3.1. Wake recovery mechanisms

We begin the results section by presenting findings from the shared cases of the Milan and Oldenburg experiments. In Appendix B we detail the close agreement between the two datasets, as well as the methodology used to compute the various terms of (2.6), which supports the combined use of both datasets. The 3-D wake fields from Milan enable the computation of the $\partial /\partial y$ and $\partial /\partial z$ terms, while the 1-D measurements from Oldenburg, with their higher resolution in the streamwise direction, are used to compute the $\partial /\partial x$ terms. Details regarding the estimation of uncertainty in the derivative calculations are also provided in Appendix B.

For the three shared motion cases (fixed-5, surge-5 and sway-5; see table 2), figure 4(a) shows the evolution of wake recovery and figure 4(b) presents the normalised recovery rate, $D \partial R_w / \partial x$ . The wake of the moving turbine recovers more rapidly – with approximately $14 \,\%$ higher recovery for the sway case and $3 \,\%$ for surge compared with the fixed case at $x = 6D$ . Correspondingly, the recovery rate (figure 4 b) is higher for both sway and surge motions up to $x \leqslant 5D$ . This enhancement in wake recovery is less pronounced than in cases with laminar inflow, as reported by Li et al. (Reference Li, Dong and Yang2022) and Messmer et al. (Reference Messmer, Hölling and Peinke2024a ). Compared with surge motion, sway leads to a $11 \,\%$ higher recovery, even though both cases share the same values of $St$ and $A^*$ . This result provides an initial indication that sway motion has a larger impact on wake recovery enhancement than surge motion – a point we investigate in more detail later in § 3.2.

Figure 4. (a) Wake recovery (2.2) for the fixed-5-O, surge-5-O and sway-5-O cases ( $St = 0.38$ , $A^* = 0.01$ ). Error bars represent the average discrepancy between the Milan and Oldenburg datasets (Appendix B). (b) Wake recovery rate. Error bars represent the estimated uncertainty based on a Monte Carlo approach (Appendix B). Both quantities are derived from the Oldenburg dataset.

In the following, we examine the mechanisms behind the enhanced wake recovery found with rotor motion. Figure 5 presents horizontal ( $y$ ) profiles of the individual terms in the recovery rate budget. At $x = 4D$ (figure 5 a–c), the black dotted line highlights a larger value of $\partial \overline {u}/\partial x$ for the sway case (figure 5 c) compared with the fixed case (figure 5 a), particularly within $y \in [-0.5D, 0.5D]$ . While the fixed and surge cases still exhibit characteristics of a developing wake – evidenced by a double-Gaussian-like shape in $\partial \overline {u}/\partial x$ – the sway case presents a developed wake profile, already resembling the single-Gaussian distribution typical of the far wake (Neunaber et al. Reference Neunaber, Hölling, Stevens, Schepers and Peinke2020). This suggests that sway motion promotes a faster transition to the far-wake regime. By $x = 8D$ (figure 5 d–f), all cases display Gaussian-like $\partial \overline {u}/\partial x$ profiles, confirming that the far wake has been reached for all cases. These findings are consistent with observations under laminar inflow conditions reported by Messmer et al. (Reference Messmer, Hölling and Peinke2024a ), where platform motion was shown to accelerate wake development.

Returning to the profiles at $x = 4D$ , the primary contributor to the positive $\partial \overline {u}/\partial x$ appears to be the term $-(1/\overline {u}) \partial \overline {u'v'} / \partial y$ , i.e. the gradient of Reynolds shear stress in the $x$ – $y$ plane. The gradient of $u'w'$ (the $x$ – $z$ component) also contributes positively to wake recovery. As discussed in Boudreau & Dumas (Reference Boudreau and Dumas2017), van der Laan et al. (Reference van der Laan, Baungaard and Kelly2023), Gambuzza & Ganapathisubramani (Reference Gambuzza and Ganapathisubramani2023), these terms are known to dominate wake recovery processes. The negative regions of $\partial \overline {u}/\partial x$ observed at the edges of the wake ( $|y| \gt 0.5D$ ) act as sinks of recovery, supplying momentum toward the wake centre where $\partial \overline {u}/\partial x \gt 0$ , consistent with the interpretation of van der Laan et al. (Reference van der Laan, Baungaard and Kelly2023). The $y$ -rotational term, $-(1/\overline {u}) \overline {v} \partial \overline {u}/\partial y$ , is positive and, thus, contributes mathematically to $\partial \overline {u}/\partial x \gt 0$ ; whereas the other rotational term, $-(1/\overline {u}) \overline {w}\partial \overline {u}/\partial z$ is negative. The sum of the two appears to be counterbalanced by the remaining residual, which is mathematically equivalent to the pressure gradient term. Finally, figure 5(d–f) shows that at $x = 8D$ the recovery rate is higher for the fixed case (figure 5 d) than for the sway case (figure 5 f), in agreement with the results shown previously in figure 4(b). At this downstream location, all cases have transitioned to the far-wake regime, where recovery is primarily driven by gradients in Reynolds shear stresses in both the $x$ – $y$ and $x$ – $z$ planes.

Figure 5. The $y$ profiles of the terms in the recovery rate budget (2.6) for the fixed-5 (a,d), surge-5 (b,e) and sway-5 (c,f) cases. Panels (a–c) correspond to $x = 4D$ and panels (d–f) to $x = 8D$ . In the legend, (M) indicates terms computed using Milan data, while (O) indicates those based on Oldenburg data.

To provide a broader view, we integrate all terms of the recovery rate budget (2.6) over the rotor diameter, following the same approach used in (2.2) for $\overline {u}$ . This integration yields a quantitative assessment of each term’s contribution (rotation, pressure, Reynolds stresses) to the overall recovery rate. Figure 6 presents the downstream evolution of these integrated contributions. Sway motion leads to a substantial increase in the Reynolds stress gradient terms in both the streamwise – spanwise and streamwise–vertical planes – up to $65 \,\%$ higher than in the fixed case (blue $\blacklozenge$ symbols in figure 6(a,c). In contrast, surge motion shows only a modest enhancement of these terms. Meanwhile, mean flow transport terms (associated with $\overline {v}$ and $\overline {w}$ ) decay more rapidly for sway – see the yellow $\blacksquare$ symbols at $x = 6D$ in figure 6(c) – compared with fixed and surge cases. The turbulent transport in the $x$ direction (grey $\times$ ) acts as a mild sink of recovery in all cases.

Notably, the residual term (green $\blacktriangle$ ) emerges as the dominant sink, suggesting that a large portion of the momentum rate budget is associated with pressure restoration or rotation-induced pressure effects. Noriega & Mazellier (Reference Fuentes Noriega and Mazellier2025) showed, using a porous disk with tilted blades, that wake rotation can induce a low-pressure core in the wake. This low-pressure region likely counteracts the potential benefits of rotational effects on wake recovery. In fact, between $x = 4D$ and $8D$ , the rotation contribution (yellow $\blacksquare$ ) appears nearly balanced by the residual, potentially indicating a negligible net impact on recovery.

Most importantly, for $x \in [2D, 6D)$ , the enhanced recovery observed for sway primarily originates from the increased Reynolds stress gradient terms: $-\langle (1/\overline {u}) \partial \overline {u' u^{\prime}_i} /\partial x_i\rangle$ with $i \in {y,z}$ . These terms represent turbulent transport in the $x$ – $y$ and $x$ – $z$ planes, and have been consistently identified as key contributors to wake recovery in previous studies on fixed wind turbines (Bastankhah & Porté-Agel 2016; Boudreau & Dumas Reference Boudreau and Dumas2017; Gambuzza & Ganapathisubramani Reference Gambuzza and Ganapathisubramani2023; van der Laan et al. Reference van der Laan, Baungaard and Kelly2023). Finally, for $x \gtrsim 8D$ , the far-wake regime is reached, where the recovery rate $D \partial R_w / \partial x$ is well approximated by the combined Reynolds shear stress gradients, i.e. $D \partial R_w / \partial x \approx -\langle (1/\overline {u}) (\partial \overline {u' v'} / \partial y + \partial \overline {u' w'} / \partial z ) \rangle$ where rotation and pressure terms are zero (Gambuzza & Ganapathisubramani Reference Gambuzza and Ganapathisubramani2023).

Figure 6. Downstream evolution of recovery equation terms (2.6) integrated over the rotor diameter for fixed-5 (a), surge-5 (b) and sway-5 (c). Panels (d–f) show $x$ – $y$ Reynolds shear stress gradients; panels (g–i) show $x$ – $z$ gradients for the same cases.

For a clearer insight into the underlying processes, we next explore the decomposition of the Reynolds shear stresses. In the $x{-}y$ plane the total term is expressed as $-\langle 1/\overline {u} \partial \overline {u' v'}/ \partial y \rangle = - \langle 1/\overline {u} \partial \overline {\widetilde {u} \widetilde {v}} / \partial y \rangle - \langle 1/\overline {u} \partial \overline {u'' v''} / \partial y \rangle$ (see figure 6 d–f) and in the $x{-}z$ plane it is written as $- \langle 1/\overline {u} \partial \overline {u' w'}/ \partial z \rangle = - \langle 1/\overline {u} \partial \overline {\widetilde {u} \widetilde {w}} / \partial z \rangle - \langle 1/\overline {u} \partial \overline {u'' w''} / \partial z \rangle$ (see figure 6 g–i). Here, the tilde terms $\overline {\widetilde {u} \widetilde {v}}$ and $\overline {\widetilde {u} \widetilde {w}}$ represent the contributions from the coherent structures induced by platform motion at a given $St$ , while the double-prime terms $\overline {u'' v''}$ and $\overline {u'' w''}$ capture the contributions from purely stochastic turbulence.

Figure 6(f) shows that the total gradient of the Reynolds shear stress in the $x{-}y$ plane is significantly enhanced for the sway case compared with the fixed case (compare the blue $\blacklozenge$ symbols in figures 6 d and 6 f). Surprisingly, for sway and $3D \lesssim x \lesssim 10D$ , more than half of the total $x{-}y$ term is covered by $\left \langle 1/\overline {u} \partial \overline {\widetilde {u} \widetilde {v}} / \partial y \right \rangle$ , i.e the coherent structure contribution. In contrast, the $x{-}z$ plane (figure 6 i) shows both a lower overall Reynolds shear stress gradient and a relatively smaller contribution from the coherent structure subterm. Although surge motion (figures 6 e and 6 h) also exhibits contributions from coherent structures in both the $x{-}y$ and $x{-}z$ planes, these are much lower than those observed for sway, reflecting its reduced impact on wake recovery.

These results conclusively show that the coherent structures induced by platform motion play a critical role in enhancing wake recovery. Consistent with previous findings on fixed wind turbines (Boudreau & Dumas Reference Boudreau and Dumas2017; van der Laan et al. Reference van der Laan, Baungaard and Kelly2023), our study shows that the gradients of Reynolds shear stress in the $x{-}y$ and $x{-}z$ planes are central to the recovery process. In the sway case, the improved recovery – evidenced by the higher $D \partial R_w / \partial x$ in figure 4(a) – is directly linked to increased Reynolds stress gradient terms: $- \langle 1/\overline {u} \partial \overline {u' v'}/ \partial y \rangle - \langle 1/\overline {u} \partial \overline {u' w'}/ \partial z \rangle$ .

Notably, a substantial portion of these gradients is attributable to the contributions of coherent structures – an observation that supports our earlier speculation in Messmer et al. (Reference Messmer, Hölling and Peinke2024a ). Previous studies, such as Yılmaz & Meyers (Reference Yılmaz and Meyers2018), Korb et al. (Reference Korb, Asmuth and Ivanell2023), Van der Hoek et al. (Reference van der Hoek, den Abbeele, Simao Ferreira and van Wingerden2024), have similarly linked the presence of coherent structures from periodic excitations to mechanisms like wake deflection and enhanced momentum entrainment. In our work, the effect of these mechanisms on recovery are quantitatively represented by the two terms: $-\langle 1/\overline {u} \partial \overline {\widetilde {u} \widetilde {v}}/ \partial y \rangle - \langle 1/\overline {u} \partial \overline {\widetilde {u} \widetilde {w}}/ \partial z \rangle$ .

Later in the paper we further decompose wake recovery by distinguishing the effects of periodic wake deflection and enhanced momentum entrainment, both of which are influenced by periodic structures, as discussed in Korb et al. (Reference Korb, Asmuth and Ivanell2023). These factors are, to some extent, hidden in our recovery rate budget analysis.

3.2. Wake profiles and recovery with increasing free-stream turbulence

So far, we have investigated wake recovery for a specific inflow condition with $\textit{TI}_{\infty } = 1.5 \,\%$ and fixed values of $St$ and $A^*$ . Next, we present the results on wake profiles and recovery for a range of inflows with different $\textit{TI}_{\infty }$ ( $\textrm{O}1.1\!-\!\textrm{O}5.8$ in table 1) and for various motion cases with $St = 0.3$ and $A^* \in \{0.01,0.017,0.024\}$ (see table 2). As detailed in § 2.1, the active grid at the University of Oldenburg was used to achieve $\textit{TI}_{\infty }$ levels in the range $[1.1, 5.8] \,\%$ . These experiments allow us to assess the combined effects of increasing inflow turbulence and platform motion on wake development and recovery.

Figure 7. The $y$ profiles of normalised wake deficit (a) and turbulence intensity (b) for fixed, surge- and sway- 3–0.01 and 0.024, i.e $St = 0.3, A^* = 0.01,\,0.024$ (see table 2) for O1.1 (a–0)–(a–4), O3.0 (b–0)–(b–4), O5.8 (c–0)–(c–4) and $x \in [2,10]D$ (0,4). Inflows in table 1.

We present in figure 7 the lateral ( $y$ ) profiles of the normalised wake deficit $\Delta u^* = (U_{\infty } - \overline {u}) / U_{\infty }$ (figure 7 a) and turbulence intensity $ TI = \sqrt {\overline {u^{\prime 2}}}/\overline {u}$ (figure 7 b). The data are organised as follows.

1. (i) Three inflow conditions, with $\textit{TI}_{\infty } \in \{1.1, 3.0, 5.8 \} \,\%$ , corresponding to cases O1.1, O3.0 and O5.8 from table 1), shown respectively in (a–0)–(a–4), (b–0)–(b–4), (c–0)–(c–4).

2. (ii) Two motion amplitudes per motion type ( $A^*=0.01,\,0.024$ ) for both surge and sway, with $St = 0.3$ .

3. (iii) Five downstream locations, $x \in [2,10]D$ denoted (.–0)–(.–4), where ‘.’ stands for a, b or c.

Figure 7(a) (a–0), (b–0) and (c–0) reveal that, in the near wake at $x = 2D$ , the wake deficit profiles are nearly identical across all motion cases for a given inflow condition. This observation, consistent with earlier findings by Fontanella, Zasso & Belloli (Reference Fontanella, Zasso and Belloli2022) and Messmer et al. (Reference Messmer, Hölling and Peinke2024a ) (in low turbulent inflows, $\textit{TI}_{\infty } \lt 2 \,\%$ ), suggests that the mean induction, i.e. the average slowing of the flow by the turbine, is effectively unchanged between cases. In other words, $\overline {C_T}$ is constant across motion cases. This equivalence is crucial as it ensures that any observed differences in wake evolution further downstream can be attributed to platform motion effects rather than discrepancies in the initial near-wake, rotor-induction conditions.

As we move downstream, figure 7(a) (a–0)–(a–4) illustrate the pronounced impact of platform motion on the wake deficit under low inflow turbulence ( $\textit{TI}_{\infty } = 1.1 \,\%$ ). Both surge and sway cases show enhanced wake recovery relative to the fixed case, with the most significant effect observed for sway motion at the highest amplitude ( $A^* = 0.024$ , green $\times$ ). This case produces a much broader and more uniform velocity deficit profile compared with the fixed case (black dots), indicating that sway motion promotes a wider wake and accelerates lateral wake expansion. Notably, by $x = 10D$ , the wake profile for this sway case deviates from the classic Gaussian shape, potentially challenging conventional analytical wake models (e.g. the Bastankhah & Porté-Agel (Reference Bastankhah and Porté-Agel2014) wake model). In contrast, the lower amplitude case (green triangle) yields a wake profile more similar to the fixed case, suggesting a weaker influence on wake growth. Comparing the two motion directions, sway induces more noticeable changes in the wake mean structure than surge, which remains closer to the fixed reference across all downstream positions.

As inflow turbulence increases, the influence of motion diminishes. At $\textit{TI}_{\infty } = 3.0 \,\%$ , the sway-induced differences are still visible (figure 7 a(b–2)–(b–4)), but to a lesser extent. For surge motion, however, this turbulence level appears to mark a threshold – no discernible difference is observed between the surge and fixed cases, even at the highest motion amplitude. Finally, under higher turbulence ( $\textit{TI}_{\infty } = 5.8 \,\%$ ), all wake deficit profiles converge, indicating that inflow turbulence dominates the wake mean profiles and suppresses any motion-induced effects.

Concordantly, the differences in the turbulence intensity profiles between the motion cases and the fixed case are most pronounced under low inflow turbulence ( $\textit{TI}_{\infty } = 1.1\,\%$ ), as shown in figure 7(b) (a–0)–(a–4).

Surge motion primarily increases the level of turbulence intensity within the rotor region ( $|y| \lt 0.5D$ ); for instance, at $x = 4D$ , the surge case with $A^* = 0.024$ (blue squares) shows a pronounced increase in turbulence intensity compared with the fixed case (black dots), as seen in figure 7(b) (a–1). This elevated turbulence accelerates the transition to the far-wake regime, as detailed in Messmer et al. (Reference Messmer, Hölling and Peinke2024a ). The influence of motion amplitude is clear: the lower amplitude case ( $A^* = 0.01$ ) induces only a slight increase in turbulence intensity, consistent with its weaker effect on the wake deficit seen in figure 7(a) (a–0)–(a–4).

Sway motion, by contrast, predominantly enhances turbulence intensity at the edges of the wake ( $|y| \gt 0.5D$ ), particularly at downstream locations ( $x \in [8D, 10D]$ ); see green crosses in figures 7(b) (a–3) and 7(b) (a–4). Similar to the trends observed for $\Delta u^*$ , lateral platform motion results in a broader and more uniform turbulence intensity distribution, with a lower intensity in the wake centre and a higher intensity at the edges – indicating a wider and more dispersed wake for the larger amplitude sway case.

As $\textit{TI}_{\infty }$ increases, the influence of platform motion on wake turbulence intensity diminishes. At $\textit{TI}_{\infty } = 3.0\,\%$ , surge motion with $A^* = 0.024$ leads to a moderate increase in near-wake turbulence for $x \leqslant 4D$ (figure 7 b (b-0)–(b-1)). Sway motion at this turbulence level still causes slight wake broadening in the far wake ( $x \geqslant 8D$ ), though the effect is noticeably reduced compared with the lower $\textit{TI}_{\infty }$ case (figure 7 b (b–3)–(b–4)). For both surge and sway, the lower amplitude case ( $A^* = 0.01$ ) shows no discernible effect on turbulence intensity profiles when $\textit{TI}_{\infty } \geqslant 3\,\%$ .

Finally, at the highest inflow turbulence ( $\textit{TI}_{\infty } = 5.8\,\%$ ), platform motion has negligible impact on turbulence intensity across all downstream positions beyond $x = 2D$ (figure 7 b (c–1)–(c–4)).

This analysis of the turbulence intensity profiles, in agreement with the wake deficit observations, confirms that the effect of platform motion weakens as inflow turbulence increases. This trend aligns well with previous CFD studies by Yılmaz & Meyers (Reference Yılmaz and Meyers2018) and Li et al. (Reference Li, Dong and Yang2022).

We next quantify the differences between the fixed case and the motion cases for multiple inflow turbulence levels $\textit{TI}_{\infty }$ and all amplitudes of motion. To do so, we compute the recovery using (2.2) for the fixed case ( $St = 0$ ), as well as for surge and sway motions at $St = 0.3$ with $A^* \in \{0.01, 0.017, 0.024\}$ (table 2). Figure 8(a–f) shows the results for the different inflow turbulence conditions $\textit{TI}_{\infty } \in \{1.1, 2.2, 3.0, 4.1, 5.0, 5.8\}\,\,\%$ .

As noted earlier in the wake deficit profiles (figure 7 a), for a given inflow condition, all cases – fixed or with platform motion – exhibit nearly identical wake recovery at $x = 2D$ . The effect of platform motion on the wake develops progressively farther downstream, driven by nonlinear interactions (Messmer et al. Reference Messmer, Hölling and Peinke2024a ). Between the different inflow cases, $R_w(x=2D)$ varies slightly but remains close to $R_w \approx 0.5$ .

The clearest differences between motion cases appear for the lowest free-stream turbulence level $\textit{TI}_{\infty } = 1.1\,\,\%$ (figure 8 a), consistent with the wake profiles in figure 7(a). Sway motion with $A^* = 0.024$ improves recovery by up to $21 \,\%$ compared with the fixed case. For both surge and sway, the impact is less with decreasing motion amplitude; for example, sway with $A^* = 0.01$ yields a $12 \,\%$ increase in recovery over the fixed case. Interestingly, even the largest surge amplitude results in a weaker recovery enhancement than sway motion at the lowest amplitude.

As $\textit{TI}_{\infty }$ increases (figure 8 b–f), the influence of platform motion diminishes. At $\textit{TI}_{\infty } = 2.2\,\,\%$ (figure 8 b), surge with $A^* = 0.024$ enhances recovery by only $3.5 \,\%$ , while sway still yields an $11 \,\%$ increase. Sway at $A^* = 0.01$ continues to improve recovery by $4 \,\%$ , whereas surge at the same amplitude has no observable effect. At $\textit{TI}_{\infty } = 3.0\,\,\%$ (figure 8 c), surge and fixed recovery curves overlap entirely, while sway continues to provide a $ 5 \,\%$ improvement. The effect of sway becomes almost negligible around $\textit{TI}_{\infty } = 5.0\,\,\%$ , and at $\textit{TI}_{\infty } = 5.8\,\,\%$ all cases, fixed, surge and sway, converge.

Figure 8 highlights three important trends. First, sway motion has a stronger influence on wake recovery than surge, a result we further quantify and discuss later in this paper. Second, larger motion amplitudes lead to greater recovery improvements, as also pointed out by Li et al. (Reference Li, Dong and Yang2022), Messmer et al. (Reference Messmer, Hölling and Peinke2024a ). Third, the effect of platform motion does not add linearly to the effect of inflow turbulence; rather, the two blend. To highlight this point, we present in figure 9 the downstream evolution of offset-adjusted wake recovery, denoted as $ R^*_w$ , for the fixed case (figure 9 a) and for the sway case with $ A^* = 0.024$ (figure 9 b), across all levels of inflow turbulence, $ \textit{TI}_{\infty } \in [1.1, 5.8]\,\%$ .

We define the adjusted recovery as

(3.1) \begin{align} R^*_w(x) = R_w(x) - R_w(x=2D) + 0.5 \end{align}

in order to align all curves at a common starting point of 0.5 at $ x = 2D$ , thereby removing small variations in $ R_w(x=2D)$ that arise from minor dependencies on $ \textit{TI}_{\infty }$ .

Figure 8. Wake recovery from (2.2) for different free-stream turbulence levels: $\textit{TI}_{\infty } = 1.1\,\% \!-\!5.8\,\%$ (cases O1.1–O5.8 in table 1). Results include fixed ( $St = 0$ ), surge and sway motions at $St = 0.3$ with $A^* \in \{0.01, 0.017, 0.024\}$ (table 2).

Figure 9. Offset-adjusted wake recovery: $R^*_w(x) = (R_w(x) - R_w(x=2D))+0.5$ ( $R_w$ defined in (2.2) for different free-stream turbulence levels: $\textit{TI}_{\infty } = 1.1\,\%\!-\!5.8\,\%$ (cases O1.1–O5.8 in table 1). Panel (a) shows the fixed case ( $St = 0$ ) and panel (b) sway with $St = 0.3$ and $A^* = 0.024$ (table 2).

In the fixed case, wake recovery increases with inflow turbulence. For example, $ R_w(x=10D)$ increases from approximately 0.7 at $ \textit{TI}_{\infty } = 1.1\,\%$ (dark blue in figure 9 a) to about 0.83 at $ \textit{TI}_{\infty } = 5.8\,\%$ (light yellow in the same figure), in agreement with trends reported in the literature, such as Bastankhah & Porté-Agel (Reference Bastankhah and Porté-Agel2014). This confirms the general observation that higher levels of free-stream turbulence lead to faster wake recovery.

In contrast, for the sway case with $ A^* = 0.024$ (figure 9 b), the recovery appears largely independent of inflow turbulence: $ R_w(x=10D) = 0.83 \pm 0.01$ across all $ \textit{TI}_{\infty }$ levels. A slight effect of turbulence is visible around $ x = 4D$ , where higher turbulence modestly enhances recovery; however, this difference diminishes farther downstream. This suggests that while higher $ \textit{TI}_{\infty }$ can slightly accelerate the onset of wake recovery, the influence of platform motion is substantially stronger. Remarkably, platform motion at low $ \textit{TI}_{\infty }$ produces recovery levels equivalent to those achieved with high turbulence in the fixed case, emphasising the dominant role of platform movements in accelerating wake recovery. This effect is examined next in relation to motion-induced coherent structures.

3.3. Motion-induced coherent structures with increasing $\textit{TI}_{\infty }$

In figure 6 we found that coherent structures formed in the wake due to platform motion contribute to enhanced recovery by increasing the gradients of Reynolds shear stresses in the $x{-}y$ and $x{-}z$ planes. However, as discussed in figures 8 and 9, this improvement diminishes with increasing $\textit{TI}_{\infty }$ . To investigate the connection between this trend and the evolution of coherent structures at higher turbulence intensities, we examine the phase-resolved wake dynamics. The Oldenburg experiments, which use an array of 21 hot wires to simultaneously measure the streamwise velocity along a vertical line, enable the computation of $\widetilde {u}(\phi ^*)$ at fixed streamwise positions. The measurements span $y \in [-1.4, 1.4]D$ and are synchronised – meaning that for a given $x$ , all probes share the same initial phase $\phi _0$ .

Figure 10. (a) Sum of the mean wake flow and the surge-induced coherent fluctuation. (b) Surge-induced coherent fluctuation alone. All velocities are normalised by the free-stream velocity $U_{\infty }$ . Columns (- .0)–(- .4) correspond to streamwise positions $x/D \in \{2, 4, 6, 8, 10\}$ . Each panel represents one full period of the motion, shown over the normalised phase $\phi ^* \in [0, 2\pi ]$ . Rows correspond to different inflow turbulence levels: (a.x) $\textit{TI}_{\infty } = 1.1\,\%$ , (b.x) $\textit{TI}_{\infty } = 3.0\,\%$ and (c.x) $\textit{TI}_{\infty } = 5.8\,\%$ , corresponding to cases O1.1, O3.0 and O5.8 in table 1. The motion parameters are $St = 0.3$ and $A^* = 0.024$ (case surge-3-0.024 in table 2).

We begin by analysing surge-induced periodic structures, illustrated in figure 10 for platform motion characterised by $St = 0.3$ and $A^* = 0.024$ (case surge-3-0.024 in table 2). Figure 10(a) shows contour plots of the combined mean and coherent streamwise velocity $(\overline {u} + \widetilde {u})/U_{\infty }$ , while figure 10(b) presents the coherent fluctuation component $\widetilde {u}/U_{\infty }$ alone. Each panel (a.-), (b.-) and (c.-) corresponds to a different inflow turbulence intensity, $\textit{TI}_{\infty } \in \{1.1, 3.0, 5.8\}\,\%$ – denoted as O1.1, O3.0 and O5.8 in table 1 – and spans one full oscillation cycle ( $\phi ^* \in [0, 2\pi ]$ ) at downstream positions $x \in [2,10]D$ . At low turbulence ( $\textit{TI}_{\infty } = 1.1\,\%$ ), the combined velocity field shows a distinct periodic pattern of alternating bulges and constrictions, especially between $x \in [6,8]D$ (see panels a.2–a.3 in figure 10 a). The corresponding fluctuation fields in figure 10(b) (a.0)–(a.4) display in-phase red and blue bands – indicative of in-phase alternating speed-up and slow-down – characterising a pulsating wake. This structure is analogous to the varicose mode observed in a planar wake excited periodically in Wygnanski, Champagne & Marasli (Reference Wygnanski, Champagne and Marasli1986). The pulsating structure is likely resulting from symmetric vortex roll-up in the shear layer due to the platform’s surging motion. Similar patterns have been reported for thrust-excited and surging floating turbines in the studies of Yılmaz & Meyers (Reference Yılmaz and Meyers2018), Messmer et al. (Reference Messmer, Hölling and Peinke2024a ), Li & Yang (Reference Li and Yang2024), Li et al. (Reference Li, Yu and Sarlak2024), Wei et al. (Reference Wei, El Makdah, Hu, Kaiser, Rival and Dabiri2024), Hubert et al. (Reference Hubert, Conan and Aubrun2025).

For $\textit{TI}_{\infty } = 1.1\,\%$ , coherent fluctuations reach amplitudes up to $15 \,\%$ of the free-stream wind speed, underscoring the strong influence of platform motion under low turbulence conditions. The downstream evolution of the mode shows an initial low amplitude near $x = 2D$ , which increases up to a peak around $x = 6D$ before decaying further downstream. This trend is consistent with observations at a fixed lateral position, which we reported in Messmer et al. (Reference Messmer, Hölling and Peinke2024a ). As $\textit{TI}_{\infty }$ increases, the amplitude of coherent fluctuations decreases substantially. At $\textit{TI}_{\infty } = 3.0\,\%$ , the pulsating pattern remains visible (see figure 10 b (b.0)–(b.4)) but is less pronounced, with amplitudes reduced by a factor of approximately three. At the highest turbulence level ( $\textit{TI}_{\infty } = 5.8\,\%$ ), coherent structures are barely discernible, with amplitudes nearly six times lower than in the low turbulence case. At this level of inflow turbulence, the development of coherent wake structures is significantly inhibited, consistent with the CFD analysis of Yılmaz & Meyers (Reference Yılmaz and Meyers2018).

Figure 11. (a) Sum of the mean wake flow and the sway-induced coherent fluctuation. (b) Sway-induced coherent fluctuation alone. All velocities are normalised by the free-stream velocity $U_{\infty }$ . Columns (-.0)–(-.4) correspond to streamwise positions $x/D \in \{2, 4, 6, 8, 10\}$ . Each panel represents one full period of the motion, shown over the normalised phase $\phi ^* \in [0, 2\pi ]$ . Rows correspond to different inflow turbulence levels: (a.x) $\textit{TI}_{\infty } = 1.1\,\%$ , (b.x) $\textit{TI}_{\infty } = 3.0\,\%$ and (c.x) $\textit{TI}_{\infty } = 5.8\,\%$ , corresponding to cases O1.1, O3.0 and O5.8 in table 1. The motion parameters are $St = 0.3$ and $A^* = 0.024$ (case sway-3-0.024 in table 2).

An equivalent analysis can be performed for sway motion. Figure 11 presents results for sideways platform motion characterised by $St = 0.3$ and $A^* = 0.024$ (case sway-3-0.024 in table 2), showing the same inflow conditions as for surge in figure 10. In contrast to surge, sway-induced platform motion causes strong lateral displacement of the wake, especially pronounced at $\textit{TI}_{\infty } = 1.1\,\%$ (see figure 11 b (a.0)–(a.4)). The phase-averaged streamwise wake velocity, combined with the mean flow, exhibits a sinusoidal shape that causes the entire wake to oscillate laterally. This lateral motion contributes to the increased wake spreading observed in figure 7. For example, at $x = 8D$ , the side-to-side amplitude of wake motion reaches approximately one rotor diameter, as shown by the $y$ displacement between the wake centre at $\phi ^* = 0$ and $\phi ^* = \pi$ in figure 11(a) (a.3). This illustrates how the relatively small lateral rotor motion ( $A_p = 0.024D$ ) is strongly amplified in the wake – a phenomenon we previously discussed in Messmer et al. (Reference Messmer, Hölling and Peinke2024a ) in terms of nonlinear dynamics.

Interestingly, the shape of the phase-averaged wake bears a resemblance to the flow past a cylinder undergoing vortex shedding, as described by Cantwell & Coles (Reference Cantwell and Coles1983), and also to the flow behind high-solidity porous disks (see the smoke visualisations in Cannon, Champagne & Glezer (Reference Cannon, Champagne and Glezer1993)). The periodic lateral wake motion shares similarities with the sinuous mode identified by Wygnanski et al. (Reference Wygnanski, Champagne and Marasli1986) in planar wakes and the coherent structures found in excited jets (Crow & Champagne Reference Crow and Champagne1971). These analogies suggest that many shear flows exhibit an intrinsic sinuous mode, which, when excited at the appropriate frequency or under favourable conditions (e.g. low turbulence and strong shear layers), can develop into large-scale coherent structures due to shear layer instabilities – even in the presence of background turbulence, as observed in our case. In the context of wind energy, this phenomenon is referred to as coherent wake meandering and studied in detail in Mao & Sørensen (Reference Mao and Sørensen2018), Li et al. (Reference Li, Dong and Yang2022). In our case, the meandering motions are triggered by the lateral turbine movements.

Accordingly – and in contrast to the surge-induced structures – the meandering patterns in figure 11(b) (a.0)–(a.4) exhibit anti-phase velocity fluctuations: $\widetilde {u}(\phi ^* = \pi , y = -R) \approx -\widetilde {u}(\phi ^* = \pi , y = R)$ . The amplitude of these streamwise meandering fluctuations reaches up to 20 % of the free-stream velocity. As with surge, increasing inflow turbulence reduces the magnitude of these coherent structures: at $\textit{TI}_{\infty } = 3.0\,\%$ amplitudes are roughly halved, while at $\textit{TI}_{\infty } = 5.8\,\%$ they are reduced by a factor of six. Similar to the results obtained for surge motion, the shape of the mode remains consistent across different turbulence intensities. Its magnitude increases with downstream distance, reaching a maximum between $x \in [4D, 6D]$ , before gradually decaying further downstream.

Later in the paper, we quantify the energy contained in these coherent structures and examine the relationship between their energy content, wake recovery enhancement and the energy input from platform excitation. A direct comparison between surge and sway motions is presented to highlight the differences in their respective impacts.

Prior to this, we address and quantify a frequently debated issue in wind energy: the effect of coherent fluctuations on the time evolution of phase-resolved wake recovery.

3.4. Coherent structures and phase-resolved wake recovery

It is often argued that the observed increase in wake recovery – computed using (2.2) and shown in figure 8 – in response to platform motion or rotor excitation could be an artefact of periodic wake displacement, as discussed by Korb et al. (Reference Korb, Asmuth and Ivanell2023). This is particularly relevant in the case of sway-induced coherent structures (figure 11 a), where the entire wake undergoes large-amplitude lateral oscillations. When such motion is averaged over time and across the rotor area, it can artificially smooth the wake profile, giving a feigned increased recovery. This raises an important question: Is the enhanced recovery a result of actual momentum exchange, or is it merely a consequence of averaging over periodic lateral displacements? In the following, we clarify this distinction and present evidence to evaluate the true contribution of coherent structures to enhance momentum exchange.

Figure 12(a) displays horizontal wind speed profiles in the wake at $x = 6D$ . Panels 12(a)(c–g) show both the time-averaged wake profile and phase-resolved wake profiles for the surging turbine operating at $St = 0.3$ , $A^* = 0.024$ and $\textit{TI}_\infty = 1.1\,\%$ . For comparison, the wake of the fixed (non-oscillating) turbine is also included. Similarly, the wake profiles for the swaying turbine under identical operating conditions are shown in figure 12(a)(h–l). We manually adjusted the phase origin $\phi _0$ such that, for $\phi ^* = 0$ , the phase-resolved wake aligns most closely with the time-averaged profile.

Figure 12. Comparison of time-averaged and phase-averaged wake recovery. (a) Time-averaged and local (phase-averaged) streamwise wind speed $y$ profiles at $x = 6D$ for selected phases $\phi ^* \in \{0, 2\pi , 4\pi , 6\pi , 8\pi \}/5$ , for both surge (a–e) and sway (f–j) motions, with $St = 0.3$ , $A^* = 0.024$ and $\textit{TI}_{\infty } = 1.1\,\%$ . (b) Local wake recovery $\widetilde {R_w}(\phi ^*)$ computed across the phase range $\phi ^*/2\pi \in [0,1]$ . The plot includes the mean of the phase-averaged recovery $\overline {\widetilde {R_w}(\phi ^*)}$ , the time-averaged recovery $R_w$ (also for the fixed reference case), shown for both surge (1) and sway (2) cases.

As $\phi ^*$ increases ( $\phi ^* \in [0, \pi ]$ ), the surging turbine’s wake initially contracts and exhibits higher wind speeds than the time-averaged case (see figure 12 a(c–e)). To better illustrate the connection between the wake profiles and the coherent dynamics, we display in figure 12(a)(c–e) the structure previously shown in figure 10(b)(a.2), and mark with a red thick line the position of $\phi ^*$ . Beyond $\phi ^* \gt \pi$ , the streamwise component of the coherent structure becomes negative, the wake expands, and the wind speed decreases below the time-averaged profile. Notably, at $\phi ^* = 8\pi /5$ , the wake speed is even lower than in the fixed case, i.e. $\overline {u}_{\textit{fixed}} \gt (\overline {u} + \widetilde {u})_{\textit{surge}}$ .

To quantify the effects of motion on recovery at different phases, we define the phase-resolved wake recovery $\widetilde {R_w}(\phi ^*)$ as

(3.2) \begin{equation} \widetilde {R_w}(\phi ^*) = \frac {1}{D U_\infty } \int _{-R + y_c(\phi ^*)}^{R + y_c(\phi ^*)} \big(\overline {u} + \widetilde {u}(\phi ^*) \big) \, {\rm d}y .\end{equation}

Here, $y_c(\phi ^*)$ is the lateral location of the wake centre at a given phase, defined as the position where $\overline {u} + \widetilde {u}$ reaches its minimum. For surge motion, this position remains approximately constant and centred: $y_c(\phi ^*) \approx 0$ .

Figure 12(b)(1) shows the evolution of $\widetilde {R_w}$ over a full motion cycle ( $\phi ^*/2\pi \in [0,\sim 1]$ ), alongside the average, $\overline {\widetilde {R_w}}$ , and the conventional time-averaged wake recovery from (2.2), both for the surging and fixed cases. In agreement with the profiles in figure 12(a)(c–g), the phase-resolved recovery initially increases (red x-. curves) when $\widetilde {u} \gt 0$ for $\phi ^*/2\pi \in [0, 0.5]$ . Then, as the wake expands for $\phi ^*/2\pi \gt 0.5$ , the recovery decreases, even falling below that of the fixed case. Overall, $\overline {\widetilde {R_w}}$ (red dashed line) is only slightly higher (by $\sim 1 \,\%$ ) than the conventional time-averaged value (blue solid line). This indicates that, for surge motion, the coherent structures do not significantly alter the mean of the phase-averaged recoveries, although they do introduce large fluctuations depending on $\phi ^*$ . As a result, downstream turbines in the wake would experience substantial power variability.

In contrast, the wake of the swaying turbine behaves differently. As seen in figure 12(a)(h–l), the wake undergoes significant lateral movement, with the wake centre $y_c(\phi ^*)$ varying between $-R$ and $R$ . This behaviour, driven by the meandering mode illustrated in figure 11(b)(a.2) and reproduced in figure 12(a)(h–l), causes the phase-averaged wake profile to shift side to side rather than contract or expand. Unlike the surge case, the minimum of $\overline {u} + \widetilde {u}$ remains almost constant across phases, but is slightly lower than that of the time-averaged wake alone. This suggests that the smoothed time-averaged profile (green solid line in figure 12(a)(h–l)) also includes a contribution from the phase displacements rather than solely from enhanced momentum transport.

Another important difference lies in the shape of the phase-resolved wake. For the surging turbine, the wake retains a near-Gaussian profile even locally, whereas the swaying turbine’s wake becomes asymmetric at different phases – indicating a distortion caused by lateral motion. This further highlights the distinct nature of wake modulation between meandering and pulsating modes.

In figure 12(b)(2) we display the phase-averaged recovery for sway motion at various $\phi ^*$ (orange squares), along with the average over all the phases, the time-averaged recovery and the fixed case for reference. Compared with the surge case, the variation in phase-resolved recovery is smaller ( ${\sim}6\,\%$ around the mean versus ${\sim}16\,\%$ for surge). Interestingly, the time-averaged recovery (green solid line) is slightly greater than the mean of the phase-resolved wake recovery, however, by only about $3\,\%$ , indicating that the enhanced recovery for sway is primarily due to increased momentum transport rather than smoothing artefacts – a result also analysed by Korb et al. (Reference Korb, Asmuth and Ivanell2023). However, the approach of Korb et al. (Reference Korb, Asmuth and Ivanell2023) differs: using CFD, they examined energy differences between the time-averaged wake and the wake in a meandering frame of reference, integrated over the full wake area at a given downstream position. They reported differences up to $15 \,\%$ , whereas our wind-speed-based line profiles show only a $3\,\%$ deviation. While the cases and metrics differ (they examined energy; we analyse streamwise velocity), both studies highlight that averaging over periodic wake motion introduces discrepancies in evaluating true recovery. If other metrics like $\overline {u^2}$ (kinetic energy) or $\overline {u^3}$ (wind power) were considered, these discrepancies would likely be more pronounced in our case. Nonetheless, we find that the dominant source of enhanced recovery remains the increased exchange between the wake and the outer flow – a point we return to in the conclusion.

3.5. Coherent structures: energy perspectives

We conclude the results section with an analysis of the energy of the coherent structures observed in the wake of the moving wind turbine, focusing on their relationship with free-stream turbulence intensity, energy of platform motion and enhanced wake recovery.

Reynolds & Hussain (Reference Reynolds and Hussain1972) provide a mathematical framework for the energy dynamics of coherent structures, describing the transfer of energy between the mean flow, stochastic fluctuations and coherent motions. They define the local kinetic energy of a coherent structure in a flow as

(3.3) \begin{equation} e_{\textit{CS}} = \frac {1}{2} \overline {\widetilde {u_i}^2} = \frac {1}{2} \big ( \overline {\widetilde {u}^2} + \overline {\widetilde {v}^2} + \overline {\widetilde {w}^2} \big ) .\end{equation}

In their work, Reynolds & Hussain (Reference Reynolds and Hussain1972) derive a transport equation for $e_{\textit{CS}}$ from the phase-averaged Navier–Stokes equations. While such an analysis would be of interest in the present context, it falls beyond the scope of this study. However, the concept of energy remains relevant, as we can compute $e_{\textit{CS}}$ from the full 3-D dataset available from Milan, and evaluate $\overline {\widetilde {u}^2}$ (a subset of $e_{\textit{CS}}$ ) across all cases with varying degrees of freedom (DoF), Strouhal number $ St$ , motion reduced-amplitude $ A^*$ and free-stream turbulence intensity $ \textit{TI}_{\infty }$ .

Hussain (Reference Hussain1986) emphasises the 3-D nature of coherent structures, which often occupy a substantial volume of a sheared flow. This motivates extending the concept of local energy to a more spatially integrated quantity, particularly across the dimensions where measurements are available. Previous CFD studies from Yılmaz & Meyers (Reference Yılmaz and Meyers2018) and Li et al. (Reference Li, Dong and Yang2022) provide visual evidence of the large-scale extent of these structures in 3-D space.

In our case, measurements are taken along a horizontal line at hub height. As observed in figures 10 and 11, the motion-induced coherent structures span a significant portion of the lateral direction. Most of the energy is concentrated in the region $ y \in [-D, D]$ , as illustrated, for example, in figure 11(b)(a.2). Based on this, and following a similar approach to Yılmaz & Meyers (Reference Yılmaz and Meyers2018), we define the spanwise-integrated kinetic energy of the coherent structures, normalised by the inflow mean kinetic energy:

(3.4) \begin{equation} E^*_{\textit{CS}} = \frac {1}{2D} \int _{-D}^{D} \frac {1}{2} \big ( \overline {\widetilde {u}^2} + \overline {\widetilde {v}^2} + \overline {\widetilde {w}^2} \big ) {\rm d}y \Big / \left ( \frac {1}{2} U_{\infty }^2 \right ) = \frac {1}{2D U^2_{\infty }} \int _{-D}^{D} e_{\textit{CS}} \, {\rm d}y. \end{equation}

To enable comparison across all datasets, we also define a streamwise-only version, $ E^*_{x,CS}$ , which uses only the streamwise component $ \overline {\widetilde {u}^2}$ :

(3.5) \begin{equation} E^*_{x,CS} = \frac {1}{2D U^2_{\infty }} \int _{-D}^{D} \overline {\widetilde {u}^2} \, {\rm d}y .\end{equation}

Similar to (3.5), we define $ E^*_{y,CS}$ and $ E^*_{z,CS}$ as the normalised energy contributions from the spanwise ( $ \overline {\widetilde {v}^2}$ ) and vertical ( $ \overline {\widetilde {w}^2}$ ) components of the coherent structures, respectively.

We begin by examining the shared dataset between Milan and Oldenburg for the cases surge-5 and sway-5 (see table 2), under inflow conditions M1.5 and O1.5 with $ \textit{TI}_{\infty } = 1.5\,\%$ (see table 1). Figure 13(a) presents the downstream evolution of $ E^*_{x,CS}$ from both the Milan and Oldenburg datasets, while figure 13(b) shows the evolution of $ E^*_{i,CS}$ , where $ i \in \{x, y, z\}$ , and the total energy $ E^*_{\textit{CS}}$ , using the 3-D data from Milan. The good agreement (i.e. a mean deviation of less than $10 \,\%$ ) between the curves marked with empty symbols (Oldenburg) in figure 13(a) and those with filled symbols (Milan) confirms the consistency of the results from the two experiments. A more detailed comparison is provided in Appendix B.

Figure 13. Integrated energy of surge- and sway-induced coherent structures for cases surge-5 and sway-5 (see table 2). (a) Comparison of the integrated energy of the streamwise component of the coherent fluctuations $E^*_{x,CS}$ (3.5) for surge and sway with $St = 0.38$ , $A^* = 0.01$ , using Milan and Oldenburg datasets ( $\textit{TI}_{\infty } = 1.5\,\%$ ). (b) Integrated energy of each component of the coherent structure: streamwise $\overline {\widetilde {u}^2}$ , lateral $\overline {\widetilde {v}^2}$ and vertical $\overline {\widetilde {w}^2}$ velocities, as well as their total noted $E^*_{x,CS},\,E^*_{y,CS}, E^*_{z,CS}$ and $E^*_{\textit{CS}}$ , respectively, for surge (b.1) and sway (b.2). Panels (b.3) and (b.4) show the same energy components normalised by their respective maximum values.

In figure 13(a) the normalised, spanwise-integrated energy of $ \overline {\widetilde {u}^2}$ for both surge- and sway-induced structures initially increases for $ x \lesssim 4D$ . This indicates that, in the near-wake to the transition region, the coherent structures are gaining energy. The energy associated with the meandering mode (sway, green curve) is approximately $50 \,\%$ larger than that of the pulsating mode (surge), correlating with the higher wake recovery for sway observed in figure 4(a). Further downstream, for $ 3.5 \lesssim x/D \lesssim 5.5$ , the energy reaches a peak and then decays. This decline indicates that the coherent structures are dissipating energy, consistent with our pointwise analysis in Messmer et al. (Reference Messmer, Hölling and Peinke2024a ). By $ x = 10D$ , the energy has dropped to about one-fifth of its peak value, showing strong dissipation and potential breakdown into smaller-scale turbulence, consistent with the classic energy cascade described by Richardson/Kolmogorov (Pope Reference Pope2001).

Figure 13(b)(1–2) provide additional insights into the downstream development of $ E^*_{y,CS}$ , $ E^*_{z,CS}$ and the total $ E^*_{\textit{CS}}$ for the Milan dataset. The differences between the surge- and sway-induced structures are much more pronounced when considering the full 3-D energy components. In particular, the spanwise component $ \overline {\widetilde {v}^2}$ (triangular markers with dashed lines) shows almost four times more energy for sway than for surge. As a result, the total energy $ E^*_{\textit{CS}}$ is nearly three times higher for sway, emphasising the energetic dominance of the meandering mode under identical excitation conditions ( $ St = 0.38,\, A^* = 0.01$ ).

Despite these differences, the overall trends are similar: $ E^*_{\textit{CS}}$ increases to a peak before decreasing downstream. This full 3-D analysis shows that a significant portion of the meandering mode’s energy resides in the spanwise direction ( $ \widetilde {v}$ ), as expected from its lateral motion characteristics. Figure 13(b).(3–4) show the same quantities as in b.(1–2), but normalised by their respective maximum values. While the normalised total energy $ E^*_{\textit{CS}} / \max E^*_{\textit{CS}}$ does not exactly match the normalised streamwise component $ E^*_{x,CS} / \max E^*_{x,CS}$ , the trends remain consistent. This supports the decision to investigate $ E^*_{x,CS}$ as a proxy for total energy across all cases – while acknowledging that it only represents a portion of the total energy. In particular, for the meandering mode (sway), a large fraction of the total energy is concentrated in the spanwise component $ E^*_{y,CS}$ , meaning that differences between surge and sway are more significant than suggested by $ E^*_{x,CS}$ alone. Nonetheless, it is reasonable to assume that a lower $ E^*_{x,CS}$ generally implies a lower total energy $ E^*_{\textit{CS}}$ .

We next analyse the $E^*_{x,CS}$ dependency to inflow conditions and motion parameters. We start with figure 14 for the normalised energy evolution of the pulsating mode from surge motion (figure 14 a) and meandering mode from sway (figure 14 b) both with $St = 0.3,\,A^*=0.024$ for the different inflows: $\textit{TI}_{\infty } \in [1.1, 5.8]\,\%$ . On both panels, we added red symbols that represent the normalised energy of the free-stream fluctuations, $\epsilon ^*_{\infty }$ , defined as

(3.6) \begin{align} \epsilon ^*_{\infty } = \epsilon _{\infty }/\big(1/2 U^2_{\infty }\big) = \big(1/2 \overline {u^{{\prime}2}_{\infty }}\big) /\big(1/2 U^2_{\infty }\big) = \overline {u^{{\prime}2}_{\infty }}/U^2_{\infty } = TI^2_{\infty } . \end{align}

Figure 14. Integrated energy of the streamwise component of the coherent structures induced by motions $E^*_{x,CS}$ for surge (a) and sway (b) both with $St = 0.3,\,A^*=0.024$ (surge- and sway-3–0.024 in table 2) for $\textit{TI}_{\infty } \in [1.1, 5.8]\,\%$ (O1.1–O5.8 in table 1). The red symbols represent the normalised energy of the free-stream fluctuations $\epsilon ^*_{\infty }$ (3.6) and the dashed line shows $\epsilon ^*_{p}$ , the normalised energy of the platform motion (3.7).

From (3.6), the higher the inflow $\textit{TI}_{\infty }$ , the higher the energy of the fluctuations. Here $\epsilon ^*_{\infty }$ can be compared with $E^*_{x,CS}$ as well as the normalised energy of the platform motion, $\epsilon ^*_p$ , which we defined and discussed already in Messmer et al. (Reference Messmer, Hölling and Peinke2024a ), as

(3.7) \begin{equation} \epsilon ^*_{p} = 2 (\pi St A^*)^2 ,\end{equation}

which quantifies the normalised energy of the platform motion. Figure 14 illustrates that, for both surge (figure 14 a) and sway (figure 14 b), increasing inflow turbulence intensity $ \textit{TI}_{\infty }$ (i.e. higher $ \epsilon ^*_{\infty }$ ) leads to a reduction in the energy of the coherent structures. This trend is also visible in figures 10 and 11, based on the spatial maps of $ \widetilde {u}$ .

For example, in the surge case, the normalised streamwise energy of the coherent structure, $ E^*_{x,CS}$ , is approximately eight times larger for $ \textit{TI}_{\infty } = 1.1\,\%$ than for $ \textit{TI}_{\infty } = 5.8\,\%$ . A similar trend is observed for sway, with the energy about five times higher at low turbulence. At a fixed inflow condition, sway-induced coherent structures exhibit higher energy levels than those induced by surge, consistent with the observations in figure 13(a).

Both surge and sway exhibit a similar evolution in coherent structure energy: an initial growth up to $ x \approx 4{-}6D$ , followed by a downstream decay. At $ x = 2D$ , the energy levels are nearly identical across all inflow turbulence intensities $ \textit{TI}_{\infty }$ , and closely match $ \epsilon ^*_p$ – the energy associated with platform motion. This suggests that at this early position, the coherent structures have not yet developed and their energy is primarily determined by the imposed motion. Further downstream, however, the growth and peak level of coherent structure energy become strongly dependent on $ \textit{TI}_{\infty }$ . For low turbulence intensities ( $ \textit{TI}_{\infty } \leqslant 2.2\,\%$ ), the structures grow substantially in energy. In contrast, for higher turbulence levels ( $ \textit{TI}_{\infty } \gt 3\,\%$ ), their growth is significantly dampened. The location of the peak energy $ E^*_{x,{CS}}$ , as well as the rate of energy increase and decay, vary depending on both the direction of motion and $ \textit{TI}_{\infty }$ . This indicates that the formation and evolution of coherent structures are highly sensitive to inflow conditions.

Under surge excitation, when the inflow turbulence intensity exceeds approximately $ \textit{TI}_{\infty } \gtrsim 3\,\%$ , the energy of the streamwise coherent structure, $ E^*_{x,{CS}}$ , generally falls below the platform motion energy threshold $ \epsilon ^*_p$ . While a slight increase above $ \epsilon ^*_p$ is observed close to the source, it quickly diminishes downstream, suggesting that higher turbulence intensity inhibits the formation and persistence of coherent structures. For sway, by contrast, coherent structure energy remains above the threshold until turbulence levels exceed approximately $ 5 \,\%$ , demonstrating that the sway-induced structures are more resilient to free-stream turbulence.

Interestingly, this energy threshold seems to correlate with the wake recovery trends shown in figure 8. In the surge case, no significant improvement in recovery is observed for $ \textit{TI}_{\infty } \gtrsim 3\,\%$ , whereas for the sway case, this limit rises to $ \textit{TI}_{\infty } \gtrsim 5\,\%$ . This suggests that $ \epsilon ^*_{p}$ may act as a critical threshold – below which the coherent structure’s energy is too weak to affect wake recovery meaningfully. Figure 14 also highlights that sway-induced structures not only carry more energy but also better resist the damping effects of increased inflow turbulence compared with surge-induced structures.

So far, we have investigated only the highest amplitude of motion ( $ A^* = 0.024$ ). We now extend the analysis in figure 15 to include all three tested amplitudes: $ A^* \in \{0.01,\,0.017,\,0.024\}$ . To facilitate the analysis of how coherent structure energy depends on $ A^*$ and inflow turbulence intensity $ \textit{TI}_{\infty }$ , we introduce a streamwise-integrated metric.

As shown in figure 14, increasing levels of $ \textit{TI}_{\infty }$ lead to a clear reduction in the overall energy of the coherent structures. On average, this energy is well above the platform motion energy threshold $ \epsilon ^*_p$ for low turbulence levels ( $ \textit{TI}_{\infty } \lt 2\,\%$ ), approximately equal to it around $ \textit{TI}_{\infty } \approx 3\,\%$ , and falls below it for higher turbulence intensities ( $ \textit{TI}_{\infty } \gt 3\,\%$ ) in the surge case (figure 14 a). For sway, the thresholds are at higher $ \textit{TI}_{\infty }$ values (figure 14 b). These observations motivate the use of a spatially integrated metric that captures the overall strength of the structure and allows direct comparison with $ \epsilon ^*_p$ . Inspired by the approach of Yılmaz & Meyers (Reference Yılmaz and Meyers2018), we define the streamwise-integrated coherent structure energy as

(3.8) \begin{align} \int E^*_{x,{CS}} = \frac {1}{8D} \int _{x=2D}^{x=10D} E^*_{x,{CS}}(x)\,{\rm d}x. \end{align}

This quantity represents a spatially averaged estimate of the coherent structure energy in the $ x$ direction, computed over the region where the structures are observed ( $ x \times y \in [2D, 10D] \times [-D, D]$ ). Figure 15 presents $ \int E^*_{x,{CS}}$ as a function of inflow turbulence intensity $ \textit{TI}_{\infty }$ for surge (a) and sway (b) excitation, at fixed Strouhal number $ St = 0.3$ , and for all amplitudes and inflow conditions considered (O1.1–O5.8 in table 1, corresponding to $ \textit{TI}_{\infty } \in [1.1\,\%, 5.8\,\%]$ ).

Figure 15. Streamwise-integrated energy of the coherent structures in the $ x$ direction, defined as $ \int E^*_{x,CS} = ({1}/{8D}) \int _{x=2D}^{x=10D} E^*_{x,CS}(x) \, {\rm d}x$ , plotted against the inflow turbulence intensity $ \textit{TI}_{\infty }$ for (a) surge and (b) sway. All cases use $ St = 0.3$ and $ A^* \in \{0.01,\,0.017,\,0.024\}$ in table 2 and inflows O1.1–O5.8 in table 1. Here red :, –,- lines represent $\epsilon ^*_p$ defined in (3.7).

A consistent trend is observed for both motion directions: for a given inflow condition, increasing $ A^*$ leads to a higher coherent structure energy. For example, at $ \textit{TI}_{\infty } = 1.1\,\%$ , the integrated energy for surge at $ A^* = 0.024$ is approximately three times higher than at $ A^* = 0.01$ (compare blue $\bullet$ and x in figure 15 a). Similarly, for sway, the increase is about 1.8 times (compare green $\bullet$ and x in figure 15 b).

A higher $ A^*$ naturally results in a higher excitation energy $ \epsilon ^*_p$ , which scales with $ {A^*}^2$ (3.7). Thus, the platform motion at $ A^* = 0.024$ carries approximately 5.8 times more energy than at $ A^* = 0.01$ . Interestingly, the relative response of the wake structures (i.e. the energy of the structures relative to $ \epsilon ^*_p$ ) is larger for smaller $ A^*$ , which aligns with classical findings in periodic excitation and synchronisation theory. As shown by Pikovsky et al. (Reference Pikovsky, Rosenblum and Kurths2001), weakly forced nonlinear systems often exhibit enhanced sensitivity, leading to more efficient energy amplification at lower excitation amplitudes. Nonetheless, in terms of absolute energy levels, our results show clearly that increasing the excitation amplitude leads to more energetic coherent structures. Since $ \epsilon ^*_p \ll 1$ , even the largest tested amplitudes still fall in the range of what can be considered infinitesimal forcing.

The further analysis of figure 15 confirms, as previously analysed in figure 14, that sway motion generates coherent structures with higher energy and greater resilience to increasing inflow turbulence intensity $ \textit{TI}_{\infty }$ , compared with surge. The connection between the energy of the coherent structures and wake recovery enhancement becomes apparent: as $ \textit{TI}_{\infty }$ increases, the energy of the structures decreases, and so does the impact of the structures on recovery. For the largest amplitude $ A^* = 0.024$ , the energy drops below the corresponding platform motion energy $ \epsilon ^*_p(A^* = 0.024)$ at approximately $ \textit{TI}_{\infty } \approx 3.0\,\%$ for surge and $ \textit{TI}_{\infty } \approx 5.0\,\%$ for sway. For the lower amplitudes ( $ A^* \leqslant 0.017$ ), $ \int E^*_{x,CS}$ remains above the respective $ \epsilon ^*_p(A^*)$ for all inflow conditions, but the absolute energy remains significantly lower than that of the structures induced by $ A^* = 0.024$ . By linking figures 8 and 15, we observe that, for any $ A^*$ , when $ \int E^*_{x,CS} \lesssim \epsilon ^*_p(A^* = 0.024)$ , the impact of platform motion on wake recovery improvement becomes negligible. Therefore, the higher energy of the meandering structures induced by sway, along with their greater robustness to increasing $ \textit{TI}_{\infty }$ , leads to an extended range of inflow turbulence conditions where sway motion improves recovery compared with surge. This observation is further quantified in figure 16.

Figure 16. Relative increase in wake recovery, $ \Delta \mathcal{R} / \mathcal{R}$ (3.10), plotted against the ratio $ \epsilon ^*_{\infty } / \epsilon ^*_p$ , which compares the normalised energy of free-stream fluctuations to that of platform motion. Data includes all motion cases with $ St = 0.3$ and inflow conditions from O1.1 to O5.8 (see tables 1 and 2). A threshold of $ 1.5 \,\%$ is shown as a red dashed line.

Figures 14 and 15 have shown that both $ \epsilon ^*_{\infty }$ (the normalised energy of the free-stream fluctuations) and $ \epsilon ^*_p$ (the normalised energy of the platform motion) are key parameters governing the energy of coherent structures and, consequently, the wake recovery enhancement. An increase in $ \epsilon ^*_p$ leads to more energetic structures for a fixed $ \epsilon ^*_{\infty }$ , while an increase in $ \epsilon ^*_{\infty }$ results in weaker structures, as they struggle to emerge and persist in a more turbulent background at constant $ \epsilon ^*_p$ . As the energy of the coherent structures decreases, so does their impact on wake recovery improvement.

This observation motivates the study of the relative increase in wake recovery, $ \Delta \mathcal{R} / \mathcal{R}$ , between a case with platform motion and the corresponding fixed-platform reference, for different energy levels of inflow and platform motion. The streamwise-integrated wake recovery $ \mathcal{R}$ is defined as

(3.9) \begin{equation} \mathcal{R} = \frac {1}{8D} \int _{x=2D}^{x=10D} \frac {1}{D} \int _{y=-R}^{y=R} \frac {\overline {u}(x,y)}{U_{\infty }} \, {\rm d}y \, {\rm d}x = \frac {1}{8D} \int _{x=2D}^{x=10D} R_w(x) \, {\rm d}x.\end{equation}

Here, $ \mathcal{R}$ represents the average recovery over the measurement region $ x \in [2D, 10D]$ . The relative increase in recovery is given by

(3.10) \begin{equation} \frac {\Delta \mathcal{R}}{\mathcal{R}} = \frac {\mathcal{R}(St, A^*, \text{Dof}, TI_{\infty }) - \mathcal{R}(St=0, TI_{\infty })}{\mathcal{R}(St, A^*, \text{Dof}, TI_{\infty })}. \end{equation}

This quantity quantifies the relative improvement in wake recovery due to platform motion, averaged over the streamwise measurement domain. The final figure 16 shows the evolution of $ \Delta \mathcal{R} / \mathcal{R}$ (3.10), plotted against the ratio $ \epsilon ^*_{\infty } / \epsilon ^*_p$ for all motion cases with $ St = 0.3$ and inflows ranging from O1.1 to O5.8. This figure complies some of the outcomes of figures 8, 14 and 15 combined.

For $ \textit{TI}_{\infty } = 1.1\,\%$ , $ \epsilon ^*_\infty = 1.2 \times 10^{-4}$ , which is lower than $ \epsilon ^*_p$ for all $ A^*$ . For example, for $ A^* = 0.024$ , $ \epsilon ^*_\infty / \epsilon ^*_p \approx 0.1$ , indicating that there is an order of magnitude more energy in the platform motion than in the free-stream turbulence. As a result, the energy of the coherent structure found in the wake and induced by the platform is significant (the largest) for both surge and sway DoFs (see the blue and green $ \bullet$ in figure 15(a) and 15(b) for quantification). Consequently, the impact on recovery is also large: about $ 16 \,\%$ more averaged recovery is observed for sway compared with $7 \,\%$ for surge, relative to the fixed case’s recovery (see the $ \bullet$ points for $ \epsilon ^*_\infty / \epsilon ^*_p \approx 0.1$ in figure 16). As $ \epsilon ^*_{\infty } / \epsilon ^*_p$ increases, the inflow turbulence either increases (if $ \epsilon ^*_p$ is fixed) or $ \epsilon ^*_p$ decreases (with fixed $ \epsilon ^*_{\infty }$ ). Regardless of the scenario, as $ \epsilon ^*_{\infty } / \epsilon ^*_p$ increases, the relative importance of the motion diminishes because the inflow turbulence becomes more dominant. For both surge and sway, small discrepancies in the impact of motion on recovery improvement can be observed at different amplitudes for a given $ \epsilon ^*_{\infty } / \epsilon ^*_p$ ratio. Specifically, the lowest amplitude ( x in figure 16) leads to larger recovery improvements compared with the highest amplitude, consistent with the statement that ‘the lower the excitation amplitude, the greater the relative impact’. Nevertheless, the overall trend remains clear: as $ \epsilon ^*_{\infty } / \epsilon ^*_p$ increases, the influence of platform motion on wake recovery diminishes, eventually falling below $ 1.5 \,\%$ – a level at which the effect becomes indistinguishable from measurement uncertainty here (Messmer et al. Reference Messmer, Peinke and Hölling2024b ).

Notably, the difference between surge and sway, discussed throughout the paper, becomes evident here. For $ \epsilon ^*_{\infty } / \epsilon ^*_p \gtrsim 1$ (marked with the transparent blue rectangle in figure 16), the impact of surge motion on wake recovery becomes negligible. In contrast, the threshold for sway is about 3.5 times higher, with no measurable impact for $ \epsilon ^*_{\infty } / \epsilon ^*_p \gtrsim 3.5$ . This key result demonstrates that surge-induced coherent structures, responsible for enhanced recovery, are more sensitive to inflow turbulence than sway-induced structures, with a factor of about 3.5. This is further discussed in the conclusion.