Interesting Aerofoil Logical Fallacy

Question

I am new to physics in general having just finished AP Mechanics and have limited knowledge of how fluid dynamics work. But just using forces and a simplified understanding of drag if have come to the following conclusion:

Drag (the simplified drag, not parasitic drag or otherwise) acts like this on a moving object:

It acts a force on the object in the opposite direction of the velocity.

However, air can also be deflected (I am using the mini collisions model of drag, like each individual air particle is colliding with the object and thus momentum is conserved):

That resultant force can be broken into two parts:

The vertical component of the force accelerates the vertical direction upwards, and the horizontal force decelerates the object to the right.

Now here is where it gets interesting:

As the foil accelerates upwards, the new resulting force decelerates the upwards velocity and accelerates the horizontal one, reverting to the initial condition, and avoiding any of the losses due to air drag.

If this were true, then why doesn't an aero foil fly like this infinitely, or at least more than other objects? And why is does it usually flutter to the ground when tested, rather than stay in what seems to be a stable equilibrium due to the feedback loop?

Answer 1

There is no possibility for perpetual motion, but you have to be careful about how you treat the drag. In your diagrams, the air particles all flow with uniform velocity, which suggests you are assuming that the wing speed is much faster than the average molecular speed of air (so that air particles can be treated as stationary in the frame of the wing). This is not the case for a typical aircraft, but we can analyze the drag in this regime nonetheless. Let the wing move with horizontal speed $v$ and let the average molecular speed be $w$.

Regime 1: Quadratic drag

Suppose $v\gg w$ so that the air particles can be treated as stationary. Your mistake is that the air particles on top of the wing still have significant horizontal velocity in the frame of the wing, so they exert negligible drag. What instead happens is that the frontal drag slows the wing and the lift deflects it upwards either until equilibrium is reached (if there's an engine) or the wing falls out of the sky (if there isn't). No perpetual motion.

In this regime, the rate of air particles deflected by the wing is about $\rho_n Av,$ and each particle exerts impulse about $mv$, so the drag is quadratic: $F\sim(\rho_nAv)(mv)\propto v^2.$

Regime 2: Linear drag

Okay, but unless the aircraft is hypersonic, we can't neglect the individual, random motion of air particles. Suppose more realistically that $v\ll w;$ in this regime, particles are constantly smashing into and being deflected off of both sides of the wing. Drag arises because more particles smash more quickly into the bottom of the wing than the top, deflecting the wing backwards and upwards. As wing accelerates upwards, the pressure exerted on the top of the wing increases which introduces a forwards force that decreases the drag (which is expected because the wing is also accelerating backwards, decreasing the relative velocity) and a downwards force that decreases the lift. This continues until lift balances weight and thrust balances drag (or the wing falls out of the sky). No perpetual motion, again.

As a rough estimate of the drag in this case, note that about $\rho_n A(w/2+v)$ particles per unit time strike the bottom of the wing with velocity about $w/2+v,$ and about $\rho_nA(w/2-v)$ particles per unit time strike the top with velocity about $w/2-v.$ So, the drag is about $F\sim\rho_nA((w/2+v)^2-(w/2-v)^2)\propto v,$ which is linear.