2.1. Set-up

The experimental set-up used to perform the investigations is illustrated in figure 2. The experiments were done in the $3\,{\rm m} \times 3\,{\rm m} \times 30\,{\rm m}$ test section of the large wind tunnel of the University of Oldenburg (Kröger et al. Reference Kröger, Frederik, van Wingerden, Peinke and Hölling2018). The set-up consists of the Model Wind Turbine Oldenburg 0.6 (MoWiTO 0.6) with a diameter of 0.58 m mounted on a six-DoF motorised platform, designed specifically for the MoWiTO 0.6 (Messmer et al. Reference Messmer, Brigden, Peinke and Hölling2022). With this system, the model turbine can be moved following predefined motion signals in the six-DoF: the three translations – surge, sway and heave; and the three rotations – roll, pitch and yaw. In this paper, we use the terminology ‘floating wind turbine’ to describe our experimental set-up, despite the fact that we consider idealised conditions (harmonic motion and laminar wind).

Figure 2. Scheme of the experimental set-up, the MoWiTO 0.6 mounted on a six-DoF platform installed in the wind tunnel (a) side view and (b) front view. The figure is not to scale.

The model turbine was equipped with a strain gauge to measure the thrust, $T$, of the ensemble (tower and rotor). For any case investigated, power produced, rotational speed and thrust of the turbine were recorded. The movements of the platform were recorded to verify the adequacy of the motions. The streamwise wind speed, $U$, was measured with an array of 19 one-dimensional hot-wires of ${\sim }$1 mm length, resolving turbulent scale of this size.

The probes were operated with multichannel constant temperature anemometer (54N80-CTA) modules from Dantec Dynamics. Data were acquired with a sampling rate of 6 kHz, gathering around $10^6$ points for each measurement. The time of measurements, $T_{meas}$, was sufficient to reach statistical convergence of any postprocessed data shown and discussed in this paper. The hot-wire probes were calibrated twice daily, and temperature, humidity and air pressure were recorded throughout the day to track any drift effects, which turned out to be negligible. The inflow wind speed was measured with a Prandtl-tube approximately 2 m in front of the model turbine.

The array was mounted on a moving motorised cart, which allowed us to measure the wind speed at a downstream position, $x$, from 1.5D to 12D (cf. figure 2a). The hot-wires were placed on a horizontal line at hub height (around 1 m above floor level) and span lateral positions, $y$ from $-2.5R$ to $2.5R$ ($R = D/2$) with respect to the hub centre (cf. figure 2b).

2.2. The MoWiTO 0.6

The MoWiTO 0.6 used for the experiments has a rotor diameter $D$ of 0.58 m. The turbine was developed at the University of Oldenburg (Schottler et al. Reference Schottler, Hölling, Peinke and Hölling2016; Jüchter et al. Reference Jüchter, Peinke, Lukassen and Hölling2022). The properties of the turbine are depicted in table 1. The model turbine was used for several measurement campaigns with a focus on wake flow; further details of the experimental procedure and wake measurements can be found in Hulsman et al. (Reference Hulsman, Wosnik, Petrović, Hölling and Kühn2020), Neunaber et al. (Reference Neunaber, Hölling, Stevens, Schepers and Peinke2020) and Neunaber, Hölling & Obligado (Reference Neunaber, Hölling and Obligado2024).

Table 1. The MoWiTO 0.6 characteristics.

For the experiments, we used wind speeds between 3 and 5 m s$^{-1}$. We fixed the tip speed ratio for all cases so that $\langle TSR\rangle = 6 \pm 0.1$. We operated the turbine at a constant blade pitch angle, around the optimum, which gives $\langle C_T \rangle = 0.86 \pm 0.05$ for $U_{\infty } = 3\,{\rm m}\,{\rm s}^{-1}$ and $\langle C_T \rangle = 0.81 \pm 0.05$ for $U_{\infty } = 5\,{\rm m}\,{\rm s}^{-1}$. The thrust coefficient account for rotor and tower aerodynamic loads (in a previous study by Neunaber et al. (Reference Neunaber, Hölling, Stevens, Schepers and Peinke2020), the load on the tower and nacelle without the blades of the MoWiTO 0.6 was measured to be 17 % of the total thrust coefficient). The power performance of the turbine depends on $Re$, we measured $\langle C_p \rangle = 0.21 \pm 0.01$ at $U_{\infty } = 3\,{\rm m}\,{\rm s}^{-1}$ and $\langle C_p \rangle = 0.3 \pm 0.01$ at $U_{\infty } = 5\,{\rm m}\,{\rm s}^{-1}$.

2.3. Non-dimensional parameters that drive the wake behaviour

Our experimental research involves numerous quantities. We briefly discuss the selection of necessary parameters for a complete characterisation. Therefore, we use the basis of the ${\rm \pi}$-theorem. For a given DoF, the system investigated involves nine independent parameters: platform motion frequency and amplitude; inflow wind speed; rotor diameter; air viscosity; air density; turbine thrust and power; and rotational speed. These are noted as $f_p$, $A_p$, $U_{\infty }$, $D$, $\mu$, $\rho$, $T$, $P$, $\omega$. The ${\rm \pi}$-theorem suggests that, a problem characterised by $m$ dimensional variables can always be reduced to a set of $m-n$ dimensionless parameters (${\rm \pi}$-groups), with $n$ the fundamental units of measure (dimensions) as depicted by Buckingham (Reference Buckingham1914). Therefore, this problem with nine variables and three dimensions can be reduced to six dimensionless parameters. These are all defined in table 2, namely the thrust coefficient, $C_T$, power coefficient, $C_P$, tip speed ratio, $TSR$, reduced amplitude, $A^*$, Strouhal number, $St$, and Reynolds number, ${\textit {Re}}$. Thus, in a laminar flow, the properties of the wake of a FOWT, $wake_{FOWT}$, are determined by these six dimensionless numbers: $wake_{FOWT} = f(C_T,C_P,TSR, A^*,St,Re)$.

Table 2. Dimensionless parameters that drive the wake of a FOWT.

The dependency on $C_T$ of wake recovery and expansion of a wind turbine wake was characterised extensively (Porté-Agel et al. Reference Porté-Agel, Bastankhah and Shamsoddin2020). The tip speed ratio also plays an important role in the development of the wake of a FOWT (Farrugia et al. Reference Farrugia, Sant and Micallef2016). Both the amplitude and frequency of motions can impact the wake of a moving turbine, as demonstrated by Chen et al. (Reference Chen, Liang and Li2022), Li et al. (Reference Li, Dong and Yang2022) and Ramos-García et al. (Reference Ramos-García, Kontos, Pegalajar-Jurado, González Horcas and Bredmose2022). As mentioned above, motion frequency has a greater influence on wake dynamics, even at low amplitudes ($A_p \sim 0.01D$). Based on these works and the set of parameters, we concluded that it is worth focusing on the impact of different $St$ values at a small amplitude of motions.

2.4. Cases investigated

Floating offshore wind turbines operate in the atmospheric boundary layer, which exhibits various turbulent conditions. Usually, the turbulence intensity in free flow, $TI_{\infty }$, is found in $[0.02, 0.15]$. (Jacobsen & Godvik Reference Jacobsen and Godvik2021; Angelou, Mann & Dubreuil-Boisclair Reference Angelou, Mann and Dubreuil-Boisclair2023). In our experiments, we used idealised conditions (laminar wind with $TI_{\infty } \approx 0.003$ and one-DoF harmonic motions). We also carried out experiments with $TI_{\infty } = 0.03$ and observed similar results as the ones shown and discussed in this paper. Our study investigates the following DoFs: surge; sway; roll; pitch (see figure 1). We examined these DoFs independently, without combining them. We imposed the following motion signal on the platform, $\xi$, for a given DoF: $\xi (t) = A_p \sin (2 {\rm \pi}f_p t) = A^*D \sin (2 {\rm \pi}(St U_{\infty }/D) t)$. Here, $A_p$ denotes the amplitude of motion (in m or $^{\circ }$), and $f_p$ represents the frequency of motion (in Hz).

In order to investigate a relevant range of $St$, we looked at the typical movements of a FOWT, which we briefly present below. The motions of a floating wind turbine are greatly influenced by the type of foundation used (tension-leg platform, semisubmersible, barge or spar). Three distinct ranges of frequencies are usually observed in the motion spectra of a floating turbine.

1. (i) Wave frequency. A floating platform is subject to ocean waves, which cause motions at frequencies around 0.1 Hz (wave period around 10 s). Although the wave-induced motions are rather low in amplitude ($A^* \sim 0.01$), they are high in frequency. For a 10 MW turbine at rated wind speed, this type of motion typically gives $St \sim 1.5$ (Messmer et al. Reference Messmer, Brigden, Peinke and Hölling2022).

2. (ii) Rotational natural frequency. When a floating turbine is displaced from its equilibrium position (due to a gust or a series of waves), it goes back to equilibrium as a damped harmonic oscillator at a frequency that depends on the DoF. For a spar or a semisubmersible, the pitch and roll natural frequencies are typically around 0.035 Hz (Robertson et al. Reference Robertson, Jonkman, Masciola, Song, Goupee, Coulling and Luan2014). For a 10 MW turbine at rated wind speed, such motions give $St \sim 0.5$ and $A^*$ up to $\sim 0.04$.

3. (iii) Translation natural frequency. As with rotational DoF, a floating turbine undergoes translational movements at its natural frequencies. These can be large in amplitude ($A^*$ up to ${\sim }0.1$) but at low frequency ( $f_p \sim 0.01$ Hz). A typical Strouhal number is $St \sim 0.1$ (Leimeister, Kolios & Collu Reference Leimeister, Kolios and Collu2018).

This discussion shows that, typically, the motions of floating turbines cover a range of Strouhal numbers, $St \in [0, 1.5]$. While high-frequency movements are relatively low in amplitude ($A^* \sim 0.01$), oscillations at lower frequencies can reach amplitudes up to 10 % of the rotor diameter. To test a wide range of $St$, we conducted experiments at frequencies ranging from $0.3$ to 5 Hz with low motion amplitudes. We covered $St \in [0, 0.97]$ (corresponding to the maximum operability range of the set-up). Table 3 details all the cases investigated in this paper.

Table 3. Motion cases investigated. For all cases, $\langle TSR\rangle = 6.0 \pm 0.1$. For rotational DoFs: $A^* = T_{length} \times \tan (A_p)$ (see table 1).

To quantify the differences between the cases, we took as reference the fixed case ($St = 0$). To make this study of our different cases comparable, it is essential to note that the mean power and thrust of the moving turbine were the same as those of the fixed turbine for a given wind speed. If this were not the case, comparing the wake flows between cases would be less meaningful, as there would be differences in mean operating conditions.

It was also confirmed by examining the wake deficit in the very near-wake ($x \approx 1.5D$), which was the same for any case with the same $Re$, $\langle C_T \rangle$, $\langle C_P \rangle$ and $\langle TSR \rangle$. With this, we concluded that the induction factor of the turbine was the same, which was also observed by Fontanella et al. (Reference Fontanella, Zasso and Belloli2022). In particular, such a result is not evident for fore–aft DoF as motions cause temporal variations in turbine power and thrust. We found that they do not affect the mean turbine output values, at least for such low amplitudes. However, the impact of the motions is directly related to their effects on the dynamics of the wake.

4.1. Basic energy concepts

In this subsection, we provide a basic mathematical framework for the energy concepts of driving motion from the platform as well as coherent structures in the wake. We start with quantifying the specific energy (i.e. energy per unit mass) of the rotor movements, which is at least partially brought to the wake. The movement velocity is $\dot {\xi }(t) = {\rm d}\xi (t)/{\rm d} t = 2 {\rm \pi}A_p f_p \cos (2 {\rm \pi}f_p t)\ \text {with}\ \xi (t) = A_p \sin (2 {\rm \pi}f_p t)$.

The specific energy, $e_m$, and specific power, $p_m$, of the movement are given by

(4.1) \begin{equation} \left.\begin{gathered} e_m(t) = \dot{\xi}^2(t) = 4 {\rm \pi}^2 A^2_p f^2_p \cos^2(2 {\rm \pi}f_p t), \\ p_m(t) = {\rm d} e_m(t)/{\rm d} t ={-} 16 {\rm \pi}^3 A^2_p f^3_p \sin(2 {\rm \pi}f_p t) \cos(2 {\rm \pi}f_p t). \end{gathered}\right\} \end{equation}

The mean specific energy of the motions, $\epsilon _p$ is expressed as follows:

(4.2)\begin{equation} \epsilon_p = \langle \dot{\xi}^2 \rangle = 2 {\rm \pi}^2 A^2_p f^2_p. \end{equation}

Expressed without dimension, this gives

(4.3)\begin{equation} \frac{\epsilon_p}{U^2_{\infty}} = 2{({\rm \pi} St A^*)}^2 .\end{equation}

Equation (4.3) shows that the mean specific energy of the driving motions is quadratically dependent on $(St A^*)$. In § 3.4, it was seen that for a given frequency of motion $St$, the wake recovery depends on $A^*$, showing that the energy of the driving motions is an important parameter. In figure 9, we mainly compared surge cases with the same amplitude $A^* = 0.007$, but different $St$, so with varying $\epsilon _p$, which cannot be omitted otherwise the comparison of different $St$ is misleading.

As shown later in the paper, the wake of the moving turbine contains coherent structures. A turbulent wind speed signal containing a number $N$ of coherent structures can be decomposed as the summation of the mean speed, the contributions of the coherent structures and the purely stochastic part (Reynolds & Hussain Reference Reynolds and Hussain1972; Baj & Buxton Reference Baj and Buxton2019). The multiple decomposition of $U(t)$ can be written as follows:

(4.4)\begin{equation} U(t) = \langle U \rangle + \sum_{n=1}^{N} \tilde{U}(\phi_n(t)) + u_s(t). \end{equation}

In (4.4), $\tilde {U}(\phi _n(t))$ is the $n$th periodic fluctuation which describes the coherent structure's velocity which oscillates with a period $1/f_n$ such that $\tilde {U}(\phi _n(t)) = a_n f(t+{1}/{f_n})$. The complex wake flow of a floating wind turbine might contain multiple coherent structures. Thus, we consider $n \in [\![1, N]\!]$. The power spectrum of $u'=U(t)- \langle U \rangle$ (noted $\varPhi _x$) contains the specific energy of each coherent structure, $e(f_n)$ as well as of the stochastic fluctuations, $e_s$. We express $\langle u'^2 \rangle$ as

(4.5)\begin{align} \langle u'^2 \rangle &= \int_{0}^{f_{K}} \varPhi_x(f)\,{\rm d} f \nonumber\\ &= \sum_{n=1}^{N} \int_{f_n -\Delta f}^{f_n + \Delta f} \varPhi_x(f)\,{\rm d} f + \int_{f \notin \bigcup_{n \in [\![1, N]\!]} [f_n - \Delta f, f_n + \Delta f] } \varPhi_x(f)\,{\rm d} f \nonumber\\ &= \sum_{n=1}^{N} e(f_n) + e_s = \tilde{e} + e_s . \end{align}

In (4.5), $f_{K}$ is the highest frequency that is physically meaningful, i.e. that contains energy from the flow (note here that we assume that turbulence is frozen (Taylor hypothesis), and so whether we consider time scale or length scale, the outcomes are the same). $f_K$ is the frequency associated with the smallest eddy in the flow, in the dissipation region of the spectrum, called the Kolmogorov scale (Pope Reference Pope2000). In (4.5), $e(f_n)$ is calculated following the methodology described in Appendix A(e).

We next define an amplification factor, $k$, to quantify the energy contained in the coherent structures that form in response to the motions with respect to the ‘specific energy of the driving motions’, $\epsilon _p$. We use as a basis the study of Fiedler & Mensing (Reference Fiedler and Mensing1985), who quantified the energy contained in coherent structures formed in a sinusoidally excited shear layer for low Reynolds ($Re < 10^3$). Although the wake of a moving wind turbine is quite different to that of a conventional shear layer, the two systems show some similarities in the dynamics. We define $k$ as the ratio of the sum of the specific energy of the coherent structures, $\tilde {e}$ (defined in (4.5)) to the ‘specific energy of the driving motions’, $\epsilon _p$ (defined in (4.3)) as

(4.6) \begin{equation} k = \frac{\sum_{n=1}^{N} e(f_n)}{\epsilon_p} = \frac{\tilde{e}}{2 ({\rm \pi} St A^* U_{\infty})^2} \end{equation}

In this study, we restrain $N \leq 4$, which is sufficient to account for the most energetic coherent structures. We use the term energy to refer to any specific energy and use the letter $e$ for this quantity. We identified in each spectrum the $N$ most energetic peaks, for which we calculate the amount of energy associated. For some cases, no periodic structures are found ($N=0$), and for some others, only $f_1$ makes sense (so $N = 1$) and fits with the formulation of (4.4)–(4.5).

4.2. Far-wake dynamics under sway motion

During a sway motion cycle, wakes are generated at various horizontal positions spanning $y \in [-A_p, +A_p]$. The mixing and interaction of these ‘multiple’ wakes affect the mean wake. For low frequencies, the wakes emitted at different horizontal positions superpose linearly (as shown in Appendix B and also observed in Meng et al. (Reference Meng, Su, Qu and Lei2022)). As in our case we are interested in small amplitudes ($A^* \sim 0.01$) we assume that this linear superposition holds if the profiles of $\Delta U / U_{\infty }$ and $TI$ coincide the fixed case. Clear deviations are interpreted as a nonlinear response. As seen in figure 5, for $St = 0.12$, the wake profiles match well the fixed case, but is not the cases for higher $St$.

To further investigate the wake generated by the swaying turbine, we computed the power spectra, $\varPhi _x$ (see Appendix A(d) for details) from the hot-wire measurements of the local velocity time series. Figure 10 displays $\varPhi _x$ computed at $y = 0$ (figure 10a–c) and $y = 0.75R$ (figure 10d–f) for the following cases: fixed (figure 10a,d); $St = 0.12$ (figure 10b,e); $St = 0.42$ (figure 10c, f) for $x \in [\kern-1pt[ 6D, 8D, 10D ]\kern-1pt]$.

Figure 10. Power spectrum, $\varPhi _x$, of the wind speed fluctuations in the wake at 6D, 8D, 10D for fixed case (a,d) and two sway cases – $St = 0.12$ (b,e) and $St = 0.42$ (c, f) – at two locations: $y/R = 0$ (a–c) and $y = 0.75R$ (d–f). Here $A^* = 0.007$, ${\textit {Re}} = 2.3 \times 10^5$. Tests C.1 and C.2 in table 3.

For $St = 0.42$, the spectra along the centreline more or less collapse for $fD/U_{\infty } > 1$, see figure 10(c). In the inertia subrange ( $fD/U_{\infty } \in [1, \sim 20]$), $\varPhi _x \propto f^{-5/3}$, which shows that turbulence in the wake centre is fully developed (Pope Reference Pope2000; Neunaber et al. Reference Neunaber, Hölling, Stevens, Schepers and Peinke2020). Thus, the far-wake is already reached at $x \leq 6D$ for $St = 0.42$, which aligns with the merging of the shear layers seen in figure 4(d).

All three spectra are unequal for $St = 0$ (figure 10a) and $St = 0.12$ (figure 10b), showing that the fully developed region takes place at $x > 8D$.

At $y = 0.75R$, the spectra of the fixed turbine display a region of high turbulent energy for $fD/U_{\infty } \in [0.1, 0.5]$, showing the presence of flow structures with characteristic frequencies in this range (figure 10d). According to Foti et al. (Reference Foti, Yang and Sotiropoulos2018), the far-wake of a wind turbine experiences meandering at natural frequencies, typically in the range of $f_m \in [0.1,0.5]U_{\infty }/D$. Gupta & Wan (Reference Gupta and Wan2019) explain that the inherently erratic wake tends to amplify small perturbations, resulting in unstructured wake meanderings far downstream. The broad peak at $f_m D/U_{\infty } \approx 0.3$ is consistent with this explanation, suggesting that such meanderings result from shear flow instabilities and differ from vortex shedding.

The spectra for $St = 0.42$ show a distinct peak at the frequency of movements for all downstream positions as well the harmonic at $2 St$, see figure 10( f). Meanwhile, for $St = 0.12$ (figure 10e), a barely distinguishable peak at $St$ is found. For the higher $St$, around 60 % of the energy contained in the range of $f D/U_{\infty } \in [0.1, 0.5]$ is concentrated in the peak at $St$ and secondarily at $2St$ (see figure 11i). The peak at the motion frequency is a clear signature of the motions in the wake, indicating that the far-wake contains a coherent structure at a scale of $St$. This was likewise observed by Fu et al. (Reference Fu, Jin, Zheng and Chamorro2019) and Li et al. (Reference Li, Dong and Yang2022). In contrast, for $St = 0.12$, the energy content is broad and very similar to that of the fixed case.

Figure 11. (a–d) Evolution of the two main (largest) frequencies in the spectra of (a) fixed case and (b–d) three sway cases at $y = 0.75R$ for $x \geq 6D$. (e–h) Evolution of the total turbulent energy, contribution of the different frequencies, energy in [0.1,0.5]$D/U_{\infty }$ for the four cases.

For $St = 0.42$, the wake flow locks on to the excitation frequency, somehow having a synchronisation-like effect. For the lowest frequency, this does not appear. This pseudo-lock-in phenomenon, as described in Gupta & Wan (Reference Gupta and Wan2019), occurs when the wake flow gets in a way synchronised with the forcing frequency imposed by the upstream periodic disturbance and amplifies those perturbations (in our case, the movements of the platform). Pseudo-lock-in depends on the frequency, $St$ of the excitation as well as the energy contained in the initial disturbances, $\epsilon _p$ (Fiedler & Mensing Reference Fiedler and Mensing1985; Karniadakis & Triantafyllou Reference Karniadakis and Triantafyllou1989; Pikovsky, Rosenblum & Kurths Reference Pikovsky, Rosenblum and Kurths2002; Gupta & Wan Reference Gupta and Wan2019). A simple representation for synchronisation in the space ($St$-$\epsilon _p$) is the Arnold tongue, which we do not represent here due to insufficient data. Based on the approach of Gupta & Wan (Reference Gupta and Wan2019), we define the following criterion of ‘synchronisation’: (i) the main frequency in the spectrum corresponds to the excitation frequency (i.e. $f_1 D/U_{\infty } = St$); (ii) the amount of energy contained in this mode is much higher than the next most energetic mode in $[0.1, 0.5]D/U_{\infty }$; (iii) independent of the periodic motion excitation and related to natural meandering (noted $St_m$), such as

(4.7)\begin{equation} \frac{e(St)}{e(St) + e(St_m)} > 0.95 \Longrightarrow \text{synchronisation} . \end{equation}

The intensity of the lock in can be measured by the amount of energy contained in the peak at $St$. Figure 11(a–d) shows, for fixed and three sway cases, the evolution of the two main frequencies in the spectrum at $y = 0.75R$ for the three downstream positions. Figure 11( e–i) display the amount of energy contained in the two main peaks of figure 11(a–d). We note that for $St = 0$ (figure 11a,e) and $St = 0.12$ (figure 11b, f), the most energetic peak (shown in blue) has low energy and for $St = 0.12$ this peak appears at a frequency different than $St$. In contrast, for higher values of $St$, the clear peak at $St$ found in figure 10( f) contains a large amount of the total turbulent energy, $\langle u'^2 \rangle / U^2_{\infty }$ (up to 20 %). With respect to the formulation of (4.4), only a coherent structure at $St$ is found for sway DoF (and weakly at $2 St$). Other frequencies are not related to any specific structure in the flow; thus, mainly, $f_1$ makes sense. Based on the criterion (4.7), we conclude that with $A^* = 0.007$, synchronisation happens for $St \in [\sim 0.2, \sim 0.5]$ (in Appendix C, we show some wind speed time series in the wake for synchronised and non-synchronised cases). Figure 11(h) highlights the spatial dependence of synchronisation. The greatest amount of energy in the peak at $St$ is observed at $x = 6D$. We can, however, expect the synchronisation to be stronger closer to the rotor and to depend on $A^*$. This is investigated later on in § 5.1.

After this energy analysis, the signals from the array of hot-wires, with 19 probes aligned horizontally (see figure 2b), are used to visualise the instantaneous wake flow of the turbine (see Appendix A( f) for more details on the methodology). Figure 12 shows the time evolution of $U(x,y,t)$ for the fixed case (figure 12a,d,g), $St = 0.12$ (figure 12b,e,h) and $St = 0.42$ (figure 12c, f,i) at the three downstream positions. For the fixed case, time ($x$-axis) is multiplied by $St_m U_{\infty }/D \approx 0.3 U_{\infty }/D$. In the case of the moving turbine, time is multiplied by the frequency of motion of the platform.

Figure 12. Instantaneous wind field in the wake at 6D, 8D, 10D for fixed case (a,d,g) and two sway cases – $St = 0.12$ (b,e,h) and $St = 0.42$ (c, f,i). Here $A^* = 0.007$, ${\textit {Re}} = 2.3 \times 10^5$. Tests C.1 and C.2 in table 3. For the fixed case, time is multiplied by $f_m = 0.3 \times U_{\infty }/D$. For moving cases, time is multiplied by $f_p$.

For the above-mentioned case of the highest lock in response ($St = 0.42$ at $x = 6D$), we see correspondingly a clear, coherent meandering pattern at the imposed frequency, $f_p$ (figure 12c). As we move downstream, the amplitude of meandering increases and the structures become fuzzier.

For fixed and $St = 0.12$, the meandering structures become more prominent for larger distances but are disordered, especially for $St = 0.12$, the frequency of the motion is not visible in the wake movements, in accordance with figure 10(e).

Overall, the clearest ordered meandering structures are obtained for $St = 0.42$. Close to the rotor ($x = 6D$), they are very coherent. The coherent structures have approximately the lateral size of the rotor diameter, ${\sim }D$, while the exciting amplitude $A_p$ of the motions is much smaller. This reveals a large amplification of the disturbances (quantified later in § 5.1). This is a remarkable difference from the other cases, which have a tendency to build up way less clear meandering structures only farther downstream.

Coming back to the profiles of wake deficit, we found that the wake expansion is the largest for $St = 0.42$ (figure 4b,c), which coincides with the biggest meandering amplitudes of the wake field (figure 12c, f,i). In figure 5, we define the wake recovery over a range of $y \in \sim [-R, R]$, which corresponds to the energy available for a virtual turbine operating in this wake. If a larger range is considered, the recovery is less significant but still higher. From our time-resolved measurements, we find that, compared with the fixed case, the platform's oscillating motions increase the wake's lateral motions, which distributes the energy more uniformly within $y \in [-2.5R, 2.5R]$.

The spectra in figure 10 only indicate the presence of some coherent structures in the wake of the swaying wind turbine, but not their type. On the other hand, the instantaneous wake flow fields in figure 12 indicate that these are coherent meandering structures. To further quantify the periodic meandering at a given frequency, we calculated the cross-correlation function (depicted in Appendix A(g)) between $U(x,-1/2R,t)$ and $U(x,1/2R,t+\tau )$ for $\tau$ varying from 0 to $3/f_p$. Figure 13 shows this quantity computed at 6D (figure 13a), 8D (figure 13b) and 10D (figure 13c) for both fixed and a few swaying cases. We plot $(U_{-1/2R} \star U_{1/2R}) (\tau )$ versus $\tau \times f_p$, except for the fixed case where we set $\tau \times f_p = \tau$. In figure 13(a), it can be seen that for $St \in [0.2,0.5]$, $(U_{-1/2R} \star U_{1/2R}) (\tau )$ is a harmonic function with a period equal to the platform's oscillation period, $1/f_p$, consistent with the pseudo-lock-in phenomenon. The function attains its minimum value (negative value) at $\tau \approx 0$. For $x = 6D$ and $St = 0.42$ (figure 13a), the minimum value of $(U_{-1/2R} \star U_{1/2R}) (\tau \approx 0)$ is approximately $-0.3$, which indicates that the signals $U(-1/2R,t)$ and $U(1/2R,t)$ are anticorrelated. This anticorrelation quantifies the side-to-side motion of the wake at a specific frequency. When the wake meanders to the left, $U(1/2R,t)$ decreases and $U(-1/2R,t)$ increases and vice versa (for illustration, see figure 13g). Consistently, the correlation is maximal at $\tau \approx 1/2f_p$.

Figure 13. Cross-correlation between $U(x,-1/2R,t)$ and $U(x,1/2R,t+\tau )$, noted $(U_{-1/2R} \star U_{1/2R}) (\tau )$, for fixed and sway cases with varying $St$ at 6D, 8D, 10D (a–c). Plot of $(U_{-1/2R} \star U_{1/2R}) (\tau \approx 0)$ versus $St$ (d–f). Here $A^* = 0.007,\ {\textit {Re}} = 2.3 \times 10^5$. Tests C.1 and C.2 in table 3. Panel (g) illustrates the cross-correlation for $\tau = 0$.

In figure 13(d–f), $(U_{-1/2R} \star U_{1/2R}) (\tau \approx 0)$ is plotted versus $St$ for the three downstream positions. In figure 13(d), the anticorrelation is significant in the range of $St \in [0.2, 0.5]$ and close to zero for other values, confirming the range of motion frequencies that initiates large coherent wake meandering.

4.3. Far-wake dynamics under surge motion

Unlike sway motions, surge motions induce variations of $C_T$ that affect the dynamics of the overall wake (Fontanella et al. Reference Fontanella, Bayati, Mikkelsen, Belloli and Zasso2021). We can write $C_T (t) = \langle C_T \rangle + \Delta C_T \sin (2 {\rm \pi}f_p t + \phi )$, with $\Delta C_T$ the amplitude of thrust oscillation and $\phi$ the phase shift with the signal of motion. In our case, following quasisteady theory, the highest value of thrust coefficient variation is $\Delta C^{max}_T \approx 0.07$ for $f_p = 5$ Hz, $A_p = 0.004\,{\rm m}$, $U_{\infty } = 3$ m s$^{-1}$. Besides the $C_T$ variations, the wake is generated behind the turbine at various $x$ locations spanning $[-A_p, +A_p]$ during one cycle of motion.

As with sway, when $St < 0.1$, the wake of the surging turbine is similar to that of the fixed turbine, even at high amplitude (see Appendix B), which is consistent with Schliffke et al. (Reference Schliffke, Aubrun and Conan2020), Meng et al. (Reference Meng, Su, Qu and Lei2022) and Belvasi et al. (Reference Belvasi, Conan, Schliffke, Perret, Desmond, Murphy and Aubrun2022). When the frequency is higher, $St \geq 0.2$, the interactions of the wakes emitted at different $x$ positions and with $C_T(t)$ become more complex.

To understand the impact of surge motions on the dynamics of the wake we carried out a similar analysis to that made for sway (§ 4.2). Figure 14 shows the power spectra of fixed (figure 14a,d), surge with $St = 0.38$ (figure 14b,e) and surge with $St = 0.81$ (figure 14c, f) for $x = 6D, 8D, 10D$ at $y = 0$ (figure 14a–c) and $y = 0.75R$ (figure 14d–f).

Figure 14. Power spectrum, $\varPhi _x$, of the wind speed fluctuations in the wake at 6D, 8D, 10D for fixed case (a,d) and two surge cases – $St = 0.38$ (b,e) and $St = 0.81$ (c, f) – at two locations: $y/R = 0$ (a–c) and $y = 0.75R$ (d–f). Here $A^* = 0.007$, ${\textit {Re}} = 1.4 \times 10^5$. Tests D.1 and D.2 in table 3.

The spectra of $St = 0.81$ at $y = 0$ all collapsed to one spectrum for $fD/U_{\infty } > 1.0$, and $\varPhi _x \propto f^{-5/3}$ (for $fD/U_{\infty } \in [1, \sim 20]$) which confirms that the far-wake is reached for $x \leq 6D$. Whereas for the lowest $St$ and the fixed case, the far-wake is found for larger distances, $x > 6D$. This behaviour is very similar to that observed for the swaying turbine.

Here $\varPhi _x$ at $y = 0.75R$ (figure 14e, f) exhibit a pronounced peak at the motion frequency, $St$, for both surge cases, in line with results from Schliffke et al. (Reference Schliffke, Aubrun and Conan2020) and Belvasi et al. (Reference Belvasi, Conan, Schliffke, Perret, Desmond, Murphy and Aubrun2022). As with sway, this indicates that the wake of the surging turbine contains coherent flow structures with a characteristic frequency of $f_p$. The peak is for both $St$ maximum at $x = 6D$ and then decreases.

When looking at the spectra of the fixed turbine at $y = 0.75R$, the frequency range with the largest amount of energy is similar ($0.1 < fD/U_{\infty }< 0.5$) to that of the other fixed case in figure 10(d). The maximum energy is located at $f_m \approx 0.3 U_{\infty }/D$, which is close to that of figure 10(d). These two cases have a ${\textit {Re}}$ from $2.3 \times 10^5$ in figure 10 to $1.4 \times 10^5$ in figure 14.

Looking at the case with $St = 0.38$, it is interesting to note that all the frequency components of the fixed case are dominated by the narrow peak corresponding to the excitation frequency, i.e. at $St = 0.38$. This is somehow similar to the sway case (see figure 10f). For $St = 0.81$, compared with the sway case, a new dynamical behaviour is observed; we find two frequencies in the spectra with apparent mixing components (see figure 14f). One peak is at the excitation frequency, i.e. $St = 0.81$. The other peak at ${\sim }0.31U_{\infty } /D$ seems to be due to a self-generated mode, $f^*$ of the wake (in $[0.1, 0.5] U_{\infty }/D$). Most interestingly, the further smaller peaks are related to linear combinations of the two frequencies, i.e. with $aSt + b(f^*D/U_{\infty })$. One small peak is around $(a,b) = (1,-1/2)$ and another one at $(a,b) = (1,-1)$. This proves nonlinear mode coupling, which is discussed later in § 4.4. For other values of $St \in [0.6, 0.9]$, we also observe a second large peak at a frequency, $f^*$ in $[0.1,0.5]U_{\infty } /D$ like the one at $0.31U_{\infty } /D$ for $St = 0.81$. We emphasise here that the value of $f^*$ does not appear to be universal.

As with sway, we are interested in the spatiotemporal structure of these oscillating modes. We show in figure 15 the instantaneous wake flows field for surge with $St = 0.38, A^*=0.017$ (figure 15a,b), and $St = 0.81, A^*=0.007$ (figure 15c,d) at $x = 6D$. For the lowest $St$, we show the case with the largest amplitude because it features the most distinct structures.

Figure 15. Instantaneous wind field in the wake at $x = 6D$ for surge, $St = 0.38, A^*=0.017$ (a,b); and $St = 0.81, A^* = 0.007$, c,d). Tests D.2 and D.$2^*$ in table 3 (${\textit {Re}} = 1.4 \times 10^5$). For $St = 0.38$, time is multiplied by $f_p$ and for $St = 0.81$ is multiplied by $f^* \approx 0.31 U_{\infty }/D$. Panels (b,d) are zoom-in of the region $y \in [-1.6R,1.6R], t^* \in [55, 62]$.

For $St = 0.38$ (figure 15a,b), the wake exhibits a clear pulsating pattern at the platform frequency, $f_p$. This coherent structure is directly linked to the peak at $St$ in figure 14(e). The wind speeds at $y= \pm R$ are in phase, and the wake motions act like a pump, pumping the outer fluid into the inner wake (see the alternation of high and low speeds in the wake in figure 15b). As with sway, the coherent structures are of the order of the rotor diameter, showing the large amplification of the turbine's motions.

The flow dynamics for $St = 0.81$ are quite different, as the wake at 6D (figure 15c,b) reveals a coherent meandering structure with a frequency of $f^* \approx 0.31U_{\infty }/D$, fully consistent with the spectrum of this case (figure 14f). The pulsating mode found for the lowest $St$ is almost not observed for the highest $St$, and the wake of the surging turbine with $St = 0.81$ unexpectedly closely resembles that of the swaying turbine (figure 12c). However, compared with sway, the coherent meandering mode of surge seems less structured, which is discussed further in § 4.5.

To investigate the pulsing and meandering modes of the wake in more detail, we computed the cross-correlation between the wind speed signals $U(x,-1/2R,t)$ and $U(x,1/2R,t+\tau )$ (cf. Appendix A(g)). We calculated $(U_{-1/2R} \star U_{1/2R}) (\tau )$ for surge cases D.1–D.4 (see table 3). Figure 16 shows $(U_{-1/2R} \star U_{1/2R}) (\tau )$ against $\tau \times f_p$. In figure 16(a), we observe that the cross-correlation is almost zero everywhere for $St < 0.2$ and $St \approx 1.0$, indicating that the wake does not oscillate at a clear frequency.

Figure 16. Cross-correlation between $U(x,-1/2R,t)$ and $U(x,1/2R,t+\tau )$, noted $(U_{-1/2R} \star U_{1/2R}) (\tau )$, for fixed case and surge cases with varying $St$ at $6D$, $8D$, $10D$ (a–c). Plot of $(U_{-1/2R} \star U_{1/2R}) (\tau \approx 0)$ versus $St$ (d–f). $A^* = 0.007, {\textit {Re}} = 1.4 \times 10^5$ and ${\textit {Re}} = 2.3 \times 10^5$. Tests D.1–D.4 in table 3. Panel (g) depicts the wake of the fixed turbine, panel (h) shows a typical pulsating motion of the wake and panel (i) displays a typical coherent meandering pattern.

For $0.2 < St < 0.55$, $(U_{-1/2R} \star U_{1/2R}) (\tau )$ is a harmonic function with a period of $1/f_p$, and the cross-correlation is maximal (positive) at $\tau \approx 0$. The velocity fluctuations between $y=-1/2R$ and $y=1/2R$ are in phase, which implies that the wake undergoes pulsating movements and not meandering.

On the other hand, for $St = 0.81$, $(U_{-1/2R} \star U_{1/2R}) (\tau )$ is a harmonic function with a period of ${\sim } D/(0.31 U_{\infty })$, and the cross-correlation is minimum (negative) at $\tau \approx 0$. This behaviour is associated with coherent meandering movements of the wake at a given frequency, as discussed for sway in § 4.2 and is consistent with the instantaneous flow fields shown in figure 15(c,d).

Next, we show that the analysis of the cross-correlation functions allows us to distinguish the different ranges of $St$ for which the dynamic behaviour of the wake changes. In figure 16(d–f), the $St$ dependency of the cross-correlation at $\tau \approx 0$ is shown. The fit of this cross-correlation allows us to identify three distinct regions, highlighted in figure 16(d) and schematically visualised in figure 16(g–i). For $St < 0.2$ and $St \approx 1.0$ (zone (g)), the motions do not significantly impact the wake's dynamic, which is similar to that of the fixed turbine (as illustrated in figure 16g). It shows disordered natural meandering. For $St \in [0.2, 0.55]$ (zone (h)), the positive correlation values indicate that the wake undergoes pulsing movements (maximum correlation at $St \approx 0.4$). A clear, coherent structure at the excitation frequency $St$ is found in the wake. For the range of $St$ in the zone (h), the criterion of (4.7) is satisfied, which shows that synchronisation is occurring (as observed with sway in figure 13d). In Appendix C, we display some wind speed time series in the wake for $St \in [0, 0.4]$ illustrating instances of synchronisation and non-synchronisation. Figure 16(h) shows a typical pulsating pattern in the wake. For $St > 0.55$, $(U_{-1/2R} \star U_{1/2R}) (\tau \approx 0)$ decreases and becomes negative, indicating that the wake undergoes coherent meandering, as shown in figure 16(i).

Farther downstream, figure 16(e, f) indicate that pulsating wake movements, as well as the periodic meandering ones, gradually vanish.

4.5. Comparison of coherent meandering between sway and surge

We saw that for surge motion, when the quasiperiodic state appears, the wake tends to generate coherent meandering structures at a frequency $f^* \in [0.1, 0.5]D/U_{\infty }$ (figures 15c,d and 17f–h). It is of interest to compare this mode with the coherent meandering caused by sway motions. We make a comparison, in figure 18, of the wake from the turbine moved in surge at $St = 0.81$ which generates a coherent meandering mode at $f^*D/U_{\infty } \approx 0.3$ with the coherent meandering structures due to swaying turbine movements at $St = 0.29$ (in both cases $A^* = 0.007$).

Figure 18. Comparison of coherent meandering from sway motion ($St = 0.29, A^* = 0.007$) with surge ($St = 0.81, A^* = 0.007$) induced wake meandering at $x = 6D$. Plot of wake deficit and $TI$ profiles (a,d). Instantaneous flow at $x = 6D$ for surge (b) and sway (c). Power spectrum at $y = 0.75R$ for the two cases, surge (e) and sway ( f). Cases C.1, C.4 and D.2 in table 3 (${\textit {Re}} = 1.4 \times 10^5$).

In figure 18(a), we show the wind speed deficit profiles at $x = 6D$, which follow very closely for surge and sway, giving a very similar recovery (also observed in figure 8). The wake of the swaying wind turbine, however, expends more, which is also seen in the $TI$ profiles (figure 18d). Moreover, the swaying motions generate more turbulence. The instantaneous flow fields are shown in figure 18(b) for surge and figure 18(c) for sway and the spectra at $y = 0.75R$ in figures 18(e) and 18( f), respectively. Coherent meandering is made very clear for sway, driven by the excitation frequency $St$. In contrast, for surge, the structures are not as clear. The instantaneous flow contains the main structure at $f^* D / U_{\infty }$ together with the interacting modes ($St - f^*D/U_{\infty }$ for instance), making in contrast to sway the whole wake less coherent. A closer look at the pattern in figure 18(b) shows that a temporal change between meandering ($t f^* \in [45, 48]$) and pulse-like structures ($t f^* \in [48, 50]$) occur, which supports the idea of interacting modes. The energy contained in the motions is approximately eight times greater for surge compared with sway in this example (see $\epsilon _p$ in § 4.1), but the effect on recovery is very similar for both DoFs. We conclude that for sway, the wake tends to better use the energy of the motion than for surge (at least when comparing sway, $St \approx 0.3$ with surge, $St > 0.55$). We also highlight the more complex structures of the wake with surge motion.

5.1. Development of coherent meandering and amplitude dependency for sway motion

We have seen in figure 8(a) that for two sway cases with the same motion frequency but varying amplitudes, differences in recovery occur (around 14 % more with the largest amplitude). We also noticed that for a higher amplitude ($A^* \approx 0.02$), the most favourable $St$ for the recovery is smaller, around $St \approx 0.3$ here, which is in good agreement with the CFD results from Li et al. (Reference Li, Dong and Yang2022). In the following, we look at the wake development for $St = 0.29$ with two amplitudes and show the results for $x \in [\![4,6,8]\!]D$.

Figure 19 compares the amount of energy contained in the wake at $y = 0.75R$ for fixed case (figure 19a) and for sway with $St = 0.29$; $A^* = 0.007$ (figure 19b) and $A^* = 0.017$ (figure 19c). Figure 19(b,c) also display the energy contained in the most energetic mode related to the coherent meandering motion of the wake at $St$ (blue thick line) as well as the amplification factor, $k$ (defined in (4.6)) (red solid line).

Figure 19. Evolution of turbulent energy, $\langle u'^2 \rangle / U^2_{\infty }$, energy of the coherent mode, $e(St)$, and energy contained in the range of frequency $[0.1,0.5]U_{\infty }/D$ at $y = 0.75R$ for fixed (a), sway with $St = 0.29, A^* = 0.007$ (b) and sway with $St = 0.29, A^* = 0.017$ (c). For panels (b,c) the amplification factor is represented with solid red line (defined in § 4.1). Instantaneous flow at $x = 4D$ is displayed in (d) for $A^* = 0.007$ and in (e) for $A^* = 0.017$. Tests C.3 and C.4 in table 3 (${\textit {Re}} = 1.4 \times 10^5$).

First, the motions largely enhance the total turbulent energy, $\langle u'^2 \rangle / U^2_{\infty }$ at $x = 4D$. Here $\langle u'^2 \rangle / U^2_{\infty }$ is more than two times higher for $A^* = 0.017$ (figure 19c) compared with the fixed case (figure 19a). A large amount of this turbulent energy is contained in the coherent structure at $St$ (thick blue line in figure 19b,c). For $A^* = 0.017$, the maximum energy of the mode at $St$ is found at $x = 4D$ and around $6D$ for $A^*=0.007$. This shows that the spatial development of the structures depends on the energy of the motions. With the same excitation frequency, $St = 0.29$, the higher amplitude accelerates the build-up.

A visualisation of the instantaneous flows for the two swaying cases at $x = 4D$ is given for $A^* = 0.007$ in figure 19(d) and for $A^* = 0.017$ in figure 19(e). In line with the energy content of the structures, for the highest amplitude (figure 19e), a very clear pattern of coherent meandering is observed, while for the lowest amplitude, the structures are still developing at $4D$ (figure 19d). The most prominent synchronisation is observed for the largest amplitude at $x = 4D$. Here, 90 % of the energy in the frequency range $[0.1, 0.5]D/U_{\infty }$ is contained in the coherent mode (see the blue thick line and the dotted dashed line in figure 19c).

In § 4.1 we derived a formula for the energy of motion, $\epsilon _p$ (4.3) and defined an amplification factor, $k$ (4.6). Between the two sway cases presented here, $\epsilon _p$ is approximately six times bigger for the highest amplitude than the lowest. In figure 19(b,c), $k$ is plotted against $x/D$ in red for the two sway amplitudes. Here, only the energy of the most energetic structure is taken into account, the other modes being insignificant and irrelevant. We find that the structure's energy at $St$ is 90 times greater than that of the rotor motions (for $A^* = 0.007$ at $6D$ (figure 19b)). This indicates that the initial periodic perturbations are strongly amplified in the wake, forming large, coherent, meandering structures. It is a typical effect of nonlinear synchronisation, which strongly amplifies the energy of the motions. This is somewhat similar to the observations made by Fiedler & Mensing (Reference Fiedler and Mensing1985) for a periodically excited shear layer (even tough at much lower $Re$ of approximately 10$^3$). The results in figure 19(b,c) also highlight the amplitude dependence of the amplification, where lower amplitudes give a ‘stronger’ response (i.e. a higher $k$), even though, in real terms, the higher amplitude provides more energy to the wake movements.

5.2. Development of coherent structures under surge motion

Depending on the frequency of the surge motion, two different types of coherent structures have been found in the wake (figure 16). On the one hand, we observed a synchronisation-like effect of the wake with $St \in [0.2, 0.55]$, which leads to the formation of coherent pulsating structures in the wake. On the other hand, for higher $St$, we saw that the wake not only responds directly to excitation with a peak at $St$ but also tends to generate a coherent meandering mode with a frequency $f^*$ in the range $[0.1, 0.5]D/U_{\infty }$. In this case, additional structures are found combining $St$ and $f^* D/U_{\infty }$. In this subsection, we detail the spatial development of the different structures. We start with the synchronisation cases and then move on to the quasiperiodic state.

In figure 20, we display the evolution of the two primary frequencies in the power spectrum of the wind speed fluctuations at $y = 0.75R$ (i.e. in the shear layer) for $x \geq 3D$ for the fixed case (figure 20a) and surge cases with $St = 0.19, A^* = 0.007$ (figure 20b), $St = 0.38, A^* = 0.007$ (figure 20c), $St = 0.38, A^* = 0.017$ (figure 20d). Figure 20(e–h) shows the evolution of the energy associated with each of these frequencies, as well as their sum and the total turbulent energy (following the formulation of § 4.1). In direct relation to the results presented previously in figure 16, for $St < 0.2$ (figure 20b, f), the wake does not synchronise with the periodic platform excitation. Here, up to $x = 7D$ the main peak in the spectrum differs from $St$. Beyond ($x \geq 7D$), the primary mode becomes $St$ but contains almost the same energy as the platform's motion-independent second peak, see figure 20( f). The energy of the mode is too low to classify the structure as a pseudo-lock-in mode (criterion (4.7) is not fulfilled). Moreover, $\langle u'^2 \rangle / U^2_{\infty }$ is slightly higher but still very close to the energy of the fixed case, by comparing figures 20(e) and 20( f).

Figure 20. Evolution of the two main frequencies in the power spectrum at $y = 0.75R$ (a–d) and associated energies as well as turbulent energy, $\langle u'^2 \rangle / U^2_{\infty }$, and energy contained in $[0.1,0.5]U_{\infty }/D$ (e–h); for $St = 0$ (a,e), surge with $St = 0.19, A^* = 0.007$ (b, f), $St = 0.38, A^* = 0.007$ (c,g) and $St = 0.38, A^* = 0.017$ (d,h). Cases D.1, D.2 and D.$2^*$ in table 3 (${\textit {Re}} = 1.4 \times 10^5$).

For increasing $St$, synchronisation is made clear by the main peak in the spectrum; see figure 20(c,d) at $St = 0.38$. The main peak in the spectrum contains much more energy than the secondary peak and satisfies our synchronisation criterion (4.7). Moreover, the turbulent energy, $\langle u'^2 \rangle / U^2_{\infty }$, is much higher than for $St = 0$. The coherent pulsating mode contains more energy for $A^* = 0.017$ (figure 20h) than for $A^* = 0.007$ (figure 20g), and the structure develops earlier for the larger amplitude. As we shall see later, this provides a reasonable explanation for the enhanced recovery observed in figure 8(b) between the two cases of various amplitude. As for sway, when synchronisation happens, most of the energy in $[0.1, 0.5]D/U_{\infty }$ is concentrated in the peak at $St$. The wind turbine's wake tends to amplify the initial disturbances leading to the formation of coherent structures, a phenomenon that we quantify by the amplification factor, $k$ (4.6). In figure 21, we plot $k$ for some surge cases as a function of $x/D$. The blue lines without markers represent the cases where synchronisation occurs (to calculate $k$, we have only taken the first mode, $f_1 = f_p$, since the other frequencies are not linked to the platform excitation). We can see that the strongest amplification is obtained for the smallest $St$, i.e. when the input energy is the lowest. Accordingly, it is consistent with the results obtained for sway. Here, at $St = 0.29$ (maximum), the coherent structure contains almost 40 times more energy than the driving movements provide. With increasing $St$, the position where $k$ reaches its maximum moves closer to the rotor, showing the spatial dependency of the synchronisation. For $St = 0.38$, the highest amplitude exhibits a lower amplification (maximum of 10 for $A^* = 0.017$ against 25 for $A^* = 0.007$).

Figure 21. Downstream evolution of the amplification factor, $k = e(St)/\epsilon _p$ at $y = 0.75R$ for the cases where synchronisation occur (here only $St \in [0.29, 0.38]$ are shown, displayed in blue without markers). Evolution of $k = \sum _{n=1}^{4} e(f_n)/\epsilon _p$ for the cases where quasiperiodic state is found ($St \in [0.58, 0.81]$ here, represented in red with markers). Cases D.1, D.2 and D.$2^*$ in table 3 (${\textit {Re}} = 1.4 \times 10^5$).

Figure 22 shows results for cases where $St > 0.5$ and $A^*=0.007$, i.e. when quasiperiodic states occur. Similar to the previous figure 20, we plot in figure 22(a–d) the evolution of the four main frequencies in the spectrum ($N = 4$) at $y = 0.75R$ for $x \geq 3D$; fixed case (figure 22a) and surge cases with $St = 0.58$ (figure 22b), $St = 0.81$ (figure 22c), $St = 0.97$ (figure 22d). The energy content of the modes and the total turbulent energy are displayed in figure 22(e–h). The evolution of the energy contained in the first four most energetic peaks for $St \in [0.55, 0.9]$ evolves erratically; at a given position, one mode has more energy than the other and vice versa. This is a typical feature of energy exchange between modes in a quasiperiodic system (Peinke et al. Reference Peinke, Parisi, Rössler and Stoop2012). When quasiperiodicity occurs, the four frequencies are linked to coherent structures and contain close amounts of energy. Nevertheless, the peak at $f^*$ is always the most energetic, at least for $x > 5D$ (see the blue thick line in figure 22f,g).

Figure 22. Evolution of the four main frequencies in the spectrum at $y = 0.75R$ (a–d) and associated energies as well as turbulent energy, $\langle u'^2 \rangle / U^2_{\infty }$, and energy contained in $[0.1,0.5]U_{\infty }/D$ for $St = 0$ (a,e), surge with $St = 0.58$ (b, f), $St = 0.81$ (c,g) and $St = 0.97$ (d,h). Cases D.1 and D.2 in table 3 (${\textit {Re}} = 1.4 \times 10^5$ and $A^* = 0.007$).

The surge case with $St = 0.58$, figure 22(b, f), is particularly interesting because it lies at the boundary between synchronisation and quasiperiodicity. The path to quasiperiodicity begins with weak synchronisation before moving towards a higher DoF. The amount of energy contained in the first four modes represents approximately 30 % of the total turbulent energy, again showing the large amount of energy contained in coherent structures with turbine motions. For motions with $St \in [0.5, 0.9]$, the amount of turbulent energy is much higher and develops closer to the rotor (figure 22f,g) than in the fixed case (figure 22e). For $St = 0.97$, the total turbulent energy is lower than in cases where $St$ is smaller, even though the amount of energy introduced by the motions is the highest for $St = 0.97$ (cf. (4.3)). This result suggests that for $St > 0.9$, the platform excitation frequency is too far away from the natural wake frequencies, and consequently, the effects on wake dynamics are less marked.

For the quasiperiodic cases, we calculate $k$ taking into account the energy contained in the four most energetic modes, which are combinations of $St$ and $f^*D/U_{\infty }$. We plot in figure 21 in red the evolution of $k$ for some surge cases with $St > 0.55$. The maximal value of $k$ is found for $St = 0.58$. Compared with the cases where synchronisation occurs (blue lines), $k$ decreases less sharply than for synchronised cases. This suggests that structures that interact and exchange energy with each other remain in the wake for longer, although gradually decaying. For the highest $St$, a very small value of $k$ is found, showing that the wake reacts or, respectively, synchronises weakly to this frequency.

6.1. Summary of the results

First, our results demonstrate the equivalence between surge and pitch, as well as between sway and roll on the wake generated (see figure 3 in § 3.1). Therefore, we focused our research on sway for side-to-side and surge for fore–aft motions.

Second, the experimental results are in excellent agreement with the CFD simulations from Li et al. (Reference Li, Dong and Yang2022) about the wake of a swaying turbine in a laminar wind (figure 5). This suggests that the wake of a floating turbine does not depend on the Reynolds number, at least for ${\textit {Re}} > 10^5$.

Third, our findings indicate that both types of turbine movement (side-to-side and fore–aft) significantly enhance wake recovery (up to 25 % more than for the fixed case), especially for $St \in [0.2, 0.6]$ with sway (figures 5, 8) and $St \in [0.3, 0.9]$ for surge (figures 7, 9). Amplitude and frequency of the driving motion are important parameters impacting wake recovery (figures 8 and 9).

Fourth, the wake recovery curves for different surging cases (various $St$ and $A^*$) can be rescaled based on the definition of a virtual origin $x^*_0$ that corresponds roughly to the position where the far-wake is reached (§ 3.4). With such renormalisation, the recovery curves appear to merge into one, demonstrating the universality of the wake once the far-wake starts. This idea is further supported by the close similarity of the power spectrum at $x = x^*_0$ in the wake centre (see figure 24b).

Fifth, we worked out different coherent structures in the wake. For $St \in [0.2, {\sim }0.5]$, sway motion leads to the formation of coherent meandering structures (see figures 12c, f and 19d,e). For surge, with $St \in [0.2, {\sim }0.55]$, the wake shows clear pulsing structures (cf. figure 15b). Astonishingly, for surge motion with $St \in [{\sim }0.55, 0.9]$, the wake features coherent meandering patterns coupled with a mild pulsing (see figure 15d).

Sixth, these structures are linked to special low-order nonlinear dynamics, which are particularly visible in the shear layer region ($y \in [0.5, 1.3]R$). Unlike the complex broadband frequency range observed in the wake of the fixed turbine (figures 10d and 14d), the wake of the moving turbine gets slaved to the excitation frequency $St$ (figure 10f for sway and figure 14e for surge). Further, for surge, the wake dynamics feature a combination of $St$ and a self-generated mode $f^*$, along with coupling components (figures 14f and 17f–h). This organised, low-dimensional (monochromatic or oligochromatic) response is much easier to describe than the complex fixed case.

On one hand, for both DoF and $St \in [0.2,\sim 0.5]$, the wake synchronises (pseudo-lock-in) with the motion frequency $St$, directly linked to the formation of coherent structures. The energy of these structures, quantified at a point in the shear layer ($y=0.75R$), is much higher than that of the driving motions, $\epsilon _p$ (as defined in § 4.1). For sway, we found that a meandering structure contains up to 90 times more energy than the driving motions (cf. figure 19b). Similarly, for surge, up to 40 times more energy than $\epsilon _p$ is found in the coherent pulsating structure (see figures 20g,h and 21 blue curves). This shows the nonlinear amplification of the energy of the driving motions.

On the other hand, for $St \in [\sim 0.55,0.9]$, the surging turbine wake initially synchronises with $St$. Moving downstream, a self-generated mode $f^*$ emerges and falls into the natural frequency range of the fixed turbine wake ( $f^* \in [0.1, 0.5]U_{\infty } / D$). The mixing components of $St$ and $f^*D/U_{\infty }$ indicate nonlinear interactions, characteristic of quasiperiodic systems. These quasiperiodic modes reach energies up to 20 times that of the driving motion (see figure 21; in red). The erratic evolution of the energy of the coupling modes, as depicted in figure 22(b,c, f,g), underscores the unpredictable nature of the interactions between $St$ and $f^*D/U_{\infty }$ characteristic of nonlinear dynamics.

In summary, we worked out one dynamic mode for sway (meandering) and two dynamic modes for surge (pulsating and meandering).

6.2. Enhanced recovery due to nonlinear dynamics

These nonlinear dynamical behaviours play an important role in the recovery process. We showed and discussed in § 5 how the wake dynamics develop in the transition region. The synchronised and quasiperiodic modes grow by amplifying the energy of the driving motion. At a position downstream, the modes become unstable. The structures then break down and decay, feeding the turbulent cascade (see figure 23). The movements of the platform enable a faster transition to the far-wake, meaning that the recovery processes start closer to the rotor, around $x^*_0$. In addition, the coherent modes likely transport momentum and thus improve recovery but also periodically deflect the wake outwards, resulting in greater recovery on average, similar to the effect of blade pitching control strategy on wake dynamics and recovery (Korb et al. Reference Korb, Asmuth and Ivanell2023).

For surge cases with $St > 0.9$, our interpretation is that the periodic fluctuations in the near wake destabilise the tip-vortex system closer to the rotor and enhance the generation of small-scale turbulence, which seems to be a reasonable explanation of the high turbulence found close to the rotor (see figure 22h for $x = 3D$). This might explain why the recovery remains high even though no clear coherent mode is found.

6.3. Application to wind energy

The results of this paper show that wake excitation due to platform motion is carried downstream, amplified and interacts nonlinearly, resulting in large coherent meandering or pulsating structures. At the same time an increase in wake recovery compared with a fixed wind turbine (up to 25 % more) is found. As the power is proportional to the cube of the wind speed such a faster recovery of the wind speed in the wake is important for the power production of a wind farm. This potentially compensates partially the effect that the power produced in a wind farm decreases from the first to the following rows, a drop of 40 % is reported in Barthelmie et al. (Reference Barthelmie2009) for the worst wind direction. However, it should be noted that large coherent structures could contribute to increasing loadings on downstream turbines. Further studies are needed to work out the balance between the favourable increase in power and the unfavourable increase in loads.

A further aspect concerns turbulent inflow conditions. To verify the relevance of our study, we performed measurements with $TI_{\infty }$ up to $0.03$. These represent typical conditions for stable atmospheric situations where $TI_{\infty }$ is low, cf. Angelou et al. (Reference Angelou, Mann and Dubreuil-Boisclair2023). We found that the effects of platform motion remain, in good agreement with the CFD results of Li et al. (Reference Li, Dong and Yang2022) (see Appendix D, up to 8 % more recovery for the best sway case is found with $TI_{\infty } = 0.03$). Beyond these stable conditions of low turbulence, other situations including higher turbulence intensity, shear and six-DoF stochastic platform motions may lead to different dynamics. Based on the results of Li et al. (Reference Li, Dong and Yang2022), for $TI_{\infty } > 0.05$, the impact of the motions seems to be more or less cancelled out by the effect of the inflow turbulence. The impact of moving turbines on the wake with broader conditions remains to some extent open. Nevertheless, we have shown that for stable meteorological conditions our results are of importance.

6.4. Routes to turbulence

The classical picture of a transition to turbulence is that a laminar flow becomes unstable with increasing parameters, like the Reynolds or Rayleigh numbers. Periodic oscillations occur, which due to further instabilities become higher-dimensional like quasiperiodic, and subsequently chaotic. The chaotic range leads to noisy power spectra, which then evolves, like in the Rayleigh–Bénard system, to weak and finally hard turbulence, cf. Castaing et al. (Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989). The final turbulence is characterised by its fractal ordering of many different modes, typically expressed in the K41 law of the inertia subrange by the $f^{-5/3}$ power spectra. Interestingly, some large scales or superstructures may reappear in the turbulent state as discussed recently by Hartlep, Tilgner & Busse (Reference Hartlep, Tilgner and Busse2003) and Pandey et al. (Reference Pandey, Chu, Laurien and Weigand2018).

For a wake, like behind a wind turbine, the transition to turbulence differs from a closed system. Given the experimental set-up, the evolution to turbulence occurs with distance to the turbine, as the flow is carried away. Similar to fractal grid, see for instance (Hurst & Vassilicos Reference Hurst and Vassilicos2007; Vassilicos Reference Vassilicos2015), the turbulence builds itself up to a point of maximal turbulent energy (see figure 23a) and then a transition to a universal, or fully developed decaying turbulence is seen, which we denoted as the far-wake where the recovery of the whole wake takes place (see figure 24b). Here again a turbulence with its common fractal properties is found (see figure 23c,d).

For the fixed turbine, we found that this transition region has a quite complex multimode structure, which tends to form some erratic meandering structures (see figure 12g) like the so-called re-establishment of a turbulent vortex street or the just-mentioned superstructures. An interesting finding is that this changes fundamentally if the turbine moves periodically in a special range of $St$ numbers. Even very small exciting motions of the turbine lead to large periodic and quasiperiodic structures in the wake. The re-establishment or superstructures get slaved or synchronised to a low-dimensional coherent dynamics, like an excitation of a weakly damped mode. The astonishing finding is that this deviating path to turbulence via nonlinear dynamics is ultimately a short cut, as we find a faster recovery.

There are different open details for this scenario of transition to turbulence.

1. (a) Is it the nonlinear growth of the low-dimensional nonlinear dynamics that makes these structures finally unstable and leads to accelerated transition? Is there an intrinsic $Re$ number for the growing structures with a threshold, $Re_c$ where they become unstable or collapse like breaking water waves?

2. (b) Does chaos play a role? From nonlinear dynamics it is well known that periodically driven damped modes, or quasiperiodicity are good candidates for chaos. A low dimensional dynamic which becomes chaotic would be accompanied by a spectral broadening of the initially well localised frequencies.

3. (c) Is the increase in the background turbulence's inertial subrange towards lower frequencies destabilising the periodic structures? This would require some kind of backward cascade leading to larger structures, as known from two-dimensional turbulence.

For the first two scenarios, the transition is marked by the periodic structures’ instability that drives the far-wake's turbulence. The last scenario would be a more complex interplay between low-dimensional dynamics and turbulence. However, it is astonishing that the enhanced recovery, or the faster transition to turbulence, goes along with simple and clear phenomena of low-dimensional nonlinear dynamics.