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 technically 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.
Aspect ratios are typically in the range between 1.5 and 35. The exact formula to determine aspect ratio is:
\begin{equation} AspectRatio=\frac{Wingspan^{2}}{WingArea} \end{equation}Which can also be expressed using standard aerospace nomenclature as:
\begin{equation} AR=\frac{b^{2}}{S} \end{equation}Why does Aspect Ratio Matter?
We can find aspect ratio in the standard aeronautical formula for the induced drag coefficient:
\begin{equation} C_{D_{i}}=\frac{C_{L}^{2}}{\pi AR\epsilon} \end{equation}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:
\begin{equation} D_{i}\propto\frac{1}{AR} \end{equation}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:
\begin{equation} D_{i}=\frac{L^{2}}{\frac{1}{2}\rho V^{2}S\pi AR\epsilon} \end{equation}Therefore we could expand our formula above to say induced drag is inversely proportional to wing area and aspect ratio:
\begin{equation} D_{i}\propto\frac{1}{S.AR} \end{equation}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:
\begin{equation} D_{i}\propto\frac{1}{b^{2}} \end{equation}This formula is unambiguous: wings of greater span have lower induced drag, irrelevant of aspect ratio or wing area.
Conclusions
Our analysis 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 possible to increase wingspan while decreasing aspect ratio and vice versa.
This misunderstanding probably came about because during initial design studies it may be the case that the wing designer first determines the needed wing area, and only then determines the wing shape and aspect ratio. In that specific situation (constant wing area), induced drag and aspect ratio are indeed proportional. However expecting 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.