CL = 2 * pi * angle
the angle being expressed in radians. Is this correct?
- question from Chris Warner
The equation depends on three key quantities:
That leads us to the specific equation you have referenced. What you have stumbled upon is an equation from a simplified approximation called the Thin Airfoil Theory. This theory is handy for learning some basic aerodynamic relationships, but it isn't terribly useful in the real world of engineering because it ignores many of the ugly realities of how the air flowing over a wing really behaves. The particular equation that you mention is an ideal approximation of how the lift coefficient behaves with angle of attack. To be more exact, it is an ideal approximation of the slope of the lift curve, a quantity referred to as CLα (pronounced "C-L-alpha").
If you remember back to your basic algebra, a straight line can be represented by the equation:
where m is the slope of the line and b where it intercepts the y-axis. The lift coefficient curve behaves in a similar manner and can be represented by an equation of the same form:
where CLα is the slope of the curve and CL0 where it intercepts the y-axis. As you mentioned in your question, α (pronounced "alpha") is the angle of attack in radians.
According to the ideal aerodynamics of the Thin Airfoil Theory, the y-intercept (CL0) is 0 and the slope of the lift curve (CLα) is equal to 2π.
Plugging in those values produces the equation you describe:
The problem with this theory is that it assumes the wing extends to infinity. In other words, the lifting surface has no wingtips. Wingtips introduce a form of drag called induced drag. The stronger the induced drag is, the lower the slope of the lift curve (CLα) becomes. In addition, Thin Airfoil Theory doesn't account for the fact that the lift coefficient eventually reaches a maximum and then starts decreasing. The angle of attack at which this maximum is reached is called the stall angle. According to Thin Airfoil Theory, the lift coefficient increases at a constant rate--as the angle of attack α goes up, the lift coefficient (CL) goes up. But in real life, the angle of attack eventually gets so high that the air flow separates from the wing and the wing stalls.
You can better understand the effects of induced drag and stall by studying the following graph.
The conclusion to draw from this explanation is that the Thin Airfoil Theory equation can be used to estimate the
lift coefficient so long as you understand its limitations. The equation can only be used for aircraft with
medium to large aspect ratio wings and only up to the stall angle, which is usually between 10° and 15° for typical
- answer by Jeff Scott, 10 August 2003
I have been trying to prove that the equation for lift is accurate. What is the lift coefficient for a Boeing 747? How do you calculate the lift coefficient of an aircraft given values for the other variables? I have tried to work it out but all I have is bad luck.
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