


However, another basic theory does provide a reasonable, firstorder approximation for the drag coefficient. This technique is called Prandtl's Lifting Line Theory. Thin Airfoil Theory is derived assuming that a wing has an infinite span, but lifting line theory applies to a finite wing with no sweep and a reasonably large aspect ratio. In simple terms, the wing is modeled as a fixed vortex with a series of trailing vortices extending behind it. These trailing vortices have the effect of reducing the lift produced by the wing and creating a form of drag called induced drag.
According to Lifting Line Theory, the lift coefficient can be calculated in the following way:
where
For example, the value 2π is used in the following graphs comparing experimental lift coefficients for two aircraft as measured in a wind tunnel against predictions from both Thin Airfoil Theory and Lifting Line Theory. The first comparison shows the Cessna 172 with its relatively high aspect ratio of 7.37. Note that the Lifting Line prediction is only a slight improvement over Thin Airfoil Theory when compared to the Cessna wind tunnel data, though the slope of the Lifting Line equation does better match that of the actual data. Also note that like Thin Airfoil Theory, the Lifting Line model is not capable of predicting stall and only provides a good estimate of the lift up to the stall angle.
where
where
where
Minimum Drag Coefficients  

Aircraft  Type  Aspect Ratio  C_{Dmin}  
RQ2 Pioneer  Single pistonengine UAV  9.39  0.0600  
North American Navion  Single pistonengine general aviation  6.20  0.0510  
Cessna 172/182  Single pistonengine general aviation  7.40  0.0270  
Cessna 310  Twin pistonengine general aviation  7.78  0.0270  
Marchetti S211  Single jetengine military trainer  5.09  0.0205  
Cessna T37  Twin jetengine military trainer  6.28  0.0200  
Beech 99  Twin turboprop commuter  7.56  0.0270  
Cessna 620  Four pistonengine transport  8.93  0.0322  
Learjet 24  Twin jetengine business jet  5.03  0.0216  
Lockheed Jetstar  Four jetengine business jet  5.33  0.0126  
F104 Starfighter  Single jetengine fighter  2.45  0.0480  
F4 Phantom II  Twin jetengine fighter  2.83 
0.0205 (subsonic) 0.0439 (supersonic) 

Lightning  Twin jetengine fighter  2.52  0.0200  
Convair 880  Four jetengine airliner  7.20  0.0240  
Douglas DC8  Four jetengine airliner  7.79  0.0188  
Boeing 747  Four jetengine airliner  6.98  0.0305  
X15  Hypersonic research plane  2.50  0.0950 
The efficiency factor, e, varies for different aircraft, but it doesn't change very much. As a general rule, highwing planes tend to have an efficiency factor around 0.8 while that of lowwing planes is closer to 0.6. A reasonable average to use for most planes is about 0.75.
The equation we have derived is also sometimes expressed in the following form, where the factors in the denominator of the C_{L}² term are combined into yet another new constant called K.
Assuming a typical value for aspect ratio of around 6 and an efficiency factor of 0.75, the value of K turns out to be about 0.07.
We now have equations to estimate the lift as a function of angle of attack and equations to estimate drag as a function of lift. It is simple to combine the two to produce an equation for drag as a function of angle of attack. Regardless of whether we use the Thin Airfoil approximation for the lift coefficient or the Lifting Line method, we get an equation of the form:
When graphed as a function of angle of attack, the drag coefficient tends to look like a parabola. It therefore makes sense that drag increases with the square of angle of attack in the above equation.
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