Discussion
Consider a conventional subsonic fixed-wing airplane. \(e\) is the Oswald efficiency factor and \(AR\) is the aspect ratio. Corresponding to the minimum \(\left(\frac{C_{D}}{C_{L}^{3/2}}\right)\), which of the following relations is true?
Answer: B
The relation is \(\frac{C_{D}}{C_{L}^{2}} = \frac{4}{3\pi eAR}\).
With the parabolic drag polar \(C_{D} = C_{D0} + KC_{L}^{2}\), \(K = 1/(\pi eAR)\), minimise \(C_{D}/C_{L}^{3/2}\) by setting its derivative with respect to \(C_{L}\) to zero:
\[ \begin{aligned} C_{L}^{3/2}(2KC_{L}) &= \tfrac{3}{2}C_{L}^{1/2}(C_{D0} + KC_{L}^{2}) \\ C_{D0} &= \tfrac{1}{3}KC_{L}^{2} \\ C_{D} &= \tfrac{4}{3}KC_{L}^{2} \end{aligned} \]
Dividing by \(C_{L}^{2}\) gives \(4/(3\pi eAR)\). This is the condition for maximum endurance of a propeller airplane, where induced drag is three times zero-lift drag. \(C_{D}/C_{L}^{2} = 1/(\pi eAR)\) would mean zero parasite drag, and the other two forms do not follow from the polar.
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