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My question is the following. Wing-tip vortices induce a downward flow on the root side of the wing. So, if this induction happens at the wing, and these vortices are then shed downstream, why is there a downward flow component at the leading edge of the wing? Shouldn't the flow come straight and bend downward at the rear of the wing? This way, the wing just sees the straight airflow, as demonstrated below, causing no induced drag.

enter image description here

My guess is, for a subsonic flow, the downward flow "information" is transmitted upstream by pressure, causing the flow to already have a downward component when it hits the leading edge, despite the downwash taking place later on.

Thanks

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  • $\begingroup$ your arrow seems backwards to me $\endgroup$
    – Tiger Guy
    Commented Feb 15 at 15:07
  • $\begingroup$ The wing-tip vortex arrow you mean? $\endgroup$ Commented Feb 15 at 16:18
  • $\begingroup$ on second look, no, that's right. But overall flow goes down just due to the forces involved. Air goes down to keep the plane up. Vortices are just an artefact of lost lift over the edge of the wing, it's why newer planes have winglets. $\endgroup$
    – Tiger Guy
    Commented Feb 15 at 21:21

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why is there a downward flow component at the leading edge of the wing?

There isn't. There is upwash in front of all subsonic lifting bodies.

There is no cause and effect between and among the fluid. The cause is that there is a lifting body passing through the air. All of the fluid perturbations and all of the forces on the body are one single result. There is only one physical solution to the fluid motion which matches to the airplane's shape, attitude, and velocity. And there is only one distribution of pressure on that airplane that corresponds to those motions.

If you have a lifting body, there will always be uplift just in front, downwash behind, and trailing vorticity in the transverse plane cutting the wake. The downwash velocity just behind the lifting surface from wingtip to wingtip is equal to twice the downwash velocity between the trailing vortices far in the wake.

Put another way, if you know the airplane's shape, attitude and velocity, you can write an expression for the velocity of the entire fluid domain surrounding the craft. You may not be able to calculate the result, but the point is there exists a system of equations which define the entire velocity field at a single instant in time. Therefore, you can calculate the pressure everywhere in the fluid - including along the fluid boundary with the airplane.

As far as induced drag is concerned, the classic (and hopelessly non-intuitive) way to explain this is to look at the wake far behind the airplane. If there wasn't any airplane, and the air was just cruising past us without us disturbing it, there would be no vorticity in the wake's transverse plane - the air's velocity would be the same in front of us and behind us - no perturbations due to its passing. So balance the kinetic energy. If the transverse velocities started at zero and ended at zero behind us, the axial velocity of the fluid must remain the same after we pass - so zero drag. But now there is an airplane; and it is experiencing lift and creating these vortices (but doing no work on the fluid because we are treating it as stationary and having the air flow past it). We now have transverse velocities and their associated kinetic energy in the wake. Where did the energy come from from? Well, there's only one source of energy - the transverse vortical energy was borrowed from the fluid's axial kinetic energy, meaning that the fluid's axial velocity is slower behind the airplane than in front of it. And that means drag - the only thing there that can slow the axial flow down is the airplane. Now why on earth do we bother with such an odd way of looking at this? Because we can directly calculate the induced drag force this way. If we shift perspective to one where the airplane is moving through the air, the energy lost to induced drag is exactly the transverse vortical kinetic energy that appears in the wake.

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