Expert: Paul Soderman - 3/31/2011

Question
QUESTION: Hello Paul,

I understand that Total Drag= Profile Drag(Cd)+Induced Drag(Cl^2/PieAR)

I am testing an aircraft model (1:200) in a low speed wind tunnel. I have obtained Coefficient of Drag and Lift values for AoA 0-18degrees.

-If I calculate Induced Drag=(Cl^2/PieAR)will that give me an accurate value for the Cdi?

-Would a Cd Vs Cl^2 graph give me a relation ship between induced drag?

ANSWER: Peter
Aircraft drag can be classified a number of ways.  I prefer: total drag = friction drag + pressure drag + compression drag + drag due to lift.  Often the first two, friction + pressure drag, are combined and called profile or form drag. And compressibility effects are small at low speed.  We usually plot total drag versus lift or total drag versus airspeed to find the minimum drag and the variation with flight condition.  In a plot of drag versus airspeed, the descending part of the curve (low airspeed) is dominated by induced drag and the ascending portion (high airspeed) is dominated by form drag.  But if you are trying to find the true wing induced drag you can plot drag coefficient versus lift coefficient squared.  You should find that the initial slope of the curve is fairly linear and follows the slope of the classic equation cd = cl^2*(1+sigma)/(pi AR), where sigma is a small number related to wing plan form.  But as you increase cl^2, the curve will deviate upward because there are other drag components that increase with lift such as fuselage drag and tail drag.  S. Hoerner (Fluid Dynamic Drag) does a nice breakdown of the various components of drag due to lift for the Me-109 and also sailplanes (chapter on Complete Aircraft).  He shows that the wing induced drag of the Me-109 was 69% of the total induced drag at Cl = 1. The relative values (wing, fuselage, tail, etc.) change with flight condition.  At low cl (high airspeed) the wing induced drag due to lift dominates other induced drag components and is not far from your equation although it is small compared to form drag.

I suggest getting Hoerner's book.  His curves are probably clearer than my explanation.
Paul

---------- FOLLOW-UP ----------

QUESTION: Hello again Paul,

I have CL against alpha graphs. I have worked out the lift curve slope (i.e. the gradient y=mc+c)and obtained experimental a_w values such as 4.3,4.4,4.9. I am told to compare the experimental a_w with the one predicted in the lifting line theory for a wing with elliptical loading using:

a_w=  a/(1+a/πAR)    (a= 5.7rads, AR=7.5) (Making allowance for any viscous effects): a_w=4.59

What sort of meaningful comparison can I give between the thoertical 5.49 and the values I have obtained from the experiment?

Thanks

Answer
The equation I have for the lift coefficient slope of an elliptically loaded wing is:

m = 2 pi/(1 + 2/AR)

Using an aspect ratio of 7.5 I get m = 4.96, which is not far from your experimental data.  Any deviation from the theoretical value means that your wing is not exactly elliptically loaded, which is to be expected.  We know that the lift slope of an arbitrary planform is always less than the ideal elliptically loaded wing.  The arbitrary planform equation is:

m = 2 pi/(1 + 2(1+tau)/AR)  where tau is a small number that can be found from Glauert's approximation of the circulation distribution using a Fourier series.

Why did you use a = 5.7 in your equation instead of 2 pi ?

Paul