Expert: Paul Soderman - 11/14/2011

Question
QUESTION: Sir,
There is a small correction. By "exam" I meant a national level entrance exam ( Like GRE in your country!) for PG entrants in India called - GATE( Graduate Aptitude Test in Engineering). Further, I am already a graduate and not a student in any university. Thus, answering of that question( which appeared in GATE 2010 Aerospace paper) will not only benefit me, but also all the other fellow students looking for the same doubt.
I am hereby wording the question:
1.From Fundamental equation in prandtl's lifting line theory( pg 409,Anderson" Fundamentals of Aerodynamics"), how is rate of change of circulation with alpha ( dgamma/dalpha) dependant on angle of attack (alpha)?
2.How is lift curve slope of finite wing(CL) dependant on alpha if we know that CL is proportional to gamma( circualtion around the finite wing?

Hope you'l help me out.

Thanks,






















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ANSWER: OK Sangeeth - Once again you are taking me back to graduate school.  If I understand the problem, a wing with an elliptical loading distribution is a lifting-line model that satisfies the terms of the problem.  In that case we know that:

G(y) = Gmax (1 - (y/(b/2))^2)^1/2  where G = gamma,  spanwise bound circulation, y = span location, b = span length

Following Katz and Plotkin:  Low-Speed Aerodynamics, the maximum circulation can be written:

Gmax = 2bU(alpha - alphaLO)/(1+pi (AR/m))  where U = free stream velocity, AR = aspect ratio, m = sectional lift curve slope (constant spanwise), alphaLO is the angle of attack for zero lift (usually negative)

Therefore it follows that the derivative dG/dalpha = constant (i.e., it is independent of alpha).

We can also write for the lift coefficient:

CL = 2 pi(alpha - alphaLO)/(1+2/AR) so the derivative

dCL/dalpha = constant (i.e., it is independent of alpha).

This makes sense to me because both wing loading and lift coefficient vary linearly with angle of attack until near stall.  The slope must be a constant in the linear range. Do you agree ?
Paul

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

QUESTION: Sir,
I think this conclusion can be brought fore by simply differentiating the fundamental equation wrt Alpha.But in general lift distribution( in which the circualtion is written as a fourier sine series) how is the linear effect of alpha included?



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Answer
You are referring to Glauert's solution for the spanwise distribtion of airloads which can be written as a Fourier series:

Gc = Sigma(Bn sin ntheta)  

where Gc is the wing sectional circulation, Sigma is a summation from n =1 to infinity, and Bn are the constants which depend on sectional chord, lift curve slope, and flight speed.  

Because the only terms relating to alpha are the lift curve slope terms, and the lift curve slope is constant in the linear range, a derivative of dG/dalpha will be constant.

An easier way to think of it is to consider that the circulation can also be written:

Gc = m alpha c V/2

where m = 2 pi per radian, c = chord, V = flight speed

Again we see that dGc/dalpha = constant.