What dictates the aspect ratio of an aircraft's wing?

Question

When designing an aircraft, there has to be a decision as to the aspect ratio of a wing. It's been said that having a higher aspect wing will reduce drag for the same wing area, however most of the time wings are shorter than they can be. So my question is, what exactly dictates the aspect ratio of a wing, and why don't they make them as long as possible?

Answer 1

Now I am going to commit a heresy, but keep reading to get an explanation:

Increasing the aspect ratio of a wing will not change its induced drag. Increasing the span will.

The induced drag coefficient of a wing is $c_{Di} = \frac{c_L^2}{\pi\cdot AR\cdot\epsilon}$, and this seems to indicate that a bigger aspect ratio AR would lower the induced drag coefficient $c_{Di}$. But only at the same lift coefficient $c_L$!

Now let's look at the real numbers and compare two wings of the same span, but different aspect ratios. For simplicity, wing 1 has an AR of 5 and wing 2 has an AR of 10. Let's further assume that both wings have the same mass. Since both wings have the same span, wing 1 has twice the wing area of wing 2. To create the same lift, wing 1 needs only half the lift per area than wing 2! This means its $c_L$ is only half as big as that of wing 2, and now lets look at the induced drag again: $D_i = q_\infty\cdot S\cdot c_{Di}$

Wing 1: $D_{i_1} = q_\infty\cdot S_1\cdot\frac{c_{L_1}^2}{\pi\cdot AR_1\cdot\epsilon}$

Wing 2: $D_{i_2} = q_\infty\cdot S_2\cdot\frac{c_{L_2}^2}{\pi\cdot AR_2\cdot\epsilon} = q_\infty\cdot 0.5\cdot S_1\cdot\frac{4\cdot c_{L_1}^2}{\pi\cdot 2\cdot AR_1\cdot\epsilon} = D_{i_1}$

If both have the same span efficiency $\epsilon$, both have the same induced drag at the same lift. To reduce induced drag requires a span increase, regardless of aspect ratio.

However, a higher aspect ratio wing does have advantages:

• Lower surface area means less friction drag
• Lower surface area also means less mass, at least at moderate aspect ratios.
• Smaller pitching moments, requiring a smaller tailplane

but also disadvantages:

• Less internal volume for fuel or the landing gear
• Needs more complex high lift devices for the same landing speed

In the end, the wing chord is chosen to minimize wing mass and to yield the minimum required fuel volume, and the aspect ratio is just a consequence of the selected wing span. Driving wing mass down is also reducing induced drag, and the $\epsilon$ of modern airliner wings is only 0.75 to 0.8, which shows how little importance the induced drag coefficient has for finding an overall optimum.

Answer 2

Aspect ratio is the ratio based on the span and chord of an aircraft's wings. The span is the length of the wings measured wingtip to wingtip; the chord is the 'depth' of the wing from the leading edge to the trailing edge, measured in a straight line.

Because very few aircraft have constant chord planforms, this requires a not-very-fancy formula to calculate (source: NASA), so that we can effectively average the chord:

$$AR=\frac{b^2}{S}$$

Where:

$AR=$ Aspect ratio

$B=$ Wingspan

$S=$ Wing's area

Math aside, aspect ratio is chosen based on an aircraft's role or requirements. A need for agility dictates a low aspect ratio, as does a need for compactness. In both cases, fighter aircraft and bush aircraft benefit from agility and small size. High aspect ratios provide great cruise efficiency but can have poor landing characteristics (high drag at low speeds or high angles of attack due to frontal area) that are often offset by high-lift devices like flaps and slats.

To the second half of your question: even when a high aspect ratio is desired, wings are not made as long as possible for two reasons.

The first is structural; the bending forces associated with wings of extreme length are, well, extreme, and the materials required are pretty space-age. See high-performance gliders, or at the crazy end, solar- or human-powered aircraft, for examples of this. It's just hard to do at the size of an airliner.

The second reason is more practical: space is expensive. An extremely high-aspect ratio wing takes up a ton of space relative to the rest of the aircraft. In an attempt to offset this, early 777s (which had a larger span than 767s and 747s) were offered with folding wingtips, but nobody bought that option and it got dropped.

Answer 3

To answer your question it's maybe helpful to remember why a higher aspect ratio generates less drag. A higher aspect ratio causes less induced drag at the same lift than an aerofoil with a lower aspect ratio. Okay, we need a certain amount of lift and our aim is to gain this lift as efficient as possible. Lets do it: Higher aspect ratio -> less drag, less drag less fuel burn, less fuel burn -> higher efficiency - perfect, but there are maybe other ways to reduce drag and safe for example space, or weight - a longer leading edge creates more form drag and where do you park this giant aircraft and it means a lot of weight to get sufficient strength for this giant wing - weight needs lift to fly and more lift causes more drag.

Okay, I think it's clear now, that it's not sufficient to consider only one way of tuning the efficiency of your aircraft. There are also nice possibilities like winglets to reduce induced drag for only a little extra weight and also just a little extra interference drag or a good range of possible CGs which requires left negative lift on the tail = less lift required at the wing. New materials and design techniques also allow to work on the all in all shape of the wing which increases the efficiency rapidly. Building a good aircraft is finding a good balance and so you can't just concentrate on only one possible solution.

I hope my wired talk can help you a little bit, sorry I fly this stuff and I guess all my colleagues are happy that I don't build it ;)