2.1. Numerical method

The nonlinear evolution of wind turbine wake is simulated with the VFS-Wind code (Yang et al. Reference Yang, Sotiropoulos, Conzemius, Wachtler and Strong2015), assuming the air is a Newtonian fluid with a constant density and viscosity. The flow is governed by the filtered incompressible Navier–Stokes equations, written in Cartesian coordinates as

(2.1)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \tilde{\boldsymbol{u}} = 0 , \end{gather}

(2.2)\begin{gather}\frac{\partial \tilde{\boldsymbol{u}}}{\partial t}+(\tilde{\boldsymbol{u}} \boldsymbol{\cdot} \boldsymbol{\nabla}) \tilde{\boldsymbol{u}} ={-}\boldsymbol{\nabla} \tilde{p} + \nu {\nabla}^2 \tilde{\boldsymbol{u}} - \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau} + \boldsymbol{f}, \end{gather}

where $\boldsymbol {u} = \{u_x, u_y, u_z\}$ is the velocity vector in the Cartesian coordinates, $p$ is the pressure and $\nu$ is the fluid kinematic viscosity. Here $\tilde {{\cdot }}$ denotes the spatial filtering process; $\boldsymbol {\tau }$ is the subgrid-scale stress resulting from the filtering of the nonlinear convection term and is closed with the dynamic Smagorinsky model (Smagorinsky Reference Smagorinsky1963; Germano et al. Reference Germano, Piomelli, Moin and Cabot1991); $\boldsymbol {f}$ denotes the body force term. In the equation, bold symbols represent vectors (e.g. $\boldsymbol {u}$) or two-dimensional tensors (e.g. $\boldsymbol {\tau }$), the rest are scalar variables.

In wind turbine wake simulations the body force term $\boldsymbol {f}$ represents the effect of wind turbines on the flow and is computed using a well-validated actuator surface model (Yang & Sotiropoulos Reference Yang and Sotiropoulos2018) for both the rotor and nacelle. This model simplifies each rotor blade as a zero-thickness rotating surface that exerts aerodynamic forces on the surrounding flow. The aerodynamic forces on each blade element are computed individually using two-dimensional blade aerodynamic coefficients as

(2.3)\begin{gather} \boldsymbol{F} = \boldsymbol{L} + \boldsymbol{D}, \end{gather}

(2.4)\begin{gather} \boldsymbol{L} = \tfrac{1}{2}c C_{L}(\alpha, Re_c)|V_{ref}|^2 \boldsymbol{e_{L}}, \end{gather}

(2.5)\begin{gather} \boldsymbol{D} = \tfrac{1}{2}c C_{D}(\alpha, Re_c)|V_{ref}|^2 \boldsymbol{e_{D}}, \end{gather}

where $\boldsymbol {F}$ is the aerodynamic force per unit span, with $\boldsymbol {L}$ and $\boldsymbol {D}$ the lift and drag coefficients. Here $C_L(\alpha, Re_c)$ and $C_D(\alpha, Re_c)$ are the lift and drag coefficients, depending on the local airfoil and the angle of attack ($\alpha$) and the Reynolds number ($Re_c$) defined with the chord length ($c$); $V_{ref}$ is the relative flow velocity with respect to the rotating blade; $\boldsymbol {e}_{L}$ and $\boldsymbol {e}_{D}$ are unit vectors defining the lift and drag directions. Three-dimensional effects (Du & Selig Reference Du and Selig1998) and the tip loss (Shen et al. Reference Shen, Mikkelsen, Sørensen and Bak2005) are corrected before computing the momentum source in (2.2)

(2.6)\begin{equation} \boldsymbol{f} = \boldsymbol{F}/ c = (\boldsymbol{L}+\boldsymbol{D})/ c. \end{equation}

The actuator surface model for the nacelle enforces the non-penetration boundary condition in the nacelle surface's normal direction and applies frictional forces with empirical friction coefficients (Yang & Sotiropoulos Reference Yang and Sotiropoulos2018). The smoothed discrete delta function of Yang et al. (Reference Yang, Zhang, Li and He2009) is employed to map these forces to the background grid nodes to avoid singularity issues.

The governing equations are discretized on a structured staggered grid using the finite differencing method. Spatial discretization utilizes the second-order central differencing scheme. The temporal integration employs a second-order fractional step scheme. The Jacobian-free Newton–Krylov approach (Knoll & Keyes Reference Knoll and Keyes2004) is used to solve the nonlinear momentum equation. The generalized minimal residual approach (Saad Reference Saad1993) with multigrid as a preconditioner is used to solve the Poisson equation derived from the continuity equation to enforce incompressibility and to determine the pressure. The reader can refer to Ge & Sotiropoulos (Reference Ge and Sotiropoulos2007) for a detailed description of the numerical implementation.

2.2. Set-up of LES cases

The 10 MW reference wind turbine of the International Energy Agency (Bortolotti et al. Reference Bortolotti, Tarres, Dykes, Merz, Sethuraman, Verelst and Zahle2019) is simulated in this work to reflect recent large-scale horizontal-axis offshore wind turbines. The turbine has a three-blade rotor with a diameter of $D = 198$ m and a cylindrical-like nacelle with a diameter of 10 m and a length of 5 m. In the simulation the turbine is controlled with a fixed tip speed ratio $\lambda = 9$ defined as the ratio between the blade tip speed and the wind speed. The wind speed is set to $U_\infty = 10\,{\rm m}\,{\rm s}^{-1}$, close to the rated condition. Under a uniform inflow condition and a fixed wind turbine, the LES predicts the thrust and power coefficients of $C_T = 0.75$ and $C_P = 0.48$, being close to the rotor's design state. The corresponding axial induction factor is $a = 0.25$, computed with $C_T = 4a(1-a)$. The thrust and power coefficients are computed as

(2.7)\begin{gather} C_T = \frac{T}{\bigg(\dfrac{1}{2} U_\infty^2 {\rm \pi}\left(\dfrac{D}{2}\right)^2\bigg)}, \end{gather}

(2.8)\begin{gather} C_P = \frac{Q \omega_r}{\bigg(\dfrac{1}{2} U_\infty^3 {\rm \pi}\left(\dfrac{D}{2}\right)^2\bigg)}, \end{gather}

where $T$ denotes the rotor thrust, $Q$ is the rotor torque and $\omega _r$ is the angular velocity of the rotor. These values are averaged over 500 rotor revolutions beyond the initial transient stage.

Due to the dynamic and complex environmental conditions in which they operate, the floating wind turbines exhibit motions with six degrees of freedom (Lyu, Zhang & Li Reference Lyu, Zhang and Li2019). In this study we focus on three representative motions, namely the sway (in-plane translation, along the $y$ axis), the surge (out-of-plane translation, along the $x$ axis) and the yaw (out-of-plane rotation, along the $z$ axis) as illustrated in figure 2. When assuming an inflow without shear and an axisymmetric wake, yaw can be considered equivalent to pitch; sway can be considered equivalent to heave, following a coordinate transformation. At small amplitude, the roll motion of FOWT can be approximated by the sway motion since the centre of gravity is far below the rotor hub centre (Ribeiro, Casalino & Ferreira Reference Ribeiro, Casalino and Ferreira2023a). Here, we assume that the FOWT is forced to oscillate harmonically for each motion.

Figure 2. Schematics of the computational domain (a); the surface mesh of the actuator surface model and the Cartesian grid for the flow solver, with the three types of motion considered (b).

For the surge motion, the rotor translates in the streamwise direction $(x)$, with a displacement of

(2.9)\begin{equation} \delta x_{h}(t) = A \sin{\left(\omega t \right)}. \end{equation}

For the sway motion, the rotor translates in the spanwise direction $(y)$, with a displacement of

(2.10)\begin{equation} \delta y_{h}(t) = A \sin{\left(\omega t \right)}. \end{equation}

For the yaw motion, the rotor rotates around the vertical axis $(z)$, with an angular displacement of

(2.11)\begin{equation} \delta \gamma_{h} (t) = A/R \sin{\left(\omega t \right)}. \end{equation}

The angle $\gamma _{h}$ is equal to zero when the rotor is perpendicular to the streamwise direction.

The above definitions control the motions with $A$ the linear displacement of the rotor tip and $\omega$ the motion's angular frequency. For a FOWT, the platform motion can be excited due to a combined load from turbulent wind and waves and the motion response falls in a large frequency range depending on the environment and the platform's natural frequency (Jonkman & Musial Reference Jonkman and Musial2010; Robertson et al. Reference Robertson2014). Previous studies (Li et al. Reference Li, Dong and Yang2022; Messmer et al. Reference Messmer, Hölling and Peinke2023) have estimated the Strouhal number generally falls in the range of $St \in [0.1, 1.5]$ for utility-scale wind turbines. The motion amplitude $A$ has been found to be in the order of $1\,\%D$ by a recent field measurement on a spar-type floater (Angelou et al. Reference Angelou, Mann and Dubreuil-Boisclair2023), which is known for small motion; larger motion amplitudes are found for semi-submersible floaters (Schliffke, Conan & Aubrun Reference Schliffke, Conan and Aubrun2023). As a detailed investigation on how the amplitude and the frequency affect the wake response has been presented for sway motion in our previous work (Li et al. Reference Li, Dong and Yang2022), the present study employs only representative values to showcase the wake response due to different motion types and to validate the predictive capacity of the proposed M2W model. These specific values will be presented later in the relevant context. An extra case without rotor motion is also included as the baseline.

Both laminar and turbulent inflows are considered. For all the cases, the simulation employs a rectangular computational domain as shown in figure 2. The length in the streamwise direction ($x$) is $14D$ and is $7D$ in both the transversal ($y$) and the vertical ($z$) directions. The wind turbine is placed on the domain centreline, and the rotor axis is aligned with the $x$ direction. The distance from the inlet to the turbine is $3.5D$. The origin of the coordinate system coincides with the rotor hub centre.

The computational domain is discretized by the Cartesian grid. The grid has a uniform spacing in the $y$ and $z$ directions of $\Delta y = \Delta z = D/40$ in the region $-2D< y<2D$ and $-2D< z<2D$ and is gradually stretched out of this region. The grid spacing in the streamwise direction is $\Delta x = D/20$. The rotor blade is discretized with an unstructured triangular mesh (see figure 2). Previous work has demonstrated that such a spatial resolution is sufficient to obtain mesh-independent results for both first- and second-order turbulence statistics in the far wake (Li & Yang Reference Li and Yang2021).

We impose the inflow velocity at the inlet boundary ($x = -3.5D$). We neglect the wind shear and set a uniform wind speed of $U_\infty = 10\,{\rm m}\,{\rm s}^{-1}$ in the streamwise direction for the cases with laminar inflow. The periodic boundary condition is imposed in the lateral directions. The Neumann condition for the velocity ($\partial \boldsymbol {u}/\partial x = \boldsymbol {0}$) is imposed at the outlet ($x=10.5D$).

The wake response to different floating wind turbine motions is first investigated under uniform inflow conditions. The parameter space is summarized by table 1. All three motion types (surge, sway, yaw) are considered. The wind turbine motion amplitude is kept small. Our previous study has shown that the wind turbine wake is very sensitive to the rotor's side-to-side motion and that even small amplitude motion ($A=0.01D$) can lead to large meandering amplitude in the far wake, especially for cases with uniform inflow (Li et al. Reference Li, Dong and Yang2022). Provided the present M2W model is based on the linear assumption, we set the motion amplitude to $A=0.01D$ in LES to keep the nonlinearity at a minimal level to facilitate the comparison between the model and the LES. To treat this small motion in LES, the aerodynamic force of the blade is first computed on the Lagrangian nodes of the actuator surface and is then distributed to the Eularian grid for the flow solver, employing the discretized smoothed delta function as the kernel (Yang et al. Reference Yang, Zhang, Li and He2009). It considers the distance between the Lagrangian and the Eularian nodes, so the motion of the rotor can be reflected even when the displacement is smaller than the grid size. The results of our previous LES study that employed the same numerical approach (Li et al. Reference Li, Dong and Yang2022) have been confirmed by Messmer et al. (Reference Messmer, Hölling and Peinke2023) using a wind tunnel experiment. Among the frequencies selected, $St \in \{ 0.1, 0.25, 0.5 \}$, our previous study (Li et al. Reference Li, Dong and Yang2022) has shown that $St = 0.25$ is close to the most sensitive forcing frequencies for the sway motion, while sway motions at $St=0.10$ and $St=0.50$ lead to less amplified wake meandering for the 5 MW reference wind turbine of the National Renewable Energy Laboratory. We will verify if the same sensitive frequency range is valid for all three motion types.

Table 1. Parameter space for investigating the wake response to typical floating wind turbine motions under uniform inflow conditions.

To consider the effect of turbulence, we impose synthetic turbulence generated by the spectral tensor method of Mann (Reference Mann1994) at the inlet boundary of the computational domain, in addition to the uniform streamwise flow of $U_\infty = 10\,{\rm m}\,{\rm s}^{-1}$. The control parameters of the synthetic turbulence are (i) $\alpha \epsilon ^{2/3}$ a scalar to control the TIs; (ii) $\varGamma$, a non-dimensional number for defining the anisotropy, setting to 3.9 according to the wind energy standard (IEC 2019); (iii) $L_{Mann}$, a length scale proportional to the size of the most energetic eddy, varies in $L_{Mann} \in \{D/8, D/4, 3D/8, D/2\}$, which are referred to as L1 to L4 in an increasing order. Taking $D=198$ m into account, $L_{Mann}$ falls into the range of 24.75 m to 99 m, containing the recommended value, $L_{Mann} = 33.6$ m for $z_{hub} \ge 60$ m, by IEC (2019). We generate a total of 12 turbulent inflows, with three TIs (shown in table 2) and four length scales. The obtained inflow turbulence length scales $L_{u_i}^{x_j}$ for different $L_{Mann}$ are shown in figure 3, which represents the spatial scales of turbulent eddies with the most energy for the three fluctuating wind velocity components $u_i = \{u_x, u_y, u_z\}$ in the streamwise ($x$), transverse ($y$) and vertical ($z$) directions, respectively (Nandi & Yeo Reference Nandi and Yeo2021). The integral lengths are computed with

(2.12)\begin{equation} L_{u_i}^{x_j} = \int_0^\infty \frac{\langle u_i(\boldsymbol{x})u_i(\boldsymbol{x}+\xi\boldsymbol{e_j})\rangle}{\left\langle u_i^2\right\rangle} \,\text{d} \xi. \end{equation}

Our previous study (Li et al. Reference Li, Dong and Yang2022) compared the wake of a fixed wind turbine and a sway wind turbine under inflow with different TIs but only a fixed $L_{Mann}$ close to the L4 case, leaving the effect of inflow turbulence length scale unexplored. For this reason, the present work uses $L_{Mann}$ to control the inflow turbulence length scale, which has been found to be critical for the wake meandering of a fixed wind turbine (Espana et al. Reference Espana, Aubrun, Loyer and Devinant2011; Gambuzza & Ganapathisubramani Reference Gambuzza and Ganapathisubramani2023).

Table 2. Componentwise TI of the synthetic turbulence imposed at the inlet for the turbulent inflow cases.

Figure 3. Integral length scale of velocity components $u_i$ in directions $x_j$. Results are shown for the (a) streamwise, (b) transversal and (c) vertical directions.

For cases with turbulent inflow, we consider only the sway motion since it leads to the largest wake meandering (as shown in § 3) and has the potential to enhance the wake recovery (Li et al. Reference Li, Dong and Yang2022; Messmer et al. Reference Messmer, Hölling and Peinke2023). Three different sway amplitudes are investigated, i.e. $A \in \{0.01D, 0.02D, 0.04D\}$, and are referred to as A1, A2 and A3, in increasing order. Increasing the motion amplitude is to strengthen the wake response, such that the motion-induced wake response can be distinguished from the background turbulence. The reduced frequency of the sway motion is equal to $St = 0.25$, falling near the most sensitive frequency for the wake's shear layer. The parameter space employed for investigating the joint effect of inflow turbulence and turbine motion is listed in table 3, containing 48 cases for LES.

Table 3. Parameter space employed for investigating the joint effect of inflow turbulence and turbine motion on the wake evolution of a FOWT.

3.1. Instantaneous wake

The wake behaviour for different rotor motions are first examined by comparing the instantaneous velocity deficit ($\Delta u_x = u_x - U_\infty$) on the hub-height plane, as depicted in figure 4.

Figure 4. Instantaneous streamwise velocity deficit in the wake on the hub-height plane for a fixed rotor (a), a FOWT rotor with surge (b–d), sway (e–g) and yaw (h–j) motions. The frequency of rotor motion is $St = 0.10, 0.25$, and $0.50$ for the first, second, and last column, respectively. All contours are plotted at the same time $tU_\infty /D = 60$. Inflow: uniform. Method: LES.

As seen, the wake of a fixed wind turbine remains nearly straight when travelling downstream due to the absence of incoming flow disturbances. Only small-scale fluctuations are observed in the far wake due to the wake-generated turbulence.

At $St=0.1$, none of the three motions is able to generate remarkable changes to the wake, indicating the wake is nearly insensitive to the disturbances at this frequency regardless of the rotor motion types.

As the motion frequency increases to $St=0.25$, the wake contour shows significant differences from the fixed turbine case. Firstly, the wake response is found to be dependent on the motion type. Specifically, the wake for cases with surge motion remains almost straight, but an alternating wind speed is found in the wake, where the wake is separated by regions with relatively higher and lower wind speeds. This accordion-like wake movement (Ramos-García et al. Reference Ramos-García, González Horcas, Pegalajar-Jurado, Kontos and Bredmose2022) is referred to as wake pulsing by Messmer et al. (Reference Messmer, Hölling and Peinke2023). In contrast, both sway and yaw motions induce obvious lateral wake motions, i.e. wake meandering, starting from $x\approx 5D$. Notably, the wake meandering caused by sway and yaw exhibits strong similarity, i.e. both have a wavy form with almost identical wavelengths, and both amplitudes gradually increase with downstream distance.

In figure 5 we compare the wake meandering caused by sway (a) and yaw (b) for cases with $St = 0.25$. The wake trajectory is outlined by the centreline $y_c(x,t)$, computed as the weight centre of the velocity deficit as

(3.1)\begin{equation} y_c(x,t) = \frac{\int_{-\infty}^{+\infty} \Delta u_x(x,y,t) y\, \text{d}y}{\int_{-\infty}^{+\infty} \Delta u_x(x,y,t) \,\text{d}y }. \end{equation}

At this instant, it is found that the lateral motion of the wake trajectory $y_c(x)$ is opposite in direction for sway (a) and yaw (b), when the motions are defined with (2.10) and (2.11) with $A=0.01D$ and $St=0.25$. This opposite wake motion is confirmed by computing the correlation coefficient of the instantaneous wake centre position time history at $x=8D$, which yields $\rho = -0.93$. Such an opposite wake centre displacement indicates that if the sway and the yaw motions are combined with different phase lags, the wake meandering can be either strengthened or weakened. To this end, a quick proof of concept is illustrated in figure 5, for cases with in-phase (c) and opposite-phase sway–yaw combinations (d). Here, in-phase refers to a direct combination of sway and yaw motions defined by (2.10) and (2.11), whereas the opposite-phase case employs a yaw motion with a $180^\circ$ phase lag. As seen, the wake meandering is significantly mitigated in figure 5(c) but enhanced in figure 5(d). The standard deviations of the instantaneous wake position $y_c(x=8D,t)$ are $0.24D$ (sway only), $0.20D$ (yaw only), $0.11D$ (sway yaw motion in-phase), $0.31D$ (sway yaw motion opposite-phase). However, such an interesting phase relation between sway and yaw can be neither explained straightforwardly using the rotor displacement nor with the LSA (Li et al. Reference Li, Dong and Yang2022) since an end-to-end M2W relation is not established in LSA. Such a phase-resolved prediction capability will be fulfilled by the model proposed by this work, as shown in § 4.3.

Figure 5. Instantaneous wake for cases with sway (a) and yaw (b). The mitigating case (c): the sway and the yaw motions are combined directly. The enhancement case (d): the yaw motion is lagged by half a period. The red dash-dot lines plot the wake centreline $y_c(x)$ and the dashed lines approximate the wake boundary using $y_c(x)\pm 0.5D$. Inflow: uniform. Method: LES.

When further increasing the motion frequency to $St=0.50$, figure 4 shows that the wake response becomes weaker for all motion types. The wake pulsing is still observed for the case with surge but its wavelength is reduced. For the cases with sway and yaw motions, the meandering of the wake is less apparent, while more velocity fluctuations at smaller scales are found compared with the wake of a fixed wind turbine.

3.2. Time-averaged wake

To investigate the wake recovery enhancement due to rotor motion, we compare the time-averaged velocity deficit $\Delta \bar {u}_x = \bar {u}_x - U_\infty$ on the hub-height plane, as depicted in figure 6. Consistent with the findings for instantaneous wakes, the fastest wake recoveries are observed for the cases with sway and yaw motions at $St=0.25$. In these cases, the expansion rate of the time-averaged wake is significantly increased from $x \approx 5D$, where the onset of meandering is observed in the instantaneous wake, as shown by the counterparts of figure 4. Furthermore, the velocity profiles deviate noticeably from the Gaussian shape, which is caused by the non-Gaussian distribution of the instantaneous wake centre locations when motion-induced wake meandering amplitude is significant (Li et al. Reference Li, Dong and Yang2022). However, for the sway and the yaw motions at other frequencies, and for the surge motion at all three frequencies, the time-averaged wake closely resembles that of a fixed wind turbine. This observation is consistent with the wind tunnel measurements of Schliffke et al. (Reference Schliffke, Aubrun and Conan2020), which found that surge motion has only a weak effect on the time-averaged wake recovery.

Figure 6. Time-averaged velocity wake deficit contour on the hub-height plane and profiles at $x = 2D, 4D, 6D, 8D$. See figure 4 for a detailed caption. Inflow: uniform. Method: LES.

Moreover, the wake recovery enhancement due to wind turbine motion is compared quantitatively in figure 7, by plotting the space–time-averaged velocity deficit $\Delta \tilde {U}_x$. This space–time-averaged velocity deficit is obtained by averaging the time-averaged velocity deficit $\Delta \bar {u}_x$ over the wake region, $y \in [-0.5D, 0.5D]$, on the hub-height plane. As seen, at $St=0.1$, the curves for cases with different types of motion collapse, indicating that the effect of motion is negligible at this frequency for all motion types. However, at $St=0.25$, the velocity deficit is recovered by approximately $0.2U_\infty$ more at $x=10D$ for both sway and yaw motions compared with the fixed wind turbine case. In contrast, the effect of surge motion is still minimal. As the motion frequency increases to $St=0.5$, the wake recovery rate for sway and yaw motions is reduced, while that for the surge motion is slightly enhanced. At $x=10D$, the motion-enhanced extra wake recovery is 3.8 %, 8.3 % and 6.3 % of the free-stream wind speed $U_\infty$ for the surge, the sway and the yaw motions, respectively. It is worth noting the above comparison showcases the effect of the wake instability triggered by rotor motion and is based on a small rotor motion amplitude $A=0.01D$. Motions with large amplitudes that directly modify the wake, without triggering flow instability, are not considered in this study.

Figure 7. Space–time-averaged velocity deficit in the wake on the hub-height plane over $-0.5D< y<0.5D$. Results are shown for motions at a reduced frequency of $St=0.10$ (a), $St=0.25$ (b) and $St=0.50$ (c). Inflow: uniform. Method: LES.

4.1. Forcing-to-wake model

The resolvent analysis is a powerful tool for predicting the response of a linear fluid system to forcing (Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993), either from external sources (Jovanović & Bamieh Reference Jovanović and Bamieh2005; Sipp et al. Reference Sipp, Marquet, Meliga and Barbagallo2010) or from the nonlinear term of the Navier–Stokes equations (McKeon & Sharma Reference McKeon and Sharma2010; Wu & He Reference Wu and He2023). To derive the resolvent-based F2W model, we apply the Reynolds decomposition to the variables of the Navier–Stokes equation as

(4.1)\begin{gather} \boldsymbol{u} = \bar{\boldsymbol{u}} + \delta \boldsymbol{u}, \end{gather}

(4.2)\begin{gather}p = \bar{p} + \delta p, \end{gather}

(4.3)\begin{gather}\boldsymbol{f} = \bar{\boldsymbol{f}} + \delta \boldsymbol{f}, \end{gather}

where $\boldsymbol {u}$, $p$, $f$ represent the velocity, pressure and rotor forcing, respectively. The time-averaged part is denoted by the $\bar {{\cdot }}$ symbol and the perturbation part is denoted by the $\delta$ symbol. The time-averaged wake of a fixed wind turbine, obtained with LES, is employed as the time-averaged part.

The first assumption of the present model is the linear assumption, following which the evolution of small coherent perturbations due to rotor forcing can be described by the LNSEs as

(4.4)\begin{gather} \frac{\partial (\delta \boldsymbol{u})}{\partial t}+ (\boldsymbol{\bar{u}} \boldsymbol{\cdot} \boldsymbol{\nabla}) (\delta \boldsymbol{u}) + (\delta \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{\bar{u}} + \boldsymbol{\nabla} (\delta p) - \nu_{eff} \nabla^2 (\delta \boldsymbol{u}) = \mathcal{B} (\delta \boldsymbol{f}), \end{gather}

(4.5)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} (\delta \boldsymbol{u}) = 0, \end{gather}

where the matrix $\mathcal {B}$ defines the spatial restriction such that forcing $\delta \boldsymbol {f}$ is confined in the rotor region. The effect of the turbulent Reynolds stresses is approximated by $\nu _{eff} \nabla ^2 (\delta \boldsymbol {u})$ (Morra et al. Reference Morra, Semeraro, Henningson and Cossu2019), using an effective viscosity $\nu _{eff}$ (Del Alamo & Jimenez Reference Del Alamo and Jimenez2006; Symon et al. Reference Symon, Madhusudanan, Illingworth and Marusic2023).

Secondly, to ensure computational efficiency, the model assumes the axisymmetry of the wake and is derived in the cylindrical coordinate system. The state variables of the flow are expressed in the cylindrical coordinate system as

(4.6)\begin{equation} \boldsymbol{q}(x,r,\theta,t) = [p, \boldsymbol{u}]^{\rm T}= [p, u_x, u_r, u_\theta]^{\rm T}(x,r,\theta,t), \end{equation}

with $u_x, u_r, u_\theta$ the velocity components in the axial $(x)$, radial $(r)$ and azimuthal $(\theta )$ directions, and $p$ the pressure. Similarly, the volume forcing to the flow is expressed as

(4.7)\begin{equation} \boldsymbol{f}(x,r,\theta,t) = [f_x, f_r, f_\theta]^{\rm T}(x,r,\theta,t). \end{equation}

With these notations, (4.4) and (4.5) can be formulated into a linear system of the fluctuation part $\delta \boldsymbol {q}(x,r,\theta,t)$ as

(4.8)\begin{equation} \frac{\partial\delta \boldsymbol{q} }{\partial t}=\mathcal{L} \delta \boldsymbol{q} + \mathcal{B} \delta \boldsymbol{f}, \end{equation}

where the concrete form of $\mathcal {L}$ is provided in Appendix A, following the standard procedure for linearizing the Navier–Stokes equations (Kaplan et al. Reference Kaplan, Jordan, Cavalieri and Brès2021).

Thirdly, we employ the effective viscosity (Symon et al. Reference Symon, Madhusudanan, Illingworth and Marusic2023) to model the effects of small-scale turbulence on the large turbulent coherent structures (Del Alamo & Jimenez Reference Del Alamo and Jimenez2006) in the wake, due to the large Reynolds number of the wind turbine wake. Looking for the optimal effective viscosity for resolvent analysis is still an ongoing topic in turbulence research (Gupta et al. Reference Gupta, Madhusudanan, Wan, Illingworth and Juniper2021; Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021). In this work, we employ a basic mean flow consistent effective viscosity. The effective viscosity is approximated by a linear fitting of the Reynolds shear stress $\overline {u'_x u'_r}$ and the streamwise velocity gradient ($\mathrm {d} \bar {u}_x / \mathrm {d} r$) from LES of a fixed wind turbine wake as

(4.9)\begin{equation} \nu_{eff} ={-}\frac{\overline{u'_x u'_r}}{\mathrm{d} \bar{u}_x / \mathrm{d} r}. \end{equation}

In the present model, the effective viscosity is assumed to be dependent only on the streamwise location $(x)$, i.e. $\nu _{eff} = \nu _{eff}(x)$ is computed at each downstream location to account for its streamwise variation.

Due to the linear nature of the model, the solution is sought in the frequency domain. To this end, the state variables and the forcing are decomposed using the Fourier series, by assuming the homogeneity in time and in the azimuthal direction as

(4.10)\begin{gather} \delta \boldsymbol{f}(x, r, \theta, t) = \sum_{\omega ={-}\infty}^{\infty} \sum_{m={-}\infty}^{\infty} \hat{\boldsymbol{f}}_{m,\omega}(x, r) \exp \left(\mathrm{i}(m \theta-\omega t)\right), \end{gather}

(4.11)\begin{gather}\delta \boldsymbol{q}(x, r, \theta, t) = \sum_{\omega ={-}\infty}^{\infty} \sum_{m={-}\infty}^{\infty} \hat{\boldsymbol{q}}_{m,\omega}(x, r) \exp \left(\mathrm{i}(m \theta-\omega t)\right) , \end{gather}

with $\mathrm {i} = \sqrt {-1}$ the imaginary unit, $m$ the azimuthal wavenumber and $\omega$ the angular frequency. Substituting this ansatz to the linearized equation (4.8) results in an input–output system for each $(m,\omega )$ pair as

(4.12)\begin{equation} (-\mathrm{i} \omega \mathcal{I}-\mathcal{L}_{m})\hat{\boldsymbol{q}}_{m,\omega}(x,r) = \mathcal{B} \hat{\boldsymbol{f}}_{m,\omega}(x,r), \end{equation}

or equivalently,

(4.13)\begin{equation} \hat{\boldsymbol{q}}_{m,\omega}(x,r) = \mathcal{R}_{m,\omega} \,\hat{\boldsymbol{f}}_{m,\omega}(x,r), \end{equation}

with $\mathcal {R}_{m,\omega } = (-\mathrm {i} \omega \mathcal {I}-\mathcal {L}_{m})^{-1}\mathcal {B}$ the resolvent operator that transfers the forcing mode $\hat {\boldsymbol {f}}_{m,\omega }(x,r)$ to the response mode $\hat {\boldsymbol {q}}_{m,\omega }(x,r)$. Here $\mathcal {I}$ is the identity matrix and $\mathcal {L}_{m}$ is provided in Appendix A.

The above equation is discretized in a two-dimensional domain of $-2D< x<10D$ and $0< r<3.5D$. The grid spacing and the discretization scheme in the streamwise direction are the same as LES with a number of grid points $N_x = 240$. In the radial direction, Chebyshev polynomials are employed to discretize the equation with $N_r = 100$ collocation points. The same mapping technique employed by Li et al. (Reference Li, Dong and Yang2022) is adopted to set half of the points inside the wake region $0< r<0.5D$.

4.1.1. Optimal rotor forcing and wake response

As the resolvent operator $\mathcal {R}_{m,\omega }$ connects the rotor forcing to the wake response, the optimal rotor forcing leading to the strongest wake response can be found by analysing this linear operator. To this end, we define the energy gain of the resolvent operator as

(4.14)\begin{equation} G_{opt}^2\left(\omega, m\right)=\max _{\hat{\boldsymbol{f}}_{\omega,m}} \frac{\|\hat{\boldsymbol{q}}_{\omega,m}\|_E^2}{\|\,\hat{\boldsymbol{f}}_{\omega,m}\|_f^2}, \end{equation}

which denotes the energy amplification in the wake to the rotor forcing. The energy norm of the wake response and of the rotor forcing is defined as

(4.15)\begin{gather} \|\hat{\boldsymbol{q}}\|_E^2 = \hat{\boldsymbol{q}}^H \boldsymbol{W}_q \hat{\boldsymbol{q}} = \int_\varOmega \big( \hat{u}^2_x + \hat{u}^2_r + \hat{u}^2_\theta \big) r \,\text{d}\theta \,\text{d}r \,\text{d}\kern0.06em x, \end{gather}

(4.16)\begin{gather}\|\,\hat{\boldsymbol{f}}\|_f^2 = \hat{\boldsymbol{f}}^H \boldsymbol{W}_f \hat{\boldsymbol{f}} = \int_\varGamma \big( \hat{f}^2_x + \hat{f}^2_r + \hat{f}^2_\theta \big) r \,\text{d}\theta\, \text{d}r, \end{gather}

where $\boldsymbol {W}_q$ and $\boldsymbol {W}_f$ are the weight matrices of the discretized energy norm. The integration domains for the energy norm definition expand $0< x<10D$, $0< r<3.5D$, $0<\theta <2{\rm \pi}$ for $\varOmega$ and $x=0$, $0< r<3.5D$, $0<\theta <2{\rm \pi}$ for $\varGamma$.

The optimization problem defined in (4.14) is equivalent to finding the largest eigenvalue of the following problem (Garnaud et al. Reference Garnaud, Lesshafft, Schmid and Huerre2013):

(4.17)\begin{equation} \boldsymbol{W}_f^{{-}1} \boldsymbol{R}^H \boldsymbol{W}_q \boldsymbol{R} \hat{\boldsymbol{f}} = \lambda \hat{\boldsymbol{f}}. \end{equation}

Here the largest eigenvalue is equal to the square of the optimal gain ($\lambda _{max} = G^2_{opt}$), and the eigenvector corresponds to the optimal rotor forcing mode ($\hat {\boldsymbol {f}}_{opt}$). The optimal response mode is obtained by solving (4.13) once the optimal forcing mode is obtained.

One issue related to the computational cost is that the explicit construction of the bi-global resolvent operator $\mathcal {R}_{m,\omega } = (-\mathrm {i} \omega \mathcal {I}-\mathcal {L}_{m})^{-1}\mathcal {B}$ gives rise to a prohibitive memory cost due to the storage of a large dense matrix of $(4\times N_x \times N_r) \times (4\times N_x \times N_r)$. This problem is circumvented by employing the Arnoldi algorithm, as described by Martini et al. (Reference Martini, Rodríguez, Towne and Cavalieri2021) and Kaplan et al. (Reference Kaplan, Jordan, Cavalieri and Brès2021).

Figure 8 depicts the computed optimal gain $G_{opt}(St, m)$ as a function of the forcing frequency $St$ and the azimuthal wavenumber $m$. The curve of the optimal gain for the rotor forcing with $m = 0$ is found to have a flat plateau in $0.2< St<0.4$ where the gain is elevated. For $m \neq 0$, the optimal gains of the opposite azimuthal wavenumbers show a similar variation via $St$, but differ slightly as a result of the non-zero time-averaged rotational velocity of the wake. For the gains of the rotor forcing with $m = \pm 1$, they both have a distinct peak near $St \approx 0.25$, which is the largest among all azimuthal wavenumbers considered. Further increasing the wavenumber beyond $|m| \ge 2$ reduces the magnitude of the optimal gain. From an application aspect view, the rotor forcings with azimuthal wavenumbers $m=0$ and $m=\pm 1$ are important because they can be induced by floating wind turbine motions, as will be shown in § 4.2. Bottom fixed turbines can also generate these rotor forcings with wake control techniques such as the dynamic induction control (DIC, Goit & Meyers Reference Goit and Meyers2015) $(m=0)$ and the helix wake control approach $(m=\pm 1)$ (Frederik et al. Reference Frederik, Doekemeijer, Mulders and van Wingerden2020). We note that the most unstable modes have an azimuthal wavenumber of $m = \pm 1$, which agrees with the result obtained by Viola et al. (Reference Viola, Iungo, Camarri, Porté-Agel and Gallaire2014) using local LSA with eddy viscosity.

Figure 8. Optimal gain as a function of the forcing frequency $St$. Each branch represents a different azimuthal wavenumber $m$. Inflow: uniform. Method: F2W model.

It is also worth noting that the optimal gain is dependent on the definition of the energy norm of the wake response and rotor forcing. One may change the definition of (4.15) to focus on other wake properties. One example of using the energy of streamwise velocity fluctuation to define the energy gain is shown in Appendix B.1.

Furthermore, Appendix B.2 investigates the sensitivity of the optimal gain to the effective viscosity ($\nu _{eff}$), because the present model employs a simple $\nu _{eff}$ modelling that does not account for the possible variation of $\nu _{eff}$ due to the dynamic wake response. The results show that, even $\nu _{eff}$ is augmented or diminished by a factor of two, the present model is able to predict the peak frequency leading to the maximum gain reasonably well for cases with $m=1$, which leads to the strongest wake response, where the peak frequency predicted by the model consistently falls in the range $0.2 < St < 0.3$. However, it is noteworthy that the magnitude of the optimal gain demonstrates sensitivity to $\nu _{eff}$. An increase in effective viscosity tends to stabilize the flow, thereby diminishing the magnitude of the optimal gain. As the result suggested, the present model can be employed to effectively narrow the focus of the frequency range for flow control purposes, such that a more detailed investigation could be realized through methods like model experiments or LES, where the nonlinear effects leading to the uncertainty of $\nu _{eff}$ can be incorporated properly. Moreover, advanced effective viscosity modelling can also be developed to enhance the predictive capability of the proposed model.

The spatial modes of the optimal rotor forcing are plotted in figure 9 for the case with $St=0.25$. The streamwise forcing is larger than the radial and azimuthal forcing components, indicating that the wake is more sensitive to forcing in the streamwise direction. The streamwise and radial components of the forcing modes for $m=\pm 1$ are similar, except the rotational directions which are the opposite. Larger discrepancies are observed in the azimuthal component, due to the non-zero time-averaged rotational velocity of the wake.

Figure 9. The real part of the spatial mode of the optimal rotor forcing predicted by the resolvent analysis at $St = 0.25$ for the azimuthal wavenumber $m=0$ (a–c), $m=1$ (d–f), $m=-1$ (g–i). The forcing modes are normalized such that $\|\hat {\boldsymbol {f}}\|_{f} = 1$, as defined by (4.16). Inflow: uniform. Method: F2W model.

The optimal forcing is also dependent on the forcing frequency as depicted by figure 10, where the forcing modes for $St=0.10$, 0.25 and 0.50 are compared for the same azimuthal wavenumber $m=1$. As seen, the spatial patterns are found to be similar for different motion frequencies. However, increasing the Strouhal number generally moves the forcing region toward the edge of the disk, and the forcing structure becomes narrower. A similar trend is also found for other azimuthal wavenumbers but is not shown for conciseness.

Figure 10. Same as figure 9 but for different $St$ with the azimuthal wavenumber $m=1$.

The wake response under the optimal forcing is depicted in figure 11 for cases with $m=0$ and $m = \pm 1$, $St = 0.10, 0.25, 0.50$. These modes are plotted in the $rOx$ plane where $r=0$ corresponds to the wake centreline. For all cases, the wake response shows a wavy pattern in the $x$ direction, as the fluctuating velocity field behaves as a wave travelling downstream. The wavelength of the coherent structures is found to be negatively related to the forcing Strouhal number $St$, as one can expect. Besides, the streamwise development of the fluctuation magnitude is also dependent on the Strouhal number. For cases with $St=0.10$ and $0.25$, the velocity fluctuation generally increases with downstream location, while increasing the frequency to $St=0.5$ dampens the far wake fluctuation for $x>5D$.

Figure 11. The real part of the optimal response of the wake velocity predicted by the resolvent analysis for the azimuthal wavenumbers $m=0$ (a–c), $m=1$ (d–f) and $m=-1$ (g–i) with forcing frequencies of $St = 0.10, 0.25, 0.50$. The modes are plotted in a two-dimensional $rOx$ plane, where $\hat {u}_x$, $\hat {u}_r$ and $\hat {u}_\theta$ represent the velocity fluctuation in the streamwise, radial and azimuthal directions. The response modes are normalized such that $\|\hat {\boldsymbol {q}}\|_{E} = 1$, as defined by (4.15). Inflow: uniform. Method: F2W model.

The above method provides an efficient way to relate the wake response due to rotor forcing by the resolvent operator. To find the optimal forcing and response it takes approximately several minutes for each pair of wavenumber and frequency $(m, St)$ on a desktop computer. However, since the rotor forcing is not directly related to the motion of FOWT, one cannot predict the wake response in a motion-specific manner yet. This shortcoming necessitates the development of a M2F model of the floating wind turbine in the next subsection.

4.2. Motion-to-forcing model

In this section we derive the second part of the M2W model, i.e. the M2F model. The idea of this modelling is to replace wind turbine motion with equivalent force. The model proposed here is an actuator disk fixed at the time-averaged position of the moving wind turbine, but exerting unsteady motion-dependent forces. To remain consistent with the resolvent F2W model, we decomposed the force on the actuator disk $\boldsymbol {f}(r,\theta,t)$ into a time-averaged thrust and an unsteady part as follows:

(4.18)\begin{equation} \boldsymbol{f}(r,\theta,t) = [-\bar{f}_T(r)+\delta f_x(r,\theta,t), \delta f_r(r,\theta,t), \bar{f}_\theta(r)+ \delta f_\theta(r,\theta,t)]^{\rm T}. \end{equation}

The time-averaged force on the rotor is simplified as a normal thrust $\bar {f}_T(r)$ and rotational force $\bar {f}_\theta (r)$. The radial variation of the thrust force is maintained to account for the blade design, which has been shown to be important for the wake recovery (Dong et al. Reference Dong, Qin, Li and Yang2023). Here $\bar {f}_T(r)$ is computed from the time-averaged blade thrust $\bar {F}_T(r)$ as

(4.19)\begin{equation} \bar{f}_T(r) = \frac{m\ {\cdot}\ \bar{F}_T(r) \delta r}{ 2{\rm \pi} r \delta r} = \frac{m\ {\cdot}\ \bar{F}_T(r) }{2 {\rm \pi}r }, \end{equation}

where $m=3$ is the number of blades and $\delta r \rightarrow 0$ is the width of an annulus at $r$. Furthermore, the thrust coefficient $\bar {C}_T(r)$ of the actuator disk is computed as

(4.20)\begin{equation} \bar{C}_T(r) = \frac{\bar{f}_T(r)}{\dfrac{1}{2} U_\infty^2}. \end{equation}

Figure 12 shows the time-averaged blade forces (a) and force coefficients of the rotor disk (b) obtained from the LES for different turbine motions. As seen, $\bar {F}_T(r)$ and $\bar {C}_T(r)$ are almost unaffected by the rotor motion, because the oscillatory motion is relatively small, which is consistent with the experiments with model wind turbine rotors (Fontanella et al. Reference Fontanella, Zasso and Belloli2022; Meng et al. Reference Meng, Su, Guo, Qu and Lei2022a). From the theoretical aspect, recent studies of Wei & Dabiri (Reference Wei and Dabiri2023) and Heck, Johlas & Howland (Reference Heck, Johlas and Howland2023) have established relations between the thrust force for wind turbines with surge motion velocity amplitudes and yaw angles. For example, Wei & Dabiri (Reference Wei and Dabiri2023) has shown that the time-averaged rotor thrust of a periodically surging wind turbine ($\bar {F}_T$) is related to that of a stationary wind turbine $\bar {F}_{T,0}$ as follows:

(4.21)\begin{equation} \frac{\bar{F}_T}{\bar{F}_{T,0}} = 1 + \frac{1}{2}\frac{{u}^2_h}{U^2_\infty}. \end{equation}

Here $u_h = A\omega$ is the motion amplitude of the rotor. The above equation indicates that the motion-induced thrust variation is small (${u}_h \approx 0.016U_\infty$, ${u}^2_h \approx 0.0003 U^2_\infty$ for $A=0.01D$ and $St=0.25$; ${u}_h \approx 0.13 U_\infty$, ${u}^2_h \approx 0.017 U^2_\infty$ for $A=0.04D$ and $St=0.5$). For yaw motion, Heck et al. (Reference Heck, Johlas and Howland2023) also demonstrates a quadratic dependence of the thrust change with the yaw angle, which can also be ignored for $\gamma \ll 1$. For this reason, we assume that the thrust force and the thrust coefficient $\bar {C}_T(r)$ for all motions with small amplitudes can be approximated by those of a stationary wind turbine. The rotational force coefficient $\bar {C}_\theta (r)$ can also be computed by the same approach.

Figure 12. Comparison of time-averaged blade forces (a) and force coefficients of the rotor disk (b) for different motion types. The normal forces ($\bar {F}_T$) and coefficients ($\bar {C}_T$) are plotted in colour and the rotational ones ($\bar {F}_\theta$ and $\bar {C}_\theta$) are plotted in grey. The reduced frequency of the motion is equal to $St = 0.25$ and the amplitude is $A=0.01D$. Inflow: uniform.

In the following we derive a linear M2F model to compute the unsteady force on the actuator disk induced by surge, sway and yaw motions, taking only the time-averaged force coefficients ($\bar {C}_T(r)$ and $\bar {C}_\theta (r)$) as input and assuming that the motion amplitude is small.

4.2.1. Surge

The surge motion refers to the displacement in the streamwise direction ($x$) as shown in figure 13(a). This motion generates a streamwise velocity of the rotor, denoted as ($u_{h}(t) = \delta \dot {x}_{h}(t)$), which alters the relative velocity between the free stream and wind turbine. To account for this variation, the thrust of the actuator disk is computed with a corrected wind speed $U_{rel} = U_\infty - \delta \dot {x}_{h}(t)$, while the thrust coefficient is assumed to equal $\bar {C}_T(r)$. This assumption is validated by examining the blade force results from LES with small $u_{h}(t)$. Based on these assumptions, the unsteady part of the rotor forcing is computed as

(4.22) \begin{align} \delta f_x(r,\theta,t) & ={-} \left( \tfrac{1}{2} U_{rel} ^2 \bar{C}_T(r) - \tfrac{1}{2} U_\infty^2 \bar{C}_T(r) \right) \nonumber\\ & = \tfrac{1}{2} \bar{C}_T(r) ( 2 U_\infty \delta \dot{x}_{h}(t) - \delta \dot{x}^2_{h}(t))\nonumber\\ & \approx U_\infty \bar{C}_T(r) \delta \dot{x}_{h}(t). \end{align}

The minus sign at the first step reflects that the thrust force points in the opposite direction of $x$ of the coordinate system. In the final step of (4.22), linear approximation is employed to truncate the high-order terms of $\delta \dot {x}_{h}(t)$, assuming $\delta \dot {x}_{h}(t) \ll U_\infty$. Substituting the harmonic surge displacement in (2.9)–(4.22) yields the unsteady streamwise forcing for harmonic surge motion as

(4.23)\begin{equation} \delta f_x(r,\theta,t) \approx \omega A \cos(\omega t) U_\infty \bar{C}_T(r) . \end{equation}

Similarly, the azimuthal component can be computed as

(4.24)\begin{equation} \delta f_\theta (r,\theta,t) \approx{-}\omega A \cos(\omega t) U_\infty \bar{C}_\theta(r) . \end{equation}

The resulting unsteady forcing is depicted in figure 14(a–c).

Figure 13. Schematics of floating wind turbine motion for surge (a), sway (b) and yaw (c). The red solid line outlines the instantaneous rotor position and the black dashed line shows the time-averaged rotor position. Red arrows illustrate the motion. The coordinates of $O=(0,0,0)$ and $O'=(\delta x_{h}(t),\delta y_{h}(t),\delta z_{h}(t))$. The derived unsteady force is applied only inside the black dashed circle.

Figure 14. Spatial mode of the linearized unsteady forcing for surge motion (a–c), sway motion (d–f) and yaw motion (g–i). Plots (a,d,g), (b,e,h) and (c,f,i) show the streamwise, radial and azimuthal components of the rotor forcing, respectively. The forcing is computed for the cases with the reduced motion frequency $St = 0.25$. All modes are normalized by $A U^2_\infty$. Method: M2F model.

4.2.2. Sway

The sway motion refers to the side-to-side translation of the rotor as shown in figure 13(b). Compared with surge motion, sway does not affect the axial relative velocity, indicating that the rotor thrust remains almost constant in the rotor-following frame of reference. However, since the thrust moves with the wind turbine, an unsteady variation of the thrust is generated in the ground-fixed frame of reference, which can be expressed as

(4.25)\begin{align} \delta f_x(r,\theta,t) & ={-}(f_T(r,\theta,t) - \bar{f}_T(r)) \nonumber\\ & ={-}\left(\frac{1}{2} U_\infty^2 \bar{C}_T(r'(t)) - \frac{1}{2} U_\infty^2 \bar{C}_T(r)\right)\nonumber\\ & \approx{-}\frac{1}{2} U_\infty^2 \frac{\partial \bar{C}_T(r)}{\partial r} (r'(t)-r)\nonumber\\ & \approx \frac{1}{2} U_\infty^2 \frac{\partial \bar{C}_T(r)}{\partial r} \cos{\theta} \delta y_{h}(t). \end{align}

Here, $r$ and $r'(t)$ denote the distance from a point $P(0,y,z)$ to the time-averaged rotor centre $O = (0,0,0)$ and to the instantaneous wake centre $O'(t) = (0,\delta y_{h}(t),0)$, respectively, as illustrated in figure 13(b). The distance to the moving hub centre is computed as $r'(t) = \sqrt {(y- \delta y_{h}(t))^2 + z^2}$. Assuming small $\delta y_{h}(t)$, the last step in the equation is achieved by using the approximation

(4.26)\begin{align} r'(t) -r & = \sqrt{(y- \delta y_{h}(t))^2 + z^2} - \sqrt{y^2 + z^2} \nonumber\\ & \approx{-}\frac{y}{\sqrt{y^2 + z^2}}\delta y_h(t) \nonumber\\ & ={-} \cos {(\theta)} \delta y_{h}(t). \end{align}

Substituting $\delta y_{h}(t)$ defined in (2.10) into (4.25) yields an unsteady streamwise forcing as

(4.27)\begin{equation} \delta f_x(r,\theta,t) \approx \frac{A}{2} \sin{(\omega t)} U_\infty^2 \frac{\partial \bar{C}_T(r)}{\partial r} \cos{\theta} . \end{equation}

Similarly, the unsteady forces in the azimuthal direction are modelled as

(4.28)\begin{equation} \delta f_\theta(r,\theta,t) \approx{-}\frac{A}{2} \sin{(\omega t)} U_\infty^2 \frac{\partial \bar{C}_\theta(r)}{ \partial r} \cos{\theta}. \end{equation}

Figure 14(d–f) illustrate the spatial mode of the unsteady forcing for the sway motion.

4.2.3. Yaw

The yaw motion corresponds to the rotational motion along the $z$ axis, which induces a misalignment between the rotor-normal direction and the streamwise direction as shown in figure 13(c). The direction vector of the thrust depends on the yaw angle $\gamma _{h}(t)$ as follows:

(4.29)\begin{equation} \boldsymbol{n} = [-\cos{\gamma_{h}(t)}, - \sin{\gamma_{h}(t)},0]. \end{equation}

The leading-order effect of this directional variation is a side force in the transverse direction,

(4.30)\begin{align} \delta f_y(r,\theta,t) & ={-} \bar{f}_T(r) \sin{\left( \delta \gamma_{h}(t) \right)} \nonumber\\ & \approx{-}\tfrac{1}{2} U_\infty^2 \bar{C}_T(r) \delta \gamma_{h}(t), \end{align}

with Taylor expansion applied in the last step, assuming a small yaw angle $|\delta \gamma _h(t)| \ll 1$. The lateral force $\delta f_y(r,\theta,t)$ is projected in the radial ($r$) and azimuthal ($\theta$) directions of the cylindrical coordinates, yielding

(4.31)\begin{align} \delta f_r(r,\theta,t) &= \delta f_y(r,\theta,t) \cos \theta \nonumber\\ & \approx{-}\tfrac{1}{2} U_\infty^2 \bar{C}_T(r) \delta \gamma_{h}(t) \cos \theta, \end{align}

(4.32)\begin{align} \delta f_\theta(r,\theta,t) &={-}\delta f_y(r,\theta,t) \sin \theta \nonumber\\ & \approx \tfrac{1}{2} U_\infty^2 \bar{C}_T(r) \delta \gamma_{h}(t) \sin \theta. \end{align}

Substituting the prescribed harmonic yaw motion defined in (2.11) into (4.31) and (4.32), the unsteady forcing caused by harmonic yaw motion is expressed as

(4.33)\begin{gather} \delta f_r(r,\theta,t) \approx{-}\frac{A}{2R}\sin{(\omega t)} U_\infty^2 \bar{C}_T(r) \cos \theta, \end{gather}

(4.34)\begin{gather}\delta f_\theta(r,\theta,t) \approx \frac{A}{2R}\sin{(\omega t)} U_\infty^2 \bar{C}_T(r) \sin \theta. \end{gather}

The unsteady forcing in the streamwise direction, mainly contributed by the time-averaged rotational force, is expressed as

(4.35)\begin{equation} \delta f_x(r,\theta,t) ={-} \frac{A}{2R} \sin(\omega t) U_\infty^2 \bar{C}_\theta(r) \sin\theta. \end{equation}

The spatial modes of the unsteady forcing are illustrated in figure 14(g–i).

4.2.4. Summary of M2F model

Table 4 summarizes the unsteady forcing for a FOWT undergoing harmonic surge, sway and yaw motions. Their spatial modes are plotted in figure 14. As seen, the forcing appears in different directions and has different azimuthal dependencies. The forcing for surge motion is axisymmetric, i.e. the azimuthal wavenumber is zero ($m=0$). For sway and yaw motions, the forcing shows a side-to-side pattern, which can be decomposed into $m=\pm 1$ components. Moreover, we note that the magnitude of surge forcing is directly proportional to the motion frequency $\omega$. For sway and yaw motion, the motion frequency does not explicitly influence the forcing magnitude.

Table 4. Summary of the unsteady forcing generated by the proposed actuator disk model for harmonic surge, sway and yaw motions, as defined by (2.9)–(2.11).

It is important to note that under the current uniform inflow condition, vertical and lateral motions (heave and sway) can be considered equivalent, differing only in their direction of motion. Similarly, the equivalence between pitch and yaw can be achieved through a change of coordinates. Roll motion around the turbine's centre of gravity results in a side-to-side motion at the rotor's height, which can be approximated as lateral motion (sway). Furthermore, the forcings of these elementary motions can be employed to compose forcings of complex, realistic motions.

4.3. Prediction of the M2W model for uniform inflow

In this subsection we show that the wake response of a FOWT can be predicted in an end-to-end manner by connecting the M2F and F2W models.

Firstly, the motion-induced forcing is projected to the basis functions ${\rm e}^{\mathrm {i}(m\theta -\omega t)}$, as defined in (4.10), to obtain the corresponding forcing modes $\hat {\boldsymbol {f}}_{m, \omega }(x,r)$. Secondly, these forcing modes are then substituted into (4.13) of the F2W model to solve for the response modes $\hat {\boldsymbol {q}}_{m, \omega }(x,r)$. Finally, the wake response is reconstructed with (4.11). It is worth noting that the above procedure is not limited to the rotor forcings derived in § 4.2. The unsteady rotor forcing obtained by other computational methods can also be employed as input.

Taking the streamwise forcing induced by the sway motion as an example, it can be projected on four basis functions as

(4.36) \begin{equation} \delta f_x(r, \theta, t) ={-}\frac{A U^2_\infty}{8\mathrm{i}}\frac{\partial \bar{C}_T(r)}{\partial r} \left({\rm e}^{\mathrm{i}(\theta-\omega t) } - {\rm e}^{-\mathrm{i}(\theta-\omega t)} +{\rm e}^{\mathrm{i}(-\theta-\omega t)} - {\rm e}^{-\mathrm{i}(-\theta-\omega t)} \right). \end{equation}

The coefficient of each basis function, ${\rm e}^{\mathrm {i}(m \theta - n \omega t)}$, forms the spatial modes as follows:

(4.37)\begin{equation} \hat{f}^x_{ m, n \omega}(x=0,r) ={-} n \frac{A U^2_\infty}{8\mathrm{i}}\frac{\partial \bar{C}_T(r)}{\partial r} \end{equation}

with $m = \pm 1$ and $n=\pm 1$. The forcing components in the radial and azimuthal directions can be decomposed similarly. By substituting all forcing modes into (4.13) and solving for the wake response mode-by-mode, the final wake response is

(4.38)$$\begin{gather} \delta \boldsymbol{q}(x, r,\theta, t) = \hat{\boldsymbol{q}}_{1, \omega}(x,r) {\rm e}^{\mathrm{i}(\theta-\omega t) } + \hat{\boldsymbol{q}}_{{-}1, -\omega}(x,r) {\rm e}^{-\mathrm{i}(\theta-\omega t) } \nonumber\\ + \,\hat{\boldsymbol{q}}_{{-}1, \omega}(x,r) {\rm e}^{\mathrm{i}(-\theta-\omega t) } +\hat{\boldsymbol{q}}_{1, -\omega}(x,r) {\rm e}^{-\mathrm{i}(-\theta-\omega t) }. \end{gather}$$

The wake response for FOWT subject to the surge or the yaw motions are computed with the same procedure.

Figure 15 depicts the three-dimensional coherent structure computed with the M2W model for the surge, sway and yaw motions. All the motions have the same frequency $St = 0.25$ and the same amplitude $A=0.01D$. For surge motion, the velocity fluctuation is distributed uniformly in the azimuthal direction, while for sway and yaw motions, the streamwise velocity fluctuation mainly resides on the lateral sides of the wake. These spatial modes are consistent with the wake pulsing pattern for the surge motion and the wake meandering phenomenon for the sway and yaw motions.

Figure 15. Streamwise velocity fluctuation predicted by the M2W model for surge (a), sway (b) and yaw (c) at $St=0.25$. The red and blue surfaces correspond to the isocontour of $|\delta u_x| = 0.05 U_\infty$, respectively. The base flow is depicted by a light blue shadow. Inflow: uniform. Method: M2W model.

The wake response is further analysed on the hub-height plane. Figure 16(a) displays the spatial modes of the velocity fluctuation ($\hat {u}_x$ and $\hat {u}_y$) predicted by the M2W model at $St=0.25$, and is compared with the Fourier modes obtained with LES in figure 16(b). It is worth noting that all contours are plotted with the same magnitude allowing a direct comparison. As seen, the proposed M2W model predicts correctly the response modes compared with the LES in terms of flow structures. For surge motion, the pulsing mode of the wake observed in the instantaneous wake contour (figure 4) is revealed as a symmetric alternation of the streamwise velocity along the centreline. Furthermore, the lateral velocity is opposite in sign across the wake centreline, indicating a varicose deformation of the wake. For the sway and yaw motions that lead to wake meandering, the model predicts an anti-symmetric mode for streamwise velocity fluctuation, while the lateral velocity fluctuation spans throughout the entire wake. Moreover, the M2W model is capable of predicting the difference in the near wake for the sway and yaw motions. For the sway motion, an obvious anti-symmetric streamwise velocity fluctuation is found immediately behind the tip and the root of the blade, coincident with the rotor forcing distribution (see figure 14d). In comparison, the yaw motion induces a relatively smaller streamwise velocity fluctuation in the near wake, since the yaw motion does not result in a streamwise forcing near the rotor tip (see figure 14g–i). Thanks to the motion-specific modelling, the M2W model provides a phase-resolved prediction of the wake response that captures very well the phase opposition in the far wake for the sway and yaw motions. The predicted spatial modes are also comparable to the Fourier modes of the LES results. However, when comparing the model's prediction quantitatively with the LES, we notice some discrepancies in the far wake. The linear model systematically underestimates the wavelength of the velocity fluctuation, e.g. the coherent structures of the LES travel approximately one-quarter of a wavelength faster than those predicted by the M2W model at $x=10D$ for all motion types. This difference should be attributed to the base flow employed by the M2W model, i.e. the time-averaged wake of a fixed wind turbine. With such a base flow, the increase of wind speed in the far wake enhanced by turbine motions (in figure 7) is not accounted for. This underestimated wind speed in the base flow that might also explain the diamond pattern of $\hat {u}_x$ in the far wake for cases with sway and yaw, predicted by the M2W model, since the velocity near the wake centre is apparently slower than in the free stream for the base flow employed. In contrast, these structures are almost square in the result of LES, because these flow structures are convected by the actual wake velocity taking into account the enhanced wake recovery in LES.

Figure 16. Real part of the spatial mode of the fluctuating velocity on the hub-height plane, predicted by the M2W model (a) and the Fourier modes of LES (b). Here $\hat {u}_x$ and $\hat {u}_y$ represent the modes of streamwise and lateral components, respectively. The motions have an amplitude of $A=0.01D$ and a reduced frequency of $St = 0.25$. Inflow: uniform.

The predictive capability of the M2W model is further verified at $St=0.10$ and $St=0.50$ in figures 17 and 18. At $St=0.10$, the lower magnitude of wake response is well captured by the linear model. Interestingly, the velocity fluctuation is found to manifest principally in the streamwise component, leaving the transverse velocity mode less obvious for both the model prediction and the LES results. Such a lower transverse velocity is in agreement with the weaker meandering motion observed in the instantaneous wake field (figure 4). It is worth noting that the phase opposition for yaw and sway motions is still observed. At $St=0.50$, the wavelength of velocity fluctuation is significantly shortened in the streamwise direction. Moreover, the fluctuation magnitude is found to reach a peak at approximately $x=4D$ and then decreases with downstream distance. The above wake patterns are successfully captured by the proposed M2W model and the predicted velocity fluctuations show generally good agreement with LES, despite some underestimation of the velocity fluctuation magnitude, which might be due to the rather basic mean flow consistent effective viscosity employed in the model.

Figure 17. Comparison of spatial modes predicted by the M2W model and LES at $St = 0.10$. The M2W model (a) and the Fourier modes of LES (b). Inflow: uniform.

Figure 18. Same as figure 17 at $St = 0.50$.

Overall, the above comparison has demonstrated the success of the M2W model in predicting the wake response. While the model shares some aspects with stability analysis (Iungo et al. Reference Iungo, Viola, Camarri, Porté-Agel and Gallaire2013) or resolvent analysis (Towne et al. Reference Towne, Schmidt and Colonius2018) that focuses on the optimal growth of perturbations, the present model excels by its capability to predict the wake response in a motion-specific manner through the additional M2F modelling. For this reason, the present model can be further employed to compute the wake response under a combination of rotor motions by simply summing the wake responses of elementary ones, thanks to the linearity of the model.

5.1. Large-eddy simulation results

5.1.1. Instantaneous wake

A visualization of the wind turbine wake is provided in figure 19, showing the contour of the instantaneous streamwise velocity deficit $\Delta u = u_x-U_\infty$ on the hub plane for the fixed wind turbine cases (a) and the cases with A3 motion (b).

Figure 19. A snapshot of instantaneous wake at the hub-height plane plotted by the velocity deficit $\Delta u$. Cases with a fixed wind turbine (a) and rotor sway motion A3 (b). The red dash-dot lines plot the wake centreline $y_c(x)$ and the dashed lines approximate the wake boundary using $y_c(x)\pm 0.5D$. Inflow: synthetic turbulence. Method: LES.

It is worth noting that the snapshots in each counterpart in subfigures (a,b) are synchronized, i.e. the same turbulent inflow is employed and the figures are compared at the same temporal instant, so the additional effect due to sway can be directly observed. Compared with the uniform inflow condition, figure 19(a) indicates that the inflow turbulence triggers the wake meandering that increases in magnitude with inflow TIs. However, the effect of turbulence integral length is more difficult to tell based on this visualization and will be analysed quantitatively in § 5.1.3. Figure 19(b) indicates that the rotor sway motion results in extra deformation of the wake. Such deformation is particularly pronounced for the cases with the lowest TI (T1) but becomes less notable when the TI is increased, which is in agreement with previous findings (Li et al. Reference Li, Dong and Yang2022). It is worth noting that the attenuation effect due to ambient turbulence is rather strong, provided that the sway motion amplitude in this figure 19(b) is $A=0.04D$, being four times as large as that in figure 4 for the uniform inflow.

5.1.2. Spectral analysis of motion-induced velocity fluctuation

To further reveal the flow dynamics, we conduct a spectral analysis of the wake velocity with the spectral proper orthogonal decomposition (SPOD), using the approach of Towne et al. (Reference Towne, Schmidt and Colonius2018). The in-plane velocity components on the hub-height plane are analysed, with spatial restriction in the region of $-1D< x<10.5D$, $-1.5D< y<1.5D$. The time sequences of the snapshots are divided into six segments, each with a size of $T_{w} = 20 D/U_\infty$, containing $N_{w} = 1600$ snapshots. A Hamming window with $50\,\%$ overlap is employed. This configuration results in a frequency resolution of $\Delta St = 0.05$ and a Nyquist frequency of $St_{max} = 40$.

Figure 20 compares the wake velocity spectra by plotting the leading SPOD eigenvalue $\lambda _1$ that contains the largest modal energy as a function of frequency. In figure 20(a) it is found that the turbulence energy for cases with the same inflow condition has approximately the same spectrum for cases with and without rotor motion, except a prominent peak at the motion frequency ($St=0.25$). In all inflow conditions, the level of this peak is found to be positively related to the motion amplitude, but inversely related to the inflow TI. A minor peak at approximately double forcing frequency $St \approx 0.5$ can also be identified, especially for cases with low inflow TI, reflecting a weak nonlinear evolution of the motion-induced wake meandering (Li et al. Reference Li, Dong and Yang2022). At higher frequencies, peaks at the blade passing frequency ($St \approx 8.6$) and its multiples also exist but are not plotted. Figure 20(b) rearranged the same data to further probe into the effect of the inflow turbulence length scale. The figure clearly reveals that increasing the inflow turbulence length scale leads to a stronger velocity fluctuation at the lower frequency range $St<0.25$, reflecting the stronger low frequent motion of the wake induced by the inflow. At higher frequencies, $St>0.25$, the effect of the length scale becomes less apparent. Interestingly, at the frequency of the sway motion $St=0.25$, the free-stream turbulence length scale in the considered parameter space has a relatively weak effect on the motion-induced wake response compared with the TI.

Figure 20. Comparison of the energy spectra of the leading SPOD mode. The same data are employed in both subfigures to demonstrate the effect of sway amplitude (a) and inflow turbulence length scale (b). A dotted vertical red line represents the sway motion frequency $St=0.25$. Inflow: synthetic turbulence. Method: LES.

5.1.3. Statistics of wake centre displacement

In figure 21 a statistical view of the instantaneous wake trajectories is provided, by plotting wake centrelines $y_c(x)$ at 60 random instants, accounting for approximately 0.2 % of the entire simulation. The red dashed lines represent the standard deviation of the wake positions $\sigma _{y_c}(x)$ computed for the entire time series. For the cases with the fixed wind turbine, i.e. the wake meandering is induced by free-stream turbulence alone, previous studies (Espana et al. Reference Espana, Aubrun, Loyer and Devinant2011; Porté-Agel et al. Reference Porté-Agel, Bastankhah and Shamsoddin2020; Gambuzza & Ganapathisubramani Reference Gambuzza and Ganapathisubramani2023) generally consent that the wake meandering amplitude increases with both TI and length scale. The simulation results in figure 21(a) show that the present simulation set-up captures well this trend of variation for cases without wind turbine motion, where the expansion rate of the wake trajectory envelope is found to increase with both TI and length scale. However, figure 21(b) shows a less apparent trend with TI and length scale due to the co-existence of motion-induced and free-stream-induced contributions, where the envelope of the wake centrelines is found to expand at comparable rates for all cases.

Figure 21. Instantaneous centreline of the wake $y_c(x)$. Wake trajectories at 60 random instants are plotted in each case. Red dashed lines represent the standard deviation $\sigma _{y_c}(x)$ of the wake centre position. Wake meandering is induced by free-stream turbulence alone (a), free-stream turbulence plus A3 sway motion (b). Inflow: synthetic turbulence. Method: LES.

To further probe into the wake meandering behaviour when both inflow turbulence and sway motion exist, we plot the histogram of the instantaneous wake centre in figure 22. For cases with the lowest TI (T1), the sway motion significantly alters the distribution of $y_c(t)$ (see left panels in figure 22), suggesting there is a significant contribution from the sway-induced harmonic motion of the wake, as has been revealed by Li et al. (Reference Li, Dong and Yang2022). This feature disappears for cases with higher inflow TIs, indicating less apparent harmonic wake meandering motions.

Figure 22. Histogram of the wake $y_c(x=8D,t)$ for a wind turbine with sway motion ($St=0.25$, $A=0.04D$). Inflow: synthetic turbulence. Method: LES.

5.2. Motion-to-wake model considering ambient turbulence

In this section the M2W model proposed in § 4 is extended to account for the effects of free-stream turbulence. As observed in § 5.1.3, the amplitude of wake meandering induced by rotor motion is reduced when the ambient TI is increased, indicating an overall decrease in the energy in the wake.

The effect of inflow turbulence on the motion-induced forcing is first scrutinized. However, the LES results demonstrate that the change in the time-averaged thrust for computing the motion-induced forcing is always less than 2 %, even for the cases with the strongest ambient turbulence (T3L4 case), as shown in figure 23, indicating that the M2F part is almost unaffected by the ambient turbulence and is not responsible for the reduction of wake meandering. Therefore, the influence of ambient turbulence can only take place in the F2W stage. In the following we propose two corrections at this stage regarding the mean flow and the effective viscosity, respectively.

Figure 23. Comparison of time-averaged blade force (a) and thrust coefficient (b) for different inflow conditions. The sway motion has an amplitude of $A=0.04D$ with $St=0.25$. The error bar in the inset indicates a range of $\pm 1\,\%$. Method: LES.

As the time-averaged wake is employed as the base flow for deriving the resolvent operator, the effects of free-stream turbulence on the mean flow are investigated first. Figure 24 displays the time-averaged wake deficit under various inflows with different TIs and length scales. As shown in figure 24(a), increasing TI significantly modifies the time-averaged wake, characterized by a thickening of the shear layer in the near field starting from $x=2D$. In contrast, figure 24(b) compares the wake deficits for cases with four inflow turbulence length scales at the same TI (T2). As seen, the velocity profiles for different cases nearly collapse in the near wake region ($0< x<4D$), with only remarkable differences in the far wake ($x>5D$). Overall, the above observation suggests that the inflow turbulence length scale has a weaker effect on the mean flow for the considered parameter space compared with TI.

Figure 24. Comparison of time-averaged wake deficit $\bar {u}_x - U_\infty$ of a fixed wind turbine due to inflow TI (a) and inflow turbulence integral length scale (b). The black line in (a) represents the laminar inflow case. Curves with lighter colours represent stronger TIs with the same length scale L2. In (b) curves with lighter colours represent a larger length scale with the same TI (T2). Inflow: synthetic turbulence. Method: LES.

A second factor that affects wake evolution is the effective viscosity, as defined in (4.9). To investigate this effect, we computed the effective viscosity using the LES results of a fixed wind turbine wake and compared it for different inflow conditions in figure 25. As seen, free-stream turbulence increases the magnitude of the eddy viscosity for all inflow conditions compared with the uniform inflow case, and $\nu _{eff}$ is positively related to both the free-stream TI and length scale. The streamwise variation of $\nu _{eff}$ shows a similar trend for all cases, starting at $\nu _{eff}(x=0) = 0$ and developing with downstream distances until it saturates on a plateau at $x \approx 4D$ to $6D$ depending on the TI. In the near wake, higher TI induces a faster increase of $\nu _{eff}$ and, in general, a higher plateau level in the far wake. In contrast, the effect of the integral length scale is less evident in the near wake, but increasing the length scale results in an elevated $\nu _{eff}$ on the far wake plateau for a same TI. Beyond the plateau, $\nu _{eff}$ decreases in cases with high turbulent intensities and length scale, e.g. T3L4, being consistent with Scott et al. (Reference Scott, Martínez-Tossas, Bossuyt, Hamilton and Cal2023).

Figure 25. Mean flow consistent effective viscosity for cases with different free-stream turbulence. Inflow: synthetic turbulence. Method: LES. Results are shown fro (a) T1, (b) T2, (c) T3.

The M2W model is modified to incorporate these effects, by using the base flow and the effective viscosity informed from the wake of a fixed wind turbine with the same inflow from LES. The effects of the mean flow and effective viscosity modifications are first analysed by checking the optimal gain of the resolvent operator, shown in figure 26, where the optimal gain for the meandering mode, i.e. mode with azimuthal wavenumber $m=1$, are compared for cases with different TIs and a fixed inflow turbulence length scale L2. Figure 26(a) shows the result when only the mean flow correction is applied. As seen, the optimal gain decreases drastically when the TI is increased, indicating that the time-averaged wake generated by higher inflow turbulence becomes more stable to the forcings associated with motion. In figure 26(b) the wake was found to be further stabilized when both mean flow and effective viscosity corrections were applied. According to this prediction, the optimal gain for case T1L2 is reduced to approximately 40 % of that for uniform inflow.

Figure 26. Effects of mean flow correction alone (a) and both mean flow correction and inflow dependent effective viscosity (b) on the optimal energy gain of the rotor forcing for synthetic turbulence with length scale L2 and TIs T1, T2, T3. Method: F2W model. (a) Mean flow, (b) mean flow and eddy viscosity.

Finally, figure 27 compares Fourier modes of LES in figure 27(a) with that predicted by the M2W model in figure 27(b) for cases with A3 sway motion and with different inflow TIs T1–T3 and a fixed length scale L2. As seen, the Fourier modes are very similar to those for uniform inflow in 16(b), indicating that the motion-induced wake dynamics is still a dominant part at the motion frequency for the considered cases without inflow large-scale coherent motions. In figure 27(a) flow structures in the near wake are more apparent since the sway motion has an amplitude of $A=0.04D$, which is four times as large as that in the uniform condition. However, these modes grow at a slower rate downstream showing a weaker amplification effect of the wake shear layer. Overall, the proposed M2F model captures the above features of the wake satisfactorily. However, some differences between LES and the linear model can still be identified, including the discrepancy near the hub, which may be related to the nacelle, which is accounted for only by LES. Moreover, the effective viscosity employed here is a rather simple one, improved effective viscosity models (Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021) can be considered to further improve the accuracy. Direct modelling of the nonlinear term of the Navier–Stokes equations as an extra forcing term may also be helpful to improve the accuracy in predicting these coherent turbulent structures (He, Jin & Yang Reference He, Jin and Yang2017).

Figure 27. Spatial mode in the wake obtained with Fourier transform of LES velocity field at the motion frequency (a) and predicted by the M2W model (b). Sway motion with $A=0.04D$, $St=0.25$. Inflow TIs T1, T2 and T3 with length scale L2. Inflow: synthetic turbulence.