The mechanism behind stable aggregations of active swimmers is not fully understood. In particular, the in-depth quantitative explanations are notably lacking. To address this, a vorticity-based force expression is proposed to study the collective behaviors of two two-dimensional tandem flapping plates. The hydrodynamic force is directly related to the generation of vorticity on the plate and viscous forces resulting from its subsequent evolution. For the first time, the physical process by which the interactions between the rear plate and the wake vortices of the leader affects the hydrodynamic force is quantitatively elucidated and not based on simplified theory. The wake vortices of the leader influence the hydrodynamic force on the rear plate indirectly by inducing an additional oncoming flow. The flow affects the generation of vorticity, which mainly occurs on the head of the rear plate. The results show that, if the trajectory of the rear plate passes through the vortex cores, vorticity generation on the rear plate is suppressed and the thrust effect is weakened. If the rear plate slaloms between the vortices, the vorticity generation and the thrust effect are enhanced. We also found that the wake vortices of the leader has a certain ability to trap the rear plate into orderly configurations—no matter the rear plate has a dissimilar flapping amplitude or is applied an external horizontal loading—by adjusting the equilibrium position of the rear plate in it. The findings may shed some light on the understanding of collective behaviors in swimming animals.

The propulsive performance and vortex dynamics of a two-dimensional model for the jellyfish-like propulsion with burst-and-coast strategy are investigated using a penalty-immersed boundary method. The simplified model comprises a pair of pitching flexible plates with their leading edges connected. The effects of two key parameters are considered, i.e., the duty cycle (DC, the ratio of the closing phase to the whole period) and the bending stiffness (K). Three different wake patterns, i.e., periodic symmetric, periodic asymmetric, and chaotic wakes, are identified in the DC–K plane. Numerical results indicate that a significant fast-close-slow-open motion is more likely to achieve higher speed, efficiency, and stability than a slow-close-fast-open motion, and proper higher bending stiffness is conducive to improving efficiency. A force decomposition based on the weighted integral of the second invariant of the velocity gradient tensor is performed to gain physics insight into the self-propulsive mechanism. It is found that the repulsive force induced by the strain-rate field between the body and the previous vortex pair is the main driving force of the jellyfish-like motion and that capturing the previous vortex pair during the closing phase can significantly enhance the strain rate as well as the thrust. This clarifies why the jellyfish can achieve thrust by pushing back vortex pairs. This study provides inspiration for the design and control of flexible jet propulsion devices.

Intermittent swimming and schooling individually offer the possibility of energy economy for fish. To study the role of the simultaneous use of both behaviors, the intermittent swimming of two self-propelled plates in tandem configuration is investigated numerically. Two intermittent swimming modes, i.e., the multiple-tail beat (MT) mode and the half-tail beat (HT) mode, are considered. For solitary plate swimming intermittently while maintaining fixed bout period, the propulsion velocity and energy consumption decrease monotonically as the duty cycle increases, consistent with the prediction of the reduced-order model. A pair of plates swimming with independent duty cycle can form orderly configurations, without separating or colliding, when their duty cycles are similar. For the MT mode, the asymmetric wake of the leader enhances the drag on the following plate by inducing an additional oncoming flow against it, making the follower harder to follow. For the HT mode, the symmetric wake of the leader reduces the drag on the follower, making it easier to follow. No significant difference was found in the propulsive performance of intermittent swimming between the leading plate in orderly configurations and solitary plate. The results of efficiency indicate that the HT mode is more economical than the MT mode for the follower. Usually, the smaller the equilibrium gap spacing is, the more efficient the follower is. The results of the cost of transport show that the follower achieves better energy economy for higher propulsion velocity. The results provide some insight into the collective intermittent swimming of fish and are helpful for bionic design.

The self-organization of active swimmers is interesting but not fully understood. Lighthill conjectured that the orderly configurations may emerge passively from the hydrodynamic interactions rather than active control mechanism. To further test Lighthill's conjecture, the effect of active control on the propulsive performance of two self-propelled flapping plates in tandem configuration is studied. Different types of external horizontal forces are applied at the leading edge of the following plate. It is found that the collective dynamic and propulsive performance of the two-plate system are mainly affected by the mean value of the external horizontal force rather than its specific form. The two-plate self-propelled system has certain ability to counteract a limited external intervention and maintain the orderly configuration by adjusting the gap spacing between two plates. For a stable configuration, the external intervention hardly affects the propulsion velocity but has a significant monotonic effect on the gap spacing and input work. Further, a simplified model is proposed to relate the external horizontal force to the gap spacing between two plates and verified to be reliable by the numerical results. Moreover, the momentum and energy transferred to fluid are investigated in terms of local vortical structures. It is revealed that the impulse of the wake vortex pair is hardly affected by the external horizontal force, while its kinetic energy and the local dissipative energy vary monotonically with it. These results may shed some light on the understanding of collective behaviors of living swimmers and robotic fish.

Intermittent swimming behaviour is commonly observed in larval zebrafish, often attributed to energy-saving mechanisms. In this study, we utilize a hybrid approach combining deep reinforcement learning and the immersed boundary–lattice Boltzmann method to train a larval zebrafish-like swimmer to reach a target with minimized energy expenditure. We find that when the tail-beat period is fixed, continuous swimming emerges as the optimal strategy. However, when the tail-beat period is allowed to vary, intermittent swimming proves superior in energy performance, achieved through reductions in tail-beat amplitude and frequency. Our detailed analysis reveals that intermittent swimmers employ rapid backward tail flicks to attain high speeds, coupled with slower forward tail flicks and coasting phases to conserve energy. Furthermore, we derive scaling laws governing the swimming performance of trained fish. These results offer valuable insights into the intermittent swimming patterns of fish, with implications for understanding bio-inspired locomotion and informing the design of energy-efficient aquatic systems.

Dynamic mode decomposition (DMD) is a widely used method to extract dynamic information from sequential flow data, aiding our comprehension of fluid dynamics and transport processes. While DMD can unveil internal system laws and predict unsteady flow phenomena, the connection between DMD modes and the nonlinear hydrodynamic behavior of solid bodies remains unexplored. This study investigated the internal relationship between DMD modes and their impact on hydrodynamic forces. We employed a penalty-immersed boundary method to simulate the behavior of a flapping flexible plate in a uniform incoming flow, generating extensive datasets of vorticity fields. By applying DMD to these datasets, we identified key modes governing the flow dynamics, including the shear layer, symmetric vortex street, and antisymmetric vortex street. Furthermore, we utilized the impulse theory to analyze the force characteristics of the plate based on the corresponding DMD modes. The net force is determined by the combined contributions of the impulse force and the vortex force. Our findings reveal that the net horizontal force is primarily influenced by the first two modes. Specifically, mode 1, characterized by a dimensionless frequency of f*=0, contributes to thrust, whereas mode 2, with f*=1, contributes to drag. This physical investigation holds relevance for fluid–structure systems involving the interaction dynamics of flexible structures with unsteady wake vortex systems.