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GATE 2022 AE – Question 40

Flight Mechanics & Space Dynamics · Airplane performance: drag polar, take-off and landing, climb and descent, ceilings, range and endurance, load factor, turning flight, V-n diagram, winds · 2 marks · Multiple choice

For a conventional subsonic aircraft, e is Oswald factor and AR aspect ratio. At minimum $C_D/C_L^{3/2}$, which relation holds?

  1. $C_D/C_L^2=1/(\pi eAR)$
  2. $C_D/C_L^2=4/(3\pi eAR)$
  3. $C_D/C_L=1/(\pi eAR)$
  4. $C_D/\sqrt{C_L}=1/\sqrt{\pi eAR}$

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Correct answer: (B) $C_D/C_L^2=4/(3\pi eAR)$

Explanation

**Parabolic polar:** $C_D=C_{D0}+KC_L^2$ with $K=\dfrac{1}{\pi e\,AR}$.

**Minimise $\dfrac{C_D}{C_L^{3/2}}$** (the condition for minimum power required, i.e. maximum endurance of a propeller aircraft).

$$f(C_L)=\frac{C_{D0}+KC_L^2}{C_L^{3/2}}=C_{D0}C_L^{-3/2}+KC_L^{1/2}.$$
$$f^\prime=-\tfrac32C_{D0}C_L^{-5/2}+\tfrac12KC_L^{-1/2}=0\;\Rightarrow\;3C_{D0}=KC_L^2 .$$

**Drag coefficient at this point:**
$$C_D=C_{D0}+KC_L^2=\frac{KC_L^2}{3}+KC_L^2=\frac43KC_L^2 .$$

$$\frac{C_D}{C_L^2}=\frac{4K}{3}=\frac{4}{3\pi e\,AR}\quad(\text{option B}).$$