Fundamentals of Lift: Differential Pressure & Wing Shape

[pivoxa15]

This is interesting stuff but I probably need to read a book with lost of pictures in order to fully understand the basics of this. Thanks for the introductions though.

 

[arildno]

First:
The "effective angle of attack" concept.
This comes from the flat-plate approximation, i.e, finding that angle a flat-plate wing would have, in order to produce the same lift as the actual wing in question.

A better concept, more in tune with reality, would be the "angle of escape"-concept (which, of course for the flat plate is the same as the angle of attack).

That is, a downwards angle of escape is directly seen to be related to the downwards deflection of air (and a corresponding lift, as in a normal action-reaction pair of forces).

Secondly:
Let us look at the proper setting of forces, namely Newton's 2. law of motion.
What is the ACCELERATION most readily associated with the lift force?

To answer that, simply look at the streamlines about the wing foil, as seen in the wing's rest frame.

Consistent with Newton's 3.law, the "majority" of these streamlines bend DOWNWARDS, i.e, i.e, the air has experienced a downwards force, and hence, the wing an upwards force (the lift).

BEND downwards..what sort of motion does this imply that the air has experienced?
Answer:
The air has undergone a CURVILINEAR motion; it was at the beginning moving strictly horizontally, but has, by passing by the wing gained a vertical component.

But, curvilinear motion is first and foremost associated with CENTRIPETAL acceleration, NOT tangential acceleration!

Thus, the force component properly related to the centripetal acceleration is the force component NORMAL to a fluid particle's trajectory, NOT the force component along the fluid particle's trajectory!

But Bernoulli's equation is merely the integral of F=ma ALONG a stream line (i.e, in the stationary case along a particle trajectory)...

But from this, it follows that the force as given by the pressure difference along the trajectory is not the force we should focus on!

Rather, we should focus on force given by the pressure difference ACROSS the streamlines, rather than along them (Crocco's theorem).

This is what I've done previously somewhere.

In general, we can replace ideally the effective angle of attack, with the effective CURVATURES of the wing foil, which display the intimate connection between foil shape and centripetal acceleration.
It should be emphasized that tipping a wing will change its EFFECTIVE curvature, even though its geometrical curvature remains unchanged.


As a simple illustration, consider what is done in the low-veløocity take-off phase:
Flaps go down, so that air following the underside of the wing experiences a centripetal acceleration with its centre of curvature way below the wing.
But that typically means that the pressure AT the underside must be GREATER than along the ground.
At the beginning of the take-off phase, we can say that the pressure ABOVE the wing is roughly equal to the GROUND pressure, that is, we have set up a lift-yielding presssure difference across the wing.

Once the plane is in the air, these flaps are no longer needed, since a low-pressure zone has been established on the upper side of the wing.