Question

What is/are the use(s) of the single horseshoe vortex model of finite wing aerodynamic theory?

• 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

Hint:

A single horseshoe vortex only carries total circulation, no loading detail, so it fails for the wing's own pitching moment or induced drag but works for far-field effects like tail downwash or formation flight.

Answer 1

The Correct Option is C, D

Step 1: Recall what a single horseshoe vortex represents.
The single horseshoe vortex model replaces a finite wing with one bound vortex line of constant strength lying along the span (usually placed at the quarter-chord), plus two trailing vortices that leave the wingtips and stretch downstream to infinity. It was the earliest and simplest version of finite-wing vortex theory, before Prandtl's full lifting-line model added many horseshoe vortices to represent a realistic, elliptic-like spanwise loading.

Step 2: Check the wing pitching moment coefficient.
Pitching moment depends on how the lift is distributed along the chord of the wing, not only on the total lift or its spanwise position. A single horseshoe vortex is placed at one chordwise location and carries the wing's total circulation, so it has no information about how the load is spread across the chord. It cannot approximate the pitching moment coefficient. Statement (A) is not correct.

Step 3: Check the wing's own induced drag coefficient.
A single horseshoe vortex assumes constant (uniform) bound circulation across the span, which corresponds to a rectangular-like loading, not the elliptical loading that real wings approach and that minimizes induced drag. Because the induced drag of a wing is very sensitive to how the loading is distributed across the span, one horseshoe vortex is too crude to give a good estimate of the wing's own induced drag; that needs many horseshoe vortices (a vortex lattice) to capture the loading shape properly. Statement (B) is not correct.

Step 4: Check the effect of the wing on a downstream tail.
For estimating the downwash that the wing induces at a horizontal tail located some distance behind and possibly below it, only the wing's overall circulation and the rough geometry of its trailing vortex sheet matter, fine details of the spanwise loading wash out quickly with distance. A single horseshoe vortex, carrying the wing's total circulation, gives a good approximation of this induced downwash, and hence of how the wing changes the tail's effective angle of attack and its own local induced drag. Statement (C) is correct.

Step 5: Check the formation flight application.
The same reasoning applies to a trailing aircraft flying in the wingtip vortex system of a leading aircraft in formation. Each aircraft can be modeled as a single horseshoe vortex, and the upwash induced by the leading aircraft's trailing vortices on the trailing aircraft's wing can be estimated well enough this way to predict the induced drag saving (or penalty) of flying in formation compared to flying alone. Statement (D) is correct.

Final Answer:
The single horseshoe vortex model is good for estimating the wing's effect on a downstream tail and the benefit of formation flight, but not for the wing's own pitching moment or its own induced drag. \[ \boxed{\text{C, D}} \]