Why does the aerodynamic center exist?

Question

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?

Answer 1

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

$$ Moment \space about \space x_{ac} = c_m.q.c^2 + c_l.q.c.(x_{ac}-x_A) $$ $$= m_o(\alpha-\alpha_{M=0}).q.c^2 + a_o(\alpha-\alpha_{L=0}).q.c.(x_{ac}-x_A)$$ $$ = (m_o.q.c^2+a_o.q.c.(x_{ac}-x_A)).\alpha + ...$$ (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.

Answer 2

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.

Answer 3

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?

Answer 4

All the forces generated on an airfoil are balanced near the 25% chord position, called the aerodynamic center. This location is only effected by airfoil camber, at 0 degree Angle of Attack (AoA), the Center of Pressure (CoP) moves rearward because of its camber producing lift. With positive AoA, below the stalling AoA, the CoP moves near 25% location. The forces are balanced near the 25% location on airfoil/flat plate with positive AoA, because the amount of forces is greater near the leading edge and gradually decrease to the trailing edge.