The flow is governed by the spatially filtered incompressible Navier–Stokes equations, which are solved using the SOWFA libraries within the OpenFOAM framework . The subgrid-scale stresses are modelled using a one-equation eddy viscosity approach for the ABL case and a Smagorinsky formulation for the laminar and low-turbulence cases. The AD forces are included as source terms in the momentum equations.

While the core numerical framework follows the standard SOWFA implementation, two main modifications were introduced to suit the current study. First, temperature variations were disregarded, focusing solely on the fluid mechanical behaviour of the flow. Second, the driving mechanisms were adapted depending on the flow case. ABL flow simulations are driven by a forced pressure gradient. For this case, a precursor region is defined upstream of the AD where a specific reference velocity is forced at hub height. Only the average velocity at hub height in this region is taken into account to calculate the pressure gradient, which is then applied to the entire domain. This approach ensures that the forcing mechanism is not biased by the local flow perturbations induced downstream the disc while still maintaining the desired boundary-layer development. Conversely, the laminar and low-turbulence flow simulations are driven by an inlet boundary condition instead of a pressure gradient. In both cases, the same reference velocity value is set. However, in the low-turbulence case, a perturbation is added to achieve a low-turbulence inflow with length scales close to the ABL case, as described in Sect. . In contrast, an unperturbed inflow is considered for the laminar case.

To represent the effect of the porous disc an AD approach is used, which is similar to the one presented in . The AD is represented by a set of nodes arranged on a planar disc that are independent of the background fluid mesh. Nodes are arranged as rings and the separation between rings is set according to . To ensure numerical stability, the nodal forces are later spread to the surrounding cells through a regularisation kernel, which relies on a three-dimensional Gaussian distribution .

First, a calibration table is constructed, for which the motionless AD is simulated with different fixed inlet wind speeds and uniform force distribution, in order to establish the induction relation between the unperturbed wind speed and the velocity at each AD node. During the computational fluid dynamics (CFD) simulation, the AD is subjected to a sinusoidal surge motion where the position and velocity are given by 1Δx(t)=Asurgesin⁡(2πfsurget),2Δv(t)=2πfsurgeAsurgecos⁡(2πfsurget). In the above expressions, Asurge is the surge amplitude, which is considered D/8 based on and fsurge the corresponding frequency, which differs across cases. Considering the fluid velocity at each node Ui, the local velocity sensed by the AD node is obtained in each time step as 3Ui|rel(t)=Ui(t)-Δv(t) which is then used to enter the calibration table and obtain the unperturbed wind speed relative to the AD, U∞|rel. Finally, the force is calculated as 4Δfi=12ρCtU∞|rel2ΔSi, where ρ is the air density, Ct is the disc thrust coefficient and ΔSi is the area corresponding to the particular AD node. The thrust coefficient of the AD is set to CT=0.65 to match the experimental measurements of the porous disc . This value is kept constant under platform surge motion. Although the platform motion induces variations in the apparent inflow velocity of approximately ±25% relative to the upstream velocity, the corresponding Reynolds number changes do not significantly affect the aerodynamic drag characteristics of the rigid porous mesh, as previously verified experimentally for varying inflow speeds .

While this constant CT approach accurately replicates the passive aerodynamic behaviour of a physical porous disc undergoing surge motion, it introduces a loss of generality when extending the results to a real three-bladed wind turbine rotor. In a physical floating wind turbine, the instantaneous thrust coefficient varies due to changes in tip-speed ratio, pitch control actions and dynamic inflow effects induced by the platform motion. Consequently, the present AD framework isolates the purely kinematic interaction between the surge motion and the wake dynamics, serving as a fundamental baseline for more complex aeroelastic scenarios.

In the case of laminar and low-turbulence flows, a 25 D long domain is built, with a 10 D × 10 D cross section. This cross-sectional area yields a blockage ratio of less than 1 %. Since this value is well below the typical 5 % threshold where confinement effects become significant , the numerical domain is wide enough to avoid artificial acceleration around the AD. The starting mesh is uniform and various mesh refinements are carried out. In order to capture the surge motion, a higher resolution is required. Therefore, the cells near the AD are twice as large in the cross directions as they are in the flow direction. To carry out a mesh convergence study, four meshes are built under these conditions and compared for the case of a motionless AD under laminar flow, analysing the wake at 4 D, 6 D and 8 D downstream the AD. The characteristics of the four meshes are presented in Table  and mesh no. 2 is shown in Fig. . This mesh contains 12.8 cells D-1 in the cross direction and 25.6 cells D-1 in the flow direction and is the one selected for the rest of the study, as will be shown in Sect. . Also, the chosen mesh leaves 8 D upstream the AD inside the more refined mesh in order to give the flow enough space to develop correctly before arriving at the AD. Finally, 12 D are left behind with the same cell size to cover the wake from 2 D till 10 D.

To achieve a converged ABL flow, many authors have opted for the precursor technique, where the flow is developed by recirculation until convergence, and only then it is presented with the wind farm . In this work, the authors choose to include the precursor region within the simulation domain, as it is done by , and the inlet is set by mapping the end of the precursor region. The zone of analysis is located downstream this region. A schematic of the numerical domain is shown in Fig. , where the flow at 75 D from the inlet is mapped into the domain inlet boundary condition (BC). For the ABL case, the domain dimensions are 175 D long and 10 D wide, but the height is reduced to 5 D. This cross section yields a blockage ratio of approximately 1.5 %, which, similarly to the uniform flow cases, is well below the 5 % threshold, ensuring that confinement effects remain negligible. The decision to restrict the domain height is intrinsically linked to the numerical modelling of the atmospheric surface layer (ASL). At the real scale, the ASL refers to the lowest portion of the ABL where turbulent quantities and fluxes display a near-constant behaviour . Simulating a deep domain for a neutrally stratified ABL without resolving temperature gradients, and thus lacking an inversion capping layer, can lead to an artificial decay of turbulence in the upper portion of the domain. To avoid this issue in the present scaled configuration, the domain height is limited to 5D. This reduced height is combined with a top stress boundary condition, which provides the required forcing to maintain realistic turbulence levels and preserve a consistent ASL profile over the downstream distance considered . The mesh originally contains 12.8 cells D-1 in the vertical direction and 3.2 cells D-1 in the horizontal direction. Also, a mesh height gradient is imposed near the bottom in order to correctly model the flow near the surface, and the same gradient is applied near the top. After the precursor region, based on the previously mentioned mesh convergence study, two mesh refinements are carried out only in the horizontal direction, reaching 25.6 cells D-1 in the region near the AD for both horizontal directions and 12.8 cells D-1 in the vertical direction. This mesh refinement continues up to 25 D downstream the AD. The resulting total number of cells is 4.84 M.

Cyclic lateral BCs are imposed in all cases. For the laminar and low-turbulence cases, the other two boundaries (top and bottom) are set to slip. For the ABL case, stress boundary conditions are prescribed at both faces to sustain the neutral ASL. The bottom boundary follows Schumann’s model to account for the physical surface roughness of the wind tunnel floor, while the top boundary applies a shear-stress condition based on the target friction velocity u*, similar to what is done by and . The inlet boundary condition is set to a velocity fixed value without any perturbation in the laminar case, while a filtered noise boundary condition is applied for the low-turbulence case. This type of inflow, as originally proposed by , consists of imposing the desired length scales on a random Gaussian white noise through a filtering operation and subsequently scaling to achieve the desired Reynolds stress tensor. Following this, the targeted mean flow is superimposed on the signal. For the low-turbulence case, the targeted mean flow is entirely uniform (i.e. it contains no vertical shear) and only the turbulent fluctuations are superimposed. For further details regarding this process, we refer the reader to . The turbulent length scales used for the filtering process in the inlet flow are set equal to the values obtained in the ABL flow, but the same length scale is set in both cross directions. In order to achieve an average uniform flow with a lower turbulence, the same values are specified in the entire inlet surface. Regarding the physical characteristics of the chosen ABL, it is modelled to represent the scaled wind tunnel boundary layer experimentally studied by rather than a specific physical site. Accordingly, the target velocity and turbulence intensity profiles were configured following VDI guideline 3783 for atmospheric boundary layer wind tunnel modelling. This provides a standardised, strongly sheared and highly turbulent baseline to contrast with the uniform inflow cases without shear.

Table summarises the cases analysed in this work. The reference velocity Uref=2.71 m s⁻¹ is taken at the inlet for the uniform flow cases and at the AD original position in a simulation without AD for the ABL case. The specific test cases are selected following those in , whose experimental setup was originally designed to replicate the Floatgen project, featuring a 2 MW Vestas V80 wind turbine with a 60 m hub height and an 80 m rotor diameter.

Although these initial experiments were scaled based on a 2 MW turbine, the selected Strouhal numbers hold direct physical relevance to modern 10 to 15 MW floating wind turbines. For such full-scale turbines, characterised by rotor diameters of 180 to 240 m and near rated wind speeds of approximately 10 to 11 m s⁻¹ , the St range of 0.12 to 0.47 corresponds to physical oscillation frequencies between 0.005 and 0.03 Hz. This frequency range is representative of the rigid-body platform motions typically observed in spar and semisubmersible floating wind turbines, whose natural frequencies generally lie between approximately 0.005 and 0.05 Hz for large-scale systems . Specifically, a motion amplitude of D8 represents a bounding scenario for extreme resonant responses in these large structures. Consequently, this study focuses strictly on these low-frequency, large-amplitude dynamics, whereas higher reduced frequencies (St>0.5), typically associated with first-order wave excitation, fall outside the current scope.

The laminar inflow case is considered in the initial stage in order to establish a reference for the subsequent cases. Figure  shows the energy content for each mode, which is obtained as the square of the singular value corresponding to the mode in Eq. (), normalised by the total of this quantity. Results are shown for St=0, St=0.24 and St=0.32, in Fig. a, b and c, respectively. These frequencies are representative of all the cases observed (see Table ). In all figures, the most energetic modes include pairs with similar energy content. Particularly for the cases with surge motion (St=0.24 and St=0.32), this pairing provides a clear baseline to understand how POD captures flow dynamics. In the context of POD, this mode pairing is the mathematical signature of a travelling wave or a convecting coherent structure, such as the vortex rings generated by the AD surge motion. Because these structures continuously change their downstream position over time, a single stationary spatial mode is insufficient to describe them. Consequently, the POD algorithm decomposes this advection into a pair of orthogonal spatial modes that exhibit similar spatial patterns but present a π/2 phase shift in the streamwise direction. When these paired spatial modes are multiplied by their corresponding harmonic temporal modes, which oscillate at the same frequency but are also phase-shifted in time, their linear combination accurately reconstructs the continuous downstream advection of the vortices. Following this interpretation, Fig. a, b and c show the first four spatial modes for St=0, St=0.24 and St=0.32, respectively. The resulting modes are displayed using vectors indicating the direction of the local velocity fluctuation and are coloured by the fluctuation magnitude, normalised by the maximum value in each case. It is evident that all three cases confirm the presence of pairs of opposite modes (1 and 2, 3 and 4). Also, all modes displayed in pairs present the same Fourier spectrum of the time component (not shown here), thus confirming that they belong to the same vortex structure. The frequencies present in modes 1 and 3 for each case are shown in Fig. a, b and c.

In the absence of motion, that is, when St=0, the spatial modes are found to be symmetric with respect to the AD axis (z=0 in Fig. a). This observation indicates that the situation is not one of alternating vortex shedding. Furthermore, the frequencies exhibited in the modes, as illustrated in Fig. a, show a distributed energy content across a specific band of frequencies (St≈0.2-0.55). While a broadband frequency distribution could theoretically indicate the presence of physical mechanisms whose exact frequency and lateral expansion slowly evolve over time (such as wake meandering), the spatial modes here remain strictly symmetric (Fig. a), contradicting an alternating meandering behaviour. Instead, these modes most likely correspond to the advection of negligible residual fluctuations in the wake, as the maximum velocity magnitude reconstructed from them is 6 orders of magnitude smaller than the mean velocity field. This confirms the absence of any dominant coherent structure under stationary laminar conditions.

On the other hand, besides presenting the energy content in pairs (Fig. b and c), cases St=0.24 and St=0.32 exhibit a discernible pattern for spatial modes 1 and 2, associated with the advection of the vortex ring structure visualised in Sect.  (Fig. b and c). In addition, modes 3 and 4 correspond to the first harmonic of this configuration, as illustrated by their spectra in Fig. b and c. Also, both pairs of modes present symmetry with respect to the AD axis (z=0) in terms of velocity direction. For modes 1 and 2, the only prevailing frequency is the surge frequency, whereas for modes 3 and 4, it is the double of this frequency. It is noticeable that St=0.32 exhibits higher energy in the first pair of modes and lower energy in the harmonics, compared to St=0.24. In both cases approximately 94 % of the total energy is contained within the first four modes, which is consistent with the findings of , where these modes accounted for about 95 % of the total energy under uniform flow. It is worth mentioning that modes 5 to 10, not shown in this work, behave in a similar way, displaying pairs of different harmonics of the vortex ring structure. Also, in all cases analysed, more than 99.9% of the energy is accumulated within the first 10 modes, showing that there is no other strong coherent structure present. These observations suggest that the vortex ring structure is a prevailing feature under laminar flow within the specified frequency range. Furthermore, it agrees with the observations made in previous studies, which reported the visualisation of the vortex ring structure for a broad range of frequencies, ranging from St=0.08 to St=1.22, under uniform inflow conditions, as discussed in Sect. . In addition, a visual inspection of the spatial modes in Fig.  reveals that the coherent pulsation associated with the first pair of modes maintains its structural integrity as it is convected downstream. This persistent coherence is a direct consequence of the complete absence of background turbulence, allowing the structures generated by the surge motion to evolve without external disruption.

The analysis continues with the application of POD to the low-turbulence flow results, with the objective of investigating how the behaviour changes in the presence of turbulence. The energy content for each mode is presented in Fig. a, b and c for the cases of St=0, St=0.24 and St=0.32, respectively. These results are followed by the first six spatial modes for each case in Fig. a, b and c. In contrast with the laminar no-motion case, where mode pairing was exhibited, the St=0 case presents six distinctly different energy contents for the first six modes characterised by a linear decay (Fig. a). Also, the spatial modes in Fig. a show asymmetry in terms of velocity direction, as evidenced by the arrows above and below the AD axis (z=0), which point in directions that are not congruent. This suggests an alternating behaviour, such as vertical meandering. Considering the frequency spectrum of modes 1 and 3 (Fig. a), it can be seen that the frequency ranges neither match between modes nor show a harmonic relation. The results obtained indicate the presence of meandering phenomena throughout the wake, which is distributed along six modes. A higher energy allocation is observed in the tail of the wake, due to increased wake movement in that sector.

In the St=0.24 case, it can be observed in Fig. b that the first two modes show similar energy content, corresponding to a mode pairing situation. Modes 3 and 4 show a similar pattern. This is confirmed in Fig. b, where modes 1 and 2 show pairing corresponding to the vortex ring structure and modes 3 and 4 show a pattern related to the first harmonic of the same structure. These patterns are symmetric with respect to the AD axis (z=0), as it was the case for laminar flow in Fig. b. Furthermore, the Fourier spectra of modes 1 and 3 (Fig. b) show a single peak corresponding to the surge frequency and twice the surge frequency, respectively. A novel feature is that now, spatial modes 5 and 6 show good agreement with the first two modes in the no motion case. This indicates that, within the most energetic modes, two new modes emerge that are related to the wake meandering rather than the vortex ring structure. The range of frequencies in these modes, not shown here, matches closely with modes 1 and 2 from the St=0 case. Also, the energy distribution presented in Fig. b shows that these modes are not paired, resembling the behaviour exhibited for St=0.

For St=0.32, a similar situation is evident for modes 1 and 2, where a clear mode pairing can be seen in Figs. c and c. This corresponds to the vortex structure shown in Fig. b. However, instead of the second harmonic, spatial mode 3 resembles the first mode corresponding to the no-motion case. This means the second harmonic signature is slightly less energetic than for the previous motion frequency, as the corresponding modes fall behind the wake meandering mode. The second mode pairing related to double the surge frequency appears in modes 4 and 5. Finally, mode 6 shows the second mode corresponding to the no-motion case, that is, mode 2 for St=0. In addition, the frequency spectra of modes 1 and 4 displayed in Fig. c agree with these observations.

To summarise the results of the POD analysis of the eight cases in Table , Fig. a and b show the energy content of the first 10 modes for each case. The filled markers connected by a line indicate mode pairing corresponding to the surge frequency, while harmonics are connected with dotted lines. The mode pairing is clear in all cases, where the modes have a similar energy and frequency spectrum. However, the harmonics in St = 0.18 present an arrangement of three modes. In addition, a local maximum is exhibited by the energy content of the pairing modes. This is more clearly seen in Fig. c, where the combined energy in both modes and harmonics corresponding to the vortex structure is shown against the St number. The total energy captured by the vortex ring structure for the laminar cases has been added to Fig. c for comparison. In the laminar regime, the absence of background turbulence makes the structures generated by the surge motion accumulate nearly 100 % of the kinetic energy across all frequencies. Conversely, under inflow conditions with low turbulence, the energy allocation drops significantly. It is apparent that for the turbulent case at St=0.30, the structure attains its maximum relative energy and subsequently begins a decline as the movement frequency increases. This direct comparison demonstrates that the presence of background turbulence, along with induced phenomena such as wake meandering, severely limits the energy that the vortex rings can retain. Furthermore, contrasting the spatial modes from both regimes (Figs.  and ) reveals that, while the coherent pulsation in the laminar case maintains its amplitude as it convects downstream (first two modes in Fig. b), the equivalent structures in the turbulent cases visibly start dissipating in the far wake region. This confirms that even a low level of background turbulence is sufficient to trigger the disruption of the pulsating structures. Ultimately, these results indicate that a characteristic frequency of approximately St=0.30 appears to be the most stable for the propagation of the vortex ring structure under the evaluated conditions with low turbulence.

In the case of ABL flow, the results differ significantly from those of the previous cases. In Fig. a, b and c the energy content for each mode is displayed for the cases of St=0, St=0.24 and St=0.32, respectively, followed by the corresponding first six spatial modes in Fig. a, b and c. In the no-motion case, the two most energetic modes exhibit an uneven distribution of energy, with the first mode having a markedly higher energy content than the second (Fig. a). Also, these two modes exhibit a new spatial distribution (Fig.a) and a frequency spectrum that spans lower frequencies (Fig. a). Instead of a mode pairing situation, the modes can be associated with the inlet ABL flow and its interaction with the AD. Subsequently, modes 3 to 6 manifest distinctive characteristics analogous to those observed in the no-motion case under low-turbulence flow, that is, a quasi-linear behaviour in the energy content decay and a non-symmetric behaviour in terms of velocity direction, as evidenced by the arrows in Fig. a. Then, these modes can also be associated with a wake meandering phenomenon divided in four modes, where the energy content is once again higher at the tail of the wake due to a stronger oscillation.

Regarding the St=0.24 case, Fig. b exhibits significantly higher energy in the first mode in comparison to the other modes, as observed in the no-motion case. The vortex pairing can be identified due to the similarity in energy content between modes 3 and 4. Also, it can be noted that the spatial modes 1 and 2 in Fig. b show a high degree of agreement with the same modes from St=0. This similarity is absent in modes 3 and 4, where St=0.24 exhibits a pair of opposite modes, as was previously observed in the context of laminar and low-turbulence inflow conditions. Furthermore, the Fourier spectrum of both modes, as illustrated in Fig. b, reveals a clear peak at the St that corresponds to the surge movement, with no other frequencies observed. Consequently, it can be deduced that the vortex ring structure is present in this case and that the first two modes are related to the inlet flow and an interaction with the AD similar to that of the no-motion case. Additionally, the Fourier spectrum of these modes, not shown here, exhibit a close match with the ones for modes 1 and 2 in St=0. In this instance, the spatial modes corresponding to the vortex ring structure show no symmetry with respect to the AD axis (z=0) in terms of velocity direction, unlike the laminar and low-turbulence cases. This indicates that, under ABL conditions, the vertical meandering and vortex ring structures are strongly coupled, as the surge motion interacts with the ambient shear flow to induce a synchronised, meandering phenomenon. In relation to the remaining two modes depicted in Fig.b, mode 5 has not been previously observed and is characterised by an absence of a peak related to the surge frequency in its Fourier spectrum (not shown). Finally a certain degree of similarity is observable between mode 6 and mode 4 from the no-motion case, which can also be related to vertical meandering.

The energy content for St=0.32, displayed in Fig. c, shows a single first mode less energetic than for St=0.24, followed by a mode pairing between modes 2 and 3. This indicates that the energy associated with this structure is higher at this frequency than at the previous one, thus explaining the significant reduction in the energy content of the first mode. The energy of the structure was found to vary with the surge frequency for low-turbulence flow, although the vortex pairing shifted to higher modes in this instance. This is corroborated by the spatial modes displayed in Fig.c, where a first mode similar to the previous cases is shown, followed by a mode pairing between modes 2 and 3. Furthermore, the Fourier spectra depicted in Fig. c reveals the absence of any other frequency except that corresponding to the movement. This finding suggests that these modes are exclusively associated with the vortex ring structure. Once again, the modes show no symmetry with respect to the AD axis (z=0) in terms of velocity direction, indicating a vertical meandering phenomenon which is driven by the combination of surge motion and shear flow. Additionally, the spatial mode 4 bears a strong resemblance to mode 2 of the St=0 case, indicating that these modes are not associated with the AD motion but rather with the inlet flow. In a manner analogous to that observed in the St=0.24 case, two novel modes emerge which do not contain the motion frequency in their spectra (5 and 6). Nevertheless, these modes demonstrate the impact of the vortex ring structure on wake behaviour.

Figure a and b show the energy share of the first 10 modes for all the cases in Table . The filled markers connected by a line indicate mode pairing. Two points merit consideration. On the one hand, the energy associated with modes in the same structure is almost uniform in all cases except for St=0.12 and St=0.18. In the first one, the surge frequency manifested in other modes but no new mode emerged due to the surge motion. In the second, both pairing modes have additional minor peaks of varying frequencies apart from the movement frequency on their spectrum (not shown here), which may have led to alterations in the energy content. This indicates that the convective structures within this range may not yet have attained sufficient strength, consistent with the threshold of St>0.19 reported by for a clear signature at 8 D downstream, under ABL flow. On the other hand, a local maximum is exhibited by the energy content of the pairing modes. This phenomenon is more clearly seen in Fig. c, where the combined energy of both modes is shown against the St number. It is apparent that for the case of St=0.32 the structure attains its maximum energy and subsequently begins to decline as the movement frequency increases. Compared with the low-turbulence case in Fig. c, a similar pattern is observed in the energy of the structure, which varies with frequency. In this instance, the peak manifests at St=0.32 instead St=0.30 and with a marked increase in the energy content, indicating that a characteristic frequency in proximity to this value favours the propagation of the vortex ring structure for the surge motion conditions studied. Furthermore, the decline in the energy content as the frequency increases is more pronounced in the case of the ABL flow. The presence of more turbulent, complex structures appears to have led to a further reduction in the energy present in the vortex ring structure, given that its energy content is much lower than in the low-turbulence case. In addition, also reported that the highest energy levels occur close to St≈0.30 for ABL, which is in agreement with the values obtained in this work. This comprehensive comparison with laminar and low-turbulence inflows has revealed distinct behaviours of the vortex ring structure and the wake in general within the investigated range when ABL flow is considered.

The POD analysis of a vertical plane revealed, among other details, the presence of vertical meandering for the cases under low-turbulence and ABL inflow conditions. In the low-turbulence scenario, the dynamics of the vortex ring structure and the wake meandering behaved as decoupled phenomena, captured by distinct POD modes with separated frequencies. Furthermore, in neither of the temporal modes did the surge frequency appear within the meandering modes. However, under ABL conditions, these phenomena became highly coupled. The POD mode associated strictly with the surge frequency revealed a spatial structure that inherently includes vertical meandering. This indicates that, in a highly turbulent and sheared environment, the surge motion itself acts as an additional mechanism that triggers induced meandering, phase-locked to the platform's oscillation. In order to delve deeper into this phenomenon, a similar analysis is carried out in this section but on a horizontal plane, with the intention of identifying side-to-side meandering. The data are obtained in the same periods analysed before, with the same sampling frequency. Considering the moving AD at St=0.32, Figs.  and show the energy content and spatial modes, respectively, resulting from applying POD analysis to a horizontal plane at hub height under low-turbulence and ABL flow.

Considering the low-turbulence inflow conditions, Fig. a shows an energy distribution across modes that closely resembles that of the vertical plane (displayed in Fig. c), with a clear mode pairing situation between modes 1 and 2, followed by three modes with similar energy content. When looking at the spatial modes displayed in Fig. a, the results show two mode pairing situations (modes 1–2 and modes 4–5), along with two other modes related to side-to-side meandering (modes 3–6). This is the exact same behaviour as the one presented by the spatial modes on a vertical plane in Fig. c. Finally, a similar situation happens when comparing the Fourier spectra of the temporal modes for vertical and horizontal planes (not shown here). These observations allow one to conclude that under low-turbulence inflow conditions, the meandering is present in both vertical and horizontal directions. Given the averaged uniformity of the inflow, it is expected that meandering will occur in the same manner in all directions, as there are no preferred directions.

In contrast, a distinct scenario emerges when considering a horizontal plane for the ABL inflow conditions. Firstly, the energy content of the modes is displayed in Fig. b. In this figure, a mode pairing situation can be distinguished between modes 4 and 5, as opposed to modes 2 and 3, as was the case for the vertical plane (Figs. c and c). In addition, the first mode exhibits a significantly higher energy level in comparison to the subsequent modes, a phenomenon that bears a notable resemblance to the observations presented in Fig. c. However, the remaining modes demonstrate a divergent energy content pattern. A detailed analysis of the spatial modes depicted in Fig. b reveals significant disparities when compared to the vertical plane modes illustrated in Fig. c. The two most energetic modes, which in the previous case were related to the inlet flow, show a different pattern in terms of the spatial modes. Upon analysis of the Fourier spectra in Fig. a, it is evident that both modes exhibit a comparable frequency range to the initial two modes observed in the vertical plane case (illustrated in Fig. a). This observation suggests that the underlying cause may be the same in both cases. In addition, spatial modes 4 and 5, which are associated with the vortex ring structure, demonstrate a symmetrical pattern with regard to velocity direction until 7 D downstream, as illustrated by the arrows. Beyond this point, the mode begins to manifest signs of asymmetry. This finding suggests that, in a significant proportion of the wake, the vortex ring structure does not exhibit lateral meandering, in contrast to the vertical meandering observed in modes 2 and 3 of Fig. c. As demonstrated in Fig. b, the Fourier spectra of both modes indicate the absence of any external signal, confirming the presence of a signal exclusively belonging to this structure. In this instance, the modes appear to be more analogous to those observed in the low-turbulence flow case.

Finally, spatial modes 3 and 6 demonstrate a complex pattern that can initially be difficult to identify. At an initial stage, spatial modes 2 and 3 appear analogous; however, a discrepancy in their frequencies prevents a valid basis for the grouping of mode 3 with the inlet flow. A thorough examination of the spatial mode depicted in Fig. b reveals that, in the initial phase, the mode undergoes a clockwise rotation during the first half of the wake, followed by an anti-clockwise rotation in the subsequent phase. As is evident in Mode 6, a similar behaviour is exhibited, yet it is divided into three distinct sections. It can thus be theorised that both modes are related to side-to-side meandering. As demonstrated in Fig. c, an analysis of the frequency spectra indicates that both modes exhibit a range of frequencies below St=0.2. However, the second mode demonstrates a higher frequency range than the first, due to presenting smaller structures in the spatial mode. Furthermore, in mode 6, a minor peak can be observed at the surge frequency, indicating a minimal presence of surge motion within the mode. However, in contrast to the vertical behaviour, the overall lateral meandering remains physically decoupled from the dynamics of the surge-induced vortex rings.

As demonstrated in Sect. , while the surge motion generates decoupled vortex ring structures under uniform inflows, the ABL shear flow couples with these structures to induce a synchronised vertical meandering. Furthermore, distinct spatial patterns emerge depending on the inflow, ranging from symmetric wake modulation to a noticeable structural inclination of the velocity fluctuations in the ABL scenario (as illustrated in Fig. c). In this section, a more profound examination of these varied wake configurations is proposed. To this end, the velocity deficit will be subjected to phase averaging in place of velocity fluctuations. This approach is intended to provide a more precise depiction of the wake's configuration during surge motion. The process is analogous to the previous analysis but rather than subtracting the mean flow values, the inflow profiles displayed in Figs. a and a are subtracted considering the corresponding case. Once again, 25 planes are obtained, containing in this instance the velocity deficit, Δũ(x,t). In this analysis, the focus was exclusively on the streamwise velocity component. Figure  illustrates one of the frames for St=0.32 in the laminar, low-turbulence and ABL flow cases. The results are presented as the percentage of Uref.

In the laminar and low-turbulence cases, illustrated in Fig. a and b, the averaged velocity deficit exhibits a wake modulated by an expansion and contraction due to the surge motion. This behaviour aligns closely with the synchronised coherent pulsing experimentally observed by under laminar inflow at a comparable Strouhal number (St=0.38 vs. St=0.32). Furthermore, recent ABL wind tunnel tests by confirmed that these periodic wake expansions persist even at a lower frequency (St=0.11), while numerical evaluations attribute this overall dynamic to strong variations in the rotor's axial induction . As demonstrated in the case of fluctuations in Fig. , there is a minimal difference between laminar and low-turbulence inflow with regard to this aspect. The only discernible discrepancy manifests towards the culmination of the wake, in proximity to 10 D, where the contraction is more pronounced in the laminar inflow scenario. Conversely, in the context of ABL flow, as illustrated in Fig. c, the phase average deficit reveals a wake characterised by traces of contraction and expansion, coupled with a vertical deformation induced by the background wind shear. Red lines in the three panels connect the expansion tops and bottoms for the first three cycles in each case. For the first two scenarios, the lines maintain a parallel configuration throughout the wake. It is evident that, in the initial line on the left, the maximum deficit is situated to the right of the red line. For the second and third lines, the maximum deficit is observed to the left of the line. This phenomenon is attributable to a disparity in advection velocity, whereby maximum deficit structures travel slower within the wake in comparison to the expansion and contraction observed at the periphery of the wake. The lines in the ABL case, in contrast to the first two cases, show an inclination towards the right. Furthermore, it has an increasing tendency further away from the AD original position. This phenomenon is attributable to the effects of shear flow, which modify the translation velocity of the vortex structure above and below the AD centreline. As was evidenced in the preceding cases, the expansion regions situated above the wake appear to travel faster than the maximum deficit structures. However, it is now evident that the expansion regions situated below the wake travel slower. Consequently, this disparity in translation velocities causes the surge motion to couple with the shear flow, inducing a synchronised vertical meandering. This phenomenon was also observed by , who conducted a SPIV analysis at 8.125 D from the position of a porous disc under surge motion. In addition, the configuration of the maximum deficit structures appears to undergo a transformation throughout the wake, as evidenced by the iso-deficit lines. Furthermore, an increase in velocity above the AD centreline results in a greater force acting on the top than on the bottom, leading to greater fluctuations over the AD axis.