Abstract
Inspired by the forward swimming of long-tailed crustaceans, we study an underwater propulsion mechanism for a swimming body with multiple rigid paddles attached underneath undergoing cycles of power and return strokes with a constant phase-difference between neighboring paddles, a phenomenon known as metachronal propulsion. To study how inter-paddle phase-difference affects flux production, we develop a computational fluid dynamics model and a numerical algorithm based on the immersed boundary method, which allows us to simulate metachronal propulsion at Reynolds numbers (RE) ranging from close to 0 to about 100. Our main finding is that the highest average flux is generated when nearest-neighbor paddles maintain an approximate 20%–25% phase-difference with the more posterior paddle leading the cycle; this result is independent of stroke frequency across the full range of RE considered here. We also find that the optimal paddle spacing and the number of paddles depend on RE; we see a qualitative transition in the dynamics of flow generated by metachronal propulsion as RE rises above 80. Roughly speaking, in terms of average flux generation, a tight paddle spacing is preferred when RE is less than 10, but a wider spacing becomes clearly favored when RE is close to or above 100. In terms of efficiency of flux generation, at RE 0.1 the maximum efficiency occurs at two paddles, and the efficiency decreases as the number of paddles increases. At RE 100 the efficiency increases as the number of paddles increases, and it appears to saturate by eight paddles, whereas using four paddles is a good tradeoff for both low and intermediate RE.
Highlights
Long-tailed crustaceans, including crayfish, krill, prawns and lobsters, swim forward by beating four or five pairs of limbs called swimmerets rhythmically through cycles of power-strokes (PS) and return-strokes (RS) [1]
Our results here show that the optimality of the approximate 25% phase-difference between neighboring paddles is robust to changes in Reynolds number, and it is robust to changes in the paddling frequency and the swimmer size
Notable is that the small copepods that paddle at very low Reynolds to large long-tailed crustaceans that operate at much larger Reynolds numbers generally have three to five limbs
Summary
Long-tailed crustaceans, including crayfish, krill, prawns and lobsters, swim forward by beating four or five pairs of limbs called swimmerets rhythmically through cycles of power-strokes (PS) and return-strokes (RS) (see Figure 1) [1]. The natural variation in swimmer size and stroke frequency leads to Reynolds numbers (RE) ranging from about 10 to 1000, in which both viscous and inertial effects are relevant Under this setting, they found that, over RE ranging from 50 to 800, the tail-to-head metachronal rhythm with an approximate 25% inter-paddle phase-differences led to a 60% increase in average flux compared to the in-phase rhythm and a 500% increase compared to the head-to-tail metachronal rhythm. They found that, over RE ranging from 50 to 800, the tail-to-head metachronal rhythm with an approximate 25% inter-paddle phase-differences led to a 60% increase in average flux compared to the in-phase rhythm and a 500% increase compared to the head-to-tail metachronal rhythm They used a simple geometric argument to illustrate that the asymmetric, 25%-period delayed arrangement of neighboring limbs during PS and RS creates an advantage by enclosing a larger volume of fluids underneath the paddles during PS and trapping a smaller volume of fluids during RS (see Figure 4 in [8]). We perform a systematic numerical study to investigate the effects of phase-difference, paddle spacing, and the number of paddles on metachronal propulsion over the RE range of 0.1–100
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