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Eliptical wings

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HLundberg

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Hello fellow rocketeers!
Am new to this forum and already have a question I would like to find an answer to:
Does anybody know if there is an ideal with to length ratio for eliptical wings as to drag performance?
 

powderburner

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First of all, welcome to TRF! We hope you come back often, and join in the discussions.

Are you asking this qstn for help in designing a rocket for competition, and absolute maximum altitude? Sport models (non-competition) usually need more robust fin designs, and the following comments do not apply to them.

If you are looking at using an elliptical fin pattern because you think it will save drag or optimize lift, you have to realize that you really only get these benefits if:
1) there is “clean” airflow onto the fin/wing (no turbulence landing on the fin leading edge due to upstream launch lugs, or body joints, or front fins, or....)
2) the thickness of the fin/wing also follows an elliptical spanwise distribution, so that the local airfoil thickness-to-chord ratio remains constant or decreases slightly.
3) the fin leading edge is not only round but very smooth, as in: you have a LOT of work to do to seal, smooth, and finish the fin until it is so shiny that you can begin to see yourself in the reflection
4) likewise, the fin sides are also very smooth and very well-finished, and don’t forget:
5) the fin trailing edge thickness is tapered smoothly down over the aft 50 to 60 percent of the chord to a paper-thin trailing edge---that will splinter on the first or second landing
6) all this fussing over “perfect” aerodynamics kind of falls apart at the extremely low Reynolds numbers where our model rockets typically operate, and all bets are off (the wings of many insects are flat planks, with no camber, and with protruding veins and hairs and other surface roughness, the exact opposite of what classical aerodynamics would indicate is needed)

Wait, there’s more: a properly-designed elliptical fin shape will still only give you this “optimum” aero performance at one condition: a near-zero angle of attack. As soon as the rocket turns and the fin begins to experience any significant (non-zero) angle of attack, there will be airflow distortions at the root and tip, which in turn will influence the subsonic airflow everywhere between. Your optimum aero quickly becomes distorted to something else.

Hoerner’s book (Fluid Dynamic Lift) indicates that you can get most (99 percent?) of the aero benefit of an elliptical planform by using a taper ratio of around 0.4. What’s a taper ratio? It is the ratio of tip chord to root chord, for a trapezoidal fin planform with straight leading and trailing edges. Such a fin is much, much, easier to physically make (ever actually try to make a proper elliptical fin, where every airfoil from the root to the tip is different?). This website is not Hoerner’s book but has a very similar plot of aerodynamic efficiency versus taper ratio:
http://www-scf.usc.edu/~tchklovs/Proposal.htm
Scroll down a bit into the beginning of the paper to Figure 1-6 and see how closely a taper ratio of 0.5 approximates the spanwise lift distribution of an elliptical shape. You can probably see that a taper ratio of 0.4 would fit even more closely.

Remember that you still need to taper the thickness of trapezoidal-shaped fins for optimum aerodynamics (but this is a lot easier to do for a linear taper). If the tip chord is only 40 percent of the root chord, the tip thickness also needs to be 40 percent (or slightly less) of the root chord. And you will still need to put a nice rounded leading edge across the full span, and a nice trailing edge thickness that is tapered down to a paper-thin edge. (Imagine your fin airfoil section to look like a stretched-out teardrop shape.) And you will still need to create a very smooth surface, with filled wood grain, primer-finished exposed surfaces, and mirror-smooth paint jobs.

The way to compare fins is to hold fin planform size constant (to try to keep drag constant) and to vary the fin shape by extending the span. If you keep fin planform area constant, and increase the span (and reduce the chord as the span increases) you get a more aerodynamically effective fin design. Note that the Barrowman equations (for rocket stability) do not give you any credit for a more efficient fin shape. In real-world aerodynamics, a longer fin with shorter chords is somewhat more aerodynamically efficient due to reduced tip airflow losses---if it stays in one piece, and if it does not twist or wobble under the airloads of oncoming airflow. These longer fin shapes are said to have higher “aspect ratio” shapes;

AR = (span2) / (area)

Fins with higher aspect ratios are better, but if you push the span too far you quickly run into problems.

Fins with very low aspect ratios (long chord, short span) begin to function aerodynamically in ways different than fins/wings because low AR fins introduce three-dimensional airflow effects that are not usually beneficial.

You can make the span of your fin just as long as you want---until you make it so long that it experiences aerodynamic flutter (to the point of structural failure). Unfortunately, we do not have any analytical means of predicting the key airspeed when a balsa fin will “blow up”, but for any single fin material, these are the main factors in how long, thin fins survive flight forces:
-- fin thickness; more thickness is better for stronger fins, but begins to add a little drag again; thickness would be added mostly at the root and less thickness would be added outboard
-- fin leading edge sweepback: moderate angles make for more survivable fins, but highly swept fins begin to experience a whole different aero effect (boundary layer buildup and spanwise flow effects combine to reduce fin aero efficiency); unswept fins need to be kept a little shorter
-- rocket speed: higher speeds mean higher airloads, and if you find the “speed of balsa” the event usually occurs quickly, dramatically, and totally (you usually don’t get a rocket back with partial damage; the fins will just plain be gone)
-- rocket mass versus fin mass; there can be some pretty squirrely dynamic interactions and 6DOF effects when fins start trying to flutter; trying to “tune” these masses requires a whole lot of high-powered computational methods and a bank of Crays, and even then still requires a healthy dose of dumb luck
-- fin root attachment; the structural configuration and the adhesives used may play a small role in dissipating some of the dynamic energy of fin flutter---or may aggravate and concentrate the loads
-- use of mass-balance booms (projecting forward of the fin); this goes way beyond anything that I have seen any model rocketeers experiment with

Are you still awake? The shorter version of all this would be to take some time to study the fin designs and building techniques for other successful competition models, and try to understand why they work. If you start out with those approaches and experiment with small changes, you might just come up with something that works well for you. Elliptical shapes might not be the best for practical reasons, but they are still pretty close to the best shape and some people just think they look cool.

You can search the old TRF 1.0 threads for more comments on optimum fin shape, or you can search the web for discussions of wing efficiency factors, induced drag, or optimum planform shapes.
 
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hardinlw

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The minimum induced drag, the drag due to producing lift, is obtained with an elliptical lift distribution. This can be achieved at a single angle of attack by a combination of wing taper and twist. Airplanes are often designed this way to approximate an elliptical lift distribution at cruise. It is also possible to obtain an elliptical distribution at ALL angles of attack by using an elliptical planform like the British Spitfire fighter of WWII. This is hard to manufacture and is not often used for that reason.

For any wing (or fin) planform, the induced drag goes down as you increase the aspect ratio (span to chord), but there comes a point where you are only getting tiny additional benefits. Furthermore, the long skinny wings are prone to flutter and may need strengthening which drives up weight. Some sailplanes have an aspect ratio on the order of 40:1, but I doubt that you'll ever see a rocket with 40:1 aspect ratio fins because of flutter problems.

At low angles of attack, there is little induced drag and it just doesn't matter. The only time a stable rocket is not flying at a low angle of attack is just coming off the launcher in windy conditions. Once it finishes weathercocking (basically aligning itself so that it is at zero angle of attack) there is little induced drag. All you are left with is the parasite drag (drag due to skin friction, etc., that does not result from producing lift) which is minimized by smooth finish, rounded leading edges, and a sharp trailing edge. In fact, if you are not super-finishing the rocket, the parasite drag will probably be so much more than the induced drag that you will never see a benefit of the elliptical planform.
 

powderburner

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In fact, if you are not super-finishing the rocket, the parasite drag will probably be so much more than the induced drag that you will never see a benefit of the elliptical planform.
This is so true, and so many of us have a cockeyed, over-inflated opinion of our own finishing skills and our ability to achieve a "super" surface finish. My wise-crack about being able to see yourself reflected on the surface is actually not far off the mark. You should also be able to drag your fingernail down the length of the rocket without being able to detect (eyes closed) the joint between the NC and BT.

For extreme competition flying, you leave off the launch lug and use a tower to guide the rocket's first four or five feet. You feather the aft end of the body tube down to (or close to) zero thickness where it wraps around the aft end of the motor.

You do all this crazy cr$p and you might get another five or ten feet of altitude out of your rocket (OK, maybe 20 feet).
 

JStarStar

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This is so true, and so many of us have a cockeyed, over-inflated opinion of our own finishing skills and our ability to achieve a "super" surface finish. My wise-crack about being able to see yourself reflected on the surface is actually not far off the mark. You should also be able to drag your fingernail down the length of the rocket without being able to detect (eyes closed) the joint between the NC and BT.

For extreme competition flying, you leave off the launch lug and use a tower to guide the rocket's first four or five feet. You feather the aft end of the body tube down to (or close to) zero thickness where it wraps around the aft end of the motor.

You do all this crazy cr$p and you might get another five or ten feet of altitude out of your rocket (OK, maybe 20 feet).
And then, your actual performance may end up being dependent on motor manufacturing variations (NAR standards do allow nominal variation in thrust and total impulse) -- whether you're lucky enough to get a "hot" motor or not. (I know some of the real competition gurus weigh their motors down to tenths of grams before launch, trying to select the heaviest one. But of course you might go with the heaviest motor, and then find out it's carrying an extra gram of delay element, not propellant. :eek: )

Others (especially powderburner) have explained the issue in much greater technical detail, but almost all the sources I have seen maintain there is little operational advantage to elliptical fins over simpler shapes such as the clipped delta or trapezoidal.

They do look cool, though. ;)
 

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