This work presents a predictive framework for energy harvesting of two tandem and staggered flapping foils based on four canonical modes of vortex–foil interaction. The role of the incoming vortex generated by the leading foil in modulating the hydrodynamic load of the trailing foil is systematically analysed. Four canonical interaction modes are classified by the vortex rotation and its interaction position relative to the leading-edge vortex (LEV). The most effective configuration occurs when the foil encounters a counter-rotating vortex on the pressure side, which strengthens the LEV and consequently enhances the lift magnitude, with maximum efficiency achieved when vortex merging occurs near stroke reversal. A second constructive mode occurs when a co-rotating vortex on the suction side promotes LEV roll-up through favourable induced velocities. Force decomposition reveals that in both constructive modes, the incoming vortices improve the efficiency of the trailing foil by enhancing the unsteady lift through altering the local velocity to strengthen the LEV or promote its roll-up, while their low-pressure cores contribute marginally to the unsteady force. Two destructive modes are also observed: direct interaction of a counter-rotating vortex on the suction side leads to only a transient lift increase; a co-rotating vortex on the pressure side reduces the effective angle of attack and leads to the poorest performance. Building on these insights, a mechanism-based predictive framework is established to rapidly identify high-performance configurations without exhaustive parametric exploration. The framework applies broadly to different wake conditions and trailing-foil kinematics and guides the design of multi-foil energy-harvesting systems.

We experimentally study a scallop-like swimmer with reciprocally flapping wings in a nearly frictionless, cohesive granular medium consisting of hydrogel spheres. Significant locomotion is found when the swimmer’s flapping frequency matches the inverse relaxation time of the material. Remarkably, the swimmer moves in the opposite direction compared with its motion in a cohesion-free granular material of hard plastic spheres. At higher or lower frequencies, we observe no motion of the swimmer, apart from a short initial transient phase. X-ray radiograms reveal that the wing motions create low-density zones, which in turn give rise to a hysteresis in drag and propulsion forces. This time-dependent effect, combined with the swimmer’s inertia, accounts for locomotion at intermediate frequencies.

Flow around a submerged cylinder near a free surface reveals that adjusting the Froude number and gap ratio influences the underwater jet pattern, vortex shedding frequency and free-surface deformation. The jet typically separates near the trough, leading to vorticity concentration and breaking waves that dissipate wave energy. Antarctic orcas collaborate to generate deep depression waves, breaking ice and washing seals from floes. Orcas raise their heads and tap their tails downward when approaching ice, which may benefit strong wave generation. We investigate the wave-generating hydrodynamics using a towing tank and particle image velocimetry. A scaled model with an elliptical body and wedge-shaped tail was tested under Froude number similarity. Experiments covered towing speeds of $0.3- 0.7\,\textrm{ms}^{-1}$, combining different body ($10^\circ$/$0^\circ$/$-10^\circ$) and tail angles ($30^\circ$/$0^\circ$/$-30^\circ$), at chord-based Reynolds numbers of $17\,030- 40\,506$. Four wake regimes are identified: small-scale vortex emergence triggered by capillary waves; extensive wave breaking due to flow separation at the trough; smooth depression wave caused by jet reattachment and downward advection of wake vortices; and large-scale vortex impingement generated by wake vortex perturbations. Under the pitched posture, the jet attaches successively to the solid surface and the trough via the Coandâ effect, suppressing flow separation, creating the most pronounced wave. The strong jet maintained a low-potential-energy state of the wave and led to large ice floes flipping and fracturing through the bending effect, while smaller ice floes were overwashed. This study suggests a novel flow-control strategy for objects near the free surface through jet attachment.

Birds extend their flight envelope and adapt to time-varying aerodynamic demands by actuating the shoulder and wrist joints to morph the local sweep angles of the inner and outer wings. However, the local sweep morphing may cause unfavourable unsteady lift on the lifting surface. We investigate these unsteady lift responses using an avian-inspired wing with local sweep morphing under different morphing strategies. The unsteady lift is computed through numerically solving the incompressible Navier–Stokes equations. The results show that the local sweep morphing wing experiences a substantial maximum lift overshoot when forward sweeping and a notable maximum lift undershoot when backward sweeping. These lift over/undershoot phenomena can be alleviated by three measures: adopting smooth nonlinear morphing kinematics, initiating morphing from lift extrema opposite to the over/undershoot direction and prolonging the morphing duration. For the lift over/undershoot, the component obtained by subtracting the extended pre-morphing lift is attributed to the modification induced by local sweep morphing. To predict such lift over/undershoot, we develop a reduced-order unsteady model for the sweeping-modification lift coefficient, where Prandtl’s lifting-line theory is extended to the regime of variable translational velocity. The proposed model captures the symmetry of the sweeping-modification lift coefficient, dominated by the horizontal morphing velocity. Additionally, the vertical morphing velocity is found to correlate with the asymmetry of the sweeping-modification lift coefficient by modulating the leading-edge vortices. This study is expected to improve the understanding of surging-wing flow physics and support the design of bio-inspired multifunctional aircraft.

Because of the high dimensionality and geometric complexity of the circular-pipe problem, formulating and implementing boundary conditions are challenging, and most existing theoretical studies either neglect boundary effects or impose purely specular-reflection boundary conditions. To address this gap, we devise and explore an analytical model for microswimmer dispersion in a cylindrical pipe flow under a diffuse-reflection boundary condition, extending our earlier studies (Jiang & Chen, J. Fluid Mech., vol. 899, 2020, A18; Zeng et al., J. Fluid Mech., 1018, 2025, A27). We derive a well-posed Laplacian eigenvalue problem under diffuse reflection and obtain a complete basis formed by products of Bessel functions and spherical harmonics. The moment equations are solved by the Galerkin spectral method, and the computation is simplified by decomposing the operator and basis functions, together with an analytical treatment of the orientational integrals. The study follows the entire transport process by examining the local and radial distributions, the drift velocity and the dispersivity, and we assess the effects of key parameters with comparisons to the specular reflection conditions. Our results show that diffuse reflection drives microswimmers away from the wall more efficiently and promotes downstream alignment and cross-stream migration. When swimming is strong, non-gyrotactic microswimmers can develop centre accumulation, whereas gyrotaxis promotes near-wall accumulation that counteracts the effect of diffuse reflection, in contrast to classical behaviour. Distinct mechanisms dominate different stages of the transient evolution, leading to different temporal trends in the radial distribution and dispersivity. Overall, diffuse reflection yields a larger drift velocity and a smaller dispersivity, while both gyrotaxis and elongation increase dispersivity.

The influence of hydrodynamic interactions on the schooling behaviour of fish is still poorly understood. This paper numerically investigates the collective motion of two parallel fish that move freely in both the longitudinal and lateral directions, focusing on the effects of wavelength and phase difference on their stable formations, swimming speed and energy efficiency. It is found that two parallel fish can achieve stable formations in both longitudinal and lateral directions, only via the hydrodynamic interactions. Three distinct modes are classified based on the cycle-averaged longitudinal speed, i.e. the steady slow mode, the steady fast mode and the fluctuating fast mode; which mode occurs depends on the wavelength and phase difference. Compared to a single fish, two fish in the steady slow mode swim slower, whereas they swim faster in both the steady fast and fluctuating fast modes, with a maximum speed increase of 12 % observed in the latter mode. Moreover, the fish school exhibits higher propulsive efficiency than a single fish in most cases. Furthermore, the power consumption and propulsive efficiency of each fish in different modes are discussed in detail. Finally, the mechanism behind the stable formations has been analysed. These results may shed some light on understanding the underlying mechanisms of fish schooling behaviour.

The propulsion of a flapping wing or foil is emblematic of bird flight and fish swimming. Previous studies have identified hallmarks of the propulsive dynamics that have been attributed to unsteady effects such as the formation and shedding of edge vortices and wing–vortex interactions. Here, we show that several key features of heaving flight are captured by a quasi-steady aerodynamic model that aims to predict stroke-averaged forces from wing motions without explicitly solving for the flows. We address the forward dynamics induced by up-and-down heaving motions of a thin plate with a nonlinear model which involves lift and drag forces that vary with speed and attack angle. Simulations reproduce the well-known transition for increasing Reynolds number from a stationary state to a propulsive state, where the latter is characterised by a Strouhal number that is conserved across broad ranges of parameters. Parametric, sensitivity and stability analyses provide physical interpretations for these results and show the importance of accounting for the flow regimes which are demarcated by Reynolds number and angle of attack. These findings extend the phenomena of unsteady locomotion that can be explained by quasi-steady modelling, and they broaden the conditions and parameter ranges over which such models are applicable.

The hydrodynamic interactions involved in the self-organisation phenomenon in biological systems are not fully understood and have attracted significant attention. A previous study (Peng et al. 2018 J. Fluid Mech., vol. 853, pp. 587–600) found that, arranged in an unbounded fluid, the largest cluster of self-propelled bodies in tandem, capable of spontaneously forming an ordered configuration, consists of eight swimmers. Here, we numerically investigate the collective behaviour of multiple self-propelled plates in tandem within a channel of width $H$, confined by two parallel walls. These plates are driven by harmonic flapping motions of uniform frequency and amplitude. Results demonstrate for the first time that the channel confinement significantly enhances group cohesion, with up to 72 individuals self-organising into ordered configurations at an optimal channel width. We observe two stable configurations: a hybrid mode with subgroups (typically at smaller channel widths) and a regular mode with sparse configuration. In large regular-mode groups, the vortex fields downstream exhibit spatial periodicity, conforming to Rosenhead’s stability criterion for confined vortex streets (vortex spacing $L_{v\textit{or}} \geqslant 1.419H$). This theoretical alignment explains both the observed upper channel-width limit ($H \approx 4.0{-}4.5$) for large-scale cohesion and the robust order in the regular mode. The plates may adopt spontaneously a ‘vortex-slalom’ path, reducing the drag force and energy consumption while maximising stability. Deviation from this path results in a spring-like restoring force, promptly returning the plate to equilibrium.

The propulsion speed of spheroidal squirmers was obtained by Keller & Wu (J. Fluid Mech., 1977, vol. 80, p. A31). It has become the benchmark to investigate the effect of shape on the propulsion of ciliated microorganisms. However, their study focused on translational motion whereas many biologically relevant organisms also experience rotational (or swirling) motion. We derive an analytical expression for the angular velocity of a swirling spheroidal squirmer. Our analysis reveals that spheroidal squirmers rotate faster than their spherical counterparts in Newtonian fluids. We also determine the contribution of the second azimuthal mode to the power dissipation generated by a spheroidal squirmer, and uncover a behaviour uniquely distinct from the power dissipation of a strictly translating swimmer.

This work studies the aerodynamics of two tandem foils flapping near a boundary layer (BL). The interaction between the tandem foils and the BL is modulated by the foil-to-wall distance rescaled by the foil chord length ($H_0/c$) and the Reynolds number based on free stream velocity (${\textit{Re}}=U_{\infty }c/\nu$). Taking the single foil under the same configuration as the reference, difference reductions are observed in forces between either of the tandem foils and the single foil, due to the weakened coupling between tandem foils in the presence of the BL. For a similar reason, it is revealed that, as ${\textit{Re}}$ increases, the force evolutions of the hind foil increasingly resemble those of the fore foil, being a second difference reduction. When examining the system’s evolution, we find that, in some cases, the evolution period of the force and the wake flow doubles that of the flapping cycle. From the Lagrangian coherent structures, it is indicated that this period doubling occurs because, in these cases, only one of two trailing edge vortices, shed in two successive cycles, is convected downstream, while the other is trapped and eventually dissipated in the BL. This interpretation has also been well confirmed by the frequency response from the modal analysis. In the cases with period doubling, the effect of the BL is relatively weak, corresponding to a coupling-dominated mode of interaction. Additionally, BL-dominated mode (low ${\textit{Re}}$ and/or low $H_0/c$) and foil-dominated mode (high ${\textit{Re}}$ and/or high $H_0/c$) are also identified, where the period doubling is not present anymore, respectively because of the strong BL effect and its absence. Finally, a bifurcation analysis is conducted to explore the dynamical nature of the system’s evolution. As $H_0/c$ increases, the system first undergoes a flip bifurcation, leading to the period doubling due to the decaying BL effect; and then an inverse period doubling bifurcation occurs, corresponding to a transition from coupling-dominated to foil-dominated interaction mode. If taking ${\textit{Re}}$ as the bifurcation parameter, a flip bifurcation is also first observed, sharing the same physical picture as the flip bifurcation identified when increasing $H_0/c$. Further increasing ${\textit{Re}}$, the system will undergo a Neimark–Sacker bifurcation due to the nonlinear nature of the convective flow, and the evolution of the system transitions from period doubling to quasi-periodic state.

The vertical, tip-to-tip arrangement of neighbouring caudal fins, common in densely packed fish schools, has received much less attention than staggered or side-by-side pairings. We explore this configuration using a canonical system of two trapezoidal panels (aspect ratio ${\textit{AR}}=1.2$) that pitch about their leading edges while heaving harmonically at a Strouhal number $St=0.45$ and a reduced frequency $k=2.09$. Direct numerical simulations based on an immersed-boundary method are conducted over a Reynolds-number range of $600\leq {\textit{Re}}\leq 1\times 10^{4}$, and complementary water-channel experiments extend this range to $1\times 10^{4} \leq {\textit{Re}}\leq 3\times 10^{4}$. Results indicate that when the panels oscillate in phase at a non-dimensional vertical spacing $H/c\leq 1.0$ with $c$ denoting the panel chord length, the cycle-averaged thrust of each panel rises by up to 14.5 % relative to an isolated panel; the enhancement decreases monotonically as the spacing increases. Anti-phase motion instead lowers the power consumption by up to 6 %, with only a modest thrust penalty, providing an alternative interaction regime. Flow visualisation shows that in-phase kinematics accelerate the stream between the panels, intensifying the adjacent leading-edge vortices. Downstream, the initially separate vortex rings merge into a single, larger ring that is strongly compressed in the spanwise direction; this wake compression correlates with the measured thrust gain. The interaction mechanism and its quantitative benefits persist throughout the entire numerical and experimental Reynolds-number sweep, indicating weak ${\textit{Re}}$-sensitivity within $600\leq {\textit{Re}}\leq 3\times 10^{4}$, and across multi-panel systems. These results provide the first three-dimensional characterisation of tip-to-tip flapping-panel interactions, establish scaling trends with spacing and phase, and offer a reference data set for reduced-order models of vertically stacked propulsors.

Many species of fish, as well as biorobotic underwater vehicles (BUVs), employ body–caudal fin (BCF) propulsion, in which a wave-like body motion culminates in high-amplitude caudal fin oscillations to generate thrust. This study uses high-fidelity simulations of a mackerel-inspired caudal fin swimmer across a wide range of Reynolds and Strouhal numbers to analyse the relationship between swimming kinematics and hydrodynamic forces. Central to this work is the derivation and use of a model for the leading-edge vortex (LEV) on the caudal fin. This vortex dominates the thrust production from the fin and the LEV model forms the basis for the derivation of scaling laws grounded in flow physics. Scaling laws are derived for thrust, power, efficiency, cost-of-transport and swimming speed, and are parametrised using data from high-fidelity simulations. These laws are validated against published simulation and experimental data, revealing several new kinematic and morphometric parameters that critically influence hydrodynamic performance. The results provide a mechanistic framework for understanding thrust generation, optimising swimming performance, and assessing the effects of scale and morphology in aquatic locomotion of both fish and BUVs.

The wake dynamics of a circular cylinder oscillating in the streamwise direction within a stably (density) stratified fluid is investigated using two-dimensional numerical simulations: Floquet stability analysis and dynamic mode decomposition. At a fixed Reynolds number ($ \textit{Re}=175$) and forcing frequency ratio ($f_d/f_{St}=1.6$), we examine the effects of the oscillation amplitude ($0.1 \leqslant A_D \leqslant 0.6$) and the stratification strength ($1 \leqslant \textit{Fr} \leqslant \infty$) on the wake structure and its symmetry breaking. In unstratified (homogeneous) flow ($ \textit{Fr} = \infty$), the wake transitions from an asymmetric vortex street at low amplitudes to a symmetric state at higher amplitudes. This transition occurs via a Neimark–Sacker bifurcation, with Floquet analysis identifying a critical amplitude of $A_D = 0.455$. In stratified flow, buoyancy forces improve symmetry and suppress vortex shedding for $A_D=0$. At $ \textit{Fr} = 1$, symmetry breaking first occurs at a threshold of $A_D = 0.246$, associated with a period-doubling bifurcation and subharmonic antisymmetric vortex shedding, and persists only within a finite amplitude window ($0.246 \lt A_D \lt 0.560$), beyond which the wake restabilises into a symmetric pattern. At a fixed small amplitude ($A_D = 0.1$), a secondary critical transition is observed at $ \textit{Fr} = 1.52$, marked by quasiperiodic antisymmetric shedding through a near-resonant Neimark–Sacker bifurcation. Stratification also influences force production: moderate stratification ($ \textit{Fr} \approx 2$) minimises drag through enhanced pressure recovery and suppressed wake asymmetry. These results highlight the dual role of stratification in promoting or delaying symmetry-breaking instabilities and modifying wake dynamics. Critical transition thresholds are established, providing insight into buoyancy-modulated flow control strategies relevant to geophysical and engineering applications involving oscillating bodies in stratified environments.

The hydrodynamic performance of oscillating elastic plates with tapered and uniform thickness in an incompressible Newtonian fluid at varying Reynolds numbers is investigated numerically using a fully coupled fluid–structure interaction computational model. By leveraging the acoustic black hole effect, tapered plates can generate bending patterns that vary from standing wave to travelling wave oscillations, whereas plates with uniform thickness are limited to standing wave oscillations. Simulations reveal that although both standing and traveling wave oscillation modes can produce high thrust, travelling waves achieve significantly higher hydrodynamic efficiency, and this advantage is more pronounced at higher Reynolds numbers. Furthermore, regardless of the oscillation mode, tapering leads to greater hydrodynamic performance. The enhanced hydrodynamic efficiency of travelling wave propulsion is associated with the reduced amount of vorticity generated by tapered plates, while maintaining high tip displacements. The results have implications for the development of highly efficient biomimetic robotic swimmers, and more generally, the better understanding of the undulatory aquatic locomotion.

The cell body of flagellated microalgae is commonly considered to act merely as a passive load during swimming, and a larger body size would simply reduce the speed. In this work, we use numerical simulations based on a boundary element method to investigate the effect of body–flagella hydrodynamic interactions (HIs) on the swimming performance of the biflagellate Chlamydomonas reinhardtii. We find that body–flagella HIs significantly enhance swimming speed and efficiency. As body size increases, the competition between the enhanced HIs and the increased viscous drag leads to an optimal body size for swimming. Based on the simplified three-sphere model, we further demonstrate that the enhancement by body–flagella HIs arises from an effective non-reciprocity: the body affects the flagella more strongly during the power stroke, while the flagella affect the body more strongly during the recovery stroke. Our results have implications for both microalgal swimming and laboratory designs of biohybrid microrobots.

Recent theoretical and experimental investigations have revealed that flapping compliant membrane wings can significantly enhance propulsive performance (e.g. Tzezana & Breuer J. Fluid Mech., 2019, vol. 862, pp. 871–888) and energy harvesting efficiency (e.g. Mathai et al. J. Fluid Mech., 2022, vol. 942, p. R4) compared with rigid foils. Here, we numerically investigate the effects of the in-plane stretching stiffness (or aeroelastic number), $K_{\!S}$, the flapping frequency, ${\textit{St}}_c$, and the pitching amplitude, $\theta _0$, on the propulsive performance of a compliant membrane undergoing combined heaving and pitching in uniform flow. Distinct optimal values of $K_{\!S}$ are identified that respectively maximise thrust and efficiency: thrust can be increased by 200 %, and efficiency by 100 %, compared with the rigid case. Interestingly, these optima do not occur at resonance but at frequency ratios (flapping to natural) below unity, and this ratio increases with flapping frequency. Using a force decomposition based on the second invariant of the velocity gradient tensor $Q$, which measures the relative strength between the rotation and deformation of fluid elements, we show that thrust primarily arises from $Q$-induced and body-acceleration forces. The concave membrane surface can trap the leading-edge vortex (LEV) generated during the previous half-stroke, generating detrimental $Q$-induced drag. However, moderate concave membrane deformation weakens this LEV and enhances body-acceleration-induced thrust. Thus, the optimal $K_{\!S}$ for maximum thrust occurs below resonance, balancing beneficial deformation against excessive drag. Furthermore, by introducing the membrane’s deformation into a tangential angle at the leading edge and substituting it into an existing scaling law developed for rigid plates, we obtain predictive estimates for the thrust and power coefficients of the membrane. The good agreement confirms the validity of this approach and offers insights for performance prediction.

We investigate the unsteady lift response of compliant membrane wings in hovering kinematics by combining analytical inviscid theory with experimental results. An unsteady aerodynamic model is derived for a compliant thin aerofoil immersed in incompressible inviscid flow of variable free-stream velocity at high angles of attack. The model, representing a spanwise section of a hovering membrane wing, assumes small membrane deformation and attached flow. These assumptions are supported by experiments showing that passive membrane deformation suppresses flow separation when hovering at angles of attack up to $55^\circ$. An analytically derived expression is obtained for the unsteady lift response, incorporating the classical Wagner and Theodorsen functions and the membrane dynamic response. This theoretical expression is validated against experimental water-tank measurements that are performed on hovering membrane wings at angles of attack of $35^\circ$ and $55^\circ$. Data from membrane deformation measurements is applied to the theoretical lift expression, providing the theoretical lift response prediction for each of the available experimental scenarios. Results of the comparison show that the proposed theory accurately predicts unsteady lift contributions from membrane deformation at high angles of attack, provided the deformation remains small and the flow is attached. This agreement between inviscid theory and experimental measurements suggests that when flow separation is suppressed, the unsteady aerodynamic theory is valid well beyond the typical low-angle-of-attack regime.

Unsteady aerodynamic forces in flapping wings arise from complex, nonlinear flow structures that challenge predictive modelling. In this work, we introduce a data-driven framework that links experimentally observed flow structures to sectional pressure loads on physical grounds. The methodology combines proper orthogonal decomposition and quadratic stochastic estimation (QSE) to model and interpret these forces using phase-resolved velocity fields from particle image velocimetry measurements. The velocity data are decomposed in a wing-fixed frame to isolate dominant flow features, and pressure fields are reconstructed by solving the Poisson equation for incompressible flows. The relationship between velocity and pressure modes is captured through QSE, which accounts for nonlinear interactions and higher-order dynamics. We introduce an uncertainty-based convergence criterion to ensure model robustness. Applied to a flapping airfoil, the method predicts normal and axial forces with less than 6 % average error using only two velocity modes. The resulting model reveals an interpretable underlying mechanism: linear terms in the QSE model the circulatory force linked to the formation of vortices on the wing, while quadratic terms capture the nonlinear component due to added-mass effects and flow–vorticity interactions. This data-driven yet physically grounded approach offers a compact tool for modelling the unsteady aerodynamics in flapping systems with potential to generalise to other problems.

Microswimmers and active colloids often move in confined systems, including those involving interfaces. Such interfaces, especially at the microscale, may deform in response to the stresses of the flow created by the active particle. We develop a theoretical framework to analyse the effect of a nearby membrane on the motion of an active particle whose flow fields are generated by force-free singularities. We demonstrate our results on a particle represented by a combination of a force dipole and a mass dipole, while the membrane resists deformation due to tension and bending rigidities. We find that the deformation either enhances or suppresses the motion of the active particle, depending on its orientation and the relative strengths between the fundamental singularities that describe its flow. Furthermore, the deformation can generate motion in new directions.

Swimming and flying animals demonstrate remarkable adaptations to diverse flow conditions in their environments. In this study, we aim to advance the fundamental understanding of the interaction between flexible bodies and heterogeneous flow conditions. We develop a linear inviscid model of an elastically mounted foil that passively pitches in response to a prescribed heaving motion and an incoming flow that consists of a travelling wave disturbance superposed on a uniform flow. In addition to the well-known resonant response, the wavy flow induces an antiresonant response for non-dimensional phase velocities near unity due to the emergence of non-circulatory forces that oppose circulatory forces. We also find that the wavy flow destructively interferes with itself, effectively rendering the foil a low-pass filter. The net result is that the waviness of the flow always improves thrust and efficiency when the wavy flow is of a different frequency than the prescribed heaving motion. Such a simple statement cannot be made when the wavy flow and heaving motion have the same frequency. Depending on the wavenumber and relative phase, the two may work in concert or in opposition, but they do open the possibility of simultaneous propulsion and net energy extraction from the flow, which, according to our model, is impossible in a uniform flow.