A single horseshoe vortex is the crudest way to represent a finite wing: one bound vortex carrying the whole wing's lift, plus two trailing vortices running downstream from the tips. Its accuracy depends on whether the question needs the fine details of the wing's own loading (near field) or only its overall effect on something else, further away (far field).
- Wing pitching moment coefficient: this depends on how lift is spread along the chord, which a single bound vortex sitting at one chordwise station simply cannot represent. Not a valid use.
- Wing's own induced drag coefficient: this is sensitive to the spanwise shape of the loading, and a single horseshoe vortex forces a uniform loading, far from the elliptical loading real wings have. It gives a poor estimate of the wing's own induced drag. Not a valid use.
- Effect of the wing on a horizontal tail's induced drag: the tail sits well behind the wing, in the far field of the wing's trailing vortex system, where only the total circulation matters, not the fine spanwise details. A single horseshoe vortex captures this downwash well enough to be useful. Valid use.
- Formation flight benefit or penalty: this is the same far-field situation, a following aircraft sits in the upwash field of the leading aircraft's trailing vortices, and a single horseshoe vortex per aircraft is the standard simple model used to estimate the induced drag change from flying in formation. Valid use.
The pattern is clear: a single horseshoe vortex fails whenever the wing's own detailed loading matters, and works whenever only its overall induced effect on something further downstream is needed.
Let's summarize:
- Not reliable for the wing's own pitching moment or its own induced drag, both need the real spanwise/chordwise loading shape.
- Reliable for far-field induced effects, such as downwash on a tail or interference in formation flight.
The correct uses are (C) and (D).