Subtopic: Range and Endurance of Jet & Propeller Aircraft
Question Statement:
What is/are the use(s) of the single horseshoe vortex model of finite wing
aerodynamic theory?
Options:
(A) It can approximate the wing pitching moment coefficient
(B) It can approximate the wing induced drag coefficient
(C) It can approximate the effect of the wing on the induced drag coefficient of a
typical horizontal tail
(D) It can approximate the aerodynamic benefit/penalty of formation flight compared
to isolated flight
Official Answer Key:
C, D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.48)**
**1. Intuitive Physical Setup & Given Parameters**:
The question asks about the capabilities and limitations of the **single horseshoe vortex model** (bound vortex of span $b$ with two trailing semi-infinite vortices) in finite wing aerodynamic theory (MSQ).
**2. Step-by-Step Aerodynamic Derivation**:
1. **Single Horseshoe Vortex Limitations**:
- A single horseshoe vortex assumes uniform circulation $\Gamma$ across the span, which gives infinite downwash at the wing tips and fails to capture the realistic spanwise circulation distribution $\Gamma(y)$.
- Because induced drag is fundamentally related to the integral of downwash across the wing ($C_{Di} = \frac{1}{V_\infty S} \int \Gamma w dy$), a single horseshoe vortex **cannot** accurately calculate the wing's induced drag coefficient (it predicts infinity or requires ad-hoc cutoffs). Statement B is **FALSE**.
- Similarly, it models the chord as a single bound line, so it cannot capture chordwise pressure distribution or wing pitching moment $C_m$. Statement A is **FALSE**.
2. **Valid Applications of Single Horseshoe Vortex**:
- In the **far field** (at distances much larger than the chord, such as at the horizontal tail or behind the aircraft):
- The downwash induced by the main wing at the horizontal tail location can be very well approximated by treating the wing as a single horseshoe vortex placed at the aerodynamic center. Statement C is **TRUE**.
- In **formation flight**, the upwash field experienced by a trailing wing flying outboard and behind the lead aircraft's wingtip is dominated by the trailing vortex pair, which is accurately captured by the horseshoe model. Statement D is **TRUE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) It can approximate the wing pitching moment coefficient**: ❌ **INCORRECT**. Requires chordwise vortex distribution, not a single spanwise line.
- **(B) It can approximate the wing induced drag coefficient**: ❌ **INCORRECT**. Produces singular/inaccurate self-induced downwash.
- **(C) It can approximate the effect of the wing on the induced drag coefficient of a typical horizontal tail**: ✅ **CORRECT**. Far-field downwash at the tailplane is well-predicted.
- **(D) It can approximate the aerodynamic benefit/penalty of formation flight compared to isolated flight**: ✅ **CORRECT**. Outboard upwash fields from trailing vortices are accurately captured.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C, D**
- **Exam Takeaway**: A single horseshoe vortex is invalid for self-induced near-field properties (induced drag, pitch moment) but highly effective for far-field aerodynamic interactions (tail downwash, formation flying upwash).
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