# Wing Span Efficiency Factor

1. Nov 28, 2013

### truffaldino

Hello Everybody,

Span efficiency factor appears in the lifting-line theory of Prandtl describing Lift and vortex drag of a finite wing. According to this theory, the most efficient wing is an elliptical one and, roughly speaking, the span efficiency factor defines an efficiency of a given wing planform relatively to elliptic wing.

In the framework of the lifting line theory the span efficiency cannot exceed 1.

My question is: can this efficiency factor be more than 1, when going beyound lifting-line approximation ? (assuming that induced drag remains to be proportional to square of lift in less rudimentary theories). In other words, does it exist such a wing whose induced drag is less than that of elliptic wing in the lifting-line theory (provided aspect ratios and lift coefficients are the same and viscous effects are neglected).

Thanks
Truffaldino

2. Nov 28, 2013

### Astronuc

Staff Emeritus
Basically efficiency is usually expressed as a ratio of actual obtain to the theoretical maximum. Since one cannot exceed the theoretical maximum, the ratio will never exceed unity (1).

One could perhaps devise a form with greater theoretical efficiency, but still the ratio will never exceed unity.

An efficiency greater than unity would indicate something wrong with the formulation.

3. Nov 28, 2013

### truffaldino

Dear Astronuc,

I would be pleased to get some non-trivial answer to my question, rather than discussion of wording.

I know what efficiency is, and at the end of my previous post I explained quite clearly what I meant (at least for people who is familiar with foundations of aerodynamics).

I can reformulate once more my question: For given lift coefficient and aspect ratio of wing minimal theoretical value of induced drag in Lifting-line theory is given by a Prandtl equation with efficiency factor 1. This theory is a crude approximation. Can this drag value be smaller in more sophisticated theories? (provided lift coefficient and aspect ratio are unchanged)

Truffaldino

Last edited: Nov 28, 2013
4. Nov 29, 2013

### rcgldr

From a practial standpoint, there are other wing shapes that have the same amount of induced drag as an elliptical wing, wiki articles:

http://en.wikipedia.org/wiki/Elliptical_wing

http://en.wikipedia.org/wiki/Lifting-line_theory

http://en.wikipedia.org/wiki/Oswald_efficiency_number

Last edited: Nov 29, 2013
5. Nov 29, 2013

### truffaldino

I am well aware of all that, but this does not give answer to my question.

These wiki articles do not go beyound lifting-line theory except one of them says: "the pure elliptical shape as a superior planform may be a myth.[citation needed]" with no citation or proof.

Let me reformulate the question once more for better clarity: Suppoce I do some high-precision inviscid CFD 3d calculations for a wing inside a big domain, then go to the Trefftz plane and get drag. Since all drag is induced could it happen that for some planform it will be smaller than that obtained for an elliptical one in the framework of the lifting-line theory (provided the lift and aspect ratios are the same for both planforms).

Or let us put it differently: take very high reynolds number incompressible flow a such that skin friction is almost zero in comparison with induced drag, separation is absent etc. Could it happen that we will have oswald factor exceeding unity.

Last edited: Nov 29, 2013
6. Nov 29, 2013

### rcgldr

e is defined to be 1 for elliptical circulation distribution, e > 1 would just mean that there's a circulation distribution better than elliptical circulation distribution (as opposed to an overall efficiency greater than unity). I don't know how to prove or disprove if this is possible.

From a pratical standpoint, a wing of finite size has to have an effective angle of attack to produce downwash and lift, so any wing of finite size producing lift has a profile drag component. I think the wiki article is implying that for the overall drag (profile + induced), that other planforms may be just as good as elliptical. You could look at the edit history of the wiki article to find who added that statement to the wiki article, and send a message to that person's wiki user talk page to clarify that statement.

7. Nov 29, 2013

### truffaldino

Indeed, notion of circulation distribution along the span implies that the wing itself is a line, so for a wing with finite aspect ratio all these notions are just asymptotic approximations. Within the lifting-line theory the proof is an easy excercise. However, for a wing with the finite chord the circulation cannot be reasonably attributed to particular span locations, neither trailling vortices are straight and exist in such a simple form etc.

This was actually a reason I was asking this question: I was trying to optimize a wing different from elliptical using some software, first running inviscid calculations with Vortex Lattice Method and then with 3D panel method and got efficiency slightly exceeding unity. So I was puzzled if it was it a numeric error or it is indeed possible. Since software implements methods which are improvements of the Lifting-line theory, I thougth that it is indeed possible even in inviscid cases. But as far as I understood, the gain from such improvement could be very small and perhaps not worthy of studying, but who knows?

Last edited: Nov 29, 2013
8. Nov 30, 2013

### rcgldr

For most aircraft, there's a trade off between wing efficiency and the cost to build the wing. Also, most powered aircraft cruise at a speed where parasitic drag effects are dominant, so there are different design goals.

Some of the research on making really efficient wings is done for gliders, both models and full scale. In the case of full scale gliders, the high end ones with 80+ foot wing spans like the Nimbus 4T have an overall 60:1 glide ratio (== lift / drag ratio) at over 60 knots. Since models are relatively cheap and sold in much larger numbers, there's more work being done on making contest type glider models very efficient, using hollow molded carbon / kevlar / fiberglass wings. Other examples of efficient wing design involve human or solar powered aircraft, but there are very few of such models made.

9. Nov 30, 2013

### truffaldino

This is exactly the reason for which I was studying this wing optimization. I designed and constructed several rc gliders and was trying to improve min. sink rate for the next one choosing optimal planform and sections along the span. It seems that for slow floaters with small to moderate aspect ratios (like 1:8 - 1:10) even rectangular wing with a certan amount of washout outperforms elliptical one due to low reynolds number effects (big form drag for smaller chordlenghts close to tips due to separation bubbles).

Another possible example is soaring birds: Seabirds that do not have severe limitations in wing span (like gulls and albatroses) have planform close to elliptical. Slow soaring birds with limited span, like eagles, have planform which is closer to rectangular (althouth, for eagles maneovrability and tip stall might be of bigger importance than min sink rate)..

Last edited: Nov 30, 2013
10. Nov 30, 2013

### rcgldr

One of the RC model designer's is Mark Drela, a professor in aerodynamics at MIT. He has a free "polars" program called XFOIL. Do a web search for xfoil and you'll find the link for the download.

The contest models are a compromise between minimum sink rate and a good lift to drag ratio for a range of speeds so that models can thermal downwind and then return at higher speed without the lift to drag ratio dropping too much. F3J models have to be very strong to handle the high g two man tow launch. These models generally rely on reducing camber on the control surfaces (ailerons and flaps) to reduce parisitic drag during faster flight modes.

For minimum sink, the high end 1.5 meter hand launch gliders rely on very light weight and thin wings. With wingspan limited to 1.5 meters, the aspect ratio is lower due to having a longer wing chord to provide more wing area. I recall some old "gas bag" type floaters up to 10 foot wing span that mostly relied on very light weight for minimum sink, but I haven't seen any of these since the 1990's.

11. Nov 30, 2013

### truffaldino

I am already using X-foil or XFLR5 (latter implements VLM and 3D panel in conjunction with x-foil polars). This was XFLR5 which gave efiiciency>1 for one of the planforms.

An hour ago I found interesting thread in rcgroups modelling science forum which confirms shat rectangular planform could be better than elliptic:

12. Dec 1, 2013

### rcgldr

It's my understanding that elliptical planform is supposed to be close to ideal for induced drag, as opposed to total drag. Normally it's assumed that the best lift to drag ratio occurs when half the total drag is induced drag and the other half parastic drag. As you've apparently read, the best planform for a real world situation is probably not elliptical. The best minimum sink rate occurs at some speed below the speed of best lift to drag ratio.

I'm not sure what the best planform for mininum sink would be or if that should be a goal. Most gliders have to launched at high speed, and except for indoor duration type glides, they need to be able to follow thermals downwind and be able to return upwind efficiently. Adjustable camber (being able to raise or lower the entire trailing edge, flaps and ailerons) helps. For F3J launch mode, the ailerons are cambered less than the flaps to increase the effective washout to prevent incidents during the high g launch.

13. May 12, 2014

### GuillermoH

I have been wondering this for a long time as well. After running many test with the vortex lattice method and reading books and papers, my conclusion was that, as the lifting line theory has been formulated under simple assumption (wake flatness in the direction of the stream and a a single span-wise vortex at 0.25C), the maximum associated to the elliptical condition is only valid for that model. For free force wakes having 3D shapes, anything could happen. Therefore, in the general case we cannot assure the elliptical condition provides the maximum efficiency. I think it is worth some investigation on that in order to make it clear.
Some serious investigations have already pointed that efficiencies greater than one can actually be achieved, and most text books don't say much about it. I think it is mainly useful to take the elliptical distribution as a reference, although it is actually not a maximum.

Last edited: May 12, 2014
14. May 12, 2014

### A.T.

Potentially relevant:

At 18:50 is a summary of optimal shapes based on constraints.

If I understand the video above correctly, the models resulting in those optimal shapes assume that the entire wingspan is used to generate lift. But if you can afford a large wingspan, then you can use the outer wing to create forward thrust from the vortex created by the inner wing. And that appears to be what birds do.

Last edited: May 12, 2014
15. May 12, 2014

### AlephZero

You seem to be missing the important point that all the theory you are using is approximate. Real wings and airflows don't necessarily follow those approximations.

In a sense, refining one part of the approximation (e.g. using a "better" 3-D flow calculation), but still assuming the flow is inviscid and incompressible, is shuffling the deck chairs on the Titanic, not necessarily giving you a more accurate overall solution to the real flow situation.

Of course the approximate methods you are using are very useful in practice, within their limitations, but I suspect you are trying to read too much into them.

It would be very surprising if the optimum actual shape for a real wing (including the interaction between the wings and fuselage, etc) was exactly an ellipse. But empirically, an elliptical profile is good enough to design a pretty good aircraft - like the http://en.wikipedia.org/wiki/Supermarine_Spitfire

(My bolding!)

16. May 13, 2014

### GuillermoH

Yes, that is another point. We use theories that neglect viscosity to calculate the induced drag. If our model was still perfect, viscosity would even affect that solution. An example of that is what happens on the wing tip: the separation point depends on the boundary layer and therefore on the viscosity. Even on a wind tunnel test it is hard (or even impossible) to separate both sources of drag.
I have lately been comparing results from the vortex lattice method and the lifting line theory on rectangular wings, and they differ in some degree depending on the aspect ratio. To give you an example, wings with aspect ratio equal to 4 resulted in a span efficiency of e = 1.005239 through the UVLM when the wake was only convected from the trailing edge, while the lifting line prediction is e = 0.984671. The difference is about 2.1% (based on the lifting line result), but which one is closer to reality I cannot say, because we are just comparing two different models. I wouldn't pretend to get an extraordinary accurate solution with any of both models, because both have their own deficiencies.

17. May 13, 2014

### GuillermoH

A different affair

The way birds get thrust is a different affair, which involves unsteady aerodynamics. Behind the aerodynamics, for a bird to get thrust it needs to release an amount of energy. The bird does that by putting its own structure into motion (flapping the wings), what requires some power. So there is no "magical" way of suppressing the induced drag and getting a forward thrust in its place.

Last edited: May 13, 2014
18. May 13, 2014

### A.T.

It's not about getting a net forward thrust instead of drag. It's about recovering some of the energy lost in creating the vortex, to create some forward thrust. It's not about propulsion, but efficient gliding and yaw control, without vertical surfaces.

Last edited: May 13, 2014
19. May 14, 2014

### GuillermoH

Well but that is just the point, the energy is not lost. The birds flap their wings, which certainly requires work, and it gets forward thrust in return. I totally agree that the mechanism is quite extraordinary. They actually invert the induced drag, but that requires energy. And that doesn't occur in man-made stiff wings.
Recall that Truffaldino was asking about the induced drag on stiff lifting surfaces and the minimum value associated to an specific model (an elliptical wing, modeled by a lifting line and a flat and straight vortex sheet). We were wondering if that minimum is really the minimum we can get.

20. May 14, 2014

### A.T.

It's not about flapping the wings, but about soaring without flapping. Albatrosses stay airborne for weeks without ever flapping their wings.

It's not about propulsion, but about efficient lift generation. Lift generation is more efficient if it creates less vorticity. The energy in the vortex is an energy loss.

Yes, and the point made in the video is that the ellipse is only optimal if the entire wingspan produces lift. If you drop that assumption other shapes might be better.