The WT1 platform position was measured using Wenglor PNBC107 laser transducers, with signals recorded at a sampling frequency of 2 kHz. The rotor forces of both wind turbines were measured using six-component ATI Mini45 force transducers (SI-580-20 calibration) mounted at the top of each tower, with data acquired at the same 2 kHz sampling rate.

For WT1, the recorded loads also include inertial and gravitational contributions of the rotor–nacelle assembly induced by platform motion. These contributions were removed following the method described by . A two-test subtraction procedure was adopted: for each motion case, two tests with identical prescribed platform motion were performed: (i) a no-wind, rotor-locked test and (ii) a wind-on, rotor-spinning test. The six-component tower-top signals from both tests were windowed to include the same integer number of motion periods and subsequently subtracted (wind-on minus no-wind). This procedure removes inertial and gravitational loads associated with the rigid-body motion of the wind turbine, under the assumption that the structure behaves rigidly and that the structural response is identical in both tests. The accuracy of the subtraction procedure was quantified through dedicated no-wind repeat tests and repeated wind-on measurements. The resulting uncertainty estimates are reported in Appendix .

After this post-processing, the resulting force signals are expressed in the respective hub reference frames: x1-y1-z1 for WT1 and x2-y2-z2 for WT2. The final signals contain approximately 100 full cycles of platform motion, ensuring robust statistical convergence of the quasi-periodic aerodynamic response due to platform motion.

The recorded loads of both WT1 and WT2 also include centrifugal contributions associated with rotor rotation. These loads arise primarily from rotor mass imbalance and manifest at the blade-passing frequency – 4 Hz for WT1 and between 2.2 and 3.8 Hz for WT2, depending on the farm configuration (see Table ) – as well as at its higher harmonics. Since centrifugal loads arise only when the rotor is spinning, they are not eliminated by the subtraction procedure applied to WT1.

Wake velocities reported in this paper were obtained using hot-wire anemometry. Measurements of the streamwise velocity in the turbine wake were performed in a prior experimental campaign using a single hot-wire anemometer capable of measuring both mean and fluctuating components of one-dimensional flow . The probe was mounted on an automated traversing system, enabling measurements in the crosswind (yi–zi) plane.

Velocity measurements were carried out at xi=3D, 4D, and 5D downstream of the rotor, along the hub-centered horizontal (yi axis) and vertical (zi axis) traverses. At each downstream location, the horizontal traverse comprised 35 evenly spaced measurement points, while the vertical traverse included 27 points. The spacing between consecutive measurement positions was 0.1 m (0.04D). For each measurement point, the acquisition time corresponded to 12 platform motion cycles or, in the fixed tower-base cases, to 12 rotor revolution periods of WT1. A preliminary sensitivity study conducted at the beginning of the experimental campaign showed that 12 cycles were sufficient to ensure convergence of the mean velocity, turbulence intensity, and – for the cases with platform motion – of the spectral amplitude at the platform motion frequency.

The hot-wire signals were acquired at a sampling frequency of 10 kHz and subsequently digitally low-pass filtered with a cut-off frequency of 100 Hz to remove high-frequency electrical noise.

To reconstruct phase-averaged spatial velocity fields, the velocity time series acquired at each probe location were recorded simultaneously with the platform motion signal. The signals were then phase aligned by synchronizing them to a common reference point of the motion cycle and conditionally averaged over the 12 cycles. The resulting phase-resolved velocity signals from the different spatial positions were subsequently combined to obtain the reconstructed velocity fields.

All the main tests were carried out for a constant free-stream wind speed (U) of 4.0 ms-1, which corresponds to a chord-based Reynolds number of approximately 1 × 10⁵ at blade mid-span. The free-stream wind speed was measured by a pitot tube positioned 3.13D ahead of the wind turbine on the hub axis. An additional measurement at 2.4 ms-1 was also performed to verify the performance of WT2 and is discussed later in the paper. The free-stream flow was approximately uniform across the wind tunnel test section, as demonstrated in a previous study : the mean velocity over the WT1 rotor area varied by 5 % compared to the hub mean velocity, and the average turbulence intensity across the rotor disk was 1.5 %. Large-eddy simulations of showed that this relatively low free-stream turbulence contributes significantly to dissipating the coherent flow structures generated by platform motion, limiting the gains in wake recovery compared to the laminar-inflow case, where such gains are higher.

Another pitot system was placed laterally to WT1 at xi=0.8D, yi=2.1D, and zi=1.3D to measure the wind speed influenced by blockage effects induced by the WT1 rotor, recording a wind speed of 4.2 ms-1. The increase in wind speed due to wind tunnel blockage is sufficiently small, and multi-fidelity simulations by on the wind turbine used in this study showed that blockage does not significantly alter its aerodynamic behavior. The effect can be accurately introduced in simulations by applying a corrected inflow velocity that reflects the value measured by the lateral pitot system.

Additional measurements performed at the WT2 rotor plane, at hub height, and at a lateral position yi=1.5D showed a mean velocity of 4.3 ms-1. This corresponds to a cumulative flow acceleration attributable to wind tunnel blockage effects of approximately 7 %–8 % relative to the upstream reference velocity U.

During testing, the air density (ρ) was 1.187 kgm-3.

WT1 had a fixed rotor speed of 240 rpm and a collective blade pitch set to the optimal value of 0°. The turbine operated near its rated condition, which corresponds to the peak of the thrust curve under the standard pitch-to-feather control strategy, and therefore produced the strongest wake deficit. In these conditions, the thrust coefficient (calculated by Eq. ) was Ct=0.9. Because rotor speed was held fixed at 240 rpm, the tip-speed ratio (TSR) varied with the apparent wind created by platform motion. Under the tested conditions, the TSR oscillation amplitude was 5 % of its value observed in the bottom-fixed scenario. This small modulation causes a slight dynamic shift of the operating point, without causing the rotor to enter a substantially different aerodynamic regime (the angle of attack varies by ± 1° at 70 % of the rotor radius, as shown by ).

The uniformity of the inflow generated by the wind tunnel across the region occupied by the wind farm was verified by measuring the loads on WT2 positioned at 5D downstream under free-stream conditions with WT1 removed from the test chamber, while the robotic platform remained installed on the wind tunnel floor. This configuration, in which the robotic platform remained installed, preserves flow conditions consistent with those of the subsequent wake-interaction experiments involving both turbines. This verification also ensured that WT2, when placed in the most downwind position, was not influenced by boundary effects near the end of the test section. Measurements were conducted at two wind speeds: 4.0 and 2.4 ms-1. The latter corresponds to the average wind speed over the rotor plane in the 5D full-wake condition, as estimated from the wake measurements reported by . At 2.4 ms-1, the rotor speed was set to 150 rpm, corresponding to a TSR close to the optimal value of 7.5. The resulting deviation from the ideal TSR (about 4 %) is due to the practical resolution with which the rotor speed can be prescribed and has a negligible effect on the operating conditions.

The energy available to WT2 when operating under waked conditions is reduced due to wake losses, which depend on its position relative to the upstream turbine. Outside free-stream conditions, WT2 operates in a non-uniform inflow caused by partial or total immersion in the wake of WT1, depending on the specific farm configuration. In these cases, the operating conditions of WT2 were defined in terms of the thrust-equivalent wind speed (UTE). For each configuration with fixed WT1, UTE was obtained by inverting a calibrated thrust–wind-speed–rotor-speed relation derived from a blade-element momentum (BEM) model of the rotor under uniform free-stream conditions (see Appendix ). The thrust-equivalent wind speed is introduced as an operational parameter to define a consistent operating condition for the WT2 rotor across configurations characterized by different wake interactions. Specifically, UTE denotes the uniform inflow speed that, in the calibrated free-stream thrust map, corresponds to the measured mean thrust at the prescribed rotor speed. It is not intended to reconstruct the actual heterogeneous waked inflow experienced by the rotor.

The blade pitch of WT2 was fixed at 0°, as for WT1, and was not adjusted based on UTE in order to avoid added complexity and to better isolate the effects of the wake on aerodynamic loading. The optimal TSR of the rotor (equal to 7.5), derived from the aerodynamic design, was used as the target operating condition. WT2 rotor speed was adjusted to achieve this optimal TSR based on the mean UTE at each position, except in the 3D1D and 5D1D configurations. In these two cases, the UTE for WT2 is higher than both the free-stream wind speed and the wind speed measured by the lateral pitot system, which accounts for the blockage induced by the presence of WT1 within the test section. The additional increase in velocity observed at the WT2 rotor plane is attributed to wind tunnel confinement effects associated with the presence of the second rotor, which contributes to the overall test-section blockage. This interpretation is supported by additional measurements performed at the WT2 rotor plane (at hub height and at a lateral free-stream position), which showed a mean velocity of 4.3 ms-1. In the 3D1D and 5D1D cases, the rotor speed was set to maintain the same thrust force as in the free-stream condition. As a result, the TSR was lower than the design value of 7.5 because of the blockage-induced speed-up at the rotor, but this choice ensured comparable thrust loading of WT2 across the farm configurations.

For the cases with dynamic WT1 motion, the rotor speed of WT2 was not updated instantaneously with the fluctuating inflow but kept constant at the value corresponding to the mean UTE, so that WT2 experienced small deviations from the nominal TSR during the motion cycle. The resulting rotor speed values, fixed for each configuration, are specified in Table . For completeness and to support reproducibility, the table also reports the corresponding average thrust force and thrust coefficient of WT2, with Ct being computed according to Eq. () using the measured thrust and the reference free-stream velocity (constant during the tests at 4.0 ms-1).

The system was studied under sinusoidal motion of different frequencies and amplitudes. For translational motions, the displacement was defined as 1d(t)=amsin⁡(2πfmt), where am is the displacement amplitude and fm the motion frequency. Surge motion corresponds to a translation of the platform reference point Op along the xi axis. Surge–sway motion corresponds to a translation at an angle γ relative to the xi axis (and hence to the wind). The displacement components are dx(t)=d(t)cos⁡γ along the surge direction and dy(t)=d(t)sin⁡γ in the crosswind (sway) direction. Angles γ = 15, 30, 45, and 90° were tested.

For pitch and yaw motions, the platform rotation was defined as 2θ(t)=aθsin⁡(2πfmt), where aθ is the angular amplitude. Pitch motion corresponds to a rotation around the yi axis with the rotor hub located at a distance rhub from the platform rotation point Op. When θ is small, the resulting hub displacement along the wind direction is dhub,x(t)=rhubθ(t). Yaw motion corresponds to a rotation around the zi axis (see Fig. b).

Surge and pitch motions cause along-wind movement of the nacelle and result in an apparent wind speed over the rotor given by 3ua(t)=U-d˙hub,x(t), where d˙hub,x=d˙ in the case of surge motion, d˙hub,x=d˙cos⁡γ in the case of surge–sway motion, and d˙hub,x=rhubθ˙ in the case of pitch motion. Accordingly, the amplitude of the apparent wind oscillation is Δu=2πfmam in the case of surge motion, Δu=2πfmamcos⁡γ in the case of surge–sway motion, and Δu=2πfmrhubaθ in the case of pitch motion.

During yaw motion, because the yaw amplitude is small, the apparent velocity remains predominantly aligned with the xi axis. However, a velocity gradient develops across the rotor span: the apparent wind speed perturbation is largest at the blade tips and vanishes at the hub. The local apparent velocity at position py along the yp axis (measured from the hub, with sign) is 4ua(t,py)=U-θ˙(t)py.

The oscillation amplitude of the apparent velocity therefore scales linearly with the spanwise position: Δu(py)=2πfmaθpy.

In research concerning the unsteady aerodynamics of floating wind turbines , frequencies are typically expressed in a non-dimensional form using the rotor-reduced frequency (fr), which is equivalent to a Strouhal number and is defined as 5fr=fmDU.

The sinusoidal motion conditions were characterized in terms of reduced frequency and amplitude. The selected amplitudes and frequencies were based on those used in the previous wind tunnel experiment of that characterized the wake behavior. The selected conditions were known to cause significant dynamic variations in the wake and were consequently expected to produce a substantial aerodynamic excitation of the downstream turbine. The frequencies correspond to reduced frequency values representative of full-scale floating wind turbine dynamics. Specifically, at a wind speed of 12 ms-1, fr=0.3 and fr=0.6 correspond to full-scale frequencies of 0.02 and 0.04 Hz, respectively, which are typical of rigid-body modes in 10 MW floating turbines. In contrast, fr=1.2 corresponds to a full-scale frequency of 0.08 Hz (with a period of 12.5 s), representative of platform motions at wave frequency.

The amplitudes of surge and pitch motions caused an apparent wind speed variation, expressed as a ratio of apparent wind to undisturbed wind speed (Δu/U) of approximately 5 % at all reduced frequency values considered in the experiment. In surge–sway motion, the displacement amplitude d was the same as in pure surge, but the variation in apparent wind speed decreased as the angle between the motion direction and the wind (γ) increased. For yaw motion, an amplitude of 2° was selected to investigate wake development under large dynamic yaw rotations, representative of those that may occur in real operating conditions under severe sea states. When combined with reduced frequencies of 0.3, 0.6, and 1.2, this yaw motion amplitude resulted in apparent wind speed variations at the rotor edge of approximately 3 %, 7 %, and 13 %, respectively.

Following the analysis of cases with sinusoidal motions, the experiment was extended to include motions similar to those induced by irregular waves acting on a 10 MW floating wind turbine from different directions, in order to assess whether the observations from the simplified cases remained valid under more realistic conditions.

In these tests, unlike the sinusoidal cases, the wind turbine motion was not prescribed from simple parametric inputs but was obtained from aero-hydro-servo-elastic simulations of a realistic floating wind turbine. Specifically, we modeled the SOFTWIND system, which consists of the DTU 10 MW wind turbine mounted on a spar-buoy platform , using the OpenFAST simulation tool . Table  reports the frequencies of the rigid-body motion modes of the SOFTWIND floating wind turbine. At these frequencies, the platform motion exhibits significant amplification of the excitation from wind and wave forcing.

The OpenFAST simulations were run for representative wind and wave conditions, and the resulting platform motion time series were used as inputs to the wind tunnel experiments. The waves in the simulation had a full-scale significant height of 5 m and a peak period of 12 s corresponding to a model-scale significant height of 0.067 m and a peak period of 0.48 s. The same waves were applied at heading angles of 0 and 30° relative to the wind direction. The simulations were carried out with a uniform wind field, with a full-scale wind speed of 12 ms-1, turbulence intensity of 1.5 %, and fixed rotor speed and blade pitch, matching the conditions of the experiment. The time series of platform motions in six directions were downscaled and employed to prescribe the motion of WT1 utilizing the robotic platform.

The time series of platform motions induced by irregular waves are shown in Fig.  and the corresponding spectra in Fig. . When waves are aligned with the wind direction (0°), the platform response is dominated by surge and pitch, with minimal motion in the other degrees of freedom. In contrast, waves at 30° also excite significant sway, roll, and yaw movements. Figure  shows that the frequency and amplitude of the platform motions of the SOFTWIND driven by the selected wave conditions are similar to those in the sinusoidal motion cases. Specifically, sinusoidal surge–sway and yaw motions represent movements in the sway and yaw directions caused by waves with a 30° heading.

The loads of WT1 in the fixed tower-base condition were measured in 40 repeated test runs. These repetitions were carried out throughout the experimental campaign, particularly after any changes in the test setup and following system initialization, for example, at the beginning of each testing day, to ensure consistency of the measurements. The average thrust force was 36.17 N and the average torque was 2.87 Nm. The loads were consistent during the experimental campaign, with maximum deviations from their mean values of ± 0.33 N for thrust (± 0.9 %) and ± 0.10 Nm for torque (± 3.5 %). Under these conditions, WT1 produced an average power of 72 W, corresponding to a full-scale value of 10.93 MW.

To verify the aerodynamic equivalence of WT2 with respect to WT1, WT2 was mounted on the robotic platform in the WT1 position, and its performance was compared under identical operating conditions. The measured thrust force (35.9 N, -0.7 % relative to WT1) and torque (3.03 Nm, +5.6 % relative to WT1) for WT2 were in close agreement with the values obtained for WT1. The thrust fell within the variability range measured for WT1, while the torque showed a slightly larger deviation but remained within an acceptable margin. Overall, these results confirm that the WT2 rotor exhibits equivalent aerodynamic performance.

The average WT2 loads are shown in Fig. . When WT2 operated in free-stream conditions at the 5D position (with WT1 removed) and a wind speed of 4.0 ms-1, its loads were consistent with those measured when the turbine was mounted on the robotic platform. At a wind speed of 2.4 ms-1, which is representative of the average velocity in full-wake conditions, the thrust force of WT2 matched the scaled thrust of the DTU 10 MW turbine used as the reference in rotor design. These results confirm that the wind tunnel provided a uniform inflow across the wind farm region and that the aerodynamic response and induction of the downstream turbine were not significantly influenced by proximity to the end of the test section. This comparison is not intended to reproduce waked inflow conditions but simply to verify that, in the absence of a wake, WT2 operates consistently at a downstream spacing of 5D, as it does at the other tested spacings.

The WT2 loads in farm scenarios with fixed WT1 were recorded across all farm configurations, with 2 to 9 repeated tests. In the configurations with aligned turbines and those with a lateral offset of 1R, the UTE for WT2 is below the rated wind speed and the loads follow adequately the values of the DTU 10 MW at model scale. In configurations with a lateral offset of 1D, the thrust-equivalent wind speed UTE for WT2 exceeds the rated wind speed, resulting in a TSR lower than the optimal value for the rotor. While the thrust remains equal to that of the turbine in free-stream conditions, as intended when defining the operating parameters, the torque is higher.

Large rotor motions induced by platform dynamics introduce dynamic variations in aerodynamic loads, which in turn alter the wake dynamics. This disturbance may enhance the natural mixing between the wake and the surrounding free-stream, leading to earlier wake recovery compared to a case where the turbine is fixed at the tower base. Earlier mixing implies that the flow regains free-stream characteristics closer to the rotor, thereby increasing the available energy for a downstream turbine. Wake velocity measurements obtained from hot-wire anemometry at the WT2 positions by showed only minor increases in mean wind speed relative to the fixed case. Larger improvements were observed in crosswind and yaw motion cases, with more significant gains at 5D than at shorter distances.

To assess the influence of upstream rotor motion on wake energy recovery, we measured the power output of WT2 at several downstream positions in the wake of WT1. Because the rotor speed was prescribed and constant in each farm configuration, the aerodynamic power of both turbines was obtained directly from the measured torque and the known rotational speed. In all tests, WT1 maintained a time-average power output of 72 W (10.93 MW at full scale). Figure  shows the ratio of WT2 power to WT1 power for various farm configurations and WT1 motion conditions, including the fixed tower-base case. The complete set of time-averaged power measurements for every farm configuration, including all amplitudes and frequencies tested for sinusoidal motions, is provided in Appendix . For sinusoidal platform motions, the values reported in Fig.  correspond for each farm configuration to the maximum WT2 power observed across all tested combinations of frequency and amplitude for that motion type.

When WT2 is fully aligned with WT1, it operates within the core of the wake and therefore produces the lowest power. In the fixed configuration, WT2 power increases only gradually with downstream distance, rising from 11 % of WT1 power at 3D to 15 % at 4D and 18 % at 5D, reflecting slow wake recovery.

These wake losses are larger than those typically reported in full-scale studies, where the power of a downstream turbine at 5D often reaches 40 %–50 % of the upstream turbine power . In the present experiments, the stronger deficit can be attributed to the combination of a high thrust coefficient of WT1 (average Ct=0.9), which generates a pronounced momentum deficit and a very low free-stream turbulence intensity (1.5 %), which limits turbulent mixing and reduces wake recovery. The value of Ct is consistent with the operating conditions adopted in previous wind tunnel investigations of floating wind turbine aerodynamics, where model rotors were typically operated near rated conditions and produced similarly high thrust coefficients . In addition, the proximity of the wind tunnel ceiling (with the rotor tip located approximately 0.2D below it) likely limits the vertical entrainment of high-momentum flow into the wake. Since vertical entrainment is a key mechanism governing wake recovery in field-scale wind farms, this confinement may further delay wake recovery and contribute to the persistence of a stronger wake deficit than would occur under open-field conditions.

It is important to note that the free-inflow characterization of WT2 at 5D (Sect. ) was performed with the robotic platform mounted on the wind tunnel floor, i.e., under the same geometric configuration adopted for the wake-interaction experiments. Under these conditions, WT1 and WT2 exhibited comparable power performance, indicating that the presence of the robotic platform does not introduce any significant alteration of the inflow under free-stream conditions. Furthermore, as shown in Appendix , the comparison between vertical and horizontal wake profiles at 3D indicates that the robotic platform does not introduce a significant distortion of the wake within the rotor-swept region. The reduced power levels of WT2 when operating in the wake of WT1 are therefore attributable to the combined effects discussed above, rather than to modifications of the inflow induced by the experimental setup, and in particular by the robotic platform.

As a result of the combined effects of the high thrust loading of the upstream turbine, the low free-stream turbulence intensity, and the vertical confinement imposed by the wind tunnel boundaries, the wake remains coherent and persistent over the investigated downstream distances, exerting a strong influence on the downstream turbine. Consequently, any additional recovery promoted by upstream turbine motion may have a more pronounced effect on downstream turbine performance in the present experimental setup than would be expected under open-field conditions. In an atmospheric boundary layer inflow, reduced confinement together with higher background turbulence would enhance vertical entrainment and likely accelerate natural wake recovery, thereby reducing the relative importance of motion-induced wake recovery.

As expected, the power generated by WT2 increases with lateral offset from the wake centerline. At an offset of 1R, WT2 produces approximately 55 % of WT1 power, as the rotor is only partially immersed in the wake. When the offset reaches 1D, WT2 generates more power than WT1, consistent with the local overspeed observed in the rotor-averaged wind-speed estimates. Local flow acceleration around turbines positioned near the edge of a wake has previously been reported in experimental wind farm studies . However, in the present experiments, the magnitude of this speed-up may also be influenced by wind tunnel confinement. The global blockage of the test section enhances lateral flow acceleration around the wake , potentially amplifying the velocity increase experienced by a partially waked turbine. Therefore, while the qualitative behavior of the speed-up is consistent with observations in the literature, its quantitative magnitude may not directly translate to unconfined flow conditions.

When WT1 undergoes platform motion, WT2 power increases under certain motion types relative to the fixed case. The most significant power gains were observed with sinusoidal yaw and surge–sway motions at reduced frequencies of 0.3 and 0.6. Wake velocity measurements reported by indicate that these motion types have the strongest impact on wake recovery. Motions driven by stochastic wave excitation at 0° incidence led to power gains comparable to those observed under sinusoidal pitch motion. For instance, in the 3D configuration, sinusoidal pitch motion resulted in a 19 % increase in WT2 power, while wave-induced motion at 0°, which excited platform pitch, produced a similar gain of 24 %. In contrast, wave excitation at 30° incidence, which introduced significant crosswind and yaw motions, did not lead to increased power output for WT2. This differs from the sinusoidal motion cases, where crosswind and yaw motions were generally associated with noticeable power gains.

While relative power gains are most pronounced when WT2 is fully aligned with WT1, they remain small in absolute terms due to the inherently low baseline power in this configuration. The highest relative power increases over the fixed-turbine case (+26 %) were observed in two scenarios: the 3D aligned configuration with WT1 undergoing yaw motion (2° amplitude, reduced frequency of 0.3) and the 5D aligned configuration with WT1 undergoing sway motion of 0.032 m amplitude (2.4 m at full scale), at a γ = 45° angle with respect to the wind and reduced frequency of 0.6. In the 3D yaw case, WT2 power increased from 11 % to 13 % of WT1 power, corresponding to a rise from 7.7 W (1.17 MW at full scale) to 9.7 W (1.47 MW). In the 5D surge–sway case, WT2 power increased from 19 % to 24 % of WT1 power, rising from 13.8 W (2.1 MW at full scale) to 17.4 W (2.64 MW).

While relative gains are most pronounced in aligned configurations, the absolute power increases are more substantial in partially misaligned layouts. For instance, in the 3D1R configuration, when WT1 underwent surge–sway motion at a reduced frequency of 0.6 with an amplitude of 0.032 m (equivalent to 2.4 m at full scale) and at a γ = 45° angle to the wind, the power output of WT2 increased from 52 % to 59 % of WT1 power. In absolute terms, this corresponds to a rise from 37.1 W (5.63 MW at full scale) to 42.5 W (6.45 MW at full scale). Conversely, when WT2 was laterally offset by a full rotor diameter (1D), the influence of WT1 motion on downstream power became negligible.

It should be noted that the present ranking of motion types in promoting wake recovery relative to the fixed case is derived under confined flow conditions. The limited vertical clearance between the rotor and the ceiling may constrain vertical wake deflection and potentially reduce the wake recovery induced by platform pitch motion. As a result, the comparatively stronger impact observed for yaw and crosswind motions in this study may be partly attributable to the flow confinement of the experimental setup.

Figure  shows the aerodynamic thrust and torque measured on WT1 and WT2 when WT1 undergoes surge and pitch motions at a reduced frequency of 0.6 and with Δu/U of 5 %; in these tests, the two turbines are aligned and spaced 3D apart. Surge and pitch motions are considered because they both involve nacelle translation in the wind direction, directly altering the apparent wind speed at the rotor, and are therefore expected to produce similar effects on the aerodynamic loads. In the figure, the load time series were binned and phase averaged according to WT1 platform motion to show variations at the platform motion frequency. The results are reported both in dimensional form (force and torque) and as non-dimensional coefficients. The thrust and torque coefficients were computed using Eqs. () and (), with the free-stream velocity U measured by the pitot tube located upstream of WT1 on the hub axis (constant during the tests at 4.0 ms-1).

In both cases, the thrust and torque of WT1 exhibits a clear sinusoidal variation at the same frequency as the platform motion. The peak loads occur at half the motion period, when the nacelle reaches its maximum upstream velocity. This behavior, well documented in previous experimental and numerical studies , confirms that the load oscillations are primarily driven by variations in apparent wind due to platform motion. The rotor loads follow these variations proportionally, indicating a quasi-steady response.

The pulsating loads on WT1 generate a periodically varying momentum deficit and shed vorticity in the wake; additionally, the rotor itself moves in space during the platform motion cycle. Together, these effects create flow structures that evolve with the same periodicity as the rotor motion and remain coherent with it as they are advected downstream. For surge and for small-amplitude pitch motion, these structures manifest as a pulsing of the wake velocity, with periodic acceleration and deceleration of the flow during the motion cycle . This pulsing behavior is illustrated in Fig. , which shows the velocity in the wake of WT1 at 3D downstream, spatially averaged along a horizontal line at hub height within the extent of the WT2 rotor (-D/2 < y < D/2). Wake velocity data are obtained from hot-wire measurements reported in . Since velocity fluctuations are not uniform across the wake (see spectra of Fig. ), this metric provides only a qualitative measure of the inflow over the WT2 rotor. Nevertheless, both surge and pitch motions produce periodic increases and decreases of the spatially averaged velocity, with comparable amplitude (≈ 1.5 % of U) and a phase shift of approximately -150° relative to WT1 platform motion (with the negative sign indicating that the WT2 wake is delayed relative to WT1).

The unsteady wake of WT1 therefore creates a periodically varying inflow for WT2. In response, WT2 exhibits periodic thrust and torque fluctuations at the same frequency as the motion of the upstream turbine. The amplitude of these oscillations is much smaller than for WT1: in the surge case, 0.25 N versus 1.98 N in thrust and 0.05 Nm versus 0.47 Nm in torque. The peaks in the WT2 loads occur with a delay of about 0.4 motion periods relative to the WT1 loads (i.e., a phase shift of ≈ -140°; the negative sign indicates that the WT2 response is delayed relative to WT1). This corresponds to a phase shift of approximately -230° relative to the WT1 motion.

The reduced amplitude of the WT2 load oscillations reflects the fact that the velocity fluctuations in the wake are of lower amplitude and less uniformly distributed across the rotor compared to the apparent wind variations acting on WT1, which are generated by its own motion. At the reduced frequency of 0.6, the induction response of the scale-model rotor to such inflow variations is expected to be quasi-steady , as also evidenced by the WT1 loads when its own motion modulates the inflow. A similar quasi-steady behavior can therefore be assumed for WT2, meaning that induction dynamics do not introduce additional phase shifts or attenuate the load oscillation amplitude. The observed time lag is therefore primarily associated with the advection of coherent flow structures shed by WT1. The corresponding advection velocity varies downstream as the wake recovers and the local mean velocity changes, and it is further influenced by the induction region upstream of the WT2 rotor.

Figure  presents the fast Fourier transform (FFT) of the thrust force of WT2 when aligned with WT1 and positioned at various downstream distances. In these tests, WT1 undergoes surge or pitch motions at reduced frequencies of 0.3, 0.6, and 1.2, with Δu/U of 5 %. The FFT is used to quantify load variations at the motion frequency and to identify whether additional harmonics appear at other frequencies.

The load spectra of WT2 consistently show a sharp peak at the frequency of WT1 platform motion, corresponding to the periodic load oscillations observed in Fig. . Across all tested configurations, this peak is clearly visible at the platform motion frequency, while the rest of the spectrum remains largely unchanged relative to the fixed case. In addition, all spectra exhibit a distinct peak at the 1P frequency of WT2, associated with imbalance loads generated by the spinning rotor. The frequency of this peak varies with the rotor speed, which differs across farm configurations (see Table ), whereas its amplitude remains essentially unchanged whether WT1 is fixed or subjected to motion.

Figures  and show that surge and pitch motions producing equal variations in apparent wind speed lead to dynamic loads of comparable amplitude on the downstream wind turbine. In general, load variations are slightly higher for pitch motion than for surge motion, likely due to the additional vertical wake meandering induced by pitch. WT2 exhibits a clear dynamic response at the frequency of WT1 motion when the reduced frequency is 0.3 or 0.6, confirming that the upstream rotor motion is the primary driver of its load oscillations. This response is not observed at fr=1.2, indicating that higher-frequency motions of the upstream platform do not significantly influence the downstream turbine. This observation is consistent with the findings of , who reported that wake velocity oscillations induced by platform motion decrease in strength as the reduced frequency increases from 0.6 to 1.2. At a reduced frequency of 1.2, the periodic velocity structures in the wake of WT1 at the WT2 location have reduced amplitude and rapidly become comparable to the background turbulence of a fixed-turbine wake.

At a reduced frequency of 0.6, the spectral peak in WT2 thrust is visible at distances of 3D and 4D from WT1 but disappears at 5D. This again agrees with previous wake measurements of , which showed that at larger distances, turbulence in the flow masks the periodic structures introduced by platform motion. The observed reduction in motion-induced effects with increasing spacing is consistent with a gradual loss of wake coherence due to turbulence and shear-layer instabilities. As the wake advects downstream, motion-imposed velocity patterns are progressively replaced by smaller-scale turbulence, diminishing their influence on the downstream rotor.

Interestingly, when WT1 is subjected to pitch motion at a reduced frequency of 0.3, a secondary spectral peak appears at twice the excitation frequency. The amplitude of this second harmonic increases with the spacing between the turbines. This peak is absent under surge motion at the same reduced frequency, despite producing similar variations in apparent wind speed and is of much lower magnitude in the other pitch cases with higher frequency and smaller rotation amplitude. This difference is attributed to the nature of pitch motion, which periodically tilts the rotor upward and downward, causing vertical deflection of the wake. As a result, the wake undergoes oscillatory vertical motion. The second harmonic in the WT2 load spectrum likely originates from this vertical displacement: each upward or downward shift of the wake produces a drop in thrust on WT2, creating a load fluctuation at twice the pitch motion frequency.

The effect is most evident at a reduced frequency of 0.3 because this case produces the largest rotor tilt amplitude (2.5°) and, consequently, the greatest vertical wake deflection. This interpretation is supported by Fig. , which shows phase-averaged velocity fields in the wake of the upstream turbine at 3D, measured with hot-wire probes as reported in for two pitch motions: reduced frequency of 0.3 with 2.5° amplitude and reduced frequency of 0.6 with 1.3° amplitude.

In both cases, the wake exhibits periodic acceleration and deceleration, but in the 2.5° case the upper portion of the wake also undergoes a clear periodic vertical displacement. This displacement is quantified by tracking, over one motion cycle, the vertical position of the upper wake minimum: the resulting peak-to-peak excursion is 0.24D for the 2.5° case and 0.11D for the 1.3° case. The larger displacement observed for the higher pitch amplitude is consistent, with the stronger second harmonic identified in the load spectrum of the downstream turbine.

The influence of lateral offset between the two turbines on the dynamic loading induced by the wake of a moving upstream turbine on a downstream rotor is examined for a pitch motion with an amplitude of 1.3° and a reduced frequency of 0.6. This case was selected because the motion produces pronounced velocity oscillations in the wake of the upstream turbine (see Fig. ) and induces clear dynamic loads on the rotor of the downstream turbine in aligned configurations.

Figure  presents the power spectral density (PSD) of the wake velocity at hub height, measured with hot-wire probes by at different downstream locations. The figure also shows the amplitude of thrust and torque oscillations at the platform motion frequency experienced by the downstream turbine. In the figure, the thrust and torque amplitudes for the “fixed” case represent the load oscillations experienced by WT2 when WT1 has a fixed tower base. These values are extracted from the load spectra evaluated at a reduced frequency of 0.6 by computing the FFT of the load signals and taking the magnitude of the spectral component at that frequency. The observed oscillations in this configuration result from turbulence generated by the upstream turbine and the natural meandering of its wake.

In all farm configurations, WT2 experiences greater load amplitudes when WT1 is subject to platform motion than when it is fixed. The dynamic loads on WT2 reach their maximum when it is aligned with WT1. In this configuration, WT2 operates fully within the wake of WT1, and velocity fluctuations at the platform motion frequency affect the entire rotor-swept area. In the velocity PSD, the spectral peak corresponding to the motion frequency is less prominent at 5D than at closer distances, as it is masked by turbulence that increases downstream of the rotor. As a result, the motion-induced loads on WT2 are also reduced at 5D, consistent with observations from Fig. . In the 4D configuration, the load amplitude is maximum. This trend is consistent for both thrust and torque. The motion of WT1 induces zero-to-peak variations in WT2 thrust of 0.4 N (equivalent to 20.3 kN at full scale) representing approximately 3 % of the average thrust developed by the turbine. The torque variation reaches 0.084 Nm (319.3 kNm at full scale), corresponding to about 12 % of the turbine average torque.

As WT2 is laterally offset from WT1, the amplitude of its dynamic load oscillations decreases. This reduction occurs because a larger portion of the WT2 rotor operates outside the wake. As the lateral offset increases, only part of the rotor is exposed to the unsteady wake flow and its coherent structures, while the remaining portion operates in relatively steady free-stream conditions. The steadier inflow on part of the rotor mitigates the effects of the wake-induced fluctuations, leading to a net reduction in dynamic loading. When the offset reaches 1 rotor diameter, motion-induced loads on WT2 become negligible. This conclusion, however, is specific to the streamwise spacing investigated in this study (3–5D). At larger spacings, wake spreading would increase the fraction of the rotor exposed to unsteady flow, potentially modifying this trend.

Figure  shows the amplitude of thrust and torque oscillations on WT2 (Fig. a and b) together with the phase-averaged wake velocity field at 3D downstream (Fig. c), generated by sinusoidal yaw motion of WT1 with a reduced frequency of 0.6 and an amplitude of 2°. The wake measurements were obtained from , and the loads are reported for WT2 positioned at different locations relative to WT1 (loads were not measured in the 4D configuration).

Phase-averaged velocity time series at hub height show a coherent lateral displacement of the wake. This is particularly evident at the wake center, which deviates from the xi axis and returns to it over the course of one motion period. By tracking the instantaneous position of the wake center over one platform motion cycle, a lateral peak-to-peak displacement of approximately 0.3D is obtained. This behavior is linked to the periodic change in the WT1 rotor orientation relative to the wind due to platform yaw motion, which deflects the wake laterally. As a result of this lateral displacement of the WT1 wake, the axial velocity field in the fixed frame of reference becomes asymmetric about the xi axis at each instant in time. With respect to the xi axis, one side of the flow field exhibits a velocity reduction (slowdown), while the opposite side shows a relative velocity increase (speed-up).

When WT2 is aligned with WT1 along the xi axis (in the 3D and 5D configurations), the periodic lateral meandering of the wake causes opposite sides of the WT2 rotor to experience axial velocity fluctuations of opposite sign: one side undergoes a velocity increase while the other experiences a velocity decrease. These local fluctuations largely cancel in the rotor-averaged thrust and torque, so their net oscillations remain small and comparable to those caused by inflow turbulence in the fixed-WT1 case. In this aligned configuration, the unsteady WT1 wake mainly generates asymmetric inflow conditions across the WT2 rotor, leading primarily to lateral force and moment fluctuations (not examined here), while the rotor-averaged thrust and torque remain only weakly affected.

When WT2 is laterally offset with respect to the xi axis (by 1R or 1D), this symmetry is broken: part of the rotor is exposed to the oscillatory velocity field associated with the wake meandering, while the remainder operates in relatively undisturbed inflow. The reduced cancellation of velocity fluctuations leads to stronger thrust and torque oscillations. In the 1R case, approximately half of the rotor is immersed in the fluctuating velocity field of the wake, producing substantial dynamic loads, whereas in the 1D case only the outer portion is affected, resulting in a smaller load response than in the 1R configuration.

In the 3D1R configuration, WT2 experiences zero-to-peak thrust variations of 0.4 N (equivalent to 20.3 kN at full scale) and torque variations of 0.074 Nm (281.2 kNm at full scale). These values are comparable to the load fluctuations observed in the 3D aligned configuration when WT1 undergoes pitch motion with an amplitude of 1.3° and a reduced frequency of 0.6. This indicates that a downstream turbine can experience similar dynamic loading either when fully immersed in a pulsating wake induced by pitch motion or when partially exposed to a laterally meandering wake caused by yaw motion.

Figure  shows the amplitude of thrust and torque oscillations on WT2 (Fig. a and b) together with the phase-averaged wake velocity field at 3D downstream (Fig. c), generated by sinusoidal surge–sway motion of WT1 defined in Sect. . The motions are characterized by a reduced frequency fr=0.6, a displacement amplitude am = 0.032 m (model scale), and different angles γ relative to the wind direction. The wake measurements were obtained from hot-wire data reported by , and the loads are shown for WT2 positioned at different locations relative to WT1 (loads were not measured for the 4D configuration).

When γ = 0°, the motion corresponds to pure surge, producing a pulsing behavior of the wake characterized by velocity oscillations concentrated near the wake center. By tracking the instantaneous position of the wake center over one motion cycle, a limited lateral peak-to-peak displacement of approximately 0.07D is obtained. As the motion angle increases, the wake exhibits more pronounced lateral meandering, similar to what is observed in yaw motion. The corresponding peak-to-peak displacement of the wake increases to about 0.25D for γ = 30° and reaches 0.28D for γ = 90°. For comparison, the lateral displacement of the nacelle is much smaller. In the γ = 30° case, the lateral motion amplitude is ≈ 0.014D, while for γ = 90° it is ≈ 0.026D. The resulting wake center displacement is therefore about 18 times larger than the rotor lateral motion in the γ = 30° case and about 11 times larger in the γ = 90° case. This clearly indicates that the perturbation introduced by rotor motion is significantly amplified as it develops through the wake, in agreement with the wind tunnel experiments of , who reported a pronounced lateral wake meandering induced by sway motion.

In the farm configuration with aligned turbines at a distance of 3D, the amplitude of the WT2 load oscillations at the platform motion frequency decreases as the motion angle γ increases. This trend reflects a shift from wake pulsation to lateral meandering. As explained in the yaw motion case, meandering-induced velocity fluctuations tend to cancel out across the rotor when the turbines are aligned, reducing dynamic thrust and torque variations. At 5D spacing, this behavior is less pronounced: the dynamic loads on WT2 remain nearly constant for motion angles of up to 45° but show a noticeable increase at γ = 90°. This rise in loading is likely due to pure sway motion which induces strong velocity oscillations in the wake that do not fully cancel out across the rotor.

In farm configurations where WT2 is laterally offset, the trend of loads reverses compared to aligned configurations. Here, the amplitude of dynamic loads increases with the motion angle, as only a portion of WT2 rotor is exposed to the periodic lateral wake meandering. As in the yaw motion case, the dynamic loads are higher in the 1R configurations than in the 1D configurations, reflecting the greater extent of rotor-wake interaction. In the 3D1R configuration, when WT1 undergoes motion at a 30° angle to the wind direction, WT2 experiences zero-to-peak thrust variations of 0.37 N (18.7 kN at full scale) and torque variations of 0.064 Nm (243.2 kNm at full scale). These values are similar to load fluctuations seen in the 3D aligned configuration with pitch motion and in the 3D1R configuration with yaw motion. The load amplitudes during crosswind motions reach their maximum when WT1 moves orthogonally to the wind direction (γ = 90°), with thrust variations of 0.61 N (30.9 kN at full scale) and torque variations of 0.109 Nm (418.1 kNm at full scale).