In the ever-present drive to improve the efficiency of aircraft, "higher aspect ratio wings" are frequently named as one of the reasons that new aircraft are more efficient than previous generations. Here are a couple of examples:

"... carbon fiber allowed designers to produce a higher-aspect-ratio wing for lower drag..." (Aviation Week, May 2015)

"... wings with high aspect ratio generate the lowest lift-induced drag..." (Flight International, April 2006)

"A higher aspect ratio wing has a lower drag..." (NASA, May 2015)

Indeed there seems to be a general consensus that higher aspect ratio wings are more efficient. This is however misleading, since drag and aspect ratio are independent of each other. Let us investigate.

 

What is Aspect Ratio?

In an aeronautical sense aspect ratio describes the shape of a wing; wings with high aspect ratios have a large span and short chord (leading edge to trailing edge length) and wings with low aspect ratio have a short span and large chord.

 

U-2

(Above) The high altitude and relatively slow Lockheed U-2 reconnaissance aircraft has a very high aspect ratio wing.

 

Mirage 2000

The fast and highly manoeuvrable Dassault Mirage 2000 has a low aspect ratio wing.

 

Aspect ratios are typically in the range between 1.5 and 35. The exact formula to determine aspect ratio is:

AspectRatio = (Wingspan2)/(WingArea)

Which can also be expressed using standard aerospace nomenclature as:

AR = (b2)/(S)

 

Why does Aspect Ratio Matter?

We can find aspect ratio in the standard aeronautical formula for the induced drag coefficient:

CDi = (C2L)/(πARϵ)

Aspect ratio can be found in the divisor in the formula above as "AR". Since we are unconcerned with the other terms at the moment we can simplify this as saying induced drag is inversely proportional to aspect ratio:

Di ∝ (1)/(AR)

This means that aircraft with high aspect ratios will have lower induced drag. Since fundamentally drag is "bad" and should be minimised, it seems reasonable that high aspect ratio wings are good.

 

Why doesn't Aspect Ratio Matter?

If we take the time to write out the entire induced drag formula we can see that "S" (the wing area), another property of the wing, is also present in the formula:

Di = (L2)/((1)/(2)ρV2SπARϵ)

Therefore we could expand our formula above to say induced drag is inversely proportional to wing area and aspect ratio:

Di ∝ (1)/(S.AR)

However this formula is somewhat confusing since aspect ratio and wing area are both geometric properties of a wing, and indeed aspect ratio itself is proportional to wing area. As such the formula above has unneeded terms and can be simplified as follows:

Di ∝ (1)/(b2)

This formula is unambiguous: wings of greater span have lower induced drag, irrelevant of aspect ratio or wing area.

 

Conclusions

Our analysis of basic aerodynamic theory shows that induced drag is fundamentally dependent on wingspan and not aspect ratio. Any relationship between aspect ratio and induced drag is simply a coincidence derived from the normally close relationship between wingspan and aspect ratio. However it is perfectly possible to increase wingspan while decreasing aspect ratio and vise versa.

This misunderstanding probably came about because during initial design studies it may indeed be the case that the wing designer determines the needed wing area, and only then determines the wing shape and aspect ratio. In that very specific situation (constant wing area), induced drag and aspect ratio are indeed proportional. However an expectation of constant wing area is in most situations absurd and the fundamental relationship remains that of wingspan and induced drag. As such confusing and misleading statements relating aspect ratio and drag should be avoided.

 

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