Fluid–structure interaction (FSI) poses a significant computational challenge due to the complex, multiscale nonlinearities of both fluid and structural dynamics. In this study, a novel strongly coupled FSI network is developed for accurate and efficient predictive modelling of FSI problems. Specifically, the framework architecture integrates a physics-constrained convolutional neural network autoencoder (CAE) with a strong coupling (SC) prediction module containing both fluid and structural dynamic prediction modules (DP) that perform recursive prediction simultaneously and interactively. First, the physics-constrained CAE learns low-dimensional nonlinear normal modes (NNMs) representations of the high-dimensional fluid field’s spatiotemporal dynamics. Subsequently, the fluid DP module in the SC module leverages these NNMs combined with structural states determined by embedding the motion equation into the structural DP, to predict the future-state flow fields efficiently. Such a strongly coupled FSI framework is achieved by integrating fluid NNMs and structural states within each time step to recursively correct the learned mapping of the trained CAE and fluid DP modules, thereby efficiently and accurately predicting future-state FSI dynamics simultaneously. The developed SC-CAE-NNM FSI framework is applied to the classic problem of vortex-induced vibrations of a circular cylinder, analysing both laminar and high-$ \textit{Re}$ flows. It is observed that the identified NNMs of fluid flows in association with the structural state achieve superior accuracy in the prediction of the flow fields and structural responses, indicating that the SC scheme effectively captures the dynamic flow characteristics and strong nonlinear interactions. Furthermore, the framework is found to be able to reconstruct small-scale flow structures in high-$ \textit{Re}$ flows accurately and predict structural responses efficiently. Additionally, the analysis of NNMs energy distributions reveals that the majority of the total energy of the flow field is captured by the first four NNMs, demonstrating significant advantages of nonlinear feature representation for efficient reduced-order modelling of complex flows. Overall, this novel framework shows strong capability for accurate and efficient predictive modelling of complex nonlinear dynamics of FSIs.

An experimental investigation is conducted in a wind tunnel on a NACA0012 airfoil with a partially flexible polydimethylsiloxane membrane leeward surface extended from 16.7 % to 83.3 % of the chord length. Aerodynamic forces, membrane deformation and the surrounding flow field are measured simultaneously. The results show that membrane vibration effectively reduces the extent of the recirculation zone, thereby improving aerodynamic performance. Specifically, the stall angle is delayed by $3^\circ$, and the maximum lift coefficient is increased by 12.4 % compared with that of the rigid airfoil. Novel insights into the flow–structure interaction are established from both spatial and temporal perspectives. Spatially, comparisons across angles of attack reveal that membrane vibrations driven by strong pressure fluctuations near the trailing edge can propagate upstream, accompanied by modifications in the distribution of turbulent kinetic energy and enhanced flow mixing. Temporally, a detailed analysis of the strongly periodic membrane motion and unsteady flow structures near stall further demonstrates that the membrane dynamics is tightly coupled with the evolution of the leading-edge vortex. This coupling is associated with the modulation of coherent flow structures through periodic absorption and release of mechanical energy. Overall, the flexible membrane can provide effective flow control by reorganising the spatio-temporal distribution of energy within the flow via flow–structure interaction, without external energy input. The findings provide insights into potential low-energy flow-control strategies.

This work investigates the hydrodynamics of a rigid cylindrical body with an arbitrary cross-section undergoing arbitrary translational oscillations in a viscous incompressible fluid. An asymptotic solution is derived analytically in a high-frequency, small-amplitude approximation, using a generalised curvilinear non-inertial coordinate system. The solution yields analytical expressions for the three components of the hydrodynamic force: the added mass, the Basset force and the viscous drag. These general dependencies are used to analyse the hydrodynamic loads on various oscillating cylinders, including an elliptical cylinder, a symmetric Zhukovsky aerofoil, a rounded polygonal profile (defined as a hypotrochoid) and a rectangular cylinder. For sharp-edged bodies, limitations are discussed. The range of applicability of the developed theory is assessed by comparing its predictions with numerical simulations and experimental data. The appendix provides formulas for calculating the hydrodynamic force coefficients in the Morison form for bodies of different shapes.

Blade-vortex interaction (BVI) is a significant aerodynamic phenomenon for rapidly developed electric vertical takeoff and landing aircraft, which generates high-amplitude noise when a rotor blade passes tip vortices. In this work, we experimentally investigate effects of vortex dynamics on rotor aeroacoustics during BVI in edgewise flows at a low Reynolds number (Re) range from $3.9\times 10^4$ to $1.0\times 10^5$, and over a practical advance ratio ($\mu$) range from 0 to 0.5, using anechoic wind tunnel tests. Thrust and power coefficients exhibit a linear relationship with $\mu$. The BVI process can be categorized into three flow regimes based on difference in vortex dynamics, each with distinct tonal and broadband noise characteristics. In regime 1 ($\mu \lt 0.1$), the interaction between the vortex and the blade boundary layer is dominant, with viscous effects on the blade surface, resulting in multiple harmonics tones over a wide frequency range. Amplitudes of high-order tones reduce with increasing $\mu$ due to downstream convection of small-scale turbulence. In regime 2 ($0.1\lt \mu \lt 0.23$), the blade interacts with the vortex shear layer, and the pressure gradient due to the leading edge becomes a primary factor during BVI. In regime 3 ($\mu \gt 0.23$), the blade directly passes through the vortex centre. Here, BVI-related tones at the middle frequency range further increase with increasing $\mu$, while spectra exhibit a broadband nature at high frequencies. Across various flow regimes at low Re, the broadband noise can be scaled using the law of $\mu ^kM_{\mathit{tip}}^5$, where $k$ is a constant determined in different flow regimes and $M_{\mathit{tip}}$ is the tip Mach number.

The impact of droplets, such as rain and sea sprays, on wind turbine blade surfaces can lead to significant damage due to leading-edge erosion (LEE). Traditionally, LEE on wind turbine blades has been solely attributed to material fatigue from repeated impacts; however, studies in similar engineering applications, such as steam turbines, suggest that cavitation may also play a critical role. This study develops a novel fluid–structure interaction (FSI) model that, for the first time, explicitly incorporates phase change in the liquid flow inside the droplet. Droplet impingement simulations cover a broad range of impact speeds relevant to most engineering applications, considering both cavitation and non-cavitation scenarios. The results present temporal evolutions of the pressure wave inside the droplet during its impact on the solid surface and demonstrates the triggering mechanisms of both homogeneous and heterogeneous cavitation. The developed FSI model suggests that the significant impact of cavitation, which can subject the material to two separate stress events during a single droplet impact, potentially reduces the material’s fatigue lifetime by half. Furthermore, the study explores the ability of heterogeneous cavitation to cause localised damage on the coating surface upon a single raindrop impact at speeds below $100$ $\mathrm{m\,s^{-1}}$. The comparison of two-dimensional and axisymmetric simulations shows that the intensified shock waves in the latter produce more focused cavitation near the surface centre, with minimal spreading along the surface. These findings highlight the need to incorporate heterogeneous cavitation effects in future studies, particularly as the turbine blade size and impact velocities increase.

This work uncovers a counterintuitive effect of structural damping on vortex-induced vibrations (VIV). The influence of structural damping is examined numerically in the canonical problem usually considered to study these vibrations driven by flow-body synchronisation, i.e. a circular cylinder free to oscillate within a uniform cross-current, for Reynolds numbers up to $160$, based on the body diameter and inflow velocity. For rectilinear VIV without structural restoring force, the peak-amplitude responses, which develop for low structure to displaced fluid mass ratios, are always attenuated by damping, as previously reported. Yet, the lower-amplitude responses occurring for higher mass ratios exhibit a contrasting trend: they are amplified by damping. The amplification can be substantial and leads to vibrations that become significant compared with the peak-amplitude responses. It is observed over the entire Reynolds number range of VIV, and is even accompanied by an extension of this range in the subcritical region, relative to the threshold of $47$ that marks the onset of flow unsteadiness for a fixed body. The enhancement of VIV by structural damping is interpreted in light of the distribution of fluid–structure energy transfer in the amplitude–frequency domain. Such an enhancement is a counterintuitive but general phenomenon, which is shown to persist along curvilinear trajectories and when a structural restoring force is added to the system.

It is a well-established fact that at a fixed angle of attack, with increasing Reynolds number, ${\textit{Re}}$, a laminar separation bubble (LSB) will shrink and eventually cease to exist, due to increase in the growth rate of hydrodynamic instability modes. However, during this wind tunnel test campaign, we observed that paradoxically the LSB increased in length from ${\textit{Re}}=2\times 10^5$ to ${\textit{Re}}=2.5\times 10^5$. Further diagnostics showed that this was due to a modification of the background forcing environment, often neglected in the literature. The perturbation generated from potential aero–vibroacoustic interaction unexpectedly boosted Tollmien–Schlichting modes and their faster amplification even upstream of the LSB drove the rapid secondary growth and breakdown at ${\textit{Re}}=2\times 10^5$. At ${\textit{Re}}=2.5\times 10^5$, when the lock-in with background forcing modes was not significant, a well-behaved LSB was observed with spatially amplifying Kelvin–Helmhotz modes. This finding raises important questions on the mechanisms entailing the dynamics and instability of LSBs proposed from not only experiments in similar facilities in the literature, but also in other scenarios where background perturbation due vibration, noise and confinement are non-negligible.

Historically, investigations on gust encounters have been limited to thin airfoils. In this work, we examine vortex-gust encounters by a family of airfoils at a chord-based Reynolds number ${\textit{Re}}_c=100$, which includes variations in the gust ratio, initial gust position, gust radius, angle of attack, airfoil thickness and airfoil camber. We examine differences in the flow fields, lift-element distributions and aerodynamic responses across several airfoil–gust interactions. We observe a large deviation of the flow fields and aerodynamic responses with respect to the baseline flows for increasing gust ratios and gust sizes. The initial position of the vortex gust influences the magnitude of the velocity gradients observed near the leading edge, effectively heightening or mitigating the amplitude of the lift response. Moreover, the lift fluctuation increases with the angle of attack until it flattens around $10^\circ$, reminiscent of an unsteady stall-like regime. Furthermore, we report a decrease in the amplitude of the gust-induced lift fluctuations for thicker airfoils, which we attribute to a decrease in the vorticity production levels from the leading edge, as well as a spatial redirection of the transient aerodynamic forces, which shifts the response in favour of drag fluctuations. The exploration of a sensitive subset of the parameter space uncovers relevant trends, shedding light on regions that have received limited attention in past studies, with special focus on the influence of airfoil geometry.

Stall flutter is suppressed on a NACA0012 airfoil at a low Reynolds number (Re $=$ 1000–5000) through a counter-intuitive strategy: targeted enhancement of the second leading-edge vortex (sLEV) via phase-optimised surface morphing, in contrast to the conventional paradigm of suppressing vortices. Two distinct flow topologies are identified: (I) boundary-layer-eruption-induced leading-edge vortex (LEV) shedding at large amplitudes ($A_\alpha$ ≥ 10°), accompanied by sLEV formation, and (II) bluff-body-type shedding at small amplitudes ($A_\alpha$ ≤ 5°) without sLEV. A new circulation scaling is introduced to identify topology I, which drives severe structural oscillations. Flow-structure analysis reveals a novel sLEV–trailing-edge vortex (TEV) interaction mechanism that mitigates stall flutter: strengthening sLEV triggers a chain effect that weakens the TEV, elevates the aerodynamic moment trough and disrupts the feedback loop sustaining the oscillation. An energy-based map is constructed to characterise the boundary of stall flutter across control parameters and to enable rapid prediction of control performance. By combining the clustering-based vortex-induced load partitioning method with this energy framework, an optimal phase indicator is established: actuation synchronised with the peak sLEV-induced moment consistently yields near-optimal control performance, providing a direct quantitative link between pre-control flow features and optimal control parameters. A phase-locked-loop strategy further demonstrates that surface morphing with an amplitude an order of magnitude smaller than the characteristic LEV scale can nonetheless achieve complete suppression of stall flutter within the topology I regime. These findings not only demonstrate the efficacy of surface morphing control but also offer new physical insight into stall flutter dynamics and their precision control via targeted vortex interactions.

This work demonstrates the occurrence of travelling wave flutter instabilities on viscoelastic coatings in supersonic flows, which forms an important milestone towards developing viscoelastic materials for high-speed flow applications. The flow-induced travelling surface waves were generated by elevating the air speed such that the flow dynamic pressure and the material Young’s modulus are of the similar order, their ratio defined as the compliance parameter. Experimental investigations probed the critical compliance parameter corresponding to surface instability onset across different fluid-structural non-dimensional parameters. The results revealed a strong dependence of the critical compliance parameter on the flow Mach number in addition to other flow and structural parameters that were found to influence the instability onset in earlier studies with aqueous flows. A broader parametric sweep of these parameters was performed using linear stability analysis. Consistent qualitative comparisons were obtained between the experiments and stability calculations on the most amplified wave characteristics. Further investigations revealed that the interaction mechanisms are similar to incompressible flows.

A numerical study is conducted on a self-propelled flexible plate moving above an array of wall-mounted compliant blades at Reynolds numbers of order 100, designed to mimic flatfish-like locomotion near the vegetation-covered seabed. Compared with a flat rigid wall located at the same vertical distance from the plate, the blade-covered wall enhances the propulsive performance of the plate, yielding a higher propulsive velocity and lower energy consumption per transport distance. This enhancement intensifies as the blade spacing and vertical distance between the plate and the bottom wall decrease, but diminishes when the flapping frequency approaches the natural frequency of the blades due to resonance. By accounting for the asymmetric motion of the plate during the upper and lower half-periods, a universal scaling law ${\textit{Re}}_c\sim ({\textit{Re}}_{f,\delta })^{3/2}$ is identified between the propulsive Reynolds number ${\textit{Re}}_c$ and the corrected flapping Reynolds number ${\textit{Re}}_{f,\delta }$. Under the excitation of the flapping plate, two distinct vibration patterns are identified for the blade array. The first is the flapping-driven mode, which propagates synchronously with the plate’s motion. The second exhibits forward-propagating travelling waves, which are generated by the cooperative underdamped vibrations of downstream blades, and have a phase velocity much faster than the propulsive velocity of the plate. This study may shed light on the design of bio-inspired underwater vehicles and flow sensing techniques.

Kirigami sheets, which contain cutting lines that introduce structural discontinuities, undergo significant deformation when they interact with the surrounding flow. We experimentally investigate the reconfiguration and drag characteristics of a kirigami sheet translating at a constant speed in the low-Reynolds-number regime. When initially planar cutting lines deform into three-dimensional holes due to fluid loading, thick shear layers that develop within the holes overlap, forming virtual fluid barriers. Despite the presence of holes, the flow bypasses the reconfigured sheet because of hydrodynamic blockage. By varying translation speed, sheet thickness and cutting ratio, we examine how the hydrodynamic blockage determines flow distribution, deformation and drag, in comparison with the high-Reynolds-number regime without blockage. A governing dimensionless parameter based on the balance between viscous fluid force and effective bending force, accounting for cutting lines, is proposed to characterise the reconfiguration-dependent porosity of the sheet. Despite high porosity, the reconfigured kirigami sheet exhibits drag comparable to or even greater than that of a flat rigid disk. The unique drag behaviour of kirigami sheets in response to highly viscous flow suggests potential applications to small-scale drag-based transport and propulsion systems.

Understanding vortex-induced vibrations (VIV) of long flexible structures with curvature such as catenary-type risers (CTRs) is essential for offshore engineering, where such structures are widely deployed. Here, we perform numerical simulations of the VIV responses of CTRs in uniform flow over a range of incoming velocities and initial configurations, revealing pronounced spanwise zoning characteristics. First, modal-group switching is identified as the mechanism governing the transition between mono- and multi-frequency responses. Within modal group, mono-frequency dominates, organising a monotonic phase angle and single travelling-wave direction. At the modal-transition region, drifting mono-frequency events constitute multi-frequency response, resulting in non-monotonic phase behaviour and travelling-wave reflection. A mixed standing–travelling-wave pattern emerges as a unifying feature. Furthermore, we find that the spanwise response distribution is governed by two intrinsic length scales: lower-span standing wavelength $\lambda _s$, scaling inversely with mode number $({\sim} n^{-1.0})$ and critical detuning length $\lambda _d$, a scale-dependent measure of natural frequency damping. By comparing the spanwise responses, three distinct regimes are identified: (i) flow-induced lock-in regime $(z/L\gt \lambda _d)$, where the vibration frequency locks in the vortex-shedding frequency and the energy transfer sustains positive ($C_{lv}\gt 0$), with predominantly counter-clockwise trajectories; (ii) non-lock-in regime $(\lambda _s\lt z/L\lt \lambda _d)$, where the local Strouhal frequency surpasses the vibration frequency, generating a discontinuous cellular pattern, with $C_{lv}\lt 0$ and predominantly clockwise trajectories; and (iii) structural resonance regime $(z/L\lt \lambda _s)$, where the frequency lock-in reoccurs but $C_{lv}\lt 0$ indicates flow-damped vibrations and the response is dominated by standing waves. These findings establish a new framework for spanwise zoning in CTRs, providing guidance for VIV prediction, suppression and control in catenary-type risers.

When the compression exerted on a thin elastic structure is sufficient to cause buckling instability, a regular wrinkle pattern emerges. The formation of wrinkles can be achieved and mechanically guided by transient external loading. As a novel dynamic wrinkling mechanism, we introduce the hydrodynamic coupling between elasticity and pressure induced by transient flow. A circular elastic loop immersed in an expanding fluid-filled gap undergoes buckling due to the net hydrodynamic pressure between its inner and outer surfaces, yielding a time-dependent wrinkle pattern. Because of the transient features of the hydrodynamic pressure, the apparent mode of the wrinkles reaches the maximum mode number immediately after the instant of peak pressure, followed by a cascade to lower modes. The maximum mode number of the wrinkles is determined by the magnitude of the peak pressure. In contrast to an elastic loop of uniform thickness, varying the thickness distribution of the loop and facilitating buckling at specific local points produce a distinct wrinkle pattern with a specific low mode consistently over time.

We study the collective behaviour of clusters of cylinders placed in the wake of a fixed cylinder and free to move in a direction perpendicular to that of the incoming flow, with no structural damping or stiffness. We keep the Reynolds number, defined based on the cylinder diameter, at 100 and consider five different configurations for the initial positions of the cluster cylinders: linear, rectangular, V-shaped, triangular and circular. In each configuration, we consider progressively increasing numbers of cylinders in the cluster. We show that overall, the cylinders tend to form final linear configurations, in which, after their transition, the cylinders form one or more lines. Some free-to-move cylinders might take the lead position in some of these linear formations depending on the initial configuration. These steady-state positions are achieved when the mean value of lift that acts on the cylinders becomes negligible. As a byproduct of these reconfigurations, the overall drag force that acts on the collection of cylinders reduces at their final steady-state locations in comparison with their original configurations. The complicated wakes that are observed in the fixed counterparts of these configurations are replaced by a series of vortex rows in the wake of separate lines of cylinders. Reducing the mass ratio allows the cylinders to oscillate about their mean displacement paths, but their transient paths and their final steady-state positions are not affected significantly by the decrease in the mass ratio.

The canonical scenario of two-degree-of-freedom vortex-induced vibration (VIV) of a circular cylinder is re-examined in this study through high-fidelity large-eddy simulations (LES) at a Reynolds number of 10 000. The in-line and cross-flow vibration amplitudes, frequency responses and hydrodynamic coefficients predicted by the present LES match classical experimental results better than previous numerical attempts. In particular, motivated by an inadequate study yet vital importance of the small-amplitude in-line response in offshore engineering design, we present the first numerical evidence for the existence of three in-line response regions. Furthermore, the present in-line response agrees well with the design guideline DNV-RP-F105. After validating the present results against DNV-RP-F105 and published experiments, the detailed LES datasets enable further analysis of new VIV characteristics and physical mechanisms that have not been explored previously. For example, we identify and explain (i) the existence of twin governing frequencies for several VIV branches with partial synchronisation, (ii) decoupled in-line and cross-flow vibrations in the first in-line branch with symmetric vortex-shedding pattern, where an in-line resonance may not induce a cross-flow resonance, (iii) existence of a new elliptic vibration trajectory for a perfectly in-line resonant condition, (iv) gradualness in the 2S ↔ 2T transition of the vortex-shedding pattern and thus a continuous variation in the vibration amplitudes and hydrodynamic coefficients amid this transition and (v) lowest spanwise correlation of vortex shedding in the super-upper and lower branches, which is induced by complex interactions among ≥4 shed vortices over a cylinder vibration period.

This study investigates how imposed transverse forced vibration modifies vortex-induced vibration of an elastically mounted cylinder, with a focus on uncovering the nonlinear interplay between forced and self-excited oscillations. Through carefully designed experiments spanning wide ranges of frequency and amplitude ratios under low and high mass ratios, three distinct response regimes are identified. In the dual-frequency regime, occurring at extreme frequency ratios, weak coupling allows coexistence of forced and natural frequencies, yielding alternating large/small amplitudes and wake transitions between two vortex pairs per cycle (2P) and two single vortices per cycle (2S) shedding. The frequency-switching regime, near resonance, features amplitude modulation and intermittent dominance of each excitation source, producing mixed-mode wakes. In the resonant regime, frequency and amplitude matching lead to complete synchronisation, sinusoidal motion and intensified, periodic 2P shedding with elongated shear layers. Crucially, comparative analysis reveals that the structural mass ratio fundamentally governs the regime boundaries and transient dynamics, noticeably compressing the transitional frequency-switching zone. Energy transfer and added mass coefficients reveal enhanced dissipation and inertial modifications near resonance. The observed nonlinear interactions challenge assumptions of linear superposition and offer new insight into coupled vibration control. These findings provide a foundation for designing structures that harness or mitigate flow-induced vibrations in marine, energy and fluid–structure systems.

We consider the response of a flexibly mounted square prism placed in inertial-viscoelastic fluid flow with one degree-of-freedom in the cross-flow direction. Under these flow conditions, both inertia and elastic effects are significant. We model the system numerically using a two-way coupling scheme to simulate the interaction between the fluid and the spring–mass system at a Reynolds number of $\textit{Re}=200$ for two mass ratios of $m^* = 2$ and 20, and a Weissenberg number of $\textit{Wi}=2$, across a range of reduced velocities. We demonstrate that introducing fluid elasticity suppresses vortex-induced vibrations of square prisms, consistent with prior findings for circular bluff bodies. However, we find that fluid elasticity amplifies the galloping response in comparison with the response in a Newtonian fluid, leading to larger oscillation amplitudes and the onset of galloping at lower reduced velocities. The predicted enhancement in galloping is significant, particularly at low mass ratios, where no galloping is observed over the wide reduced velocity range tested for Newtonian fluids. We show that this enhancement of galloping is likely the result of the observation that the addition of viscoelasticity increases the magnitude of the rate of change of the transverse flow-induced force on the prism with increasing angle of attack of the incoming flow.

This study presents a systematic investigation of hydrodynamic forces on a stationary circular cylinder inside relatively wide and deep trapezoidal trenches in steady flows, representing subsea power cables laid in seabed valleys or artificial trenches, with particular emphasis on the role of Kelvin–Helmholtz (K–H) instability. Using high-fidelity three-dimensional (3-D) implicit large-eddy simulation (iLES), the influences of trench geometry and the incoming flows on hydrodynamic forces were examined. The parametric space includes the trench-width-to-cylinder-diameter ratio ($ W^{*}$ $\leq$ 20), trench-depth-to-cylinder-diameter ratio ($H^{*} \leq 5$), trench side slope ($15^{\circ}\leq \theta \leq 60^{\circ}$) and the streamwise offset of the cylinder centre from the trench centreline (${L}_{p}^{*}$). The large-scale cavity-type flows generated by trench geometry and K–H shear-layer dynamics originating at the leading edge of the trench are found to be the dominant mechanisms affecting hydrodynamic forces on the cylinder. Narrow, deep and steep-sided trenches are shown to provide a sheltering effect, reducing forces on the cylinder; whereas wide, shallow trenches promote the generation of strong K–H instabilities that interact directly with the cylinder, leading to periodic force fluctuations. Notably, when the cylinder is positioned downstream of the shear-layer reattachment point in a wide, shallow trench, the mean drag and/or lift forces can exceed those experienced over a flat seabed, accompanied by substantial fluctuations. Increasing Reynolds number (Re) from 500 to 3900, as defined based on the diameter of the cylinder, enhances fluctuations of hydrodynamic forces, primarily due to the upstream migration of the onset of K–H instability and the intensified interactions of shed vortices with the cylinder within the trench. A hydrodynamic regime map, constructed from mean force reduction factors and peak fluctuation intensities, is proposed to classify the flow into four distinct regimes: no interaction, transitional, interaction and strong interaction. These findings offer new physical insights and provide practical guidance for evaluating hydrodynamic responses on unburied subsea cables in complex seabed topographies.

We examine theoretically the flow interactions and forward flight dynamics of tandem or in-line flapping wings. Two wings are driven vertically with prescribed heaving motions, and the horizontal propulsion speeds and positions are dynamically selected through aero- or hydro-dynamic interactions. Our simulations employ an improved vortex-sheet method to solve for the locomotion of the pair within the collective flow field, and we identify ‘schooling states’ in which the wings travel together with nearly constant separation. Multiple terminal configurations are achieved by varying the initial conditions, and the emergent separations are approximately integer multiples of the wavelength traced out by each wing. We explain the stability of these states by perturbing the follower and mapping out an effective potential for its position in the leader’s wake. Each equilibrium position is stabilised since smaller separations are associated with in-phase follower-wake motions that constructively reinforce the flow but lead to decreased thrust on the follower; larger separations are associated with antagonistic follower-wake motions, increased thrust and a weakened collective wake. The equilibria and their stability are also corroborated by a linearised theory for the motion of the leader, the wake it produces and its effect on the follower. We also consider a weakly flapping follower driven with lower heaving amplitude than the leader. We identify ‘keep-up’ conditions for which the wings may still ‘school’ together despite their dissimilar kinematics, with the ‘freeloading’ follower passively assuming a favourable position within the wake that permits it to travel significantly faster than it would in isolation.