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.

A novel fast-running model is developed to predict the three-dimensional (3-D) distribution of turbulent kinetic energy (TKE) in axisymmetric wake flows. This is achieved by mathematically solving the partial differential equation of the TKE transport using the Green's function method. The developed solution reduces to a double integral that can be computed numerically for a wake prescribed by any arbitrary velocity profile. It is shown that the solution can be further simplified to a single integral for wakes with Gaussian-like velocity-deficit profiles. Wind tunnel experiments were performed to compare model results against detailed 3-D laser Doppler anemometry data measured within the wake flow of a porous disk subject to a uniform free-stream flow. Furthermore, the new model is used to estimate the TKE distribution at the hub-height level of the rotating non-axisymmetric wake of a model wind turbine immersed in a rough-wall boundary layer. Our results show the important impact of operating conditions on TKE generation in wake flows, an effect not fully captured by existing empirical models. The wind-tunnel data also provide insights into the evolution of important turbulent flow quantities such as turbulent viscosity, mixing length and the TKE dissipation rate in wake flows. Both mixing length and turbulent viscosity are found to increase with the streamwise distance. The turbulent viscosity, however, reaches a plateau in the far-wake region. Consistent with the non-equilibrium theory, it is also observed that the normalised energy dissipation rate is not constant, and it increases with the streamwise distance.

Spanwise vortex instability and the growth of secondary hairpin-like vortical structures in the wake of an oscillating foil are investigated numerically at Reynolds number 8000 in a range of chord-based Strouhal number ($0.32 \le St_c \le 0.56$). The phase-offset ($\phi$) between the heaving and pitching motion is $\phi = 90^\circ$. The wake at the lowest $St_c$ (0.32) is characterized by a single system of streamwise hairpin-like structures that evolve from the core vorticity outflux of the secondary leading edge vortex (LEV) over the foil boundary. The primary LEV features spanwise dislocations, but it does not reveal substantial changes advecting downstream. Increasing $St_c$ beyond 0.32 reveals that the transition in spanwise instability characterizes the deformation of primary LEV cores, which subsequently transforms to hairpin-like secondary structures. At higher $St_c$, stronger trailing edge vortices (TEVs) grow in close proximity to the primary LEVs, which contributes to an enhanced localized vortex compression and tilting near dislocations. This phenomenon amplifies the undulation amplitude of primary LEVs, eventually leading to vortex tearing. The larger circulation of TEVs with increasing $St_c$ provides an additional explanation for an accelerated vortex compression that coincides with a faster transition of spanwise LEV instability to secondary hairpin-like structures in the wake.

This work reports an experimental study of the turbulent entrainment into the planar wake of a circular cylinder, exposed to various turbulent backgrounds, from the near- to the far-field. The background turbulence features independently varying turbulence intensity and integral length scale, thereby rendering different turbulent/turbulent interfaces (TTIs) between the background and the primary flow (wake). Combined, simultaneous particle image velocimetry and planar laser induced fluorescence measurements were conducted to quantify the entrainment characteristics across these various TTIs at an inlet Reynolds number of 3800. The primary focus was on understanding how turbulent entrainment evolves spatially in conjunction with the rapid development of the large-scale coherent vortices in the planar wake, and how such evolution is affected by the background turbulence. It is found that TTIs can establish two layers when the background turbulence is sufficiently intense, which distinguishes TTIs from the turbulent/non-turbulent interface (TNTI). The two layers are underpinned by different physical mechanisms but have the same thickness and appear to scale with the local Kolmogorov length scale after the wake spreading transition position (Chen & Buxton, J. Fluid Mech., vol. 969, 2023, A4). It is also found that the probability density functions of the entrainment velocity for both TTIs and a TNTI display power law tails, which are associated with extremely large entrainment velocities occurring more frequently than for a Gaussian process. These intermittent, extreme entrainment velocities make a remarkable contribution to the mean entrainment velocity, particularly in the near wake, which leads to a much higher mean entrainment velocity than farther downstream, for both a TNTI and the TTIs. Conditionally averaged analysis reveals that these extreme events of the entrainment velocity are directly associated with intense enstrophy structures close to the interface.

This paper explores active wake-flow control on a notchback Ahmed body using genetically inspired optimization. Hotwire and particle image velocimetry measurements record velocity data and flow structures in the wake. Pulsed jets at four actuation slots (two at the roof trailing edge, two at the side trailing edges) dynamically control the wake to minimize aerodynamic drag. The study achieves up to 9.2 % (without consideration of energy consumption) drag reduction, primarily by manipulating vortices from the roof rear end. The paper elucidates the underlying flow mechanism and evaluates various actuation strategies, highlighting how optimal control leads to reattachment of wake separation at the rear slant, diminishing the slant bubble and promoting downstream reattachment for enhanced drag reduction.

Losses due to wake interactions between wind turbines can significantly reduce the power output of wind farms. The possibility of active flow control by wake deflection downstream of yawed horizontal-axis wind turbines has motivated research on the fluid mechanics involved. We summarize the findings of a wind tunnel study (Bastankhah & Porté-Agel, J. Fluid Mech., vol. 806, 2016, pp. 506–541) of the flow associated with a yawed model wind turbine, and the insights and modelling developments that have followed this important study.

Stratified wakes past an isolated conical seamount are simulated at a Froude number of $Fr = 0.15$ and Rossby numbers of $Ro = 0.15$, 0.75 and $\infty$. The wakes exhibit a Kármán vortex street, unlike their unstratified, non-rotating counterpart. Vortex structures are studied in terms of large-scale global modes, as well as spatially localised vortex evolution, with a focus on rotation effects. The global modes are extracted by spectral proper orthogonal decomposition (SPOD). For all three studied $Ro$ ranging from mesoscale, submesoscale and non-rotating cases, the frequency of the SPOD modes at different heights remains coupled as a global constant. However, the shape of the SPOD modes changes from slanted ‘tongues’ at zero rotation ($Ro=\infty$) to tall hill-height columns at strong rotation ($Ro=0.15$). A novel method for vortex centre tracking shows that, in all three cases, the vortices at different heights advect uniformly at approximately $0.9U_{\infty }$ beyond the near wake, consistent with the lack of variability of the global modes. Under system rotation, cyclonic vortices and anticyclonic vortices (AVs) present considerable asymmetry, especially at $Ro = 0.75$. The vorticity distribution as well as the stability of AVs are tracked downstream using statistics conditioned to the identified vortex centres. At $Ro=0.75$, intense AVs with relative vorticity up to $\omega _z/f_{c}=-2.4$ (where $\omega_z$ is the vertical vorticity and $f_c$ is the Coriolis frequency) are seen with small regions of instability and they maintain large $\omega _z/f_{c}$ magnitude in the far wake. Recent stability analysis that accounts for stratification and viscosity is found to improve on earlier criteria and show that these intense AVs are stable.

A permeable disk serves as a simplified model for the conversion of wind energy by a horizontal axis wind turbine. In this study, we investigate how inflow turbulence intensity (TI), $I_\infty$, and inflow turbulence integral length scale, $L_\infty$, influence the flow recovery in the wake, the capability of a permeable disk in extracting turbulence kinetic energy (TKE) of the incoming flow, and the statistics of wake-added turbulence using large-eddy simulation. The simulated inflows include various TIs (i.e. $I_\infty =2.5\,\%$–$25\,\%$) and integral length scales (i.e. $L_\infty / D =0.5$–$2.0$) for two thrust coefficients. Simulation results show that both inflow TI and integral length scale influence flow recovery via enhanced ejections and sweeps across the wake boundary, with the former strongly affecting the position where the wake starts to recover and the latter mainly on the recovery rate. Moreover, it is shown that increasing $I_\infty$ and $L_\infty$ increases the TKE extraction by the disk, occurring mainly at scales ($s$) greater than $0.5D$ and frequencies depending on the inflow integral length scale. As for the wake-added TKE, the inflow TI mainly affects its intensity, while the inflow integral length scale affects both its intensity and the sensitive frequencies, with the spectral distributions in scale space ($s$) being similar and the peak located around $s/D=1.0$ for the considered inflows.

Pressure gradient over topography will significantly affect wind-farm flow. However, knowledge gaps still exist on how to superpose wind-turbine wakes in the wind-farm flow analytical model to account for this effect, leading to systematic errors in evaluating wind-farm wake effects. To this end, we derive an implicit momentum-conserving wake superposition method under pressure gradient (PG-IMCM) based on the total momentum deficit equation, which is linearised by the convection velocity introduced by Zong & Porté-Agel (J. Fluid Mech., vol. 889, 2020, A8). The PG-IMCM method consists of the linear-weighted sum of individual velocity deficits, the sum of the individual pressure correction terms and the total pressure correction term. Based on a sensitivity analysis, we demonstrate that the last two terms nearly cancel out and, thus, can be neglected, resulting in a simplified form, which has the same form as its counterpart under zero pressure gradient but with the single-wake quantities redefined based on the wake model under pressure gradient. This motivates us to further examine the performance of the combination of five empirical superposition methods and the stand-alone wake model under pressure gradient. Validation results based on large-eddy simulation show that PG-IMCM has an overall satisfactory performance in both the magnitude and shape of the velocity-deficit profiles, provided that the stand-alone turbine wake can be modelled accurately, which is virtually identical with its simplified form. Further comparison with empirical superposition methods shows that local linear and wind product superposition methods based on the updated base flow also have comparable performance, with only discernable differences with the PG-IMCM method in the near-wake region of downstream turbines. Therefore, they are two attractive methods for engineering applications.