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To find the velocity of the air flow at each section of the aerofoil, am I right to say that the horizonatal velocity of the air flow is constant throughout the chord while the vertical component of the velocity varies along the chord depending on the angle of chord w.r.t. horizontal?

So velocity at first plane is $\frac{U}{cos\beta}$?

If that is the case, will shifting the air upwards, not push the wing down? Or am I right to say that as the airflow around the wings, there is no net upward motion of air, thus it does not push the wing downwards?

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No, you can't say that the horizontal component of velocity is constant throughout the chord. There are few reasons why not:

1) From the formula you provided (which is correct if your assumption on velocity components is true for an aerofoil) we can see that as beta gets bigger the final velocity tends to infinity, which of course cannot happen.

2)The velocity change on aerofoil is dependant upon its pressure change, it reaches maximum at the point of maximum camber and not at the point of maximum thickness and I think that as per your theory it would than be reached at the point with maximum thickness.

3) Third and probably the most important thing is this: Lets imagine the flat plate aerofoil at some small angle of attack. The plate will produce lift, and the particles above an aerofoil will reach the trailing edge first. Since the airflow above aerofoil will be deflected by the same angle as below the aerofoil, by above stated theory, particles should reach the trailing edge at the same time because of a constant horizontal velocity. Which isn't true.

Second part of the question is correct, there is no net upward motion of air. In fact, there is a net downward motion of air behind an aerofoil called downwash. Basically how much the force wing exerts on air by deflecting the partincles down, by that amount of force the aircraft will be pushed up. Newton's third law.

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No, horizontal component of velocity varies around the wing, being quite significantly higher above it than below:

(By Kraaiennest [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], from Wikimedia Commons)

Roughly speaking, without involving math, the behaviour of air is like this:

  • The air avoids the leading edge, and due to inertia would like to continue straight away from the wing.
  • However viscosity limits the velocity gradient, forcing the air just above and below the wing to move as well¹.
  • As the air moves out, it frees up space, so pressure decreases.
  • The oncoming air is accelerated along the pressure gradient, which makes it follow the wing contour.
  • Decreasing pressure increases velocity due to conservation of energy, as the sum of kinetic and potential¹ energy needs to remain constant; this is known for fluids as the Bernoulli's principle.
  • Since the air above the wing has to expand to follow the contour, it has lower pressure and correspondingly higher velocity.

Now in mathematical terms, it is the Navier–Stokes equations. They are set of vector partial differential equations. There is no way to solve them analytically, they can only be integrated numerically with help of suitable software. The basic tool for analysing airfoil shapes, reduced to two dimensions, is XFOIL. There is more complex software that can calculate flow around whole aircraft in three dimensions, but unlike free XFOIL, it is either really expensive, or internal tools of large aircraft manufacturers.


¹ Unless the wing stalls. When the flow has low enough energy, the viscosity is no longer enough to flush out the air above the wing and a pocket of stagnant, and thus higher pressure, air forms above the aft part of the wing, which, since it has higher pressure, no longer pulls the wing up generating lift (some lift is still generated on the underside, but it is much less).

² Well, also thermal energy. In adiabatic expansion, as the pressure decreases, the temperature also decreases, which frees up even a little bit more energy to convert to kinetic, but at typical speeds can be neglected.

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