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I have found only the description of this pattern with experimental graphs.Could anyone please tell me the mathematical formula required to plot the distribution?

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    $\begingroup$ I'm a little unclear by what you the task. Do you think that you could perhaps include a pictures/figures of what you have got and what you're trying to get? $\endgroup$ – Thunderstrike Jun 6 '14 at 10:57
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    $\begingroup$ Funny - answers are pouring in, and we have yet no idea how the pattern looks like. $\endgroup$ – Peter Kämpf Jun 6 '14 at 11:40
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I have found only the description of this pattern with experimental graphs.

That's because that is the best way to have detailed and precise data: you either simulate (e.g., with Nastran-Patran or XFoil, as cited by Peter Kämpf in his answer) or build a [scale] model and use a wind tunnel. If you use a scale model, depending on the focus of the experiment, you will have to be careful (while scaling back up) not to damage your data: for this you have adymensional quantities to look after, such as the Reynolds number.

Could anyone please tell me the mathematical formula required to plot the distribution?

There is not one formula to do that, but rather a method that given the airfoil shape (and the angle of attack) will estimate the pressure distribution. This method was extensively used in the days before computers and FEM tools existed (for what I know SR71 is a notable example where this method has been used in the early phase of the wings design), but, for what I know, nowadays is seldom used.

The method is based on Conformal Mapping, in particular the Joukowsky transform or the Kármán–Trefftz for more detailed and general cases (see Sec. 4.2 of this pdf for a more in-depth explanation). This is a transformation in the complex plane that, from the airfoil profile that you want to analyse, will map it onto a circle.

You want to do this because the ideal flow around a circle is well known: enter image description here

At this point you have to know that around an airfoil the air will have a certain amount of circulation and that the total flow around it is the sum of the ideal unviscous flow plus the flow induced by the circulation:

enter image description here

Image from MIT

For this reason, you want to compute the ideal flow around a cilinder that has a rotation speed equivalent to the circulation around your airfoil[*]:

enter image description here

and it turns out that we have some formulas for that (image above taken from this page).


[*]: note that you will have different circulations for different angles of attack.


Now, why did we do all this? Because once we have the flow around the rotating cilinder and we have the mapping from the airfoil to the cylinder, we "only" have to invert the mapping:

enter image description here

and we immediately have our flow around the airfoil.

And once we have the airflow, finding the pressure distribution is quite straightforward, thanks to Bernoulli (you have the velocity distribution and you can assume that the transit of the air around the wing is fast enough to be adiabatic)

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  • $\begingroup$ Conformal mapping is the most simple way to arrive at an exact solution for the inviscid case. Joukowsky airfoils have a circular camber distribution, and the transformation becomes extremely complex when a specific airfoil is to be mapped. $\endgroup$ – Peter Kämpf Jun 6 '14 at 11:38
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XFoil uses two ways of plotting pressure distributions, one with the chord coordinate on the X axis and the negative pressure coefficient $c_p$ on the Y axis, like that: Cp plot over chord of an Eppler 502 mod

The dashed line is for inviscid flow (from the potential flow solver), the solid line for pressure with boundary layer effects.

Another version plots the pressure as arrows orthogonally on the contour of the airfoil, like that: Same airfoil, now with arrows to indicate pressure

Please note that negative values of $c_p$ produce arrows pointing away from the airfoil, and positive values point towards the airfoil. In all cases, the formula for $c_p$ is $$c_p = \frac{p - p_{\infty}}{q_{\infty}}$$ $p$ is the local pressure, $p_{\infty}$ is the pressure far away from the airfoil, and $q_{\infty}$ is the dynamic pressure far away from the airfoil.

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  • $\begingroup$ I knew you would have answered :P $\endgroup$ – Federico Jun 6 '14 at 11:19
  • $\begingroup$ And your answer touches a different aspect. Very nice! I now wait and see which pattern is actually meant. $\endgroup$ – Peter Kämpf Jun 6 '14 at 11:42
  • $\begingroup$ I wonder why yours keeps getting upvoted a mine sits there, is there a major fault in my answer? :/ $\endgroup$ – Federico Jun 10 '14 at 17:22
  • $\begingroup$ @Federico: Surprises me as well. Happens to me in reverse, too, where a flawed answer gets the votes and my answer sits at the bottom. I still wonder what pattern the question is about ... $\endgroup$ – Peter Kämpf Jun 10 '14 at 18:54
  • $\begingroup$ I start wondering if we'll ever know... $\endgroup$ – Federico Jun 10 '14 at 19:09

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