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Originally Posted by shdw
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I will, but I do not want to break down the math because it's too involved. Instead I will stick to words and offer a commentary. The math is not required for the purpose at hand anyway. Most of it is college Trig with some Calc III thrown in.
Quote:
Originally Posted by shdw
[...] From what I get out of it, it is proving that pressure difference is how we properly calculate lift.
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Well, kind of. It is a solution for finding the velocities around a flat plate in an inviscid (non-viscous) medium but with a bunch of caveats. 1) There must be enough viscosity to allow circulation to occur, but no more than that or you can't get very good results from the math; 2) the flow from the bottom (or top if it is flying at negative angle of attack) must not work its way back to the other side around the trailing edge. This is the famous
Kutta Condition, which says a flow must leave the trailing edge of a wing or plate cleanly.
Quote:
Originally Posted by shdw
[...] However, the math is far above my level of understanding (Calculus, 4 years ago) and I am getting tied up in the nomenclature.
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It took me a while to get back to speed on it too and I had the whole 6 years worth of courses. The math is downright tricky even for those often doing it, and it requires lots of practice. You begin to see how smart the great theorists were when you actually go through solutions like this and work them out.
Ok, back to the weblink. I do not recommend this weblink because it does not include most of the best reasons for having a Thin Airfoil Theory in the first place; not that this derivation is wrong of course, just that I know of a few better ones (my opinion), better in how they tie together the reasons for having the derivation with its uses and practical applications.
My recommendation is to put aside the current weblink for a bit and get a book like John J. Bertin,
Aerodynamics for Engineers, Fourth Edition, Prentice Hall, 2002 (used $83). However, since we have the web link I will first offer a few general comments, go to bed, then come back for specifics related to contents of the point by point analysis.
The purpose of Thin Airfoil Theory is to quantify and show the relationships between circulation and velocity around a very simple airfoil, actually just a flat plate. To do this simple task, you have to make a bunch of assumptions some of which are listed above. Some more are-
1) no thick boundary layer
2) only small angles of attack
3) only small amounts of camber
4) thin airfoil (preferably flat)
5) incompressible flow (this means slow, for air)
Also, because there is no third dimension in play and very little viscosity, there is no drag to be found.
The name of the game in classical Thin Airfoil Theory is simply to show what velocity field would satisfy all these preconditions and develop an expression mathematically that covers it. Such an equation will be a Governing Equation from which other solutions can be obtained. This equation must satisfy the boundary conditions-
1) the vortex sheet we place on the airfoil to represent circulation must operate as a streamline of the flow, and there cannot be any air moving perpendicular to the airfoil surface
2) the circulation at the trailing edge must be zero. This is in fact what happens on a real wing or airfoil.
Later on we may kick the results around in order to extract useful information from it like coefficients of moment, lift, center of pressure, lift curve slope, etc.