Spanwise loading distribution c.cl/cref

[Lucas G.]

Greetings!

This may seem like a rather elementary question to someone more familiar with the subject, however I can't for the life of me figure out why exactly the quantity c.cl/cref (local chord times local cl divided by the reference chord [in this case set as the MAC], I presume) is equal to the spanwise loading distribution.

Could anyone point me towards some book or resource where this result or a deduction is explained? I've scoured various aerodynamics and aircraft design books thus far and can't seem to understand where that comes from.

Any answers are greatly appreciated!

[Linda Sutrina]

This should do the trick : http://wpage.unina.it/fabrnico/DIDATTICA/PGV_2012/MAT_DID_CORSO/09_Progetto_Ala/Wing_Design_Sadraey.pdf

A good basic NACA airplane design guide is  a between the wars report from NACA which was used to size potential airplanes.

Report No. 408  "General Formulas and Charts for the Calculation of Airplane Performance" by W. Bailey Oswald

There are other reports of this period on spins etc.

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[Rob McDonald]

Any intro aero book should lay this out -- I'm not sure which text you have handy so I won't quote page numbers...

First, note some subtlety in notation.  Usually we use cl, cd for 2D airfoil coefficients and CL, CD for 3D coefficients.

I'm sure you're comfortable with:

CL = L/(q*S)

Where q = 0.5*rho*V^2

We non-dimensionalize the force on a surface by dividing by the dynamic pressure (q) and the reference area (S).

What is the force on a 2D airfoil?  If you were holding a 2D foil in your hands, what force could it resolve?

So, we talk about the load on a 2D section as being 'per unit depth'.  So, force per inch of span or per meter, or whatever you prefer.  That load per unit span is usually denoted lower case l.  With that, we have:

cl = l/(q*c)

From these, you can probably convince yourself of things like integrating the chord to get the area:

S = 2*int_0^(b/2) c(y) dy

This means the integral starting at zero _0 going to the tip (b/2) ^(b/2).  Where b is the span.

If you were to divide S by b, you could get:

cbar = S/b

l(y) is the load distribution -- the thing that we want to be elliptical for minimal induced drag.

l(y) = cl(y) * c(y) * q

q is a constant, so we can divide both sides by q and we still have a curve with the same shape.  Likewise, we can divide both sides by another constant (cbar) and we still keep the same shape.

l(y)/q = cl(y) * c(y)

This is the quantity that OpenVSP plots -- it has the shape of the load distribution.

Rob

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[Lucas G.]

Thanks a lot for the speedy reply, Rob! This clarifies it quite well!

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[skyc...@gmail.com]

Hi,

Follow up question for this, where does each section along the span resolve the moments that are output in the .lod file and the .csv file?  Each row mentions xavg, yavg, zavg, is this used as the moment center for that section?

Thanks!

Bryan 

[skyc...@gmail.com]

Sorry, follow-up to follow-up:  If xavg is the moment center, what is xavg?  Half way between LE and TE at that section cut?

Thanks,

Bryan

[Tim Swait]

Bryan, This is what I've been trying to get to the bottom of, not very successfully so far! My efforts so far are in this thread. I'm trying to find the points on the wing that the forces pass through (i.e. the points about which the moments are zero). The point (Xavg, Yavg, Zavg) is simply the centre of the panel for that row of the .lod file. The moments coefficients (Cmx, Cmy, Cmz) are NOT about the points (Xavg, Yavg, Zavg) for each row, instead they are ALL about the same point, the (Xref, Yref, Zref) point that you define when you run the analysis. While I'm trying to work this stuff out, I've set that to (0,0,0). You can see that your Cmx value increases a lot as you go further out on the wingspan, as you'd expect for this. 

Now I would have thought that since moment = force x distance, so distance = moment/force, then I ought to be able to find for example the distance in x by dividing cmy/cz. Alternatively I should be able to find it by dividing cmz/cy, so I can cross check my results. However this doesn't seem to work. I'm still trying to understand this. There's also the business about whether to use the normalised (*c/cref) values or not. I've just set cref to 1 while I'm trying to work out what's going on. For some reason it seems that if I use the normalised version of the moment coefficient and the not normalised version of the force coefficient then I get some values for the zero moment points that look sort of almost but not completely believable! Please let me know if you get anywhere with this, I've been banging my head on the wall for some time on it now!