How is the area rule applied on airliners like the A380?

Question

Wikipedia says that because of the area rule's impact on wave drag:

aircraft have to be carefully arranged so that at the location of the wing, the fuselage is narrowed or "waisted", so that the total area doesn't change much.

This much makes sense to me. Though it's a bit difficult to tell, I understand that in the picture supplied there, the green circle toward the front of the aircraft is intended to have roughly the same area as the blue circle and lines through the wing cross-section:

_{All images from Wikipedia unless otherwise noted}

This is more clear in the picture of the F-106 Delta Dart where one can easily see the Coke bottle waist where the wings near their widest:

_{That's an awesome picture! You could stare at it all day, couldn't you?}

Peter Kämpf even provides a nice explanation of the area rule in his answer here including a drawing from the Junkers patent on the subject:

_{Image posted by Peter in his answer, referenced above}

In this image, I can see how the areas through several lines across the fuselage, wings & engines are all intended to be (roughly) equal.

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What I don't understand is how this is applied to airliners like the Airbus A380 which they say is:

showing obvious area rule shaping at the wing root

in reference to this image:

To me, it appears that the cross-section at the wing root consists of the same area at the front of the plane (roughly behind the cockpit appears to be where maximum diameter is achieved), plus the area of the wing root, plus the area through the wing, plus the area through a pair of engines (depending on exactly where you make the cross section). As I see it, there's no slimming of the body to make the area through the wing cross section equal to the area of the cross section just behind the cockpit.

This shot from above of another A380 shows that there is no Coke bottle waist near the wings:

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Am I not seeing this correctly, or do I not understand the area rule correctly?

Answer 1

First of all, not the area should be equal, but the area gradient along the direction of flow should be shallow. The minimum drag with a given volume can be achieved when the area distribution is that of a Sears-Haack body. Ideally this rule applies only at Mach 1, and once you go faster, the cross sections which matter are those along a Mach cone, not those of the aircraft's cross section.

For subsonic flight, the penalty for neglecting the area rule is small; it only matters when local flow is supersonic ahead of a contraction of the contour of the aircraft. Subsonic flow would decelerate, whereas supersonic flow will accelerate further and result in a drag-intensive shock downstream. Adding something to fill up the contraction will reduce the pressure gradients and ideally avoid the shock. By this, area ruling will help to shift the onset of the Mach-related drag rise and enables airliners to cruise a little faster. Honestly, I cannot see the "obvious" area rule shaping on the A380 - to me this is classic subsonic aerodynamics which tries to avoid sharp pressure gradients in the area of the wing airfoil's rear loading. Especially the outer engine pylons could be done better, but I digress.

For airliners, it is much more important to have a constant fuselage cross section, which makes stretching the fuselage simpler and is much easier to build. Area-ruling the fuselage simply is not worth it (yet) when your maximum cruise speed is just Mach 0.85. It is enough to add some Küchemann bodies to smooth out pressure gradients.

Below is a comparison of a swept wing at Mach 0.9, on the left clean and on the right with Küchemann bodies. Note the massive flow separation on the left wing, whereas the flow pattern on the right wing shows attached flow.

Mach 0.9 wing comparison (picture source)

Answer 2

On airliners it is applied via the use of these sharp pods under wings.

A coke-bottle-like fuselage would be extremely impractical for internal arrangement and cargo loading, so it is not done on airliners even though it could probably improve the efficiency a bit.

What is done is:

• The engines shifted ahead of the wing help smooth out the increase of cross-sectional area around the wing leading edge.
• The enlarged flap track fairings, a.k.a. anti-shock bodies help smooth out the decrease of cross-sectional area around the wing trailing edge.

Answer 3

Here are a couple of oil flow visualization pictures. Presumably they are taken at the same Mach number and lift coefficient. The left hand picture appears to show the upper surface of the wing with flow separation where the shock meets the surface. The airfoil sections probably pre-date supercritical designs. The right hand picture shows the application of Kuchemann carrots (or Whitcomb fairings). The flow appears to be shockless.

The lower picture shows application of these fairings to the Convair 990. I think I'm right in saying that supercritical airfoil sections, which are designed to weaken the shock, eliminate the benefit of these fairings.

It's not clear to me that the flap track fairings on the A380 (and many other aircraft) have the same (or similar) effect seeing as the flow on the lower surface of the wing is subcritical at just about all operating conditions, as Peter Kampf points out. His analysis implies that the benefit is not due to transonic area ruling, but to other aerodynamic effects, and I'm sure he is correct.