Hi, Victor--
Thanks for your input! And nice work on your oscillating foil craft. It looks like you've done a lot of building there, and I salute you. Oscillating foils do appeal in theory, for sure, but I agree the devil can be in the details.
I'll concede I can't follow the math you reference. But I don't agree with your and the authors' findings, based on general principles and observed trials. I think, for a human-powered craft at least, the goal should be length of travel, not speed of oscillation.
To start with, if you think of fins as imitating either the up-and-down flapping of a dolphin or whale tail, or alternatively as the side-to-side swishing of a shark or tuna tail, I think you're neglecting an important aspect of what those animals are actually up to. As near as I can tell from what I've read, what distinguishes these swimmers from human and mechanical attempts to imitate them is that they rely on an undulating "leading body" to shed alternating vortices (Karman vortex streets) that the tail then interacts with. The animals essentially draw their tails through their own shed vortices at just the right timing and angles that they maximize relative velocities and thrust vectors. They can give their tails a perfectly-timed flick at just the right moment, something like a whip crack, to gain maximum thrust at the optimal orientation. What they're doing is basically this:
Swimming Upstream: Computational Hydrodynamics of Trout Locomotion
It's a fascinating motion and it obviously works very well for them, but I think we can agree that designing and fabricating an appropriately undulating leading body like this is mechanically very difficult. It has not been convincingly done, although it's been tried a few times, that I've seen. And yet if we don't have it, we don't have the shed vortices to work with. We're trying to "flap" a disembodied tail, or to wave a tail (or "tails") behind a rigid body. It tends to give disappointing results. You might get something like this:
Flapping foil propulsion 4
Which in the real world might look something like this:
Oscillating foil test
Or this:
bi-flapper boat experiment
Not real impressive. Yes, there are likely some shed vortices, or at least vortices cast off at the turn, but they're not cast by a "leading" body. By the time the foil makes the turn to try to catch its own vortices and do something with them, they're already disappearing astern. Frustrating.
But what if we just gave up on trying to get "power at the turn," or what you're referring to as the "motion swing?" What if we gave up on leading bodies and undulations and shed vortices? What if we gave up entirely on fishes, and whales/dolphins, and sharks for inspiration?
After all, people are not sharks (well, most of us.) We don't have super-flexible bodies that can deliver powerful, short strokes at high frequencies. But what we DO have, that the fishes DON'T have, is leverage. We are LONG. We have long, rangy, gangly limbs. They don't move fast, at least not back and forth, but they can move FAR, especially when you combine full extension of legs, lean of body, and contraction of arms. THIS:
No shark of comparable size can come remotely close to that "stroke length." She has at least four times the movement. Just look at her!
And when our limbs and body still don't move far enough, we make tools to give us even more leverage and mechanical advantage. Atlatls, for instance. OARS.
How about this guy?
Given that long, low-frequency strokes are what humans are good at, the longer the foils travel, the better. As a single sculler in a racing shell, this guy's oar blade tips have a wetted travel of about 4m on each stroke. Calculate it if you like, I have. Good luck matching THAT, fishies. Twelve feet, for the benighted among us. In about a second, at sprint speed.
So we can't match the frequency of sharks, and we don't have the vortices to double back on. But man, can we do amplitude. To me, that means we should focus not on the turn, but rather on getting as much thrust out of the travel BETWEEN the turns as we can. And make that travel as long as possible.
I think a helpful model is a sailboat. By pulling your foils back and forth along a transverse track, you're essentially trimming your "sails" so that they always present a more or less consistent angle of attack against the flow that they encounter (except at the turn, of course, but let's ignore that for now). You're operating in water rather than wind, and you're "making your own wind" by pulling your foils along their tracks rather than having the wind blow across you. But the principles are exactly the same. Particularly relevant is the principle of "apparent wind"--the forward shift of the wind the sail (or foil) encounters as a result of the boat's forward travel.
Say a boat is going 18.5 knots. The foils are not moving along their tracks at all--they're feathered, just trailing back. How fast are they going? 18.5 knots. Straight forward. If you now pull them transversely across the flow that is parallel to the boat's travel, at ANY speed, at ANY angle of attack below stall, you will generate lift (thrust). You do NOT turn. You do NOT "flap." You do NOT sweep. Focus only on pulling the foils at a rigid angle, transverse to the boat. Set them at the angle you want, and pull them straight out from the boat, perpendicular to its line of travel. Keep on going for as long as you can, until you are forced to turn around ("tack"). Try not to lose to much power when you do tack. Then do the exact same thing to go back to where you started. Foil angle fixed. No radial component to the travel.
Thrust may not all be directed forward, and in practice it won't--it'll be mostly athwartships, like a close-hauled sailboat heeling over. Only some (usually fairly small) component of thrust will be directed forward. You are also generating a lot of drag, and it may be that the added drag outweighs the added thrust, or that the thrust vector is oriented so nearly athwart that it isn't contributing to any added forward motion (akin to a sailboat being "in irons"). But the important thing here is that I am NOT looking at the turn--what you're calling a "motion swing." I don't care about the swing. If I could, I'd avoid it altogether.
But if travel is what it's all about, why don't we just stick with a conventional rowing configuration? There is, after all, well over a thousand years of tradition behind it, and a good century and a half of history of sliding seats. Sliding riggers, which are just a bit better, had a brief flourishing about 50 years ago. A good rowing shell powered by an elite athlete in a sprint really is pretty fast. Is there any room for further improvement?
I submit, yes. For instance, here is an "Aquaskipper" over 100m:
He's doing about 12 knots, about equal to the rower. CERTAINLY the most impressive oscillating foil configuration I've seen. Very similar, actually, to the contemporary fad of pump foiling:
https://youtube.com/shorts/TqaEfH4DlGc?si=m5JhPcg_eAcfFiRL. Some claim around 14 knots, although I haven't seen evidence. This guy might be close. Both are clearly starting to get close to my goal of 18.5 knots. And neither is anywhere near 5 cycles per second, so that's obviously just baloney. Equations to not determine the world, it's the other way around. Both are doing maybe doing 2 cycles per second, if that. And certainly nowhere close to 4m per stroke. More like .5. Is there something in between, perhaps?
I submit, yes--a rowboat. But a better rowboat. I believe that the secret is, a rowboat already IS an oscillating foil propulsion system. Even a conventional one. It's a pretty bad one, but that's exactly what it is, in effect if not appearance. I can go into this into great detail, if you need convincing. And unlike those aquaskippers and pump foils, it doesn't waste energy bobbing the pilot up and down, and it uses leverage (long oarshafts) to get twice, three times, four times the travel. Its only problem is that its "foil" (blade) travel, while incredibly long, is absolutely all over the place. Big, radial sweeps. Its angle of attack is appalling. It's at stall a good half the time. Its thrust vectors spend a lot of the time athwartships, out to sea, up, and down. It's embarrassing. That's what I want to get at. I want a better rowboat, AND a better aquaskipper/pump foil. Combine the advantages of each, and throw out their deficiencies. Forget the fishies--we are land animals with a completely different anatomy and its attendant set of both advantages and disadvantages. Stop trying to imitate something we're not, and start with a clean slate.
There's a lot more to it, but this is already getting pretty long. Still, I hope I've convinced you 5 cycles per second is nonsense. That is not a barrier. More to come, if you're interested.
.