Is the classical definition of induced drag merely a mathematical construct?

Question

I understand that the presence of vortices at the wing tip cause part of wing to see a higher angle of attack and cause more drag. But the explanation that lift vector tilts backwards seems like a mathematical construct to me since I defined lift as force along a vector of my choosing in the first place. Isn't it using the effect, namely reduction in lift and increase in drag as a cause to explain the phenomena?

I do not contradict the existence of induced drag, I just want to come up with a better explanation for the phenomena.

Answer 1

All of physics is artificial construct. Nothing of it is how the world actually works but rather our description of patterns we found in it. We do it using terms that exist only in our minds. So lift and drag, being terms, are artificial constructs.

Now there is a pattern we always see in the world that we describe as conservation of momentum and conservation of energy. Using this we can conclude that any aerodynamic device (usually airfoil, but applies equally to e.g. Flettner rotors) that creates a force perpendicular to the flow (which we call lift) also creates a force in the direction of flow (which we call induced drag).

To create the lateral force, force has to be applied to the air (via principle of action and reaction), which accelerates it sideways (downward, usually), changing its velocity. But:

• In the reference frame of the device there is nothing to increase kinetic energy of the air, so the magnitude of the stream velocity can't change, so to add sideways component the component along free flow direction must reduce and that requires forward (in flight direction) force on the air, which causes aft reaction on the aerodynamic device.
• In the reference frame of the free stream the air velocity does increase, which means the aerodynamic device must do work on it, and since work is dot product of force and path, the aerodynamic device must be applying force in the direction of its motion, and have force in the opposite direction applied by the air.

From this argument it also follows that this component of drag decreases (for given lift) with:

• Square of flow velocity as more air mass is available, so it only needs to be deflected by less, and since momentum grows with velocity, but kinetic energy with square of velocity, the work that needs to be done for the same lift decreases.
• Cross-section of the affected flow, which grows approximately with square of span, for the same reason.

That should explain how the laws of physics we observe make those specific definitions of lift and drag work well. But it's still in our minds, not in the outside world. The outside world does not care how we call things.

Answer 2

This question is as much philosophical as anything else. All our mathematical models are only approximations which predict what will happen in certain circumstances. They do not necessarily address what is "really" going on.

To take a common example from another field of engineering, is a given sound a particular waveform or a particular spectrum of frequencies? It is produced by some vibrating object and moves through the air as a wave, yet is detected by the hairs of the inner ear as a spectrum of frequencies. Mathematically we move between the two descriptions, the time and frequency domains, via the Fourier transform. Digitally, we quite often convert between the two as it suits. But which is the "real" nature of sound? Ultimately, that question is meaningless, its reality is what it is and we just look at it in the most useful way at the time.

The same can be said of induced drag. We can say that the lift vector is tilted backwards, or we can say that the induced lift is vertical and the induced drag is horizontal. There is no right or wrong, there is only utility for the current calculation.

Answer 3

All of the force things we talk about in aviation are mathematical constructs. All Aerodynamic forces, Lift, Drag, Induced or parasite, on whatever vector direction you choose, are constructs. The only real forces that an aircraft (or any body in a fluid) feels, are the forces exerted by the little molecules bouncing off the surface of the body. Each time a molecule hits the surface, and bounces off, there is a change in momentum, and a corresponding force exerted on the surface normal (perpendicular) to the surface, at that specific point where the molecule struck. All the constructs we talk about are just integral summations of all these tiny forces which are actually distributed over the entire airframe. We summarize them and pretend that an equivalent single force exists, which acting through a single point on the airframe (the aerodynamic center or AC), and chosen so that the actual effect of this constructed Psuedo force will be equivalent to the actual effect of the uncountable individual forces of all the molecular collisions discussed above.

Lift, Drag, etc. are just engineering psuedo forces chosen to add clarity or to allow engineering to perform calculations without having to deal with all the individual molecular forces, and get the same (or equivalent) results. Generally, Lift is chosen as the component of the actual aerodynamic Force which is normal or perpendicular to the aircraft flight path vector, and Drag is that component that lies parallel to the aircraft flight path vector.

Even the thrust of a propeller is in actuality, caused by the collision between air molecules and the cambered surface of the propeller, in the exact same way. Same with the thrust of a jet engine. It is caused by collisions of the high pressure combustion gasses on the turbine blades inside the engine. All the other formulae and shortcuts we use (like Bernoulli principle, or application of downwash comparisons for lift explanations), are accurate, not because they represent what is really going on, but because they take advantage of principles of physics like energy conservation of or momentum conservation.