A thought occurred that the term "induced drag" was more specialized in the
aerodynamics texts than I had realized. Checking the indices of both Batchelor
and A & vD I found that this term is only mentioned in a few places in these
books. I am now in full appreciation of what has happened. The early
aerodynamicists developed the theories of lift based purely on inviscid flow.
The result was the well known Magnus Effect idea and its circulation, seen even
today. Then the viscous nature of air and the induced drag were added as an
afterthought. In A & vD it never made it completely into the Appendix IV
empirical graphs of airfoil sections. By this I mean that the coefficient of
drag shown must be being taken at a direction slightly off of the direction of
the "line of flight", i.e. the air velocity vector ahead of the wing. This
violates its own definition, probably just due to the measurement inaccuracies
of these quite small values of drag force vector magnitudes and narrow
directions. A very small angle of force component off of the direction of lift
is enough to add plenty of induced drag.
This butts up against what is known as "D'Alembert's paradox". Succinctly stated, this is a drag found in practice where theories relating to perfect fluids indicate that none should be present. Someone should have made a better case about this than has been made. I grant that Batchelor says point blank on one of his pages that for all the assumptions in these theories of flow being "irrotational", the flow generally is not "fully" irrotational at all, but only "partly" irrotational.
My own approach has been the opposite. I started with air as a viscous fluid that is deflected downward, causing an induced drag, identified well by the phrase, "the drag penalty of lift". Then only later was the circulation concept accepted but only as an "upwash" at the leading edge coupled with an added "downwash" at the trailing edge. One advantage of this approach is that the source of the circulation is better understood. The "spinning baseball" and the "Flettner rotors" on ships had at least something present to give rise to the circulation. When the theory was transferred later to airfoils, which were not spinning, the circulation flow was kept but its source was changed. A statement was made that a "Kutta-Joukowsky" condition, under which the flow must leave the trailing edge moving exactly in parallel to its pitched surfaces, was where the circulation arises. (This is at least better than an idea occasionally still seen that the flow above the wing must move faster because it has a longer path to traverse, something about which even Albert Einstein famously made comments.) I bless Francis Weston Sears in his textbook for providing the images of flow, added with almost no connection to the text presented, in which the flow is breaking off and not so leaving the trailing edges.
My fault has been that I did not see how circulation was consistent with conservation of momentum in providing lift but now I do. So it can be said that a merging has taken place between my views and those in these texts. Lift does have something of a "magic carpet" effect to it, wherein no energy is required to produce it. It does so to the extent that air can be treated as an inviscid medium (characteristic of higher, though more turbulence-prone, Reynolds Numbers). To the extent that it is not, then, an induced drag occurs which spoils this perfect view. Some care must be taken by those being introduced to these ideas to get them right. Even among those who fully understand them, the somewhat inaccurate approach hidden in the texts relating to the coefficient of drag vector and its direction should be exposed with treatment not yet in evidence. The unquestioning acceptance of all this flight theory by wind energy, needless to say, is a matter that also adds its own flavor to it all.
It should not be thought a heresy to say that traditional aviation wing lift theory involving the stream function with the superposed gamma circulation might not be perfect. Certain conservation laws may not be quite being observed. Energy is required to establish the circulation flow. Once in place and functioning, the question arises - how is it maintained. Presumably the flow simply circulates from the trailing edge of the wing to the leading edge. Well enough but this flow must bump up against non-circulating flow around its edges. This means friction and continuing expansion of the circulation farther and farther from the wing. Energy is thus necessary, one way or another. Energy requirements mean drag on the wing.
Another possibility instead is that reverse circulations occur both before and after the wing that accept the main circulation without it necessarily expanding through space. Superposed circulations within the stream function, of course, are never seen as vortices but only as regions where the flow varies in velocity above and below a reference location in space. Such upside down flow variations can certainly arise at these two locations, never having been considered earlier. What happens is that the effects of such reverse circulations would be to counter the effects of the main circulation, including the lift force generated.
Either way, carrying investigations of the conservation laws to greater depth, it is suggested here, could very well uncover some difficulties with these traditional theories never having been suspected all along.
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