the plane itself also moves. Also in this case, the phenomenon can be studied in the reference system of the airplane, which sees the air coming at it with a speed v2. In the reference system of the tip of the propeller blade (see drawing), the air arrives at it with a speed consisting of two components perpendicular to each other, v1 due to its motion and v2 due to the forward motion of the airplane.
Let's consider the action of the propeller before the airplane starts to move (v2=0). The force L on the blade of the propeller, which provides the thrust to the airplane, is perpendicular to the motion of the blade (or almost perpendicular), and pulls the plane forward, as required.
Now let's assume that the plane flies at a moderate v2 speed. The propeller no longer perceives a frontal velocity v1, but a velocity v that hits the blade at an angle to the frontal direction (see upper drawing).
This was not a serious problem for the first planes, since their speed was rather low. For such planes v2 was always much smaller than v1, and a wooden or metal propeller, with the blades slightly inclined to adapt to v at the normal cruising speed of the plane (or a little more inclined, to provide a small angle of attack), worked equally well even at different speeds. Many small airplanes still use this type of propeller today.
The fastest planes, however, need propellers with adjustable blades, which can increase the angle ("pitch") at which they "bite" the air as the speed increases, so that they are always oriented frontally with respect to the combined speed v due to their motion and that of the airplane.
No compensation can be obtained by increasing the rotation speed v1 of the propeller, because if the tip of the blades reaches the speed of sound, the efficiency of the propeller drops sharply (and the noise produced increases!).
The adjustable blades ("variable pitch propellers"), more expensive and more complicated than the propellers all in one piece, have long been a common feature of the fastest propeller planes. But even in this way there is a limit. Let's suppose that the airplane moves at the same speed as the tip of the propeller, i.e. v2 = v1. In this case the tip of the propeller blade must be oriented by 45 degrees towards the direction of motion (see lower drawing). A destabilizing tendency becomes evident.
First of all, as we have seen from the "triangle for the sum of vectors" and from Pythagoras' theorem, the total speed v perceived by the propeller blades is considerably higher (by about 41%) than both components of the speed, so we get even closer to the speed of sound and the problems connected with it. And secondly, the holding force L on the blade is also rotated by 45 degrees! So only the L1 component pulls the airplane forward, while the other component, L2, actually opposes the rotation of the propeller and requires more power from the engine, which is not used in any useful way.
Because of these problems, propeller airplanes have never reached speeds comparable to those of jets. The fastest military World War II propeller planes flew at about 600 km/hour. The speed record for a purely propelled airplane of 745 km/h was obtained in Germany before the war (in 1939) and has remained unmatched for decades.
The current record is 528.33 mph (850 km/h), and was obtained in 1989 by the "Rare Bear" aircraft, a U.S. Navy aircraft of World War II (type 8F8 "Bearcat"), modified to reach the highest speeds. The plane crashed in 1962 and remained on a cornfield in the state of Indiana, near a runway, before Lyle Shelton found and restored it in 1969. He later replaced his 2400 horsepower engine with a 4000 horsepower one (which did less than a mile with a gallon of gasoline, at its top speed), replaced the propeller and reduced its weight. The plane is still in flight efficiency. (Thanks to Dr. Eddie Irani for this information).
Ultimately: due to the viscosity of the air and the shape and orientation of the wings, the latter force the air itself to move in a complicated way (but well known to aeronautical engineers, there are the so-called Navier-Stokes equations for this purpose) which, on average, gives rise to an "exhaust" downwards. Action. The plane goes up. Reaction.
