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GATE 2025 AE – Question 36

Aerodynamics · Airfoils and wings: coefficients, Kutta-Joukowski theorem, thin airfoil theory, finite wing theory, critical and drag divergence Mach numbers · 2 marks · Multiple choice

A lifting surface has a spanwise circulation distribution of $\Gamma(\theta) = A\sin 3\theta$ (where $A \ne 0$) over its span $-\frac{b}{2} \le y \le \frac{b}{2}$, and $y = -\frac{b}{2}\cos\theta$ is the spanwise coordinate. Furthermore, the downwash varies along the span as $w(\theta) = V_\infty\left(\frac{3A\sin 3\theta}{\sin\theta}\right)$, where $V_\infty$ is the freestream velocity. Which one of the following options represents the total lift ($L$) and induced drag ($D_i$)?

  1. $L = 0$ and $D_i = 0$
  2. $L = 0$ and $D_i \ne 0$
  3. $L \ne 0$ and $D_i = 0$
  4. $L \ne 0$ and $D_i \ne 0$

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Correct answer: (B) $L = 0$ and $D_i \ne 0$

Explanation

The lift is $L = \rho V_\infty\int\Gamma\,dy = \rho V_\infty \frac{b}{2}A\int_0^\pi \sin 3\theta\sin\theta\,d\theta = 0$, since the two sines are orthogonal. The induced drag is $D_i = \rho\int w\,\Gamma\,dy \propto \int_0^\pi \frac{3A\sin 3\theta}{\sin\theta}\,A\sin 3\theta\,\sin\theta\,d\theta = 3A^2\int_0^\pi\sin^2 3\theta\,d\theta = \frac{3\pi A^2}{2} \ne 0$.