Swim like an animal
Propellers are only one answer. Animals had several hundred million years' head start, and when engineers finally measured what fish and jellyfish do, they found a single number that nearly all of them share.
In 1936 the British zoologist James Gray did a calculation that seemed to prove dolphins couldn't exist. He took their observed speed, estimated the drag on a rigid body that shape, and worked out the muscle power needed. The answer was far beyond what he believed dolphin muscle could give. "Gray's paradox" sent engineers hunting for a hidden trick in dolphin skin. Better measurements decades later mostly dissolved it: dolphins are simply much stronger than Gray assumed. The hunt itself, though, produced a field.
In the 1990s at MIT, a team led by Michael Triantafyllou built RoboTuna, a robotic fish with a flexible body driven by cables and pulleys, to measure what a swimming body does to the water around it. The answer was in the wake. A fish's tail sheds a staggered street of swirling vortices, with each swirl pushing a jet of water backwards. How well that jet is formed depends on a single ratio: how often the tail beats, times how far it sweeps, divided by how fast the fish is going. Across many animals that swim and fly, cruising values of this ratio fall in a narrow band.
Jellyfish found a different answer. In 2013, researchers measuring moon jellies reported one of the lowest energy costs for getting around of any animal ever studied. Part of the trick is that after each pulse the bell relaxes, and the swirl of water left underneath gives it a free second push. Robots now borrow both ideas: fins that flap at the right rhythm, and soft bells that pulse. The jellyfish in this site's Builder runs on that pulse.
The Strouhal number
The ratio. The Strouhal number is St = f × A ÷ U: tail-beat frequency times the width of the tail's sweep, divided by swimming speed. It measures how hard the tail is working compared with how fast the body is moving.
The sweet spot. Below about 0.2, the tail barely disturbs the water, the vortices don't form a jet, and thrust is feeble. Above about 0.4 the tail thrashes, throwing energy into swirls that point every which way. In between, the vortices line up into a clean backward jet. Cruising dolphins, sharks, birds, bats and insects all tend to land between 0.2 and 0.4. Evolution kept finding the same efficiency peak.
Why robots care. A propeller spins steadily. A fin or bell works in strokes, so it can start and stop and turn quickly, it's quiet, and it's gentle on anything it brushes. That matters near reefs, wrecks and animals. The cost is control complexity: every stroke is a little burst of flow you have to time. That's why biomimetic vehicles are still mostly research machines, and why the ones that work are worth studying.
House style. Soft, pulsing bodies are our signature, because a vehicle that's about as stiff as the water around it barely notices depth (Tuesday's Comply), and one that swims by pulsing can hover over a wreck without blasting silt off it. Next week the vehicles stop feeling their way and start to see, with sound.
A robotic fish holding steady speed in a flume. Change how fast the tail beats and how wide it sweeps, watch the wake re-form behind it, and steer the Strouhal number into the band where the jet goes clean.
Fly to Monterey Bay, above the submarine canyon where researchers have filmed deep-sea jellies with ROVs since the late 1980s, and where much of what we know about how they swim was first seen.