Which of the following statements regarding 3D finite wing vortex aerodynamics and induced drag are **CORRECT**?

(A) For a planar wing of fixed span $b$ and total lift $L$, an elliptical lift distribution yields the absolute minimum induced drag coefficient $C_{Di} = \frac{C_L^2}{\pi AR}$

(B) Non-planar wing configurations (such as vertical winglets or C-wings) can achieve an induced drag span efficiency factor $e > 1.0$ relative to a planar wing of the same projected span

(C) Wing downwash far downstream in the Trefftz plane is exactly twice the downwash at the wing bound vortex ($w_\infty = 2 w_0$)

(D) Induced drag vanishes completely when an aircraft flies at supersonic Mach numbers

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles 1. **Elliptic Lift Distribution:** By Munk's theorem, an elliptic circulation distribution produces constant downwash across the span, minimizing induced drag for a planar wing: $C_{Di} = \frac{C_L^2}{\pi e AR}$ with $e = 1.0$. 2. **Non-Planar Lifting Systems:** Vertical winglets spread trailing vorticity vertically as well as horizontally, allowing effective span efficiency $e > 1.0$ based on planar projected span. 3. **Trefftz Plane Downwash:** In the Trefftz plane (far downstream), trailing vortex lines are effectively doubly infinite ($-\infty$ to $+\infty$), inducing exactly twice the velocity as at the bound vortex ($w_\infty = 2 w_0$). 4. **Supersonic Induced Drag:** Induced drag is vortex drag caused by trailing lift vortices; it is present whenever lift is generated, regardless of Mach number (in addition to wave drag). ### 2. Statement Evaluation - **Statements A, B, C:** **CORRECT.** - **Statement D:** **INCORRECT.** Induced drag is generated whenever lift is produced. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Supersonic aircraft experience both wave drag *and* lift-induced vortex drag ($C_{Di} \propto C_L^2$). - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 5.

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