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I am a glider pilot and instructor with a master’s degree in actuarial mathematics. I teach aerodynamics every winter for upcoming glider pilots.

As you surely know, the aerodynamic center is the location where the aerodynamic moment remains constant regardless of the angle of attack – the leverage of the lifting force acting through the center of pressure gets larger for decreasing angles of attack while the force acting through this location gets smaller.

Although I find it easy to understand the mechanisms behind the aerodynamic center, I struggle with understanding how it can exist? How can we be sure that there is such a point in the wing? I know that by definition it is a fixed point, but will it be exactly fixed in practice or just approximately fixed?

Is it not possible to construct an airfoil which will somehow have a discontinuity in center of pressure leverage but delivering the same lifting force in the two situations, yielding non-constant moment around the aerodynamic center?

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  • $\begingroup$ Rolle's theorem comes to my mind... $\endgroup$ – xxavier Nov 16 '18 at 11:28
  • $\begingroup$ Take a look at this question and see if anything there helps. Seems like there’s another question where PK explains this a little differently, but I can’t find it at the moment. $\endgroup$ – TomMcW Nov 16 '18 at 18:39
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    $\begingroup$ From my understanding the aerodynamic center only exists as a theoretical point in the linearized formulas. I think it only works because the airfoil in the normal angle of attack range has a constant lift derivative (6.28 cl per alpha) and the moment coefficient is a direct function of the lift coefficient. In practice this point is approximately fixed until you enter the stall range (non linear, flow separations,...) or when you go supersonic (shockwaves change the pressure distribution and aerodynamic center) $\endgroup$ – Jan Nov 17 '18 at 13:05
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Assume:

  • Thin airfoil
  • Vary with small angle of attacks
  • Small airspeed
  • Incompressible, attached flow

I hope you feel comfortable with 2 below observations:

$c_l = a_o(\alpha-\alpha_{L=0})$ This means the value of lift varies linearly with angle of attack (*)

$c_m = m_o(\alpha-\alpha_{M=0})$ This means the value of moment at a arbitray point (say point A) on the chord line varies linearly with angle of attack (**)

Suppose we are provied two above equations with all constants are known and the coordinate of point A. Then moment about $x_{ac}$ (unknown) is enter image description here

$$ Moment \space about \space x_{ac} = c_m.q.c^2 + c_l.q.c.(x_{ac}-x_A) $$ $$= m_o(\alpha-\alpha_{M=0}).q.c^2 + a_o(\alpha-\alpha_{L=0}).q.c.(x_{ac}-x_A)$$ $$ = (m_o.q.c^2+a_o.q.c.(x_{ac}-x_A)).\alpha + ...$$ (substitute (*) and (**)) , with q is dynamic pressure, c is chord line

Because moment about $x_{ac}$ is constant when $\alpha$ changes hence the total coefficient of $\alpha$ must be 0, then we solve the equation: $$ m_o.q.c^2+a_o.q.c.(x_{ac}-x_A) = 0 $$ $$ m_o.c+a_o.(x_{ac}-x_A) = 0$$ $$ x_{ac} = \frac{-m_o.c}{a_o}+x_A$$

We now know that the AC exists because on the way finding it, we can find it, it is the root of the above equation which is the coordinate of the AC. It is clear that it is just approximately fixed becasue two observations at the begining are just approximate.

The AC only exist as long as two above observations are true, if they are not linear anymore because the angle of attack is greater than stall angle, you can have two angles with the same lifts but the moments about AC are different.

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It exists as a mathematical abstraction for conventionally shaped aerofoils at moderate speeds. In the end it is derived from the centre of pressure. The centre of pressure is the point where the moments of the individual aerodynamic forces over the aerofoil give 0 momentum. This always exist on any aerofoil. Now, the centre of pressure is not at a constant location, it is continually shifting as a function of angle of attack, and it moves quite a lot. However the rate of movement and change in lift force is in proportion to each other, and based on that, the stable point can be found, which for conventional aerofoils at normal speeds is about 25% chord.

Note, that to get this, the proportionality between lift and CP location is a must. You can create any shaped wing cross sections (might not want to call it an aerofoil) which will not have this property and won't necessarily have an aerodynamic centre. Whether it is fit to fly is questionable though. However, they will always have a CP, as it is a simple mathematical property, and it will very likely move with angle of attack. Also note, that the AC does change location with increasing speed, for supersonic speeds it moves significantly backwards.

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In potential flow theory, lift can be calculated as the linear superposition of a contribution from camber and one from angle of attack. While the camber-related part of lift is constant, the angle-of-attack related part varies linearly with this parameter. The center of pressure of the camber part is somewhere at mid-chord (details depend on the camber line; with a Joukowski airfoil the center of pressure is precisely at mid chord). The center of pressure of the angle-of-attack dependent part is at the quarter chord (the center of the area below the chordwise Birnbaum distribution of lift). The important part is the self-similarity of Birnbaum distributions for different angles of attack: The center of pressure of the angle-of-attack dependent part is constant and at 25% of chord for 2D flow and wings of large aspect ratio.

Is it not possible to construct an airfoil which will somehow have a discontinuity in center of pressure leverage[?]

Not in non viscous flow. And you want to minimize viscous effects in order to minimize drag, don't you?

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First it may be helpful to focus on first thought "the aerodynamic moment remains constant regardless of angle of attack". This is referred to as the "nuetral point" of an aircraft. Also known as the "parachute point" it has no real value to a pilot as putting CG there would produce no nose down pitching moment in sink, which is the essence of gliding.

CG at nuetral point would most likely be directionally unstable. A pilot simply does not use it. CG belongs at or around the Center of Lift of the wing(s). Easiest way to consider is a flat plate H stab, which has no contribution to lift at 0 AOA, but helps keep TOTAL Clift near CG as AOA increases (on wing Clift tends to move forward as AOA increases). To flying wing purists, the H stab may seem like training wheels on a bicycle, but it has stood the test of time for millions of years with birds.

Leverage of lifting forces is interesting academicly, but should not overshadow the all important relationship between CG, Clift, and elevator trim.

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    $\begingroup$ The CG is usually aft of the center of lift of the wing which is essential to make gliders fly efficiently with little trim drag. This works because at positive angle of attack the tail is also producing a bit of lift and with the longer lever the nose is forced down enough. So you can have a CG of 35% of the chord even though the center of lift of the wing is at 25%... But the total center of lift is has to be aft of that CG to be stable $\endgroup$ – Jan Nov 17 '18 at 13:13
  • $\begingroup$ Notice the modern gliders put the H stab on top of the vertical stabilizer. This assymetric structure actually gives a little "down" force on the tail instead of draggy elevator trim, helping set CG a little bit more forward. But it then becomes a matter of taste for the pilot, the glider being a bit more responsive when CG is slightly aft. $\endgroup$ – Robert DiGiovanni Nov 17 '18 at 22:52
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    $\begingroup$ As CG moves aft aircraft gradually loses its margin for stability. So neutral point is the CG location where the aircraft becomes neutrally stable in pitch (hence neutral point). I don’t think it has anything to do with aerodynamic center in that terms. $\endgroup$ – Kolom Sep 23 at 19:32
  • $\begingroup$ @Kolom How about "as CG moves aft aircraft gradually loses its DIRECTIONAL stability" relative to the AERODYNAMIC neutral point in PITCH (for the entire plane) where change in AOA does not cause a pitching moment. However, notice that on your way back you are moving further and further away from the CP of the mono wing. Langley tried identical fore and aft wings, but failed due to interference. For monoplanes, CP is always ahead of (net) AC, and CG is at or near CP. Anything else and you have a bi-plane (less efficient). $\endgroup$ – Robert DiGiovanni Sep 23 at 20:17

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