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:
https://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.