# Onset of wake meandering for a floating offshore wind turbine under side-to-
side motion

Zhaobin Li1,2  Guodan Dong1,2  Xiaolei Yang1,2<EMAIL_ADDRESS>1The State
Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy
of Sciences, Beijing 100190, China 2School of Engineering Sciences, University
of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Wind turbine’s wake, being convectively unstable, may behave as an amplifier
of upstream perturbations and make the downstream turbine experience strong
inflow fluctuations. In this work, we investigate the effects of the side-to-
side motion of a floating offshore wind turbine (FOWT) on wake dynamics using
large-eddy simulation and linear stability analysis (LSA) on the NREL 5MW
baseline offshore wind turbine. Simulation results reveal that the turbine
motion can lead to wake meandering for motion frequencies with the Strouhal
number $St=fD/U_{\infty}\in(0.2,0.6)$ (where $f$ is the motion frequency, $D$
is the rotor diameter, and $U_{\infty}$ is the incoming wind speed), which lie
in the range of the natural roll frequencies of common FOWT designs. This
complements the existing wake meandering mechanism, that the side-to-side
motion of a FOWT can be a novel origin for the onset of wake meandering. The
amplitude of the induced wake meandering can be one order of magnitude higher
than the initial perturbation for the most unstable frequencies. The
probability density function of the spanwise location of the instantaneous
wake centers is observed having two peaks on the time-averaged wake boundaries
and a trough near the time-averaged wake centerline, respectively. It is also
found that the LSA can predict the least stable frequencies and the
amplification factor with acceptable accuracy for motion amplitude $0.01D$.
Effects of non-linearity are observed when motion amplitude increases to
$0.04D$, for which the most unstable turbine oscillations shift slightly to
lower frequencies and the amplification factor decreases.

##  1 Introduction

When a wind turbine generates electricity from wind, it leaves behind a wake
featured by low wind speed and high turbulence intensity. In a wind farm, the
wake can result in a decrease of energy production and an increase of fatigue
loads for downstream turbines. For this reason, to understand and to control
the wake for mitigating these negative impacts have become one of the major
challenges faced by the wind energy industry and the wind energy science
community (van Wingerden et al., 2020; Veers et al., 2019; Meneveau, 2019;
Porté-Agel et al., 2020). For floating offshore wind turbines (FOWTs), the
motion of the platform further complicates the dynamics of wind turbine wakes.
In this study, we investigate the effects of the side-to-side motion of a FOWT
on the dynamics of wind turbine wakes, and show that such motion of a FOWT can
trigger wake meandering and help the recovery of wind turbine wakes.

In the literature, a number of studies have been conducted to identify the
features of wind turbine wakes and to reveal the underlying physics using
field measurements (Méchali et al., 2006; Hong et al., 2014; Sun et al.,
2020), wind tunnel experiments (Medici & Alfredsson, 2006; Chamorro & Porté-
Agel, 2009; Espana et al., 2012; Bastankhah & Porté-Agel, 2016), and numerical
simulations (Kang et al., 2014; Yang et al., 2016, 2018; Li & Yang, 2020). An
essential feature discovered by these work is the unsteady nature of the wind
turbine wake, especially its low-frequency, large-scale lateral motion,
referred to as wake meandering (Medici & Alfredsson, 2006; Yang et al., 2018;
Li & Yang, 2020; Larsen et al., 2007). From the time-averaged point of view,
the wake meandering spreads the wake in a large region and reduces the time-
averaged velocity deficit (Li & Yang, 2021). On the other hand, the wake
meandering results in unsteady loads on downstream turbines since it
frequently alters the incoming wind of the downwind turbines between full
wake, partial wake and the freestream (Larsen et al., 2008). Two mechanisms
have been proposed to explain the origin of wake meandering: i) the large-eddy
mechanism, and ii) the shear layer instability mechanism. The first mechanism
regards the wind turbine wakes as passive scalars that are transported by
eddies larger than the rotor diameter in the atmospheric boundary layer (ABL)
so that the wake trajectory is deformed as travelling downstream. With Taylor
frozen flow hypothesis, the first mechanism is employed for developing the
dynamic wake meandering model (DWM) (Larsen et al., 2008; Madsen et al., 2010;
Keck et al., 2014). On the other hand, the second mechanism considers the wake
meandering as the intrinsic property of the wake flow, which is triggered by
the selective amplification of upstream disturbance. Recently, computational
simulations of Yang et al. (2018) and field measurements of Heisel et al.
(2018) both confirmed the coexistence of both mechanism at utility scale, and
shows that the large-eddies mechanism mainly contributes to very low frequent
wake meandering $\textrm{St}\ll 0.1$ while the shear layer mechanism
contributes to the wake meandering in the range of $0.1<\textrm{St}<0.3$,
respectively.

Examination on the two meandering mechanisms has been carried out using field
measurements, wind tunnel experiments and numerical simulations. To verify the
conjecture of the large-eddy mechanism, Trujillo et al. (2011) compared the
prediction of the DWM with lidar measurements of instantaneous wake center
positions behind a 95 kW wind turbine and found a cross-covariance about 0.58.
Moreover, Espana et al. (2011, 2012) studied the wake meandering process by
modeling the wind turbine as a porous disk in a wind tunnel and confirmed that
the wake meandering is dominated by eddies with integral length larger than
the rotor diameter. The shear layer instability mechanism is often examined by
comparing the measured/computed wake meandering frequency with the typical
frequency of bluff bodies. Medici & Alfredsson (2006) carried out wind tunnel
experiments of a miniature wind turbine (rotor diameter $D=0.18~{}$m), and
observed low-frequent periodic wake fluctuations with $0.12<\textrm{St}<0.25$
(varying with rotor thrust and yaw angle). Later, Chamorro & Porté-Agel (2010)
found different thermal stability conditions can alter the meandering
frequency in the range of $0.33<St<0.40$ for a miniature wind turbine. Barlas
et al. (2016) revealed the effect of inflow turbulence on the wake meandering
and showed that the wake meandering happens at $\textrm{St}\approx 0.25$ for
ambient turbulence intensity TI = 6.8% but is suppressed when TI is increased
to 14%. Iungo et al. (2013) performed linear stability analysis (LSA) on the
time-averaged flow from a wind tunnel experiment, and found that the measured
wake meandering frequencies match well the most amplified frequency computed
from LSA. Mao & Sørensen (2018) scrutinised the meandering of the wake behind
a porous disk triggered by atmospheric eddies and found the most amplified
perturbation falls in the frequency range of
$\textrm{St}\in\left[0.25,0.63\right]$. Gupta & Wan (2019) investigated the
far wake meandering behind a tidal turbine with inflow velocity magnitude
periodically varying using large eddy simulations, and developed a low-order
model based on the complex Ginzburg–Landau (CGL) equation.

For FOWTs, we speculate that the disturbances introduced by the platform
motion may trigger wake meandering. In the literature, distortion of near wake
structures by the surge and pitch motions of the platform was observed in
computational results obtained using the free vortex method (Farrugia et al.,
2016; Lee & Lee, 2019) and the unsteady Reynolds-averaged Navier-Stokes
(URANS) method in recent studies (Tran & Kim, 2016; Liu et al., 2016). Such
influence of platform pitch and surge motions on the wake was also observed in
several wind tunnel experiments but with controversial results (Rockel et al.,
2014, 2017; Schliffke et al., 2020; Fu et al., 2019, 2020). For instance,
Rockel et al. (2014, 2017) considered the effects of platform pitch motion on
the far wake evolution and observed a decrease of wake replenishing due to a
time-averaged upward wake deflection. Schliffke et al. (2020) reported no
apparent influence on the wake at $4.6$ rotor-diameter downstream even with
surge motions of large amplitude according to their wind tunnel measurement of
an actuator disk type wind turbine model. As seen, most work have been focused
on the surge and the pitch (_i.e_., fore-aft motions) of the platform.
However, waves may also induce side-to-side platform motions since they are
not always aligned with the wind direction (Wanninkhof, 1992), and such
oblique waves are able to induce side-to-side turbine motions as large as the
fore-aft motions (Lyu et al., 2019). Recently, Fu et al. (2019) measured the
wake of a miniature wind turbine rolling at relatively low Strouhal numbers
$\textrm{St}<0.02$ in a wind tunnel. However, they found that effect of roll
motion only resides in the near wake and become negligible at $7$ rotor-
diameter downstream, different from wake meandering. To the best of the
authors’ knowledge, no systematic studies have been carried out to examine the
possibility of wake meandering induced by the motion of a FOWT.

As a first step, this work focuses on the side-to-side motion of the platform
considering that it is in the lateral direction of the wake and may have the
largest potential to trigger wake meandering. The NREL 5MW baseline offshore
wind turbine parametrized with the actuator surface model is simulated using
large-eddy simulation. The side-to-side motion of the platform is simplified
as imposed harmonic motion. A wide range of amplitudes and frequencies are
selected in the LES together with local LSA to look for those that can induce
wake instability.

The rest of the paper is organized as follows. The numerical methods and
simulation setups are presented in Section 2. Then simulation results are
shown in Section 3, including the wake statistics and the local LSA. At last
the discussion and conclusion are given in Sections 4 and 5, respectively.

##  2 Numerical methods and simulation setups

###  2.1 Flow solver

The air flow around the wind turbine is simulated using the LES module of the
Virtual Flow Simulator code, VFS-Wind (Yang et al., 2015b). The air is assumed
to be an incompressible Newtonian fluid, which has a constant density and
viscosity. The flow is governed by the filtered incompressible Navier-Stokes
equations as follows,

$\displaystyle J\frac{\partial U^{j}}{\partial\xi^{j}}$ $\displaystyle=0,$ (1)
$\displaystyle\frac{1}{J}\frac{\partial U^{i}}{\partial t}$
$\displaystyle=\frac{\xi^{i}_{l}}{J}\left(-\frac{\partial}{\partial\xi^{j}}\left(U^{j}u_{l}\right)+\frac{\mu}{\rho}\frac{\partial}{\partial\xi^{j}}\left(\frac{g^{jk}}{J}\frac{\partial
u_{l}}{\partial\xi^{k}}\right)-\frac{1}{\rho}\frac{\partial}{\partial\xi^{j}}\left(\frac{\xi^{j}_{l}p}{J}\right)-\frac{1}{\rho}\frac{\partial\tau_{lj}}{\partial\xi^{j}}+f_{l}\right),$
(2)

where $i,j,k,l=\\{1,2,3\\}$ are tensor indices, $\xi^{i}$ are the curvilinear
coordinates which are related to the Cartesian coordinates $x_{l}$ with the
transformation metrics $\xi^{i}_{l}=\partial\xi^{i}/\partial x_{l}$. $J$
denotes the Jacobian of the geometric transformation,
$U^{i}=\left(\xi^{i}_{l}/J\right)u_{l}$ are the contravariant volume flux with
$u_{l}$ the velocity in Cartesian coordinates. $\mu$ denotes the fluid dynamic
viscosity and $\rho$ is the fluid density. $g^{jk}=\xi^{j}_{l}\xi^{k}_{l}$ are
the components of the contravariant metric tensor. The pressure is denoted by
$p$ and $f_{l}$ are body forces to account for the effects of wind turbines on
the flow. $\tau_{ij}$ in the momentum equation is the subgrid-scale stress
(Smagorinsky, 1963),

$\tau_{ij}-\frac{1}{3}\tau_{kk}\delta_{ij}=-\mu_{t}\overline{S_{ij}},$ (3)

where $\overline{S_{ij}}$ is the filtered strain-rate tensor,
$\overline{(\cdot)}$ denotes the grid filtering operation and $\mu_{t}$ is the
eddy viscosity computed by

$\mu_{t}=C_{s}\Delta^{2}|\overline{S}|,$ (4)

where $\Delta$ is the filter width.
$|\overline{S}|=(2\overline{S_{ij}}\overline{S_{ij}})^{1/2}$ is the magnitude
of the strain-rate tensor and $C_{s}$ is the Smagorinsky constant computed via
a dynamic procedure (Germano et al., 1991).

The governing equations are discretized on a structured curvilinear grid. The
second-order central differencing scheme is employed for the spatial
discretization together with a second-order fractional step scheme (Ge &
Sotiropoulos, 2007) for the temporal integration. The momentum equation is
solved with a matrix-free Newton-Krylov method (Knoll & Keyes, 2004). The
pressure Poisson equation derived from the continuity constraint, is solved
using the Generalized Minimal Residual (GMRES) method with an algebraic multi-
grid acceleration (Saad, 1993).

###  2.2 Wind turbine modeling

A class of well-validated actuator surface model (Yang & Sotiropoulos, 2018)
is employed to model the rotor and the nacelle of the wind turbine. It
represents a rotor blade with a simplified two dimensional surface defined by
the chord length and the twist angle, with distributed forces computed from
the lift and drag ($\mathbf{L}$ and $\mathbf{D}$) determined as follows:

$\mathbf{L}=\frac{c}{2}\rho
C_{\text{L}}|V_{\text{ref}}|^{2}\mathbf{e_{\text{L}}}$ (5)

and

$\mathbf{D}=\frac{c}{2}\rho
C_{\text{D}}|V_{\text{ref}}|^{2}\mathbf{e_{\text{D}}},$ (6)

where $c$ denotes the local chord length, $C_{\text{L}}$ and $C_{\text{D}}$
are the lift and the drag coefficients defined in 2D airfoil tables as a
function of Reynolds number and angle of attack, $\textbf{e}_{\text{L}}$ and
$\textbf{e}_{\text{D}}$ are the unit directional vectors of the lift and drag.
$V_{\text{ref}}$ is the local incoming velocity relative to the moving blades
averaged over the local chord width. A 3D stall delay model (Du & Selig, 1998)
and a tip loss correction (Shen et al., 2005) are applied. L and D are
employed to calculate the body force in equation (2) by uniformly distributing
the resultant force along the chord width as follows,

$\mathbf{f}=(\mathbf{L}+\mathbf{D})/c.$ (7)

In the actuator surface model for nacelle, the normal component of the force
is computed by satisfying the non-penetration boundary condition, and the
tangential component is computed using a specified friction coefficient and
incoming velocity, respectively. The transfer of quantities between the
actuator surface and the background grid nodes is achieved via a smoothed
discrete delta function (Yang et al., 2009).

###  2.3 Simulation setup

In the present cases, the NREL offshore 5MW baseline wind turbine (Jonkman et
al., 2009) is employed. It has a rotor with diameter $D=126~{}$m and a
cuboidal shape nacelle ($2.3~{}\text{m}\times 2.3~{}\text{m}\times
14.2~{}\text{m}$). The simulation represents the rated operation condition
with the inflow wind speed of $U_{\infty}=11.4~{}\text{m/s}$ and tip speed
ratio of 7. The Reynolds number based on the inflow velocity and the rotor’s
diameter is $\textrm{Re}\approx 9.6\times 10^{7}$. The simulations are
conducted with an idealized uniform inflow to reflect the low turbulence
offshore environment (Bodini et al., 2020) with the wind veer and shear
neglected so that the side-to-side rotor motion is the only source of
disturbances for the wake.

Figure 1:  Schematics of the computational domain and the rotor side-to-side
motion.

The computational domain, showing in figure 1, is a cuboid with $L_{x}\times
L_{y}\times L_{z}=14D\times 7D\times 7D$, where $x,y\textrm{~{}and~{}}z$
denote the streamwise, horizontal and vertical directions, respectively. The
turbine is placed $3.5D$ from the inlet and located at the center of the cross
section. The origin of the coordinate system coincides with the wind turbine’s
hub center. The inflow velocity is imposed on the inlet boundary at $x=-3.5D$.
Neumann condition for velocity ($\partial u_{i}/\partial x=0$) is applied on
the outlet boundary at $x=10.5D$. Free slip condition is imposed on the four
lateral boundaries. The computational domain is discretized with a Cartesian
grid. The streamwise grid is uniform with $\Delta x=D/20$. In $y$ and $z$
directions, the grid spacing is $\Delta x=\Delta y=D/40$ near the wake
centerline, _i.e_., in the region $y\times
z\in[-1.5D,1.5D]\times[-1.5D,1.5D]$. Out of this near region, the grid is
gradually stretched in both lateral directions. The total number of nodes is
$N_{x}\times N_{y}\times N_{z}=281\times 141\times 141\approx 5.6$ M. Such a
grid resolution is fine enough for the actuator model to predict the velocity
deficit and turbulence intensity in turbine wakes with reasonable accuracy
based on a previous grid refinement study (Yang et al., 2015b). Each
simulation takes about 40 rotor revolutions to fully develop the wake
initially. After this transitional stage, the instantaneous flow fields are
saved for further investigation.

###  2.4 Side-to-side motions

No. | Amplitude [$A$ (m)] | Nondimensional Amplitude [$A/D$] | Roll Amplitude [$\theta(^{\circ})$]
---|---|---|---
1 | $1.26$ | 0.01 | 0.72
2 | $2.52$ | 0.02 | 1.44
3 | $5.04$ | 0.04 | 2.88
Table 1: Amplitudes of the simple harmonic motion of the rotor hub and the corresponding roll motion amplitudes.  No. | Strouhal Number [${St}=D/TU_{\infty}$] | Period [$T$ (s)] | Note
---|---|---|---
1 | 0.10 | 110.5 | Natural sway frequency
2 | 0.20 | 55.3 | $\downarrow$
3 | 0.25 | 44.2 |
4 | 0.30 | 36.8 | ——–
5 | 0.35 | 31.6 | $\uparrow$
6 | 0.40 | 27.6 | Natural roll frequency
7 | 0.45 | 24.6 | $\downarrow$
8 | 0.50 | 22.1 | ——–
9 | 0.60 | 18.4 |
10 | 0.80 | 13.8 | $\uparrow$
11 | 1.00 | 11.1 | Ocean waves
Table 2: The Strouhal numbers and the periods of the simple harmonic motions,
the last column describes approximately the corresponding ranges of the
different physics.

In the simulations, the wind turbine is imposed with a simple harmonic motion
in $y$ direction as shown in figure 1, which can either stem from the
translation along the $y$ axis (sway motion) or the rotation along the $x$
axis (rolling motion) with respect to the center of gravity (CoG) of the
floating wind turbine and the platform. For small roll motion, the side-to-
side rotor hub displacement $\Delta y$ and the roll angle $\Delta\theta$ can
be approximated with $\Delta y\approx l\Delta\theta$ where $l\approx
100~{}\textrm{m}$ is the distance from the hub to the CoG (Robertson et al.,
2014). In the present work, we only considered motions with relatively small
amplitudes as listed in table 1 to ensure the hub acceleration are within the
mechanical design requirement (Nejad et al., 2019).

From the wave-structure interaction point of view, the motions of a FOWT can
be divided into two groups, _i.e_. those induced directly by waves and those
at natural frequencies (Jonkman & Musial, 2010; Robertson et al., 2014). For
this reason, the present work selects a rather large frequency range to
analyze the different influence of different motions from both groups, as
shown in table 2. The high frequency part corresponds to the motions directly
induced by the first-order wave forces, _i.e_., with periods approximately 10
$\sim$ 15 s at operational sea states (Toffoli & Bitner-Gregersen, 2017). On
the other hand, the natural frequencies, although change from case to case due
to different turbine and platform designs, usually falls into a lower
frequency range. For instance, the periods corresponding to the natural
frequencies of roll and sway motions are 25$\sim$30 s and 100$\sim$120 s,
respectively for currently available prototype floating platforms (Jonkman &
Musial, 2010; Robertson et al., 2014). These lower frequencies are also
considered in the present work as shown in table 2.

##  3 Results

In this section, we first present the results from a specific case in section
3.1, for which the wake meandering is triggered by the side-to-side motion of
the turbine. This case is employed to illustrate the difference between the
wakes behind an oscillating turbine and a fixed turbine in detail. Moreover,
this case is used to introduce the techniques employed for extracting the wake
characteristics that will be further applied to all the other cases for
investigating the influence of the motion amplitude and frequency in section
3.2. Section 3.2 also presents the local LSA on the time-averaged wake behind
a fixed wind turbine for predicting the wake meandering, with results compared
with those obtained from LES cases.

###  3.1 Effects of wind turbine oscillations

In the case presented in this section, the frequency and amplitude of the
turbine motion are $\textrm{St}=0.25$ and $A/D=0.04$, respectively. We will
show that the meandering motion is triggered by the motion of the turbine for
this case, and compare the wake of this oscillating turbine with that of a
fixed turbine.

####  3.1.1 Instantaneous wake characteristics

Figure 2: Instantaneous streamwise velocity deficit in the wake behind a fixed
wind turbine (a) and a wind turbine subject to side-to-side simple harmonic
motion of $\textrm{St}=0.25$ and amplitude $A/D=0.04$ (c). The wind turbines
are illustrated with black solid lines. The red dash-dotted lines and the
dashed lines indicate the center-lines and the wake width, respectively. They
are obtained with the Gaussian fit of the velocity deficit profile, as shown
in (b) and (d) for the wakes of the fixed wind turbine and of the moving wind
turbine, respectively.

The influence of turbine side-to-side motion on the instantaneous wake is
revealed in figure 2, where snapshots of the instantaneous wakes behind a
fixed wind turbine and behind an oscillating one are compared. Figures 2 (a)
and (c) compare the contours of the instantaneous streamwise velocity deficit
with centerline and width of the wake extracted with a Gaussian fit defined as
follows,

$\Delta u(x,y)=u(x,y)-U_{\infty}=\Delta
u_{c}(x)\exp{\left(-\frac{(y-y_{c}(x))^{2}}{2s^{2}(x)}\right)},$ (8)

where $u(x,y)$ is the streamwise velocity and $s(x)$ is the standard deviation
of the Gaussian distribution to be fitted with the velocity profile. The wake
centerline position is defined as the peak location of the velocity deficit
plotted with red dash-dotted lines. The wake width (red dashed lines) is
defined as $r_{1/2}(x)=\sqrt{2\ln 2}s(x)$, which gives the distance from
$y_{c}(x)$ to the position where $\Delta u=\displaystyle\frac{1}{2}\Delta
u_{c}$. This fitting technique based on self-similarity, which is often
applied to extract the features of time-averaged wakes (Bastankhah & Porté-
Agel, 2016; Xie & Archer, 2015), has been demonstrated being able to extract
features of instantaneous wakes (Li & Yang, 2021).

As seen in figure 2 (a), the centerline of the wake behind the fixed wind
turbine is approximately straight in the near wake and fluctuates mildly in
the far wake ($x>6D$). Moreover, the width of the wake behind the fixed
turbine remains almost constant at all downstream locations, suggesting that
the wake recovery is slow for the fixed turbine for the simulated case without
inflow turbulence. In contrast, the oscillating wind turbine generates a wake
with large-scale oscillations as shown in figure 2 (c), revealing an
significant impact of the side-to-side motion on turbine wakes. The meandering
amplitude is found being approximately equal to the turbine diameter
($A\approx D$) in the far wake, while the amplitude of the side-to-side motion
is only $A=0.04D$. This rapid growth shows that the initial perturbation
induced by the turbine motion is vastly amplified as traveling downstream.
Besides the wake centerline oscillation, the velocity deficit in the far-wake
is also significantly affected, being weaker behind the oscillating turbine
when compared with the fixed turbine as shown in figures 2 (a) and (c). This
difference is more evident in figures 2 (b) and (d), where the transverse
profiles of the velocity deficit at downstream location $x=8D$ are plotted.
The blue curves represent the instantaneous velocity deficit, and the red
dashed lines are their Gaussian fits defined with equation (8). It is clear
that the centerline velocity deficit $\Delta u_{c}$ is stronger and the wake
width is narrower for the fixed wind turbine when compared with the
oscillating wind turbine. As for the distribution of the velocity deficit,
wake behind the fixed turbine is symmetric and can be fitted well with the
Gaussian curve. The wake downstream of the oscillating wind turbine, on the
other hand, shows obvious asymmetry. Despite the observed asymmetry, the
essential features of the meandering wake, _i.e_., the centerline position,
the velocity deficit, and the width, are well captured by the Gaussian fitting
as shown in figures 2 (c) and (d).

Figure 3: Frequency spectra of the spanwise velocity fluctuation in the wake
behind the fixed wind turbine $(a)$ and the wind turbine subject to side-to-
side FOWT motion with $\textrm{St}=0.25,~{}A/D=0.04~{}(b)$. Data are collected
at four downstream locations ($x=\\{2D,~{}4D,~{}6D,~{}8D\\}$) along $y=0$.

The wakes downstream of the fixed and the oscillating wind turbines are
further compared by examining the frequency spectra of the spanwise velocity
fluctuation ($\phi_{v}$) in figure 3. The velocity spectra ($\phi_{v}$) is
computed with instantaneous spanwise velocities ($v$) recorded along the time-
averaged wake centerline ($y=0$) at four turbine downstream locations
($x=\\{2D,~{}4D,~{}6D,~{}8D\\}$) using the discrete Fourier transform. As
shown in figure 3 ($a$), the velocity behind the fixed wind turbine contains
only small-scale fluctuations, which are influenced by the hub vortex the wake
of the nacelle, and distributed over a wide range of frequencies. No obvious
growth can be found for the fluctuation amplitude as traveling downstream. In
contrast, the velocity spectra behind the oscillating wind turbine consist of
a distinct peak at the frequency of the imposed motion ($\textrm{St}=0.25$).
Moreover, the spectral level at $\textrm{St}=0.25$ grows significantly with
downstream distances, revealing the onset of the instability of the wake,
which significantly amplifies the disturbances initiated by the side-to-side
motion, with the spectral level at other frequencies remaining approximately
the same except for the trice of the imposed frequency. It is observed that
the spectral level at the imposed frequency ceases to grow from $x=6D$ to
$x=8D$, while the peak at the higher frequency ($\textrm{St}\approx 0.75$)
start to appear at $x=6D$, which indicates the saturation of the growth of the
imposed motion and the energy transfer from low to high frequencies via
nonlinear effects (Schmid et al., 2002; Drazin & Reid, 2004).

Figure 4: Frequency spectra of the wake center motion behind the fixed wind
turbine $(a)$ and the wind turbine subject to side-to-side FOWT motion at
$\textrm{St}=0.25,~{}A/D=0.04~{}(b)$. Data are collected at four downstream
locations ($x=\\{2D,~{}4D,~{}6D,~{}8D\\}$) along $y=0$.

Similar to above analyses, the spectra of wake center displacement is computed
and shown in figure 4. It is observed that the amplitudes of wake center
oscillations behind the oscillating wind turbine are one to two orders of
magnitude higher than the fixed wind turbine at the frequency of the imposed
side-to-side motion. For other frequencies, the spectral levels from the
oscillating wind turbine are comparable to the fixed wind turbine, which does
not have any distinct peak frequencies. Furthermore, it is observed that the
spectral level at the peak frequency grows monotonically as traveling
downstream from the turbine and is comparable to the rotor diameter in the far
wake ($x=6D,8D$).

Figure 5: The probability density function of wake center positions at turbine
downstream location $x=8D$ for the fixed wind turbine $(a)$ and the
oscillating wind turbine $(b)$.

Another interesting observation is that the spanwise distribution of
instantaneous wake center positions becomes very different for the oscillating
turbine when compared with the fixed one. These instantaneous wake centers are
sampled from instantaneous flow fields at evenly distributed time steps. In
figure 5, the PDF of instantaneous wake positions at turbine downstream
location $x=8D$ are compared. As seen, the PDF for the wake centers of the
fixed wind turbine as shown in figure 5 (a) approximately follows the normal
distribution located in a relatively narrow band with $y/D\in[-0.05,0.05]$
with a single peak located near $y=0$. In contrast, the wake centers behind
the oscillating wind turbine spread in a much wider extend with
$y\in(-0.7D,0.7D)$, being more than 10 times larger than the fixed wind
turbine. More importantly, the PDF shows a distinct shape with two peaks near
$y=\pm 0.5D$ and a trough near $y=0$, which is significantly different from
that observed for the fixed wind turbine (Li & Yang, 2021) and reflects the
fact that the wake center spends more time at $y=\pm 0.5D$ than at $y=0$. It
is noticed that this PDF is close to that of a simple harmonic motion with a
trough in the equilibrium position and two peaks at extremities (Tennekes &
Lumley, 1972), indicating the significant effect the turbine motion has on the
frequency of the meandering motion. From the practical point of view, this PDF
implies that the wake of an oscillating wind turbine may partially shelter the
downstream turbine more often than a fixed wind turbine.

Figure 6: The space and time correlation of the transverse velocity in the
wake behind the moving wind turbine: (a) the time correlation, (b) the space
correlation, and (c) the space time correlation at point $x=5D$.

To further understand the spatial and temporal evolution of the meandering
induced by the side-to-side motion, the space and time correlations of the
transverse velocity fluctuations in the wake of the oscillating wind turbine
are examined in figure 6. As seen, they are distinctively different from the
space-time correlation in fully developed turbulent flows (He et al., 2017; Wu
et al., 2017; Wu & He, 2020, 2021), with clear periodicity in both space and
time showing the dominant effects of the coherent wake meandering induced by
the side-to-side motion. The periodicity observed on the time correlation
contour at different streamwise locations as shown in figure 6 (a) agrees well
with the imposed motion. The space correlations at different downstream
locations are shown in figure 6 (b). Interestingly, it is observed that the
wavelength ($\lambda$) obtained from the space correlation gradually grows
while traveling downstream, which is probably caused by the recovery of the
streamwise velocity in the far wake, which transports the flow structures at a
higher velocity. The space-time correlation is shown in figure 6 (c). Contours
of straight lines with a nearly constant slope are observed, indicating the
validity of Taylor’s frozen flow hypothesis for the simulated cases. By
inspecting figure 6 (c) carefully, it is seen that the slope becomes slightly
steeper when traveling downstream, which indicates the increase of the
streamwise convection velocity, being consistent with the increase of
wavelength shown in figure 6 (b).

####  3.1.2 Time-averaged wake characteristics

Figure 7: Time-averaged streamwise velocity deficit contour in the wake behind
a fixed wind turbine (a) and a wind turbine subject to side-to-side simple
harmonic motion with $\textrm{St}=0.25$ and amplitude $A/D=0.04$ (b). The red
lines represents the spanwise variation of the velocity deficit at different
downstream locations.

The effects of turbine motion on the time-averaged wake are examined in this
section. First, we show the contour of streamwise velocity deficit in figure
7.  It is observed that the time-averaged velocity deficit of the fixed and
oscillating cases are close to each other in the near wake region ($x<3D$). At
further downstream locations, the wake of the fixed wind turbine as shown in
figure 7 (a) only shows a mild wake expansion and the wake shear layer
gradually grows in both inward and outward directions. The maximum deficit
along the wake centerline remains approximately $\Delta
U=U-U_{\infty}=-0.5U_{\infty}$ at $x=8D$ (a common spacing between two
turbines in a wind farm (Hansen et al., 2012; Yang et al., 2018; Ahsbahs et
al., 2020)). By contrast, the wake behind the oscillating wind turbine as
shown in figure 7 (b) shows an obvious broader wake expansion starting from
$x=4D$, as a result of large-scale meandering motion which spreads the low
speed region in an larger extend. A direct result of this broad wake expansion
is the faster recovery of momentum deficit in the wake core. As shown in
figure 8, the time-averaged velocity deficit is approximately $\Delta
U\approx-0.2U_{\infty}$ at $x=8D$, being significantly lower than that behind
the fixed wind turbine ($\Delta U\approx-0.4U_{\infty}$). From the practical
point of view, this faster wake recovery is beneficial for turbines in the
wake because of the increase of the available wind power. It is also worth
noting that the spanwise profile of the velocity deficit in the far wake is
different from the Gaussian bell shape, especially for the oscillating wind
turbine, whose wake profile is featured by a relatively flat central part and
a steep transition near the wake boundary and related with the PDF of the
instantaneous wake center positions as shown in the previous subsection.

Figure 8: Spanwise profile of the velocity deficit at $x=8D$ behind the fixed
wind turbine (a) and the oscillating wind turbine (b).

####  3.1.3 Turbine-added turbulence

Figure 9: Turbulence intensity (TI) in the wake behind a fixed wind turbine
(a) and a wind turbine subject to side-to-side simple harmonic motion with
$\textrm{St}=0.25$ and amplitude $A/D=0.04$ (b). The transverse profiles of TI
are plotted with red lines at different downstream locations.

Except for the benefit of the faster wake recovery, the wake meandering also
results in stronger unsteady loads for the downstream turbines, which is
unfavorable for the structure of the downstream turbines. To assess this
effect, the turbulence intensity (TI) in the wake of the oscillating wind
turbine, which is defined as
$\displaystyle\textrm{TI}=\sqrt{\frac{1}{3}(u^{\prime 2}+v^{\prime
2}+w^{\prime 2})}/U_{\infty}$, where $u^{\prime},~{}v^{\prime},w^{\prime}$ are
the root-mean-square of the fluctuation of the three velocity components, is
computed and compared with the fixed turbine case. The obtained contours and
profiles of the TI are shown in figure 9. As expected, the oscillating motion
of wind turbine has drastically intensified the turbulence intensity in the
wake in terms of both magnitude and spatial extent.  Specifically, it is
observed that the TI behind the hub and the tip of the rotor is intensified in
the near wake because of turbine oscillation. Starting from $x=2D$, the TI
grows in the shear layer of the wake, showing a bi-modal distribution in the
spanwise direction, and extends in both inward to the wake core and outward to
the free stream as traveling downstream. At approximately $x=5D$, the shear
layers from the tips meet at the wake center for the oscillating turbine case,
which happens at nearly $x=9D$ for the fixed turbine case. At further turbine
downstream locations, the TI ceases to grow and slightly decrease starting
from $x=7D$. At $x=8D$, the TI in the oscillating turbine case is also
significantly higher when compared with the fixed turbine case, with the
maximum TI $\approx$ 13.4%, 8.5% for the former and latter cases,
respectively. It is noticed that this level of turbulence intensity is still
within the range of the wind turbine design standard (turbulence category C as
per IEC 61400-1) and comparable to the atmospheric turbulence intensity caused
by complex terrain topology (Yang et al., 2015a, 2018; Li et al., 2021a) for
land-based wind turbines (DNV, 2016).

###  3.2 Effects of motion frequencies and amplitudes

After showing the results from a specific case, for which the wake meandering
is induced by a given side-to-side motion, in this section we perform local
LSA and carry out simulations with a series of frequency and amplitude
combinations to examine how the onset of wake meandering depends on the
frequencies and amplitudes of the side-to-side motions.

####  3.2.1 Local linear stability analysis

Previous works on wind turbine wakes in the literature have demonstrated its
convective instability (Iungo et al., 2013; Sarmast et al., 2014; Mao &
Sørensen, 2018; Gupta & Wan, 2019), _i.e_., the wake behaves as an amplifier
of upstream disturbances. For a prescribed base flow, the linear stability
theory can predict the spatial amplification rate for an infinitely small
disturbance at different excitation frequencies. In the following, the local
LSA is performed to assess the response of wake to small disturbances with
different frequencies. The time-averaged flow behind a fixed wind turbine is
simplified using locally axis-parallel and axisymmetric assumptions, _i.e_.,
the velocity is simplified as $(\overline{U}(r),~{}0,~{}0)$ in the streamwise,
radial and azimuthal directions (where $\overline{U}$ is the temporally and
azimuthally averaged streamwise velocity from the fixed turbine case), and
employed as the base flow for the LSA. The perturbation velocity components
$(u_{x},~{}u_{r},~{}u_{\theta})$ and the perturbation pressure $(p)$ are
assumed to be the form of
$(u_{x},~{}u_{r},~{}u_{\theta},p)=(\hat{u}_{x},~{}\hat{u}_{r},~{}\hat{u}_{\theta},~{}\hat{p})\left(\exp(ikx+in\theta-i\omega
t)\right)$, where $\hat{\cdot}$ dentoes the shape function in the radial
direction. Spatial stability analyses are performed with the real angular
frequency $\omega=2\pi f$ and the complex wave number $k=k_{r}+ik_{i}$.
Substituting the perturbation variables into the linearized Navier-Stokes
equations yields,

$\displaystyle
ik\hat{u}_{x}+\frac{\hat{u}_{r}}{r}+\frac{d\hat{u}_{r}}{dr}+\frac{in\omega}{r}\hat{u}_{\theta}$
$\displaystyle=0$ (9) $\displaystyle
i(k\overline{U}-\omega)\hat{u}_{x}+\frac{d\overline{U}}{dr}\hat{u}_{r}$
$\displaystyle=-ik\hat{p},$ (10) $\displaystyle
i(k\overline{U}-\omega)\hat{u}_{r}$ $\displaystyle=-\frac{d\hat{p}}{dr},$ (11)
$\displaystyle i(k\overline{U}-\omega)\hat{u}_{\theta}$
$\displaystyle=-\frac{in}{r}\hat{p}.$ (12)

The above equations are the same as those derived by Batchelor & Gill
(Batchelor & Gill, 1962) for axisymmetric jets/wakes, expect that the viscous
terms are excluded, because their contribution is less important for base flow
velocity profiles with inflection points (Schmid et al., 2002). The azimuthal
wave number is selected as $n=1$, because a side-to-side motion velocity
($u_{y}$) can be projected in the radial and azimuthal directions as
$u_{r}=u_{y}\cos\theta$ and $u_{\theta}=u_{y}\sin\theta$. The linearized
Navier-Stokes equations are discretized with Chebyshev polynomials on $128$
Gauss–Lobatto nodes $\xi\in\left[-1,1\right]$ and stretched to
$r\in[0,r_{\textrm{max}}]$ through an algebraic mapping (Schmid et al., 2002)
defined as $\displaystyle r=a\frac{1+\xi}{b-\xi}$ with $\displaystyle
a=\frac{r_{\textrm{max}}}{r_{\textrm{max}}-2}$ and $\displaystyle
b=1+\frac{2a}{r_{\textrm{max}}}$. The size of the computational domain is
chosen as $r_{\textrm{max}}=15D$. The size of the domain and the number of
discretization points are verified to be large enough to not influence the
stability analyses. The far field boundary condition is that the perturbations
vanish at $r=r_{\textrm{max}}$. On the wake centerline, $\hat{u}_{x}=0$
$\hat{p}=0$ and $\hat{u}_{r}+i\hat{u}_{\theta}=0$ are imposed following
Batchelor & Gill (Batchelor & Gill, 1962).

Figure 10: Local linear stability analysis results of the wake behind the
fixed wind turbine: $(a)$ the streamwise velocity deficit, $(b)$ contours of
local spatial growth rate $-k_{i}$, and $(c)$ the integrated amplification
factor $\displaystyle G=\exp{(-\int_{0}^{x}k_{i}\textrm{d}x)}$. The red dotted
lines in $(b)$ and $(c)$ indicate the most amplified frequency along the
streamwise locations.

Figure 10 shows the results of local LSA conducted at different downstream
locations. As seen in figure 10 (a), the velocity deficit profiles in the
region between the core of the wake and the free stream is characterized by a
shear layer. The thickness of this shear layer gradually increases in the
turbine downstream direction due to the momentum mixing between the wake and
the freestream, and consequently makes the local stability properties vary
with the downstream distance as shown in figure 10 (b). In the near wake
($x<3D$), the steep velocity variation in the shear layer results in a broad
range of unstable frequencies $\textrm{St}\in(0,0.9)$ approximately, with the
most unstable frequency at $\textrm{St}\approx 0.5$. At further turbine
downstream locations, both the most unstable frequency and the range of the
unstable frequencies decrease. This trend of decrease of the most unstable
frequency via the downstream distance is in agreement with the work of Gupta &
Wan (Gupta & Wan, 2019), where LSA is applied to the time-averaged wake of a
tidal turbine. Such downstream variations of the spatial growth rate indicates
that the most unstable perturbations in the near wake becomes stable when
propagating downstream and ceases to grow subsequently. On the other hand,
perturbations with lower frequencies may excel in the far wake since they can
grow for a longer distance. To demonstrate this accumulation effect via
downstream distances, figure 10 (c) displays the contours of the integrated
amplification factor defined as
$G=\displaystyle\exp{(-\int_{0}^{x}k_{i}\textrm{d}x)}$. It is found that the
frequency with the largest amplification factor also decreases in the
streamwise direction, being consistent with that of the local growth rate
$-k_{i}$. In the far wake, perturbations with evident amplifications fall in
the frequency range of $0.1<\textrm{St}<0.8$. The strongest amplification
factors are predicted for $0.2<\textrm{St}<0.4$, explaining the large-scale
wake meandering triggered by turbine side-to-side motion with
$\textrm{St}=0.25$ as presented in section 3.1. Furthermore, this frequency
range is rather low compared to the commonly encountered excitement
frequencies generated by wind turbines, _e.g_., the rated rotor frequency and
the blade passing frequency are approximately equal to $0.2\textrm{~{}Hz}$
($\textrm{St}\approx 2.2$) and $0.6\textrm{~{}Hz}$ ($\textrm{St}\approx 6.6$),
respectively for the NREL 5WM wind turbine (Jonkman et al., 2009) as well as
other wind turbines at utility scale. As a result, unstable modes of the wake
are unlikely to be triggered by the rotation of blades alone, being consistent
with the literature that the hub vortex at lower frequencies plays an
important role in triggering the shear layer instability and the far wake
meandering for the bottom-fixed turbine (Kang et al., 2014). Moreover, this
range is also lower than the wave frequencies at operational sea states
($\textrm{St}\geq 1.0$) as shown in table 2, indicating that floating wind
turbines moving at wave frequencies cannot trigger significant wake meandering
which has been shown by experiments conducted by Schliffke _et al_. (Schliffke
et al., 2020). On the other hand, this low frequency range ($0.2<St<0.4$)
overlaps partially with the natural frequencies of the roll or the sway
motions as shown in table 2.

Although the above LSA can provide useful information on the instability
properties of the wake, it is worth noting the analyses are carried out based
on several assumptions for the base flow and disturbances, _i.e_., i) the
axis-parallel and axisymmetric assumption for the base flow, ii) the small
perturbation assumption for linearizing the Navier-Stokes equations, and iii)
the inviscid flow assumption. Often, these prerequisites are not fully
satisfied in real applications, for which the base flow is complex and the
disturbances are of finite amplitudes, such that the nonlinear effects may
manifest. To further investigate the wake characteristics of the oscillation
wind turbine with side-to-side motions of finite amplitude and to verify the
results from the LSA, results from LES cases with the frequencies and
amplitudes with setups presented in tables 1 and 2 will be analyzed in the
following subsections.

####  3.2.2 Instantaneous wake

Figure 11: The instantaneous streamwise velocity deficit behind a wind turbine
subject to side-to-side simple harmonic motion for various frequencies and
amplitudes. Figures in the same row have the same motion frequency denoted by
the Strouhal number given in the last column. Figures in the same column have
the same motion amplitudes.

Snapshots of the instantaneous streamwise velocity deficits are plotted for
comparison in figure 11. As observed, the wake behavior varies via the turbine
oscillation frequency, being consistent with the LSA. For cases with the
lowest and the highest motion frequencies, _i.e_., for $\textrm{St}=0.1$ and
$\textrm{St}\geq 0.8$, no apparent wake meandering is observed and the wake is
almost straight. For the cases with the lowest motion frequency
($\textrm{St}=0.1$), the spanwise motion of the wake is very mild and the wake
is close to that behind a fixed turbine. Although increasing the amplitude of
the side-to-side motion to $A/D=0.04$ can deform slightly the wake into a
sinusoidal form, the amplitude of wake center displacement is still much
smaller than the wake width. For cases with the largest frequencies
($\textrm{St}\geq 0.8$), the wake centerline is approximately straight and
without significant oscillations, indicating the overall meandering motion of
the wake is negligible. On the other hand, small-scale fluctuations are
observed on the wake boundary, which, however, do not show apparent
amplifications as travelling downstream and remain locally near the boundary
without apparent interaction with the core of the wake. Moreover, it is
observed that these small scale fluctuations do not seem to grow in magnitude
when increasing the amplitude of the side-to-side motion that roughly the same
instantaneous wakes are observed for cases at $\textrm{St}\geq 0.8$ with
different amplitudes considered in the present study. In contrast, the cases
with $0.20\leq\textrm{St}\leq 0.60$ show a much stronger far wake meandering,
which is consistent with the larger amplification factor predicted by LSA.
Specifically, the cases with $0.20\leq\textrm{St}\leq 0.40$ exhibit apparent
sinusoidal form deformations with a continuous growth of wake meandering
amplitudes via the downstream distance until the very far wake. Larger
meandering amplitudes are also observed when the motion amplitudes is larger
with the same turbine motion frequency. For cases with higher Strouhal numbers
($0.4<\textrm{St}\leq 0.6$), however, the part of the wake with sinusoidal
deformation is limited to a certain downstream distance, beyond which the wake
loses this periodicity and the coherent structures cannot be identified
visually, and the effects of motion amplitude on the wake meandering in the
far wake are hard to tell directly from these instantaneous snapshots in that
region because of the complex far wake flow.

Figure 12:  Contours of the amplification factor $G$ computed from the LES
results for different turbine motion amplitudes: $(a)$ $A/D=0.01$, $(b)$
$A/D=0.02$, $(c)$ $A/D=0.04$. The amplification factor is defined as
$G=\phi_{v}/\phi_{v}^{0}$, where $\phi_{v}$ is the amplitude of spanwise
velocity in the wake obtained with discrete Fourier transform of the time
series of the spanwise velocity obtained from LES along the time-averaged wake
centerline, and $\phi_{v}^{0}$ is the velocity amplitude of the wind turbine’s
simple harmonic motion. The largest amplification factor is marked together
with its Strouhal number.

It is observed from the instantaneous results that the wake meandering is
affected by both amplitude and frequency of the wind turbine oscillation. To
further investigate such influences quantitatively, we compute the
amplification factor of the spanwise velocity in the wake for different
turbine oscillation frequencies and amplitudes, as shown in figure 12. The
amplification factor is defined as $G=\phi_{v}/\phi_{v}^{0}$, with $\phi_{v}$
the amplitude of spanwise velocity at the turbine motion frequency obtained
with discrete Fourier transform from simulation results collected along the
time-averaged wake centerline and $\phi_{v}^{0}$ is the velocity amplitude of
the wind turbine’s simple harmonic motion computed as $\phi_{v}^{0}=2\pi fA$,
with $A$ the turbine motion amplitude and $f=U_{\infty}\times\textrm{St}/D$
the turbine oscillation frequency. The amplification factors at every $0.5D$
for $x\in[0D,10D]$ are computed.

The obtained amplification factors are presented in figure 12 $(a),(b)$, and
$(c)$ for the cases with turbine motion amplitudes of $A/D=0.01,0.02,0.04$,
respectively. As seen, all the three figures show a similar pattern for the
regions with amplification factors higher than one, indicating that the
initial spanwise velocity is amplified while traveling downstream for those
cases. It is observed that the amplification of the spanwise velocity is most
intensive for turbine motion with frequency in the range of
$\textrm{St}\in[0.2,0.3]$ and gradually weakened for lower or higher
frequencies. For the cases with the smallest turbine motion amplitude, _i.e_.,
$A=0.01D$ in figure 12, the location and size of the region with high
amplification factors are closer to that predicted by the local LSA (as shown
in figure 10 $(c)$) when compared with the cases with higher oscillation
amplitudes. In terms of the magnitudes of the amplification factors, the LES
predictions are observed being in general lower than the LSA predictions,
because that the hypotheses employed in LSA, _i.e_., the infinite small
perturbation, axis-parallel base flow and the inviscid fluid, are not full
satisfied. Moreover, it is observed that the differences between LSA and LES
predictions are enlarged when increasing turbine motion amplitudes as shown in
figure 12 $(b)$ and $(c)$, which suggests that the non-linearity caused by the
finite size perturbations plays a role on reducing the growth rate, being
consistent with the nonlinear stability theory (Schmid et al., 2002) and a
previous work on tidal turbine wakes (Gupta & Wan, 2019). Furthermore, figure
12 shows that the increase of turbine motion amplitudes slightly lowers the
most amplified frequency and narrows the region with high amplification
factors.

####  3.2.3 Time-averaged wake

Figure 13: The time-averaged streamwise velocity deficit at the hub-height
plane. The wind turbine is subject to side-to-side simple harmonic motion
defined with various frequencies and amplitudes. Figures in the same row have
the same motion frequency denoted by the Strouhal number given in the last
column. Figures in the same column have the same motion amplitudes. Figure 14:
The time-averaged centerline velocity deficit for cases with different turbine
oscillation frequencies and motion amplitudes of $A/D=$ 0.01, 0.02, and 0.04
for subplots (a),(b),and (c), respectively.

To investigate the influence of turbine motion on the wake recovery, we
analyze the time-averaged velocity deficits computed from different cases. In
figure 13, we show contours of the time-averaged streamwise velocity deficits.
As seen, the recovery of the time-averaged wake for cases with
$\textrm{St}=0.1$ and $\textrm{St}\geq 0.8$ is slow and the wake is close to
that of a fixed wind turbine, being consistent with the instantaneous velocity
deficit. In contrast, the velocity deficit contours are significantly altered
for cases with motion frequencies in the range of $0.20\leq\textrm{St}\leq
0.60$, and the wake recovery is observed being dependent on both motion
amplitude and frequency. Taking the cases with $A/D=0.01$ as example, the
velocity starts to recover from the wake edge before finally reaching the wake
centerline. The velocity recovery on the wake boundaries is found being faster
with higher platform motion frequencies, which is related to the stronger
mixing induced by the steeper wavy deformation of the shear layer as seen from
the instantaneous wake snapshots (figure 11). However, faster overall far-wake
recovery is only observed for the cases in the frequency range of
$0.2<\textrm{St}<0.6$ where intense wake meandering is observed in the far
wake, indicating the large-scale wake meandering is more efficient than the
mixing induced by waves in the shear-layer.

The above observation is further confirmed by the quantitative comparison
shown in figure 14 (a), where the velocity deficit along the mean wake
centerline is displayed for cases with different turbine motion frequencies at
the same motion amplitude $A/D=0.01$. As seen, the centerline velocity for
cases with $\textrm{St}=0.6$ recovers rapidly in $4D<x<6D$ then stops growing.
On the other hand, cases with lower turbine motion frequencies
($\textrm{St}=0.3$ and $\textrm{St}=0.4$ ) are featured by the smallest
velocity deficit of $\Delta U_{c}/U_{\infty}\approx 0.2$ at $x=10D$, compared
with $\Delta U_{c}/U_{\infty}\approx 0.4$ from the fixed wind turbine case.
Another interesting phenomenon found in figure 14 is the influence of turbine
motion amplitude on the wake recovery is nonlinear, _e.g_., the turbine motion
frequency with strongest wake recovery in the far wake is different when the
motion amplitude increase from $A/D=0.01$ to $A/D=0.04$.

####  3.2.4 Turbine-added turbulence

Figure 15: The turbulence intensity at the hub-height plane. The wind turbine
is subject to side-to-side simple harmonic motion defined with various
frequencies and amplitudes. Figures in the same row have the same motion
frequency denoted by the Strouhal number given in the last column. Figures in
the same column have the same motion amplitudes. Figure 16: The turbulence
intensity on the wake centerline for cases with different turbine oscillation
frequencies and motion amplitudes of $A/D=$ 0.01, 0.02, and 0.04 for subplots
(a),(b),and (c), respectively.

In this section, we examine the characteristics of turbine-added turbulence
for cases with different side-to-side motions. Figure 15 displays the contours
of the turbulence intensity (TI) on the hub-height plane for cases with
different motion amplitudes and frequencies. For the cases with
$\textrm{St}=0.10$, both magnitude and spatial distribution of TI are very
similar to that of the fixed turbine case as shown in figure 9 (a), _i.e_.,
the TI appears first on the wake edge, making a bi-modal spatial distribution
then it gradually expands inwards to the wake centerline and outwards to the
freestream. For higher turbine motion frequencies $0.2\leq\textrm{St}\leq
0.6$, the TI is generally higher than the fixed wind turbine and spreads in a
larger extent. For $\textrm{St}\geq 0.8$, the TI becomes weaker due to the
absence of large-scale wake meandering as observed in the instantaneous wake
snapshots. Figure 16 compares the TI along the wake centerline for cases with
different turbine motion amplitudes and frequencies. In figure 16 (a), it is
found that the streamwise variations of TI for cases with $\textrm{St}=0.1$,
$\textrm{St}=0.2$ and $\textrm{St}=1.0$ are close to the case of the fixed
wind turbine. Other cases, on the other hand, exhibit a much stronger
streamwise increase of TI before reaching their peak, after which the TI
starts to decrease. The location of peak shift towards the turbine with
increasing turbine oscillating frequency. For all cases, the peak TI does not
exceed the upper limit of 0.15 and decreases to approximately TI$=0.1$ at
$x=10D$. Increasing the turbine oscillation amplitudes does not amplify the
magnitude of TI significantly, while shifts the peak of TI upstream and it
leaves a longer distance for the TI to decrease before reaching possible
neighbours in the downstream. Consequently, the final TI reaching these
downstream turbines located in the far wake ($x/D=10$) can be smaller for
cases with larger amplitudes for cases with certain side-to-side motion
frequencies, _e.g_., the cases with $0.3\leq\textrm{St}\leq 0.5$. Such
decrease is probably related to the recovered velocity deficit in the wake,
which makes the nominal velocity fluctuations induced by the wake meandering
smaller.

##  4 Discussion

The most important finding from this work is that the turbine side-to-side
motion can be a novel origin for the onset of wake meandering. This finding
complements the existing wake meandering mechanism (Iungo et al., 2013; Mao &
Sørensen, 2018; Gupta & Wan, 2019), and is of particular importance for FOWTs
because their compliant foundations are easily subject to motions induced by
waves/winds. Platform motions at certain frequencies can be amplified in the
far wake due to wake instability and leads to wake meandering with amplitude
comparable to the turbine rotor diameter.  As revealed by the space-time
correlation analysis, the frequency of the meandering motion is strongly
related with that of the side-to-side motion and well preserved in the entire
wake, being distinctively different from the meandering motion caused by
inflow large-eddies. The PDF of the instantaneous wake center positions, which
is featured by two peaks, one at each extremity, being consistent with the PDF
of the side-to-side FOWT motion, is significantly different from the normal
distribution commonly observed for wake meandering in the literature.

The wake meandering observed in the simulated cases with ideal setups is very
likely to happen for real FOWTs. The target wind turbine in this study (NREL
5WM reference wind turbine) is one of the most studied turbines for the FOWT
research. The frequencies and amplitudes considered in the present cases are
feasible in realistic applications. The turbine motion amplitudes being less
than $0.04D$, are within the design requirement for offshore wind turbines
(Nejad et al., 2019). The frequencies resulting in the significant far wake
meandering falls in the range of $0.2\leq\textrm{St}\leq 0.6$, although being
lower than the wave frequencies commonly encountered (corresponding to
$\textrm{St}>1.0$ for the present turbine design), are within the range of the
natural motion frequencies of the floating platform, indicating the importance
of considering wake meandering induced by the FOWT motion in the design of
FOWT farms. For example, the well studied reference offshore wind turbine
platform, the DeepCwind semi-submersible platform (Robertson et al., 2014),
has its natural roll period about 26 s ($\textrm{St}\approx 0.4$), which will
dominate the platform motion when second-order wave forces are considered
(Mahfouz et al., 2021). The present work shows that this design may trigger
large-scale far-wake meandering when the platform oscillates at its natural
roll frequency. On one hand, the induced wake meandering can help wake
recovery. On the other hand, wake meandering at the frequency close to the
platform’s natural frequency may even trigger large motion of the platform of
downstream turbines like a chain reaction which may eventually propagates in
the entire wind farm. This trade-off between the benefit and the risk must be
carefully considered when designing the hydrodynamic properties of the
platform.

Thirdly, the present work demonstrates that the local LSA can be employed to
develop low-order models for predicting wake meandering caused by side-to-side
motion of FOWTs.  The comparison between the predictions from LSA and LES
shows that the local LSA can predict the least stable frequency and the
amplification factor with a reasonable accuracy, especially when the imposed
turbine motion is small (_e.g_., $A/D=0.01$). Although the amplification
factors are overpredicted by LSA especially for turbine motions with high
amplitudes as a result of the nonlinear effect, the local LSA employed in this
work can at least give an upper bound of the meandering motion in the far
wake. Besides the non-linearity effects, the inviscid fluid assumption may
also lead to over predictions of the amplification factor for LSA, which can
be incorporated in future studies to further improve the accuracy of the
present LSA, such as including the non-linearity using the Ginzburg-Landau
equation (Gupta & Wan, 2019), and adding the eddy viscosity to the linearized
Navier-Stokes equations as shown by Rukes _et al_. for a turbulence jet flow
(Rukes et al., 2016), for which the eddy viscosity could be obtained using a
data-driven approach (Zhang et al., 2020).

##  5 Summary and conclusions

The effects of the side-to-side harmonic oscillations of a floating offshore
wind turbine on the wake evolution is investigated using large eddy
simulations of the NREL 5MW offshore wind turbine parameterized with the
actuator surface model. In the simulated cases, the side-to-side motion of the
turbine with different frequencies and amplitudes are prescribed.  The
simulation results show that the side-to-side motions with frequencies in the
range of $0.2\leq\textrm{St}\leq 0.6$ can trigger large-scale far wake
meandering. The wake meandering observed in the simulated cases, with its
dominant frequency and the PDF of instantaneous wake positions consistent with
the side-to-side motion of FOWTs, is different from the meandering observed in
the literature, complementing the existing meandering mechanism that the
turbine side-to-side motion can be a novel origin for the onset of wake
meandering. Moreover, this work shows that such wake meandering induced by the
FOWT side-to-side motion can be acceptably predicted using the local LSA.
Using only the streamwise velocity component of the time-averaged wake behind
a stationary turbine as the base flow, the local LSA is able to predict the
unstable frequency range with a reasonable accuracy at a negligible
computational expense. In the meanwhile, the simulation results also show that
the local LSA slightly overpredicts the most unstable frequency and greatly
overpredicts the amplification factors for high amplitudes of the side-to-side
FOWT motion because of the nonlinear effect, which can not be considered yet
in the employed local LSA.

From the practical point of view, the side-to-side turbine motion has a two-
fold influence on the downstream neighbours. On one hand, violent wake
meandering in the far wake induces strong unsteady loads. On the other hand,
the meandering enhances the mixing between the wake and the freestream and
consequently accelerates the recovery of the time-averaged velocity deficit,
which benefits the overall power output of the wind farm. Moreover, analyses
using the methodologies employed in this work, _i.e_., LES and LSA, can be
integrated in the design stage of FOWT. The present work shows that the wake’s
unstable frequencies lie over a broad range and overlaps with the floater’s
natural frequencies, and thus this kind of wake meandering is highly probable
with the present floater designs. Predicting the unstable frequency range of
the wake as done in this work can provide an important criterion for the
design of the floater’s natural frequency no matter the wake meandering
induced by the side-to-side turbine motion is considered as helpful or
harmful.

It is worth noting that the scenario considered in the present work is rather
idealized (_i.e_., only simple harmonic side-by-side turbine motion is
considered and the inflow turbulence is fully ignored) to facilitate the
analyses of the wake’s essential features. In the real-world applications, a
FOWT is subject to complex motions of six degrees of freedom induced by both
wave (Liu et al., 2016) and wind in the ocean atmospheric boundary layer
(Rockel et al., 2014; Wise & Bachynski, 2020). These interactions between the
air, the FOWT, and the waves, which can be solved using fully coupled two-
phase CFD solvers (Zhu et al., 2019; Gatin et al., 2020; Li et al., 2021b; Zhu
& Masud, 2021), are not considered in the present work.

## Acknowledgement

This work is partially supported by NSFC Basic Science Center Program for
“Multiscale Problems in Nonlinear Mechanics” (NO. 11988102).

## Declaration of Interests

The authors report no conflict of interest.

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