Drag Coefficient & Lifting Line Theory

Is there a way to estimate the drag coefficient using Thin Airfoil Theory? I know that lift coefficient is estimated as 2*pi*alpha, but does Thin Airfoil Theory even predict a value for the drag coefficient?
- question from Scott
The source of the lift coefficient equation that you've cited was discussed in a previous question about the Thin Airfoil Theory. As you imply in your question, Thin Airfoil Theory does not predict drag, only lift and pitching moment.

However, another basic theory does provide a reasonable, first-order 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.

Creation of trailing vortices due to a difference in pressure above and below a lifting surface

According to Lifting Line Theory, the lift coefficient can be calculated in the following way:

where

CL = 3D wing lift coefficient
Clα = 2D airfoil lift coefficient slope
AR = wing aspect ratio
α = angle of attack in radians
Note that this equation is of the same form as that derived from Thin Airfoil Theory. In fact, the above equation becomes identical to that predicted by Thin Airfoil Theory if we let the aspect ratio go to infinity, as it would for an infinite wing, and if we assume the lift curve slope of the airfoil section, Clα, is the theoretical maximum value of 2π. If you know the actual lift curve slope for the airfoil on a particular aircraft you wish to analyze, you can substitute that value for a more accurate estimate. However, 2π is usually a very close approximation.

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.

In contrast, the Lifting Line model is a significant improvement over Thin Airfoil Theory in predicting the lift of the Lightning. As was discussed in our previous article on Thin Airfoil Theory, that approach breaks down for aircraft with small aspect ratio wings like the Lightning, with its AR of 2.52. Even though the Lifting Line Theory assumes an unswept wing, it still produces a good approximation of the lift produced by the Lighting's highly swept-back wings.

Lifting Line Theory agrees so much better with the Lightning wind tunnel data than does Thin Airfoil Theory because of the introduction of the aspect ratio, AR. This variable makes it possible to estimate the influence of trailing vortices and their downwash on the lift of the wing. This same factor makes it possible to approximate the induced drag that downwash creates on the wing by the following equation:

where

CDi = induced drag coefficient
CL = 3D wing lift coefficient
AR = wing aspect ratio
Knowing the induced drag is useful, but it is only one component of the total drag acting on an aircraft. For subsonic aircraft, the total drag is almost entirely due to the induced drag plus another form of drag called profile drag. Combining these two forms allows us to estimate the total drag on a wing by the relationship:

where

CD = 3D wing drag coefficient
CDmin = minimum 3D wing drag coefficient
k = constant of proportionality
CL = 3D wing lift coefficient
AR = wing aspect ratio
δ = ratio of induced drag to the theoretical optimum for an elliptic wing
Since many of these variables are nearly constants, the above equation can be simplified by introducing a new constant called Oswald's efficiency factor (e) in their place:

where

CD = 3D wing drag coefficient
CDmin = minimum 3D wing drag coefficient
CL = 3D wing lift coefficient
AR = wing aspect ratio
e = Oswald's efficiency factor
We now have a useful equation for estimating the drag of an aircraft. The minimum drag coefficient, CDmin, can be estimated relatively easily. A good value to use is around 0.025 for subsonic aircraft and 0.045 for aircraft operating faster than the speed of sound. Values for a variety of aircraft in cruise configuration, as measured in wind tunnel experiments, are compared in the following table.

Minimum Drag Coefficients
Aircraft Type Aspect Ratio CDmin
RQ-2 Pioneer Single piston-engine UAV 9.39 0.0600
North American Navion Single piston-engine general aviation 6.20 0.0510
Cessna 172/182 Single piston-engine general aviation 7.40 0.0270
Cessna 310 Twin piston-engine general aviation 7.78 0.0270
Marchetti S-211 Single jet-engine military trainer 5.09 0.0205
Cessna T-37 Twin jet-engine military trainer 6.28 0.0200
Beech 99 Twin turboprop commuter 7.56 0.0270
Cessna 620 Four piston-engine transport 8.93 0.0322
Learjet 24 Twin jet-engine business jet 5.03 0.0216
Lockheed Jetstar Four jet-engine business jet 5.33 0.0126
F-104 Starfighter Single jet-engine fighter 2.45 0.0480
F-4 Phantom II Twin jet-engine fighter 2.83 0.0205 (subsonic)
0.0439 (supersonic)
Lightning Twin jet-engine fighter 2.52 0.0200
Convair 880 Four jet-engine airliner 7.20 0.0240
Douglas DC-8 Four jet-engine airliner 7.79 0.0188
Boeing 747 Four jet-engine airliner 6.98 0.0305
X-15 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, high-wing planes tend to have an efficiency factor around 0.8 while that of low-wing 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 CL² 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.

Examples comparing the experimental drag coefficients of the Cessna 172 and the Lightning against the results predicted by Lifting Line Theory are presented above. Note that the lifting line approximation matches up against the wind tunnel results quite well.
- answer by Jeff Scott, 11 July 2004

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