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Ignoring for a moment potential schematic difficulties in creating an aircraft that can land and take-off without threatening a downward vertical stabilizer, would a vertical stabilizer that extends both above and below the aircraft's centerline improve or hinder in-flight stability?

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As long as the vertical tail surface sits above the centerline (more precisely the longitudinal axis of inertia - for now we can assume both fall together), any side force will also create a rolling moment. This is undesirable because now a yaw command will not only create the intended yawing moment, but also a rolling moment.

If you want to yaw in order to turn, the vertical tail above the centerline will even roll the aircraft in the wrong direction, so you need more coordinated aileron command than with a symmetrical vertical tail.

So the short answer to your question is: Terry is right: Extending the vertical tail below is clearly a benefit.

This can be seen by the widespread use of ventral fins in combat aircraft. Adding area below the centerline helps to reduce the roll contribution of the vertical tail in a sideslip, which improves handling characteristics (more directional stability AND lower yaw-induced roll). See the picture below for an example where the effects of additional vertical tail area are compared to the effects of a ventral fin of the same area.

comparison of fin and vertical tail area effectiveness

At supersonic speeds, the low aspect ratio ventral fin has a nearly constant contribution to directional stability, whereas the wing-like vertical tail loses effectivity in proportion to the Prandtl-Glauert factor $\frac{1}{\sqrt{Ma^2-1}}$. The small ventral fin of the F-104 raised directional stability by 30% at Mach 2.

F-104 ventral fin contribution to directional stability

At high angles of attack, the vertical tail is in the wake of the fuselage which reduces local dynamic pressure and, consequently, effectivity. The ventral fin then is in ideal flow conditions, so it can help to stabilize the aircraft at high angle of attack, just when the fuselage contribution to instability is largest.

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  • $\begingroup$ but having a rolling moment with the rudder will help when over-rolled in severe turbulence $\endgroup$ – ratchet freak Jun 6 '14 at 19:39
  • $\begingroup$ Yes, there are of course cases where the top-heavy rudder is a benefit. But overall a neutral rudder would be best, wouldn't it? $\endgroup$ – Peter Kämpf Jun 6 '14 at 19:42
  • $\begingroup$ The "good" roll comes from dihedral or sweep, and is due to the sideslip that will happen anyway, regardless of the vertical position of your rudder. $\endgroup$ – Peter Kämpf Jun 6 '14 at 22:43
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Not to suggest this as an answer but as a minor factor that may be a consideration. When the vertical stab is above the aircraft's center of gravity, its drag causes a slight pitch up force assisting the elevator in supplying the pitch up force needed to counter the typical pitch down tendency due to the center of gravity being just forward of the wing's center of lift. If you put half of the vertical stab below the aircraft's CG, you will cancel this small benefit and cause the elevator to do all the compensating itself thus adding extra (albeit a small amount of) drag.

The drag vector of a horizontal stab in a tee tail configuration offers a bit of help in this regard too and if you placed half the vertical below the centerline on a tee tail aircraft, the lowered horizontal's drag vector arm would reduce the pitch up force supplied by its drag and you would get a slight overall drag increase from the elevator's extra load.

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