The Myth of Aspect Ratio

Thanks for the link and clarification. It makes sense.

 

This thread is a good example of thread drift. Very interesting discussion but somewhat removed from the original topic.

 

A positive and constructive thread drift, I'd say.

 

XFLR5 had problems when I tried it a while ago. I'll try again.
I have written several VLM's myself and most had problems with curved leading-edges and/or trailing-edges. Maybe XFLR5 has improved with those cases now. The circular planform case is a tough one for most panel methods and VLM, but it is important because there is an exact solution for the lift, moment and behaviour of the loading at the wingtips.

I don't use Matlab myself, so Tornado is out.

Thanks,
Leo.

 

I suspect that problems might arise at the outer portions of the wing, where a leading-edge vortex (due to very high leading-edge sweep "seen" by the flow) would play a major role in lift generation. If the numerical method used for the analysis doesn't have means to predict this behaviour (like leading edge suction analogy), it will be necessarily flawed. The VLM implemented in XFLR5 is a linear one, with no leding-edge vortex simulation. So I do believe that it cannot model the circular wing correctly.

However, Id' like to give it a try. Could you please give me some reference (possibly a free one) where a theoretical solution for a circular platform wing is given? I'll then try to compare results with XFLR's output.

Cheers

 

The content has been great. But it's not obvious to anyone looking at the title of the thread that it contains good information about the applicability of aerodynamic theory, etc. I would have prefered to see another thread started with an applicable title.

 

Briefly back to the point of the original post. If span/draft of a keel or rudder is fixed and the amount of lift/lateral force is fixed, the induced drag (as in the portion of drag due to trailing vorticity) is independent of aspect ratio to first or second order. Therefore the aspect ratio and planform shape should be selected based on other considerations.

As far as I'm aware this conclusion is valid irrespective of the details of the keel or rudder sections, and doesn't not depend on any particular airfoil theory.

 

But the conclusion is based on formulas valid for moderate to high AR keels. Before declaring it universal it has to be checked for the case of low AR planforms - where some other, non-linear, phenomena happen. Like, for example, the said leading-edge vortex-generated lift and the thickness effect (the dependence of induced drag on foil thickness).
And the low AR keels and rudders are very often found on live-aboard sailboats.

 

Agree that leading-edge vortex-generated liftand the resulting drag is beyond the applicability of the formulas used.

The general derivation for the lift and induced drag of a lifting surface doesn't require any assumptions about the magnitude of the aspect ratio, only that the surface meet the thin lifting surface assumptions. It does require that the flow is attached on the leading edge. See Marine Hydrodynamics by Newman (1977), section 5.10 "Induced Drag". The induced drag to lift squared ratio is shown to be a function of the shape of the distribution of circulation only. So it's valid for a low aspect ratio surface as long as there isn't separation on the lead edge. While Newman doesn't discuss it concentrated tip vortices can be handled by the use of generalized functions.

Lifting line theory does require suitably large aspect ratio, and lifting line theory is sometimes used in discussions of induced drag. Lifting line theory also provides the (relatively) simple formulas relating lift to planform shape, and these formulas can then be used to relate planform shape to induced drag. But, lifting line assumptions are not required to derive induced drag as a function of lift.

Also of interest are both experimental results and theory which show that planform shape becomes less of a factor for induced drag as the aspect ratio decreases. See the references I mentioned previously. Again, this is with flow attached at the leading edge. This suggests that for relatively low aspect ratio keels and rudders the planform shape should be selected based on considerations other than induced drag.

 

Applied mathematicians are cocooned in a bubble bath applying their own fudge factors to indeterminate boundary layers

Somewhere in all of this comes a healthy dose of reality where sub layers are dwarfed by surface roughness and even super smooth foils get linear flow for around 300mm before they trip, and that's for a foil that's only put in the water briefly between rigorous dry land polishing sessions.

The realities for even super smooth sailboat keels is that they will be in full turbulent flow most of the time.

 

Once it has been expressed as a mathematical equation, the application is irrelevant to some of us. It's just another equation that needs computing quickly and accurately.

 

I have attached a small compendium of results for some "standard" planforms that might be of interest to those in this thread. (Some graphs in this draft are without comments yet, but they are fairly self-explanatory in most cases).

Theoretical, experimental and calculated results for the circular wing are in the section on Elliptical Planforms. I'd be interested to see how results for some other codes if anyone has some lying around and has spare time.

I'll put the computer program I used to create the results in the Software Forum when I finish the manual. (I hate writing manuals).

Leo.

 

That looks like a good job, Leo. And I see that you have already included the leading-edge suction analogy, so it's useless to compare the results with the "linear" XFLR. Can't wait to see your software on-line. I have a feeling that it will follow the faith of Michlet and will become a benchmark in the field of low-AR wings, keels and rudders.
One question, if I may - can it deal with multiple lifting surfaces?
Cheers

 

Since the talk about Aspect Ratio is interesting but (all in all) pretty limited one, and the side-drifts are getting more and more numerous and interesting, maybe it would be appropriate to ask Jeff (the Admin) if he could change the name of the thread? It is becoming a discussion about general aero/hydrodynamics theory applied to sailboats, so perhaps a title should reflect this continuously developing content? What do you think DCockey (after all, it's your thread... ) ?

 

Good suggestion to change the name of the thread, though I would prefer if this didn't become another 40 page long thread on diverse and only sightly related topics.