# Why is a rectangular wing planform less efficient than others?

I read somewhere that a rectangular wing is inefficient compared to other wing planforms. Can someone clarify why this kind of wing planform is inefficient compared to other planforms and what causes this "inefficiency"?

• – ymb1 Mar 13 at 21:28
• I read this post, but still I don’t know what causes this „unefficiency”. More drag? If yes, I will be more than happy to know why there is more drag on this wing planform. – Konrad Mar 13 at 21:33
• The topic is explained in steps. From that related post you arrive at the concept of "downwash angle", and following that links leads to: Why does the elliptical wing have the lowest drag? and How is the induced drag calculated for a wing with elliptical planform? HTH. – ymb1 Mar 13 at 23:33
• So the reason is just non-elliptical lift distribution? – Konrad Mar 14 at 7:48
• @konrad Can you please post the link of the article that you have read? So we can give a clear answer based on the article you have read – sai teja Mar 14 at 17:58

The fundamental issue is the behaviour at the wing tip.

On a rectangular wing the high pressure beneath and the low pressure above cause air to spill up round the tip and spiral backwards in a wingtip vortex. This both lowers the pressure difference, reducing lift near the tip, and loses energy to the vortex, increasing drag. Both these effects reduce overall efficiency.

If the tip is made smaller while keeping the wing span and area the same, for example by tapering the wing or making it elliptical, then the tip losses and the inefficiency are also reduced.

A more detailed explanation gets more complicated, but I am not sure you want all that.

• Thank you! Now it is clear. If you have more detailed explanation I'll be more than happy to read it and fully understand the topic! – Konrad Mar 14 at 17:06
• Proper washout will make your point moot. Tip losses are identical for identical circulation distributions. – Peter Kämpf Mar 15 at 6:35
• @PeterKämpf Lots of technical variations make the point moot; a well-designed rectangular wing can be more efficient than a badly-designed elliptical one, but that does not affect the basic other-things-being-equal situation. – Guy Inchbald Mar 15 at 9:08

For the same wing span, an elliptical wing planform is the most aerodynamically efficient compared to any other flat wing planform because it has the lowest induced drag. Note that we are comparing only aerodynamics here. Things get much more nuanced once you couple structural weights, stability & control for a global efficiency (also see @Peter's answer).

## 1. Induced drag

So restricting to aerodynamics, elliptical wing planform is the most efficient because it has the lowest induced drag. Induced drag is produced because wings have finite span, and therefore one section of the wing distorts the flow on another section, creating downwash. This downwash distorts the lift vector on each wing section, thereby converting what otherwise would be lift into drag.

## 2. Downwash

Now that you understand induced drag has to do with downwash on each span section, you must appreciate that the distribution of downwash over the span is important to the induced drag generation. Distribution of downwash is dictated completely by the lift distribution over the wing (and vice-versa), which is in turn, dictated by the shape of the wing planform.

## 3. Lift distribution

For an elliptical wing planform, we can prove that it also generates an elliptical lift distribution. When you do the math of integrating all the sectional induced drag from the downwash and sectional lift, elliptical lift distribution has the lowest induced drag.

The following figures show the distributions of elliptical wing, rectangular wing (taper = 1) and tapered wing (taper = 0.5), with aspect ratio 8 and all producing a total $$C_L=0.4$$. Here, Ref Cl is the actual lift coefficient distribution that we've talking about, but multiplied by span length; Local Cl is the lift coefficient produced at each wing section, as if it's standalone airfoil; $$\alpha_i$$ is the downwash distribution in deg. The data are generated from Lifting Line Theory.

As you can see, elliptical distribution maintains a constant downwash, compare that to a wider and more varied one for a rectangular wing. Also, a taper of 0.5 gets you fairly close to the elliptical lift and downwash distributions, especially when you consider the $$\frac{C_L}{C_{D_i}}$$, which is 62.8 for elliptical and 61.8 for tapered.

• Thanks a lot! Very comprehensive and easy to understood explanation. – Konrad Mar 15 at 13:38
• It'a a bit misleading to only present results for straight wings without washout when in reality most rectangular wings use at least some washout. – Peter Kämpf Mar 16 at 8:51
• "Local Cl is the lift coefficient produced at each wing section, as if it's standalone airfoil" doesn't make sense to me. If we just looked at the 2d airfoil at each section then this would show a curve corresponding to an ellipse for the elliptical wing, a flat line for the rectangular wing and linearly decreasing line for the tapered wing, no? – xnor Jun 15 at 17:32
• @xnor What doesn't make sense to you? I don't understand the second part of your comment. What's "a flat line" for rect wing and "decreasing line" for tapered wing? – JZYL Jun 15 at 18:06
• @JZYL flat line as in constant across y, decreasing line as in linearly decrease with y. It's just not clear what Ref and Local Cl are, and the descriptions don't help. You also say Ref Cl is Cl multiplied by span length, but span length is a constant and the same for all three cases. – xnor Jun 16 at 22:41

The inefficiency is the result of a comparably high weight in relation to the possible lift. However, for small aircraft that disadvantage is not very pronounced (this is true especially for braced wings and biplanes) and is outweighed by very favorable stall characteristics. Also, a rectangular wing is very easy to build. Compare with what has been built so far - many competitive small aircraft and most biplanes use a rectangular wing. Since they often cruise at speeds much above their best range speed, induced drag optimisation is of minor importance in those designs.

A better planform has more chord at the root and less at the tip: Such a trapezoidal wing can be built lighter. If the root has more chord, the spar can be made thicker, and if the area at the tips is smaller, less bending moment is created for the same lift. Bracing takes care of the root bending moment, but in cantilevered monoplanes the rectangular wing loses most of its benefits. A taper ratio of 0.7 will still leave enough stall margin at the tips when combined with some washout.

Especially large aircraft tend to have even smaller taper ratios because they concentrate most lift near the root wich makes their wings more efficient. Even though their induced drag coefficient might be higher than that of an elliptical wing, their overall induced drag is lower because of the comparably low wing mass which lowers the overall lift requirement.

• Yes, the requirements for structural efficiency are a bit different from aerodynamic efficiency. broadly speaking, taper helps both, but the details are rather different. – Guy Inchbald Mar 15 at 9:10
• Thanks for the answer! Another great explanation from you. – Konrad Mar 15 at 9:57