Question

A lifting surface has a spanwise circulation distribution of \( \Gamma(\theta) = A \sin 3\theta \) (where \( A \neq 0 \)) over its span \( -\frac{b}{2} \leq y \leq \frac{b}{2} \), 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 \( D_i \)?

• A. \( L = 0 \) and \( D_i = 0 \)
• B. \( L = 0 \) and \( D_i \neq 0 \)
• C. \( L \neq 0 \) and \( D_i = 0 \)
• D. \( L \neq 0 \) and \( D_i \neq 0 \)

Hint:

In problems involving lift and induced drag, always check the symmetry of the circulation distribution. If the circulation distribution integrates to zero, the total lift will be zero. However, a nonzero downwash still produces induced drag.

Answer 1

The Correct Option is B

Step 1: Lift Calculation 
The total lift \( L \) on a lifting surface is related to the circulation distribution by the Kutta-Joukowski theorem: \[ L = \rho V_\infty \int_{-b/2}^{b/2} \Gamma(y) \, dy \] However, in this problem, the circulation distribution \( \Gamma(\theta) = A \sin 3\theta \) is given with odd symmetry (because of the \( \sin 3\theta \) term), and when integrated over the span, the total circulation results in zero: \[ \int_{-b/2}^{b/2} \Gamma(y) \, dy = 0 \] Therefore, the total lift \( L \) is zero. 
Step 2: Induced Drag Calculation 
The induced drag \( D_i \) is related to the downwash distribution, which is given by: \[ w(\theta) = V_\infty \left( \frac{3A \sin 3\theta}{\sin \theta} \right) \] Since the downwash is nonzero, the interaction between the circulation and the downwash will produce a nonzero induced drag. The induced drag \( D_i \) is given by: \[ D_i = \int_{-b/2}^{b/2} \frac{\Gamma(y) w(\theta)}{V_\infty} \, dy \] This results in a nonzero induced drag because the downwash \( w(\theta) \) is nonzero and varies along the span. Thus, \( L = 0 \) and \( D_i \neq 0 \).