How to relate 2D airfoil Cd to 3D wing Cd

[rbogie]

Hi, I'm trying to help my teenage daughter with a science project where we take 2D airfoil lift and drag curves and relate them to to the actual wings of real airplanes and see how close we can calculate stall speed and max speeds of the real world airplanes. We went to a lot of trouble picking out the CL's and Cd's of the airfoils used by reading off the graphs in the "Theory of Wing Sections", but now I can't find any way of relating those CL's, Cd's to 3D values. I know that I can estimate the 3D CL from Prandtl's Lifting Line theory CL=CLalpha x (AR/AR+2) x alpha and that at least uses the slope of the CL vs alpha line and the alpha at the stall speed for that airfoil. I also know from Lifting Line theory that Cd=Cdmin +(CL^2/pi x AR x e--Oswald's efficiency factor), but you see that that formula doesn't make any use of the 2D value for Cd from the charts--the only thing it relates to again is the CL vs alpha slope.
In other words what can we actually do with the wonderful numbers charted in "Theory of Wing Sections" and any other CL vs AoA or Cd vs CL graphs??

 

[Guest0527211403]

Will you get the award if she wins?

 

[rbogie]

Good tease.
I loaned her my copy of "Theory of Wing Sections", so she knows how to look up CL vs Aoa and Cd vs CL curves. I also pointed her to the Aerospaceweb.net website that pretty much lays out all the formulas. This is really a question for me, as I noticed that all those formulas were only based upon the slope of the CL vs AoA curve, and in terms of CD had no reference at all to the curves that were so meticulously plotted in the book. What do we do with the curves as graphed? Those are only for 2D airfoils. How can those be correctd for 3d induced drag effects?

 

[cpinz]

Hi Rbogie

what those nondimensional numbers tell you is "the ratio exisitng between a Force (drag) and a given planform( wing)of infinite length realized with that profile, at a specific incidence and at a given Re number ( speed) ".

A wing with infinite Aspect ratio will have, at the Re reported in those tables,a curve cd-alpha similar to the ones found in you tables ( that is 2d curves)and will experience a drag force porportional to the coeff. reported in the 2d curve, and nothing else.

As the wing becomes shorter and shorter (..more and more "3d"..), the induced drag will influence the performance of the wing,but this will be a lift dependent drag, while the other components of the drag (always present in a wing, that is now "3d" object) will still dipend by that coefficient that you find in 2d curves....

More confuse than before...?

cpinz

 

[rbogie]

Actually, I came up with something that involves a little bit of intuition. If we make a guess and say that the max airspeed will be about . . . say 300 knots and we definitely know the weight (say max gross weight of A/C), then we can work it backwards to solve for the actual max speed CL. If we plug that into the lifting line theory we can come up with an induced drag factor. We have to know what the experimental results are for the CDmin. Adding that to the induced drag gives a totat drag factor, and from that and knowing thrust (see the discussion I listed above about finding thrust), we should be able to find the calculated max A/S.

That all sounds good, but what I can't rationalize out is how do we factor in the skin friction (parasitic) drag. We know that goes up as A/S goes up. But how do you separate the skin friction portion from the initial CDmin? Surely there is skin friction buried in the CDmin? And I've seen the graphs of skin friction drag, it goes up parabolically with A/S. My guess is that it would be a factor times the CDmin, and that factor would be based on (V-Vmin)^2/Vmin, where Vmin is the velocity for best L/D, which is usually best power off glide speed.

 

[rb1957]

if A/S is aspect ratio, why do you need to calculate it ?
(if you're using a real world airplane)

my 2c is the 2D drag probably doesn't include skin friction (as the 2D section has no skin area).

 

[rbogie]

Sorry, different abbreviation, Airspeed (A/S), Aspect Ratio (AR). The AR sometimes seems hard to find as manufacturers rarely list it but it can also be found by AR = b^2/S (b-wingspan, S-surface area), which manufacturers always list.

Yeah, the 2D drag curve, especially at cruise (min drag) gives a Cd of around .005 which is way low. On aerospaceweb.net they have a list of Cdmin's for different airplanes and those range from about .024 thru .031 so I'd say Prandtl's Lifting line equations are closer using Cdmin and Cdi, but I still don't see a parasitic (skin friction) drag term there. Of course another complication is judging what reference area to use for that term. Rather than trying to actually determine the skin surface area of the whole airplane(pretty difficult), you would probably use the equivalent flat plate surface area of the average cross section of the fuselage and perhaps the slenderness ratio of the fuselage.
Like I said before, the Cdmin term probably has skin friction of the whole airplane (at an A/S just below take off) and so what we want at this point would be the delta in parasitic drag as speed increases.

 

[Sparweb]

rbogie,
Having tried this exercise in school, I can commiserate on the difficulty. What's more, you will fail at the task, because you also have to account for fuselage drag, thrust, interference drag, excressence drag, stabilizer drag.... etc. get the drift? Oh, by the way, there are also slight lift +/- contributions from stabilizer lift and fuselage lift.

If you're trying to model a simple wing with no cumbersome airplane bolted onto it, then you already have the to figures you want. The induced drag + profile drag is close to the drag of the real wing, but skin friction, as mentioned before, is also a factor. For that you need to be familiar with Reynolds number. Then once you start counting the rest of the airplane, all bets are off, the fuselage could look like a dart or a barn.

Sorry about the negative tone - that's why aerodynamics is more of a college/university topic. I do have something helpful to say: Why not work the problem backwards?

You start with the weight, wing area, engine power, and the speeds for stalling and flat-out maximum. Then work out the lift and drag coefficients that match each case. This way, the only "fuzzy" number is the propeller efficiency.

Steven Fahey, CET

 

[rbogie]

Well, I've found formulas for calculating propeller thrust and they don't look too hard, although for this level of calculation, just grabbing a number between 80%-87% is probably close enough. I'm just surprised that I can't find any formula that approximates skin friction drag. You're right Re Num. would be part of it, so would a number for Myoo (you know the Greek letter) for skin friction (depending on smoothness of the skin coating), and average cross sectional area of fuselage, and the fuselage slenderness ratio. As to figuring the friction from the vertical and horizontal tails there are probably rule of thumb ratios (ie conventional configuration, subsonic airplane that is piston vs jet will have a vertical that is 10% the area of the wing while the horizontal stab is 25% the area of the wing) vs (conventional airplane with swept wing at M .9 has a vertical that is 20% the area of the wing and a horizontal that is 20% area of the wing.
These are just rule of thumb guesses used in the initial design phase of an airplane. Them plus the performance calculations would get iterated back and forth (if we were going thru the full design function) until you get closest to the best compromise for the desired performance.

 

[rb1957]

i think the issues are too involved to detail in these forums. i'd suggest gettting hold of a design textbook ... Raymer "Aircraft Design, A conceptual Approach" or Torenbeek "Synthesis of Subsonic Airplane Design" are two i have, but there are many others.

the issues are complicated, only really long-winded.

good luck

 

[rbogie]

Yes, I need to get a copy of Raymer's Acft Design Book to compare against. I am currently using "Fundamentals of Aircraft Design", by Leland Nicilai from Univ of Dayton. A pretty good book. Lays out the formulas pretty understandably.

 

[InteroperabilityFix]

I remember for an ideal wing it being around CL3D=0.9 CL2D?

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[rbogie]

Thanks, that rule of thumb seems to be repeated in Nicolai's book as well, so that makes use of actual (2D) airfoil L & D graphs (the CL and CD values), but the lifting line equations don't use CL or CD, all they use is CLalpha (the slope of the CL vs AoA curve).

Actually, to back up just a little, that .9 factor I think is dependent on the approximate range of the Aspect ratio (AR). I think .9 factor is correct for AR 6-8. I think its lower for AR <6 and higher for AR >9.