We investigate the dynamics and the stability of the incompressible flow past a corrugated dragonfly-inspired airfoil in the two-dimensional (2-D) $\alpha {-}Re$ parameter space, where $\alpha$ is the angle of attack and $Re$ is the Reynolds number. The angle of attack is varied in the range of $-5^{\circ } \leqslant \alpha \leqslant 10^{\circ }$, and $Re$ (based on the free stream velocity and the airfoil chord) is increased up to $Re=6000$. The study relies on linear stability analyses and three-dimensional (3-D) nonlinear direct numerical simulations. For all $\alpha$, the primary instability consists of a Hopf bifurcation towards a periodic regime. The linear stability analysis reveals that two distinct modes drive the flow bifurcation for positive and negative $\alpha$, being characterised by a different frequency and a distinct triggering mechanism. The critical $Re$ decreases as $|\alpha |$ increases, and scales as a power law for large positive/negative $\alpha$. At intermediate $Re$, different limit cycles arise depending on $\alpha$, each one characterised by a distinctive vortex interaction, leading thus to secondary instabilities of different nature. For intermediate positive/negative $\alpha$, vortices are shed from both the top/bottom leading- and trailing-edge shear layers, and the two phenomena are frequency locked. By means of Floquet stability analysis, we show that the secondary instability consists of a 2-D subharmonic bifurcation for large negative $\alpha$, of a 2-D Neimark–Sacker bifurcation for small negative $\alpha$, of a 3-D pitchfork bifurcation for small positive $\alpha$ and of a 3-D subharmonic bifurcation for large positive $\alpha$. The aerodynamic performance of the dragonfly-inspired airfoil is discussed in relation to the different flow regimes emerging in the $\alpha {-}Re$ space of parameters.

This paper presents detailed analyses of the Reynolds stresses and their budgets in temporally evolving stratified wakes using direct numerical simulation. Ensemble averaging is employed to mitigate statistical errors in the data, and the results are presented as functions of both the transverse and vertical coordinates – at time instants across the near-wake, non-equilibrium, and quasi-two-dimensional regimes for wakes in weakly and strongly stratified environments. Key findings include the identification of dominant terms in the Reynolds stress transport equations and their spatial structures, the generation and destruction processes of the Reynolds stresses, and the energy transfer between the Reynolds stress and the mean flow. The study also clarifies the effects of the Reynolds number and the Froude number. Additionally, we assess the validity of the eddy-viscosity type models and some existing closures for the Reynolds stress model, highlighting the limitations of isotropy and return-to-isotropy hypotheses in stratified flows.

The laminar flow past rectangular prisms is studied in the space of length-to-height ratio ($1 \leqslant L/H \leqslant 5$), width-to-height ratio ($1.2 \leqslant W/H \leqslant 5$) and Reynolds number ($Re \lessapprox 700$); $L$ and $W$ are the streamwise and cross-flow dimensions of the prisms. The primary bifurcation is investigated with linear stability analysis. For large $W/L$, an oscillating mode breaks the top/bottom planar symmetry. For smaller $W/L$, the flow becomes unstable to stationary perturbations and the wake experiences a static deflection, vertical for intermediate $W/L$ and horizontal for small $W/L$. Weakly nonlinear analysis and nonlinear direct numerical simulations are used for $L/H = 5$ and larger $Re$. For $W/H = 1.2$ and 2.25, the flow recovers the top/bottom planar symmetry but loses the left/right one, via supercritical and subcritical pitchfork bifurcations, respectively. For even larger $Re$, the flow becomes unsteady and oscillates around either the deflected (small $W/H$) or the non-deflected (intermediate $W/H$) wake. For intermediate $W/H$ and $Re$, a fully symmetric periodic regime is detected, with hairpin vortices shed from the top and bottom leading-edge (LE) shear layers; its triggering mechanism is discussed. At large $Re$ and for all $W/H$, the flow approaches a chaotic state characterised by the superposition of different modes: shedding of hairpin vortices from the LE shear layers, and wake oscillations in the horizontal and vertical directions. In some portions of the parameter space the different modes synchronise, giving rise to periodic regimes also at relatively large $Re$.

Wind turbines operate in the atmospheric boundary layer (ABL), where Coriolis effects are present. As wind turbines with larger rotor diameters are deployed, the wake structures that they create in the ABL also increase in length. Contemporary utility-scale wind turbines operate at rotor diameter-based Rossby numbers, the non-dimensional ratio between inertial and Coriolis forces, of $\mathcal {O}(100)$ where Coriolis effects become increasingly relevant. Coriolis forces provide a direct forcing on the wake, but also affect the ABL base flow, which indirectly influences wake evolution. These effects may constructively or destructively interfere because both the magnitude and sign of the direct and indirect Coriolis effects depend on the Rossby number, turbulence and buoyancy effects in the ABL. Using large eddy simulations, we investigate wake evolution over a wide range of Rossby numbers relevant to offshore wind turbines. Through an analysis of the streamwise and lateral momentum budgets, we show that Coriolis effects have a small impact on the wake recovery rate, but Coriolis effects induce significant wake deflections which can be parsed into two regimes. For high Rossby numbers (weak Coriolis forcing), wakes deflect clockwise in the northern hemisphere. By contrast, for low Rossby numbers (strong Coriolis forcing), wakes deflect anti-clockwise. Decreasing the Rossby number results in increasingly anti-clockwise wake deflections. The transition point between clockwise and anti-clockwise deflection depends on the direct Coriolis forcing, pressure gradients and turbulent fluxes in the wake. At a Rossby number of 125, Coriolis deflections are comparable to wake deflections induced by ${\sim} 20^{\circ }$ of yaw misalignment.

An adaptable estimation technique is presented to reconstruct time-evolving three dimensional (3-D) velocity fields from planar particle image velocimetry measurements. The methodology builds on the multi-time-delay estimation technique of Hosseini et al. (2015) by implementing the finite-impulse-response spectral proper orthogonal decomposition (FIR-SPOD) of Sieber et al. (2016). The candidate flow is the highly modulated turbulent near wake of a cantilevered square cylinder with a height-to-width ratio $h/d=4$, protruding a thin laminar boundary layer ($\delta /d=0.21$ with $\delta$ being the boundary layer thickness) at the Reynolds number $Re=10600$, based on d. The novelty of the estimation technique is in using the modal space obtained by FIR-SPOD to better isolate the spatio-temporal scales for correlating velocity and pressure modes. Using FIR-SPOD, irregular coherent contributions at frequencies centred at $f_{ac1}=(1\pm 0.05)f_s$ and $f_{ac2}=(1\pm 0.1)f_s$ (with $f_s$ the fundamental shedding frequency) could be separated, which was not possible using proper orthogonal decomposition. With the FIR-SPOD bases, the quality of the estimation improved significantly using only linear terms, and the correct phase relationships between pressure and velocity modes are retained, as is required for synchronizing coherent motions along the height of the obstacle. It is shown that a low-dimensional reconstruction of the flow field successfully captures the cycle-to-cycle variations of the dominant 3-D vortex shedding process, which give rise to vortex dislocation events. Thus, the present methodology shows promise in 3-D reconstruction of challenging turbulent flows, which exhibit non-periodic behaviour or contain multi-scale phenomena.

The acoustic field radiated by a system of contra-rotating propellers in wetted conditions (with no cavitation) is reconstructed by exploiting the Ffowcs Williams–Hawkings acoustic analogy and a database of instantaneous realizations of the flow. They were generated by high-fidelity computations using a large eddy simulation approach on a cylindrical grid of 4.6 billion points. Results are also compared against the cases of the front and rear propellers working alone. The analysis shows that the importance of the quadrupole component of sound, originating from wake turbulence and instability of the tip vortices, is reinforced, relative to the linear component radiated from the surface of the propeller blades. The sound from the contra-rotating propellers decays at a slower rate for increasing radial distances, compared with the cases of the isolated front and rear propellers, again due to the quadrupole component. The quadrupole sound is often neglected in the analysis of the acoustic signature of marine propellers, by considering the only linear component. In contrast, the results of this study point out that the quadrupole component becomes the leading one in the case of contra-rotating propulsion systems, due to the increased complexity of their wake. This is especially the result of the mutual inductance phenomena between the tip vortices shed by the front and rear propellers of the contra-rotating system.

Self-similar fractal tree models are numerically investigated to elucidate the drag coefficient, non-equilibrium dissipation behaviour and various turbulence statistics of fractal trees. For the simulation, a technique based on the lattice Boltzmann method with a cumulant collision term is used. Self-similar fractal tree models for aerodynamic computations are constructed using parametric L-system rules. Computations across a range of tree-height-based Reynolds numbers $Re_H$, from 2500 to 120 000, are performed using multiple tree models. As per the findings, the drag coefficients ($C_D$) of these models match closely with those of the previous literature at high Reynolds numbers ($Re_H \geqslant 60\,000$). A normalization process that collapses the turbulence intensity across various tree models is formulated. For a single tree model, a consistent centreline turbulence intensity trend is maintained in the wake region beyond a Reynolds number of 60 000. The global and local isotropy analysis of the turbulence generated by fractal trees indicates that, at high Reynolds numbers ($Re_H \geqslant 60\,000$), the distant wake can be considered nearly locally isotropic. The numerical results confirm the non-equilibrium dissipation behaviour demonstrated in previous studies involving space-filling fractal square grids. The non-dimensional dissipation rate $C_\epsilon$ does not remain constant; instead, it becomes approximately inversely proportional to the local Taylor-microscale-based Reynolds number, $C_\epsilon \propto 1/Re_\lambda$. We find significant one-point inhomogeneity, production and transverse transport of turbulent kinetic energy within the non-equilibrium dissipation near wake region.

This paper presents a numerical investigation of the turbulence transition phenomenon in the wake of wall-mounted prisms. Large-eddy simulations are performed at $Re = 1\times 10^3 {-}5\times 10^3$ for prisms with a range of aspect ratio (height to width) from $0.25$ to $1.5$, and depth ratios (length to width) between $1$ and $4$. The results show that the wake irregularity is enhanced with increasing depth ratio, evidenced by higher turbulent kinetic energy (${\approx}90\,\%$) near the leading edge, and the onset of irregular, unsteady vortex shedding. This is attributed to interactions between Kelvin–Helmholtz instability (KHI) of the shear layer and large-scale vortex shedding, and it is induced by an unsteady shear layer, resembling flapping-like motion. These interactions elevate the flow momentum due to increased turbulence intensity and mixing, contributing to the wake transition phenomenon. To this end, this study defines the role of depth ratio in the transition phenomenon by showing that increasing depth ratio (e.g. from $1$ to $4$) leads to earlier onset of KHIs in the shear layer. These instabilities intensify with depth ratio, resulting in stronger interactions between shear layer and large-scale vortex shedding. Specifically, KHI-induced vortices interact more frequently with large-scale wake structures for higher depth ratio prisms, exciting larger flow fluctuations and irregular wake patterns. This interaction alters the frequency and coherence of vortex shedding, revealing a complex coupling mechanism that drives the transition to turbulence.

The three-dimensional flow field past an I-shaped dual-step cylinder has been obtained by numerical integration of the Navier–Stokes equations at Reynolds number ($Re_D$) 150. The I-shaped cylinder consisted of two large-diameter (D) cylinders with a small-diameter (d) cylinder in between. With a view to exploring the vortex dynamics and structural loads, simulations were performed for eight different lengths $l$ of the small cylinder, varied from $l/D=10$ to 0.2 for a fixed diameter ratio $D/d=2$. When the length of the small cylinder is sufficiently large, the wake behind the I-shaped cylinder is similar to the wake behind the single-step cylinder (Tian et al., J. Fluid Mech., vol. 891, 2020, A24). As the small cylinder length decreases, the enhanced interactions between the two steps make the present wake deviate from the wake of the single-step cylinder, leading to four different wake modes distinguished by different combinations of vortex cells. The physical formation mechanisms were analysed in terms of the vortex dynamics. Besides the wake flow, the streamwise vortices around the I-shaped step cylinder were also investigated. A pair of edge vortices and a junction vortex were identified for $l/D \geq ~1$. When the gap between the two steps becomes too small, $l/D \leq ~0.2$, the junction vortex disappears, and only a pair of edge vortices exists. Varying the distance between the two steps strongly affects the structural loads (drag and lift) along the I-shaped cylinder. The dependence of the loads on $l/D$ was readily explained by the different wake modes.

We report a comprehensive study of the wake of a porous disc, the design of which has been modified to incorporate a swirling motion at an inexpensive cost. The swirl intensity is passively controlled by varying the internal disc geometry, i.e. the pitch angle of the blades. A swirl number is introduced to characterise the competition between the linear (drag) and the azimuthal (swirl) momenta on the wake recovery. Assuming that swirl dominates the near wake and non-equilibrium turbulence theory applies, new scaling laws of the mean wake properties are derived. To assess these theoretical predictions, an in-depth analysis of the aerodynamics of these original porous discs has been conducted experimentally. It is found that, at the early stage of wake recovery, the swirling motion induces a low-pressure core, which controls the mean velocity deficit properties and the onset of self-similarity. The measurements collected in the swirling wake of the porous discs support the new scaling laws proposed in this work. Finally, it is shown that, as far as swirl is injected in the wake, the characteristics of the mean velocity deficit profiles match very well those of both laboratory-scale and real-scale wind-turbine data extracted from the literature. Overall, our results emphasise that, by setting the initial conditions of the wake recovery, swirl is a key ingredient to be taken into account in order to faithfully replicate the mean wake of wind turbines.

Fluid dynamics systems driven by dominant, near-periodic dynamics are common across wakes, jets, rotating machinery and high-speed flows. Traditional modal decomposition techniques have been used to gain insight into these flows, but can require many modes to represent key physical processes. With the aim of generating modes that intuitively convey the underlying physical mechanisms, we propose an intrinsic phase-based proper orthogonal decomposition (IPhaB POD) method, which creates energetically ranked modes that evolve along a characteristic cycle of the dominant near-periodic dynamics. Our proposed formulation is set in the time domain, which is particularly useful in cases where the cyclical content is imperfectly periodic. We formally derive IPhaB POD within a POD framework that therefore inherits the energetically ranked decomposition and optimal low-rank representation inherent to POD. As part of this derivation, a dynamical systems representation is utilized, facilitating a definition of phase within the system's near-periodic cycle in the time domain. An expectation operator and inner product are also constructed relative to this definition of phase in a manner that allows for the various cycles within the data to demonstrate imperfect periodicity. The formulation is tested on two sample problems: a simple, low Reynolds number aerofoil wake, and a complex, high-speed pulsating shock wave problem. The method is compared to space-only POD, spectral POD (SPOD) and cyclostationary SPOD. The method is shown to better isolate the dominant, near-periodic global dynamics in a time-varying IPhaB mean, and isolate the tethered, local-in-phase dynamics in a series of time-varying modes.

Flow field in the near wake of a short-finite circular cylinder at $L/D=1.0$ with an angle of attack between 0$^\circ$–15$^\circ$, where the transition from the non-reattaching flow to the reattaching flow appears, is investigated in wind tunnel tests with a supportless condition. Stereo particle image velocimetry measurements were applied to the experiments at the Reynolds number of $3.46\times 10^4$, and velocity fields in the near wake were obtained. The data was mainly analysed using spectral proper orthogonal decomposition. Characteristic large-scale wake structures of recirculation bubble pumping and large-scale vortex shedding were identified in the near wake of the cylinder regardless of the angle of attack. The phase difference of expansion and contraction of the recirculation flow appears in the recirculation bubble pumping at $\alpha \neq 0^\circ$. On the other hand, the eigenfunctions of velocity fluctuations at the vortex shedding frequency show a similar spatial pattern regardless of $\alpha$. Frequency analyses of wake position calculated from the reconstructed velocity field clarified that peak frequency is different between two in-plane directions when $\alpha \neq 0^\circ$. In addition, three vortex shedding patterns (anticlockwise/clockwise circular and flapping) are identified not only at $\alpha =0^\circ$ but also $\alpha \neq 0^\circ$. The feature of wake position in the radial direction for each pattern is observed regardless of the angle of attack. The relationship between the recirculation bubble pumping and the wake position in the radial direction is apparent in the non-reattaching flow but is weaker with $\alpha$ in the reattaching flow.

The effect of geometric twist ($\delta$) of a finite wing of various semi-aspect ratios, on the flow, aerodynamic forces and strength of wing-tip vortex, is investigated. The number of vortex shedding cells increases with increase in $\delta$. In general, the vortex shedding frequency at the root and tip of the wing is approximately the same as that for an untwisted wing. However, close to the $\delta$, where the number of cells changes, the end-cell frequency of the twisted wing undergoes a departure from the value for the untwisted wing. Dislocations at the junction of neighbouring cells are of fork-type for $\delta > -2^\circ$ and of reverse fork-type for $\delta < -2^\circ$. Additional ring-like vortex structures are observed for $\delta =-4^\circ$. Despite a significant effect of the twist on the flow and spanwise variation of the local force coefficients, low to moderate twist of the wing has a relatively minor effect on the span-integrated force coefficients. Larger positive $\delta$, however, results in a significant decrease in the time-averaged force coefficients and rolling moment at the wing root, their unsteadiness and an increase in the strength of the wing-tip vortex. Twist can be utilized as a design parameter for an air vehicle operating at low Reynolds number. Positive twist results in a decrease in unsteadiness in the flow and lower rolling moment at the wing root that can enable lowering the structural weight. Negative twist, on the other hand, weakens the wing-tip vortices that assists in formation and swarm flying by causing lower disturbance to downstream air vehicles.

This study conducts experimental investigations into wake-induced vibration (WIV) of a circular cylinder placed downstream of an oscillating cylinder. Surprisingly, it is observed that the previously identified WIV phenomenon, characterized by a sustained increase in amplitude at higher reduced velocities, does not occur when the upstream cylinder oscillates at large amplitudes. Instead, a different phenomenon, which we refer to as the ‘wake-captured vibration’, becomes dominant. The experiments reveal a negative correlation between the vortex-induced vibration amplitude response of the upstream cylinder and the WIV amplitude response of the downstream cylinder. Through a quasi-steady and linear instability analysis, the study demonstrates that the previously proposed wake-displacement mechanism may not be applicable for predicting the cylinder WIV response in the wake of an oscillating body. This is because the lift force gradients across the wake, measured through stationary cylinder experiments, decrease significantly when the upstream cylinder vibrates at higher amplitudes. Consequently, actively controlled vibration experiments are conducted to systematically map the hydrodynamic properties of the downstream cylinder vibrating in the wake of an oscillating cylinder. The findings align with observations from free-vibration experiments, and help to explain the amplitude and frequency response of WIV. Additionally, wake visualization through particle image velocimetry is conducted to provide further insights into the complex wake and vortex–body interactions.

In this work, the shape of a bluff body is optimized to mitigate velocity fluctuations of turbulent wake flows based on large-eddy simulations (LES). The Reynolds-averaged Navier–Stokes method fails to capture velocity fluctuations, while direct numerical simulations are computationally prohibitive. This necessitates using the LES method for shape optimization given its scale-resolving capability and relatively affordable computational cost. However, using LES for optimization faces challenges in sensitivity estimation as the chaotic nature of turbulent flows can lead to the blowup of the conventional adjoint-based gradient. Here, we propose using the regularized ensemble Kalman method for the LES-based optimization. The method is a statistical optimization approach that uses the sample covariance between geometric parameters and LES predictions to estimate the model gradient, circumventing the blowup issue of the adjoint method for chaotic systems. Moreover, the method allows for the imposition of smoothness constraints with one additional regularization step. The ensemble-based gradient is first evaluated for the Lorenz system, demonstrating its accuracy in the gradient calculation of the chaotic problem. Further, with the proposed method, the cylinder is optimized to be an asymmetric oval, which significantly reduces turbulent kinetic energy and meander amplitudes in the wake flows. The spectral analysis methods are used to characterize the flow field around the optimized shape, identifying large-scale flow structures responsible for the reduction in velocity fluctuations. Furthermore, it is found that the velocity difference in the shear layer is decreased with the shape change, which alleviates the Kelvin–Helmholtz instability and the wake meandering.

We investigate the effect of three-dimensionality on the synchronisation characteristics of the wake behind an oscillating circular cylinder at ${\textit {Re}} = 300$. Cylinder oscillations in rotation, transverse translation and streamwise translation are considered. We utilise phase-reduction analysis, which quantifies the phase-sensitivity function of periodic flows, to examine the synchronisation properties. Here, we present an ensemble-based framework for phase-reduction analysis to handle three-dimensional wakes that are not perfectly time-periodic. Based on the phase-sensitivity functions, synchronisability to three types of cylinder oscillations is evaluated. In spite of similar trends, we find that phase-sensitivity functions involving three-dimensional wakes are lower in magnitude compared with those of two-dimensional wakes, which leads to narrower conditions for synchronisation to weak cylinder oscillations. We unveil that the difference between the phase-sensitivity functions of two- and three-dimensional flows is strongly correlated to the amplitude variation of the three-dimensional flow by the cylinder motions. This finding reveals that the cylinder motion modifies the three-dimensionality of the wake as well as the phase of vortex shedding, which leads to reduced phase modulation. The synchronisation conditions of three-dimensional wakes, predicted by phase-reduction analysis, agree with the identification by parametric studies using direct numerical simulations for forced oscillations with small amplitudes. This study presents the potential capability of phase-reduction to study synchronisation characteristics of complex flows.

Describing the evolution of a wind turbine's wake from a top-hat profile near the turbine to a Gaussian profile in the far wake is a central feature of many engineering wake models. Existing approaches, such as super-Gaussian wake models, rely on a set of tuning parameters that are typically obtained from fitting high-fidelity data. In the current study, we present a new engineering wake model that leverages the similarity between the shape of a turbine's wake normal to the streamwise direction and the diffusion of a passive scalar from a disk source. This new wake model provides an analytical expression for a streamwise scaling function that ensures the conservation of linear momentum in the wake region downstream of a turbine. The model also considers the different rates of wake expansion that are known to occur in the near- and far-wake regions. Validation is presented against high-fidelity numerical data and experimental measurements from the literature, confirming a consistent good agreement across a wide range of turbine operating conditions. A comparison is also drawn with several existing engineering wake models, indicating that the diffusion-based model consistently provides more accurate wake predictions. This new unified framework allows for extensions to more complex wake profiles by making adjustments to the diffusion equation. The derivation of the proposed model included the evaluation of analytical solutions to several mathematical integrals that can be useful for other physical applications.

The motion of a sphere freely rising or falling in a 5d (d is the diameter of the sphere) square tube was numerically studied for the sphere-to-fluid density ratio ranging from 0.1 to 2.3 (0.1 ≤ ρs/ρ ≤ 2.3, ρs is the density of spheres and ρ the fluid density) and Galileo number from 140 to 230 (140 ≤ Ga ≤ 230). We report that Hopf bifurcation occurs at Gacrit ≈ 157, where both the heavy and light spheres lose stability. The helical motion is widely seen for all spheres at Ga > 160 resulting from a double-threaded vortex interacting with the tube walls, which becomes irregular at Ga ≥ 190 where heavy spheres act differently from their counterparts; that is, heavy spheres change their helical directions alternately while light spheres exhibit helical trajectories with jaggedness in connection with the shedding of the double-threaded vortices. This is because of the difference in inertia between the heavy and light spheres. We also checked the oscillation periods for the helical motion of the spheres. They show opposite variations with ρs/ρ for the two types of spheres. Light spheres (ρs/ρ ≤ 0.7) reach a zigzagging regime at Ga ≥ 200 where a vortex loop (hairpin-like vortical structure) is formed which may develop into a vortex ring downstream at small ρs/ρ. This might be the first time a transition from the helical motion to the zigzagging motion for heavy spheres (ρs/ρ ≥ 1.8) has been reported. Finally, we examined the dependence of both the terminal Reynolds number and the drag coefficient of the spheres on the Galileo number.

The unsteady flow physics of wind-turbine wakes under dynamic forcing conditions are critical to the modelling and control of wind farms for optimal power density. Unsteady forcing in the streamwise direction may be generated by unsteady inflow conditions in the atmospheric boundary layer, dynamic induction control of the turbine or streamwise surge motions of a floating offshore wind turbine due to floating-platform oscillations. This study seeks to identify the dominant flow mechanisms in unsteady wakes forced by a periodic upstream inflow condition. A theoretical framework for the problem is derived, which describes travelling-wave undulations in the wake radius and streamwise velocity. These dynamics encourage the aggregation of tip vortices into large structures that are advected along in the wake. Flow measurements in the wake of a periodically surging turbine were obtained in an optically accessible towing-tank facility, with an average diameter-based Reynolds number of 300 000 and with surge-velocity amplitudes of up to 40 % of the mean inflow velocity. Qualitative agreement between trends in the measurements and model predictions is observed, supporting the validity of the theoretical analyses. The experiments also demonstrate large enhancements in the recovery of the wake relative to the steady-flow case, with wake-length reductions of up to 46.5 % and improvements in the available power at 10 diameters downstream of up to 15.7 %. These results provide fundamental insights into the dynamics of unsteady wakes and serve as additional evidence that unsteady fluid mechanics can be leveraged to increase the power density of wind farms.

This work investigates the spatio-temporal evolution of coherent structures in the wake of a generic high-speed train, based on a three-dimensional database from large eddy simulation. Spectral proper orthogonal decomposition (SPOD) is used to extract energy spectra and energy ranked empirical modes for both symmetric and antisymmetric components of the fluctuating flow field. The spectrum of the symmetric component shows overall higher energy and more pronounced low-rank behaviour compared with the antisymmetric one. The most dominant symmetric mode features periodic vortex shedding in the near wake, and wave-like structures with constant streamwise wavenumber in the far wake. The mode bispectrum further reveals the dominant role of self-interaction of the symmetric component, leading to first harmonic and subharmonic triads of the fundamental frequency, with remarkable deformation of the mean field. Then, the stability of the three-dimensional wake flow is analysed based on two-dimensional local linear stability analysis combined with a non-parallelism approximation approach. Temporal stability analysis is first performed for both the near-wake and the far-wake regions, showing a more unstable condition in the near-wake region. The absolute frequency of the near-wake eigenmode is determined based on spatio-temporal analysis, then tracked along the streamwise direction to find out the global mode growth rate and frequency, which indicate a marginally stable global mode oscillating at a frequency very close to the most dominant SPOD mode. The global mode wavemaker is then located, and the structural sensitivity is calculated based on the direct and adjoint modes derived from a local spatial analysis, with the maximum value localized within the recirculation region close to the train tail. Finally, the global mode shape is computed by tracking the most spatially unstable eigenmode in the far wake, and the alignment with the SPOD mode is computed as a function of streamwise location. By combining data-driven and theoretical approaches, the mechanisms of coherent structures in complex wake flows are well identified and isolated.