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The aerodynamic model cited in this paper (pdf) as "flat plate theory" was a good first-order approximation of the aerodynamics of a perching glider. I immediately saw the similarity with the Newtonian theory that works well in hypersonics, but it is only a similarity, not the same.

Around the internet I found these other two articles: (1) and (2), which make reference to this flat plate theory in the context of wind turbines.

My questions are:

  • Where does this flat plate theory come from?
  • In what textbook or article or other bibliographic reference can I find its derivation?

Edit:

Here is another article that cites this flat plate theory, now in the context of aircraft flight dynamics. This one makes reference to (1).

Here is yet another one in the context of wind turbines.

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Ludwig Prandtl's lifting line theory produces a Taylor series as the result of the lift equation, and in almost all books this has been simplified just to the first term of the series, which then was even robbed of its sinus function to simplify it even further by using a small-angle approximation.

What you will find in books today is something like: $$c_L = c_{L\alpha}\cdot\alpha$$

$c_l$ is the lift coefficient, $c_{L\alpha}$ is the lift curve slope and $\alpha$ is the angle of attack in radians.

What the author should at least mention (but I think most are ignorant of this detail) is that a more precise version would be: $$c_L = c_{L\alpha}\cdot\sin\alpha$$

And now the books should tell you that there are more members in the solution to the lift equation; the one above is only the first one which dominates the result for small values of $\alpha$. But they don't. That is OK when the aircraft moves only through the small angle of attack region of attached flow, but for the perching motion you need to look at bigger angles, so the small-angle solution will produce noticeable errors.

The lifting-line theory assumes inviscid flow and does not know about flow separation. Perched flight of birds would be impossible without the high angle of attacks which produce flow separation and makes use of instationary effects which delay separation. All this is not covered by lifting line theory, but nevertheless it produces quite useable results. I guess, however, that the equation in the paper was found empirically.

The lift equation cited in the paper is a consequence of observing forces in a 360° polar, one where the airfoil is rotated through full 360° instead of the narrow range between maybe -8° and +12° where flow is attached and lift forces vary linearly with the angle of attack.

The plot below is from Hoerner's book "Fluid Dynamic Lift" (Scribd) and shows the results of several measurements over 180°. Apart from the spikes around 15° and 175°, the forces correspond well with $c_L = x\cdot\sin\alpha\cdot\cos\alpha$, where $c_L$ is the lift coefficient, $\alpha$ denotes the angle of attack and $x$ is a proportionality factor (the paper postulates $x$ = 2). Plot of <span class=$c_L$ over 180° angle of attack from Hörner's book">

A flat plate has no leading edge radius and, consequently, no leading edge suction, which would be responsible for the aforementioned spikes. Therefore, the simple trigonometric approximation will fit the data quite well.

Both the flat plate and Newtonian theories assume the aerodynamic force to be orthogonal to the plane of the wing or plate, and when you take the portion of that force which is orthogonal to the direction of movement, which is how lift is defined, you get above trigonometric equation. This is simple geometry and shows no deeper insight into fluid mechanics.

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  • $\begingroup$ Great answer, but I have some questions, you say the formula calculates the portion of the wing that is orthogonal to the direction of movement, but to to the value of $c_l$ would then be a simple sine function. Where does the cosine come from? $\endgroup$ – ROIMaison May 8 '15 at 7:05
  • $\begingroup$ The link you added does not contain the book in pdf directly but a download button instead, and the file to download is an executable. Is that safe? $\endgroup$ – Katerl3s May 8 '15 at 7:08
  • $\begingroup$ In other words: The coefficient of pressure is Cp=xsin(AoA). The lift, by definition, is obtained when you project this into the direction perpendicular to the flow: Cl=xsin(AoA)*cos(AoA). However, the question is: where that Cp comes from. Peter Kämpf recognizes this fits the data, but that is not the answer I am looking for (I already have that answer from a lot of papers). I'm looking for a theory that explains why Cp=x*sin(AoA) fits the data, in the same way there is an explanation why the Newtonian theory holds in hypersonic flow. A simple data fit is not normally called "theory". $\endgroup$ – Katerl3s May 8 '15 at 7:45
  • $\begingroup$ @KeyDamo: Sorry, I didn't check this. Thanks for letting me know; I replaced the link with one pointing to Amazon. And you are right about the theory. I will edit the post when I have more time. $\endgroup$ – Peter Kämpf May 8 '15 at 8:16
  • $\begingroup$ @KeyDamo: Now I have added a theory section, but I think that the equation in the paper is based on empiricism. $\endgroup$ – Peter Kämpf May 9 '15 at 12:25

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