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  1. Free, publicly-accessible full text available July 12, 2024
  2. In this study, we numerically investigate the effects of the tail-beat phase differences between the trailing fish and its neighboring fish on the hydrodynamic performance and wake dynamics in a two-dimensional high-density school. Foils undulating with a wavy-like motion are employed to mimic swimming fish. The phase difference varies from 0° to 360°. A sharp-interface immersed boundary method is used to simulate flows over the fish-like bodies and provide quantitative analysis of the hydrodynamic performance and wakes of the school. It is found that the highest net thrust and swimming efficiency can be reached at the same time in the fish school with a phase difference of 180°. In particular, when the phase difference is 90°, the trailing fish achieves the highest efficiency, 58% enhancement compared with a single fish, while it has the highest thrust production, increased by 108% over a single fish, at a phase difference of 0°. The performance and flow visualization results suggest that the phase of the trailing fish in the dense school can be controlled to improve thrust and propulsive efficiency, and these improvements occur through the hydrodynamic interactions with the vortices shed by the neighboring fish and the channel formed by the side fish. In addition, the investigation of the phase difference effects on the wake dynamics of schools performed in this work represents the first study in which the wake patterns for systems consisting of multiple undulating bodies are categorized. In particular, a reversed Bénard–von Kármán vortex wake is generated by the trailing fish in the school with a phase difference of 90°, while a Bénard–von Kármán vortex wake is produced when the phase difference is 0°. Results have revealed that the wake patterns are critical to predicting the hydrodynamic performance of a fish school and are highly dependent on the phase difference.

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  3. Abstract

    For a Legendrian link with or , immersed exact Lagrangian fillings of can be lifted to conical Legendrian fillings of . When is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom.19(2021), no. 3, 635–722], for each augmentation of the LCH algebra of , there is an induced augmentation . With fixed, the set of homotopy classes of all such induced augmentations, , is a Legendrian isotopy invariant of . We establish methods to compute based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math.374(2020), 107348, 71 pp.] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary , we give examples of Legendrian torus knots with distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when and ,every‐graded augmentation of can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of ‐graded augmented Legendrian cobordism.

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  4. Numerical studies are presented on the propulsive performance and vortex dynamics of multiple hydrofoils pitching in an in-line configuration. The study is motivated by the quest to understand the hydrodynamics of multiple fin–fin interactions in fish swimming. Using the flow conditions (Strouhal and Reynolds numbers) obtained from a solitary pitching foil of zero net thrust, the effect of phase differences between neighboring foils on the hydrodynamic performance is examined both in position-fixed two- and three-foil systems at Reynolds number Re = 500. It is found that the threefoil system achieves a thrust enhancement up to 118% and an efficiency enhancement up to 115% compared to the two-foil system. Correspondingly, the leading-edge vortex (LEV) and the trailing-edge vortex (TEV) of the hindmost foil combine to form a ‘2P’ wake structure behind the three-foil system with the optimal phase differences instead of a ‘2S’ wake, a coherent wake pattern observed behind the optimal two-foil system. The finding suggests that a position-fixed three-foil system can generate a ‘2P’ wake to achieve the maximum thrust production and propulsive efficiency simultaneously by deliberately choosing the undulatory phase for each foil. When increasing Reynolds number to 1000, though the maximum thrust and propulsive efficiency are not achieved simultaneously, the most efficient case still produces more thrust than most of the other cases. Besides, the study on the effects of three-dimensionality shows that when the foils have a larger aspect ratio, the three-foil system has a better hydrodynamic performance, and it follows a similar trend as the two-dimensional (2D) foil system. This work aids in the future design of high-performance underwater vehicles with multiple controlled propulsion elements. 
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  5. Numerical simulations are employed to study hydrodynamic interactions between two-dimensional fish-like bodies under a traveling wavy lateral motion in high-density diamond-shaped fish schools. This study focuses on two different streamwise spacings, a dense school with 0.4 body length (BL) spacing and a sparse school with 2.0 BL spacing, respectively. An immersed-boundary-method-based incompressible Navier–Strokes flow solver is then employed to quantitatively simulate the resulting flow patterns and associated propulsive performance of the schools. The results suggest that a fish in the dense school achieves higher thrust production and higher propulsive efficiency than that in the sparse school due to a strong wall effect from neighboring fishes. In addition, results from changing the lateral spacing in the dense school have shown that the wall effect is enhanced as the lateral spacing decreases. Flow analyses have shown that the wake pattern of the fish swimming diagonally behind the leading fish in a dense diamond-shaped school transfers from 2S to 2P when the lateral spacing is smaller than 0.6 BL. As a result, an angled jet is produced behind the school and brings more momentum downstream. At the same time, the appearance of the trailing fish results in a stronger pressure region behind the leading fish and leads to a higher hydrodynamic performance of the leading fish in the dense school. The insights revealed from this study will contribute to understanding physical mechanisms in fish schools and providing a new swimming strategy for bio-inspired underwater swarm robots. 
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  6. null (Ed.)
  7. Acu, Bahar ; Cannizzo, Catherine ; McDuff, Dusa ; Myer, Ziva ; Pan, Yu ; Traynor, Lisa (Ed.)
    Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known ``elementary'' building blocks for Lagrangian cobordisms that are smoothly the attachment of 0- and 1-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is ``decomposable'' into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms. 
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