# Modal Analysis of the Wake Shed Behind a Horizontal Axis Wind Turbine with
Flexible Blades

S. Salavatidezfouli  A. Sheidani  Mathematics Area, MathLab, International
School for Advanced Studies (SISSA) via Bonomea 265, I-34136 Trieste, Italy.
K Bakhshaei  Mathematics Area, MathLab, International School for Advanced
Studies (SISSA) via Bonomea 265, I-34136 Trieste, Italy.  A Safari  Department
of Mechanical Engineering, The Hong Kong University of Science and Technology,
Hong Kong, China  A Hajisharifi  Mathematics Area, MathLab, International
School for Advanced Studies (SISSA) via Bonomea 265, I-34136 Trieste, Italy.
G Stabile<EMAIL_ADDRESS>Department of Pure and Applied Sciences,
Informatics and Mathematics Section, University of Urbino Carlo Bo, Piazza
della Repubblica, 13, I-61029 Urbino, Italy  G Rozza  Mathematics Area,
MathLab, International School for Advanced Studies (SISSA) via Bonomea 265,
I-34136 Trieste, Italy.

###### Abstract

The proper orthogonal decomposition (POD) has been applied on a full-scale
horizontal-axis wind turbine (HAWT) to shed light on the wake characteristics
behind the wind turbine. In reality, the blade tip experiences high
deflections even at the rated conditions which definitely alter the wake flow
field, and in the case of a wind farm, may complicate the inlet conditions of
the downstream wind turbine. The turbine under consideration is the full-scale
model of the NREL 5MW onshore wind turbine which is accompanied by several
simulation complexities including turbulence, mesh motion and fluid-structure
interaction (FSI). Results indicated an almost similar modal behaviour for the
rigid and flexible turbines at the wake region. In addition, more flow
structures in terms of local vortices and fluctuating velocity fields take
place at the far wake region. The flow structures due to the wake shed from
the tower tend to move towards the center and merge with that of the nacelle
leading to an integral vortical structure 2.5D away from the rotor. Also, it
is concluded that the exclusion of the tower leads to missing a major part of
the wake structures, especially at far-wake positions.

††preprint: AIP/123-QED

##  I Introduction

Clean energies are considered to be a suitable substitute for conventional
carbon-intensive energy resources to decrease environmental disasters such as
climate change, global warming and acid rain. In this matter, a great portion
of the global energy demand due to the world population surge in the last
decades can be addressed with renewable energy resources (Alkhabbaz  _et al._
, 2022; Siddiqui  _et al._ , 2022). Therefore, more investments have been
drawn to the development and optimization of utilization of these energy
sources (Bajuri  _et al._ , 2022; Spyropoulos  _et al._ , 2022).

Wind energy is characterized as a feasible substitute owing to its
availability, reliability and cleanness. Wind Turbines are one of the major
sources of energy production burdening low impact on the environment and are
associated with high efficiency and performance(Majidi Nezhad  _et al._ ,
2022; Zhang  _et al._ , 2020). This led to a great rise in the installation of
wind turbines around the world and it is predicted that the share of wind
turbines in electricity production will rise to 5% by 2030 (Moghaddam  _et
al._ , 2011). To meet the increasing demand for energy production, large-scale
wind turbines are going to be clustered in more extensive areas with condensed
configurations to ensure enough energy production. In fact, two challenges are
always highlighted among the vital design factors of the large-scale offshore
wind farm (Uchida  _et al._ , 2021; Bastankhah and Portagel, 2019; Jaiswal
_et al._ , 2022): (i) the wake shed by the upstream turbines which can
decrease the amount of power generated by the downstream turbines and (ii) the
probability of failure of the wind turbine structures including nacelle and
blades. An efficient and reliable design of wind turbine heavily depends on
the identification of the dynamics of wind turbine wakes and their effects
associated with neighbouring wind turbines for optimization and development of
next-gen wind turbines and wind farms (Regodeseves and Morros, 2021; Uchida
and Gagnon, 2022). Moreover, a full analysis of the near and far wake regions
of the wind turbine is necessary to accomplish the performance assessment of
the blades and further acoustic analysis. The wake region behind a wind
turbine can exhibit flow structures ranging from millimetres to hundred meters
which correspond to the size of wind turbine components (Kabir and Ng, 2019;
Kabir  _et al._ , 2020; Purohit, Kabir, and Ng, 2021).

Although large-scale offshore wind turbines undergo severe aerodynamic and
weight forces, they have attained a lot of attention during recent years due
to their high ability in power production, less visual and environmental
impacts and abundance of wind in offshore areas (Zhang and Kim, 2018; Liu  _et
al._ , 2019). To overcome this complication, manufacturers aim to produce
lighter blades to mitigate the problem of large loads applied to the tower and
mechanical connections (Tarfaoui, Nachtane, and Boudounit, 2020; Liu  _et al._
, 2022). As a result, the larger and lighter blades lead to more flexible
structures which alter the performance of the wind turbine and flow structures
shedding from the rotor needs also to be studied (Hand, Kelly, and Cashman,
2021; Sathish, 2021). Moreover, the wind turbines are typically placed in wind
farms in tandem configuration, and therefore, the effect of blade flexibility
on the wake shed behind the rotor needs to be investigated as well. As a
result, this study, as one of the primary objectives, is intended to
investigate the phenomenon of wake shed behind a HAWT. For this purpose,
Proper Orthogonal Decomposition (POD) will be utilized to extract the hidden
structures of the fluid flow.

The employment of POD to extract different modes of flow dynamics was first
proposed by Lumely (Lumley, 1967). This idea was then nurtured and developed
into more robust methods in different studies namely, Sirovich (Sirovich,
1987) who put forward the concept of snapshot POD. Consequently, in recent
years, the advances in Reduced Order Modelling (ROM) techniques have led to
plenty of studies intended to enlighten the hidden flow characteristics by
employing methods like POD (Sheidani  _et al._ , 2023a; Lassila  _et al._ ,
2014; Verveld and Kier, 2017; Štefan  _et al._ , 2017; Sinha, Chauhan, and
Balasubramanian, 2018; Wang, Ma, and Liu, 2018; Chouak, Dufresne, and Seers,
2018; Benner  _et al._ , 2021, 2020a, 2020b; Star  _et al._ , 2021; Barra  _et
al._ , 2021). For instance, Premaratne et al. (Premaratne, Tian, and Hu, 2022)
investigated coherent vortices embedded in the turbulent wake flow behind a
prototype HAWT experimentally. A high-resolution Particle Imaging Velocimetry
(PIV) system was used to quantify the evolution of the unsteady wake vortices.
The POD analysis was followingly conducted to identify energetic vortex
patterns in the turbulent near wake. They concluded that the wake region
adjacent to the turbine accounts for a high percentage (62%) of the modal
energy. While highly stretched fluctuations, especially in the vertical
direction, were observed at the end of the wake region.

Due to the large dimensions of such turbines, the experimental study regarding
the blades’ flexibility effects is impractical. In addition, it is hard to
establish a complete similarity between the actual model and the prototype.
For instance, in the existence of sea waves, an equal value of Froude number
should be maintained to ensure hydrodynamic similarity. However, maintaining
similar aerodynamic dimensionless numbers such as the Tip Speed Ratio (TSR)
and Reynolds Number at the same time is almost impossible (Khosravi, Sarkar,
and Hu, 2016; Fontanella  _et al._ , 2021; Mancini  _et al._ , 2020; Madsen
_et al._ , 2020). As a result, the numerical methods may yield more reliable
outcomes for offshore wind turbines owing to the fact that all the details are
taken into consideration.

From the numerical perspective, to take the effect of blade flexibility into
consideration, the structural and fluid dynamics phenomena need to be modelled
by independent solvers followed by an interface to transmit the results from
one to the other (Tran and Kim, 2018; Deng, Xiao, and Yang, 2020). Obviously,
the accuracy and computational cost are the two main factors which govern the
Fluid-Structure Interaction (FSI) simulations, and one needs to strike a
balance to conduct a cost-efficient study. For instance, Blade Element
Momentum (BEM) oriented techniques are mainly employed to reduce the cost of
the computational cost associated with the flow simulation (Rafiee, Tahani,
and Moradi, 2016). The FAST code was first developed by the National Renewable
Energy Laboratory (NREL) to extend the aerodynamic calculation of horizontal
off-shore wind (Prowell, Elgamal, and Jonkman, 2010). The code utilized a
variety of packages to address fluid and structure models, and more
inclusively BEM method for simulating the fluid flow. Several research studies
were reported on the utilization of the FAST package to model FSI in wind
turbines (Tran and Kim, 2018; Borouji and Nishino, 2019; Yu  _et al._ , 2020;
Tran and Kim, 2016). In parallel, due to the simplifications related to the
BEM theory, many attempts have been made to modify and improve the accuracy of
such simulations. To illustrate, Mo et al. (Mo  _et al._ , 2015; Rajan and
Ponta, 2018) modified the BEM theory by considering the flexional fluctuations
and torsional deformations for each element. Rajan and Ponta (Rajan and Ponta,
2018) developed the Dynamic Rotor Dynamics (DRD) – BEM models to capture the
three-dimensional effects of blade deformation more accurately by constantly
altering the point of element equilibrium.

Some authors took benefits of simplified structural solvers while achieving
high-order solutions for fluid flow. Sayed et al. (Sayed  _et al._ , 2019)
conducted a numerical study on a DTU 10MW HAWT by combining Unsteady Reynolds
Average Navier-Stokes (URANS) with a structural solver called Carat++ which
calculates the deformation of the blade by Finite Element Method (FEM) under
the assumption of non-linear Timoshenko beam. In this work, an increase in the
wind turbine power due to the blade flexibility by 1% was reported. On the
other hand, the discrepancy between the rigid and flexible simulations has
been reported to be more significant for other HAWT types. For instance,
Badshash et al. (Badshah, Badshah, and Kadir, 2018) showed almost a 5%
difference in the power coefficient of a two-bladed HAWT by taking the effect
of blade flexibility into consideration. Generally, the effect of blades’
flexibility plays an important role in larger turbines.

The precise perception of flow characteristics in the wake region of the wind
turbine requires the employment of accurate numerical methods accompanied by
powerful analysis tools. In spite of the fact that cost-efficient methods like
BEM or simplified structure concluded high accuracy in terms of output power,
they are not able to accurately predict the wake region behind the wind
turbine. Hence, as the main objective of this paper, we aim to implement a
fully coupled high-fidelity FSI simulation to account for the wake
characteristics behind a large-scale HAWT. On the other hand, POD analysis was
implemented to reveal flow structures in the wake region as it is claimed to
be a great tool in this respect (Sheidani  _et al._ , 2023b). Additionally, as
another objective, in order to understand to what extent the blade’s
flexibility alternates wake properties, a comparison between the rigid and
flexible blades was carried out. NREL 5MW horizontal wind turbine was selected
for these goals based on the work of (Dose  _et al._ , 2020). At first, a
validation study for the FSI solver with respect to the numerical results of
(Dose  _et al._ , 2018) is performed. Afterwards, the POD analysis of the wake
region based on the high-fidelity simulation of rigid and flexible turbines is
presented to unfold the physics of the wake followed by a conclusion on the
results.

##  II Numerical Method

The rigid and flexible simulations of the wind turbine have been carried out
with STAR CCM+ solver. It incorporates Finite Volume Method (FVM) and Finite
Element Method (FEM) solvers, respectively, for the flow and structural
simulations. In addition, mesh motion, overset and mesh morphing techniques
were employed simultaneously to address the nodes’ movement through the
simulation. In the following section, the details of the utilized solvers will
be discussed.

###  II.1 Solver Description

#### Fluid Flow.

The incompressible fluid flow assumption is often utilized for wind turbines.
Especially, in the case of large-scale horizontal wind turbines, Mach number
is typically low which assures incompressibility conditions. In order to
perform the numerical simulation of the problem, the Unsteady Reynolds Average
Navier-Stokes (URANS) was numerically solved using Star CCM+ V12.

The URANS formulation adopted for the continuity and momentum equations of the
incompressible fluid is as follows:

$\displaystyle\frac{\partial\overline{U_{i}}}{\partial x_{i}}=0,$ (1)
$\displaystyle\rho\frac{\partial\overline{U_{i}}}{\partial
t}+\rho\frac{\partial\overline{U_{i}}\;\overline{U_{j}}}{\partial x_{j}}=$ (2)
$\displaystyle\;\;\;\;\;\;\;\;-\frac{\partial P}{\partial
x_{i}}+\frac{\partial}{\partial
x_{j}}\left[\mu\left(\frac{\partial\overline{U_{i}}}{\partial
x_{j}}+\frac{\partial\overline{U_{j}}}{\partial
x_{i}}\right)-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right],$

where $\overline{U_{i}}$ and $\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}$ are
the averaged velocity and Reynolds stress tensor, respectively. The latter can
be written as:

$\displaystyle-\rho\overline{\bm{u}_{i}^{\prime}\bm{u}_{\bm{j}}^{\prime}}=2\mu_{t}\overline{S_{ij}}-\frac{2}{3}k\rho\overline{\delta_{ij}},$
(3)

where the mean strain rate tensor is:

$\displaystyle\overline{S_{ij}}=\frac{1}{2}\left(\frac{\partial\bar{U}_{i}}{\partial
x_{j}}+\frac{\partial\bar{U}_{j}}{\partial x_{i}}\right).$ (4)

The turbulent viscosity term, $\mu_{t}$, can be addressed with common one- or
two- equation eddy viscosity models to provide turbulence closure (Safari,
Saffar-Avval, and Amani, 2018).

The selection of the suitable RANS model is vital for HAWT as separation
occurs during the blades’ rotation. Many reports have agreed on both the
accuracy and simplicity of $k-\omega$ SST model for capturing turbulent
features of the flow around the wind turbines (Muiruri, Motsamai, and Ndeda,
2019; Pinto, Franzini, and Simos, 2020; Cai  _et al._ , 2022), and hence, was
utilized for this research. All the spatial and temporal terms in Equation 2
were discretized by the second-order method and SIMPLE algorithm was employed
to solve the set of equations, as suggested by literature (Moussa, 2020;
Marzec, Buliński, and Krysiński, 2021).

#### Structural.

The deformation of the flexible blades is described using a Lagrangian
formulation and is governed by the structural momentum conservation equation,
which can be expressed as follows:

$M_{s}\ddot{d}+C_{s}\dot{d}+K_{s}d=F_{s},\\\ $ (5)

where $M_{s}$, $C_{s}$ and $K_{s}$ represent the mass, damping coefficient and
stiffness of the turbine blade, respectively. $\ddot{d}$, $\dot{d}$ and $d$
are acceleration, velocity, and displacement of the structure element,
respectively.

$F_{s}$ represents the force vector acting on the solid including pressure,
shear, Coriolis, centrifugal and mass forces:

$F_{s}=F_{P}+F_{\tau}+F_{Cori.}+F_{cent}+F_{w}.\\\ $ (6)

The solution was derived through time integration by solving the motion
equation in the Newmark scheme and the solution to the solid displacement was
formulated in terms of the time history.

#### Mesh Morpher.

For the simulation of the strongly coupled FSI problem, a two-way information
exchange across the interface is needed. The traction forces across the
interface, i.e. the pressure and wall shear, are mapped from the fluid domain
to the solid domain. This results in deformations in the solid. The solid
deformation across the interface, in turn, is mapped back to the fluid domain.
The solid deformation only specifies how the interface nodes are displaced,
and hence, the displacements of the inner nodes of the fluid domain must then
be determined. To accomplish this, Star CCM+ employs a mesh morphing technique
known as Radial Basis Function (RBF) to propagate nodes displacement
throughout the whole fluid domain. Accordingly, a couple of control points are
placed on the moving boundary, i.e. Fluid-Structure interface, at which the
displacement is known, and then, the interpolation field will be constructed
to specify the displacements of the inner nodes. A full description of the RBF
technique can be found in (Bao and Chen, 2021).

###  II.2 Two-way Coupling

Generally, there exist two methods for numerical simulations of the fluid-
structure interaction in terms of dealing with the system of equations. In the
so-called monolithic approach, the coupled problem is considered as one system
of equations, and hence, the flow and structural fields are solved
simultaneously. Whereas, in the partitioned approach, separate solvers are
employed for each of the solid and fluid domains. In the latter, there will be
a need for the transfer of forces and displacements between the solvers
through time. In fact, there is still no comprehensive discretization method
to address both fluid and structural problems accurately. Hence, utilization
of the monolithic approach is not common for real-world applications. However,
as for the partitioned method, a suitable general-purpose solver can be
utilized for each domain in which case we need to implement inter-
communication protocols between solvers.

In a general two-way FSI problem, after the solid equations are solved, the
boundary displacements are imported into the fluid solver and the mesh morpher
determines the new locations for the nodes of the fluid domain. Following the
flow solution, aerodynamic forces acting on the body are exported to the
structural solver. The data of displacement and force can be passed between
solvers with implicit and explicit methods. As for the explicit, data exchange
only happens at the end of the time step. While it occurs in the inner loop
for the implicit method until convergence takes place. An implicit partitioned
approach was utilized for the sake of accuracy and convergence.

The main drawback with the conventional partitioned FSI method is that the
convergence decreases dramatically as the density ratio of solid to fluid
decreases (so-called added mass effects (Idelsohn  _et al._ , 2009; Kassiotis,
Ibrahimbegovic, and Matthies, 2010; Stabile, Matthies, and Borri, 2018), and
divergence is observed regardless of the time step value in case of the
incompressible flow (Fernández, Gerbeau, and Grandmont, 2007; van Brummelen,
2009). To overcome this difficulty, especially in the case of strongly coupled
FSI problems, a stabilization method should be employed. Star CCM+ utilizes
Boundary Interface Added Mass technique in which the fluid domain traction
resulting from the interface deformation and consequently the displaced fluid
per unit area are predicted and taken as a force correction in the solver.
More details of this technique can be found in (Nordenström, 2022).

In addition, the starting time is typically regarded as the major bottleneck
in the convergence of the simulation of FSI problems with large deflections.
To guarantee convergence, almost rigid conditions, i.e. high Young’s modules,
were applied to the wind turbine at the start of the simulation. Then, the
elastic properties decreased to that of the actual values. This corresponds to
the oscillatory nature of the results of the generated power depicted in Fig.
7, which will be discussed later on.

Finally, to enhance the numerical stability of such an FSI problem with large
deformations, Aitken’s dynamical relaxation method is employed (Küttler and
Wall, 2008), in which the values of two previous iterations are used to
improve the result of the current iteration to obtain a better convergence.
The full procedure of the FSI simulation is shown in Fig. 1.

Figure 1: Solution strategy for the partitioned FSI solver

###  II.3 Problem Description

#### Fluid Flow.

The wind turbine model used in the present work is the onshore NREL 5 MW based
on the work of (Dose  _et al._ , 2020), as shown in Fig. 2. The detailed
specifications of the turbine model are presented in Table 1. Other
geometrical details of the turbine can also be found in (Coulling  _et al._ ,
2013; Zhang and Kim, 2018).

Figure 2: Geometrical properties of the NREL 5MW Table 1: Specifications of the NREL 5MW onshore wind turbine Rotor Orientation, configuration | Upwind, 3-Blades
---|---
Blade Sections | DU and NACA airfoils
Blade Length | 61.5m
Rotor Diameter | 126m
Tilt Angle | 5∘
Precone Angle | 2.5∘
Hub Diameter | 3m
Tower Height | 90m

Fig. 3 illustrates the computational domain of the fluid. The upstream and
sides boundaries are defined as the velocity inlet boundary condition and the
pressure outlet is considered for the downstream boundary. The no-slip wall
condition is imposed on the turbine walls and the bottom boundary. All the
boundaries were put far enough from the turbine to ensure flow uniformity.
Thereby, a hexahedral domain with the size of $3000$\mathrm{m}$\times
1000$\mathrm{m}$\times 750$\mathrm{m}$$ (length, width, height) was generated.
The rated conditions presented by (Jonkman  _et al._ , 2009) were considered
for the wind turbine. In this regard, a uniformly distributed wind velocity
and rotor angular velocity of $11.4$\mathrm{m}\mathrm{/}\mathrm{s}$$ and
12.1$\mathrm{r}\mathrm{p}\mathrm{m}$ were employed, respectively. A turbulence
intensity of 0.14 was considered at the inlet boundary. The air density and
viscosity were set to $1.2\,$\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$$
and $1.8\times{10^{-5}}$\mathrm{P}\mathrm{a}\,\mathrm{s}$$, respectively.

Figure 3: Fluid domain and boundary conditions considered for the onshore NREL
5MW

#### Solid Structure.

Finite element simulation of the structural model was carried out with Star
CCM+ as well. The advantage of using the same software for both physics is the
accessibility to the solver settings and the ability to implement advanced
post-processing techniques for the fluid and solid simultaneously.

The current research study only focuses on the effect of blade deformation on
the wake properties behind the wind turbine. Hence, there would be no need to
model the deformation of the tower and nacelle. Structural properties were
considered similar to that of (Asareh and Volz, 2013).

#### Fluid-Structure Interface.

The boundary condition at the interface of the solid and the fluid, $\Gamma$,
can be defined as follows. Assuming the no-slip condition at the wall, the
displacement of the solid, $d_{\Gamma}^{S}$ is equal to the displacement of
the fluid, $d_{\Gamma}^{F}$, at the interface:

$d_{\Gamma}^{S}=d_{\Gamma}^{F},$ (7)

This is true for the rate of change of displacement as well. Simply,
substituting the fluid displacement at the boundary with the velocity yields:

$\hat{\mathbf{n}}\cdot\mathbf{v}_{\Gamma}^{f}=\hat{\mathbf{n}}\cdot\frac{\partial\mathbf{d}_{\Gamma}^{s}}{\partial
t},$ (8)

and $\hat{\mathbf{n}}$ is the interface normal vector. Moreover, based on
Newton’s third law, a balance of forces exists on the interface between the
fluid and solid:

$\hat{\mathbf{n}}\cdot\bm{\sigma}_{\Gamma}^{S}=\hat{\mathbf{n}}\cdot\bm{\sigma}_{\Gamma}^{F},$
(9)

where $\bm{\sigma}_{\Gamma}^{S}$ and $\bm{\sigma}_{\Gamma}^{F}$ are the solid
and fluid stresses at the interface, respectively. In essence, Eq. 8 and 9
describe the kinematics and dynamics of the fluid-structure interface,
respectively.

###  II.4 Mesh Generation

Precise considerations need to be taken for the mesh generation of the large-
scale blade-resolved simulation of wind turbines. While it is required to
implement either the sliding mesh or overset technique for rigid wind
turbines, flexible FSI simulations impose an additional complexity caused by
the blade deformation. To address this difficulty, a combination of the
overset technique with mesh morphing was implemented.

As for the CFD, hexahedral elements are suited for simplified geometries which
can guarantee a better orthogonal quality, and therefore, result in better
accuracy, whereas the polyhedral elements are suited for models with
geometrical complexities such as the blade tip and leading edge (Wang, Cao,
and Okaze, 2021). For implementing the overset solver, two fluid regions
namely background and overset, as shown in Fig. 4, were created. Hexahedral
elements were utilized for the rectangular background and for the overset
region, which encompasses the blades, polyhedral elements with a refined
boundary layer mesh were considered. Fig. 4 shows the corresponding meshes for
the fluid regions.

Star CCM+ FEM solver supports hexahedral and tetrahedral elements. The latter
was utilized for the structural model of the wind turbine, due to the
mentioned reasons. A comparison between the fluid and structural meshes of the
blade is depicted in Fig. 5.

Figure 4: Mesh of the background and overset regions  Figure 5: Polyhedral and
tetrahedral meshes for the blade in the fluid and solid regions, respectively

###  II.5 Proper Orthogonal Decomposition Method

Proper Orthogonal Decomposition is employed to extract the coherent structures
both in the spatial and temporal domain (Li  _et al._ , 2013). It should be
noted that the POD method implemented in this study is based on the work of
Taira et al. (Taira  _et al._ , 2017). In order for POD to be implemented, the
Singular Value Decomposition (SVD) method was employed owing to its robustness
against round-off errors (Lengani  _et al._ , 2014). In order to perform SVD,
the ’matrix of snapshots’ denoted as ’$\bm{X}$’ needs to be formed. The rows
of $\bm{X}$ include the value of the variable of interest corresponding to its
position in the computational domain and the columns correspond to the time
instance at which the data has been extracted. Having formed the $\bm{X}$
matrix, SVD is applied as follows:

$\displaystyle\bm{X}=\Phi_{\mathrm{U}}\Sigma\Phi_{\mathrm{V}}^{T}.$ (10)

Where $\Phi_{\mathrm{U}}$ and $\Phi_{\mathrm{V}}^{T}$ are called the left and
right eigenvectors, respectively. Also, $\Sigma$ is a diagonal matrix
containing the eigenvalues organized in descending order. Therefore, each
element of the $\Sigma$ matrix demonstrates the importance of each column of
$\Phi_{\mathrm{U}}$. It should be noted that the left eigenvector corresponds
to the spatial coherent structures which are in other words, the modes of the
flow. On the other hand, the right eigenvector indicates the evolution of each
mode in the time domain.

###  II.6 Verification and Validation

The mesh sensitivity analysis needs to be carried out for the output
parameters to asses meshing-related numerical error. To do so, coarse, medium
and fine meshes with the total number of $2.4\times 10^{6}$, $4.6\times
10^{6}$ and $5.9\times 10^{6}$ elements were created and examined. The
corresponding number of elements for each region is presented in Table 2.

As for the dimensionless wall distance index, $y^{+}$, the average value was
kept below 40 for all cases, as instructed by (Siddiqui  _et al._ , 2017),
with at least 10 prism layers (Lillahulhaq and Djanali, 2019) to capture the
near-wall phenomena and optimize the element number, simultaneously.

A comparison between edgewise deflection for three meshes is shown in Fig. 6.
There exist a negligible difference between the results of the second and
third mesh, and hence, the rest of the post-processing including POD
extraction was carried out for the second mesh.

Table 2: Number of elements for different regions of the model | Mesh #1 | Mesh #2 | Mesh #3
---|---|---|---
Solid | 305915 | 593197 | 794005
Fluid Overset | 788520 | 1436501 | 2391163
Fluid Background | 1362236 | 2561289 | 2752772
Figure 6: Comparison of the edge-wise deflection in one cycle for three meshes

To validate the flow solver the generated power of the wind turbine for the
rigid and FSI cases was compared with the results of (Dose  _et al._ , 2018),
which is shown in Fig. 7. As mentioned earlier, the oscillatory nature of FSI
results corresponds to the change of elastic properties of the blade at the
initial times (see section II.2. After approximately three revolutions (i.e.
$T=15$\mathrm{s}\mathrm{e}\mathrm{c}$$), the numerical results almost reached
that of (Dose  _et al._ , 2018).

The flap-wise deflection and tip torsion were selected to assess the FEM
solver. The average tip deflection as indicated by (Dose  _et al._ , 2018) is
$6.03$\mathrm{m}$$ which is comparable to our results, as depicted in Fig. 8.
Moreover, the tip portion through one revolution of the turbine is shown in
Fig. 9. These conclude that the FSI solver could accurately capture the blade
deformation through time.

Figure 7: Evolution of the turbine power over the simulation time for rigid and FSI simulations Figure 8: The flap-wise deflection of a blade through one rotation cycle Figure 9: The tip torsion of a blade through one rotation cycle Table 3: Comparison of the results under the rated working condition of NREL 5MW wind turbine | Flap-Wise | Edge-Wise | Rotor | Thrust
---|---|---|---|---
| Deflection (m) | Deflection (m) | Power (MW) | Force (kN)
Sabale and Gopal (Sabale and Gopal, 2019) | 4.41 | -0.57 | 4.97 | 490
Jonkman et al. (Jonkman  _et al._ , 2009) | 5.47 | -0.61 | 5.28 | 814
Yu and Kwon (Yu and Kwon, 2014) | 4.72 | -0.63 | 5.22 | 656
Ponta et al. (Ponta  _et al._ , 2016) | 3.85 | -0.56 | 5.19 | 660
Li et al (Li  _et al._ , 2020) | 4.49 | -0.57 | 5.3 | 678
Present Study | 4.02 | -0.58 | 5.01 | 636

##  III Results and discussion

(a) View normal to the Y-coordinate

(b) View normal to the Z-coordinate

Figure 10: Illustration of the planes for the wake analysis Figure 11:
Cumulative modal energy of the rigid and flexible turbines at the crosswind
(ZX) plane behind the nacelle Figure 12: Cumulative modal energy curve at
different spanwise (ZY) planes behind the turbine Figure 13: Iso-surfaces of
Q-criterion coloured by velocity magnitude Figure 14: Contours of velocity
magnitude at different streamwise planes Figure 15: POD modes of velocity at
different sections behind the turbine

(a) The crosswind (ZX) plane with $+D/2$ distance from the nacelle

(b) The streamwise plane with $+D/2$ distance from the nacelle

Figure 16: Accumulative modal energy on (a) crosswind and (b) streamwise
planes

(a) mode 1

(b) mode 2

(c) mode 3

Figure 17: POD modes on the streamwise plane on the blade tip ($+D/2$)

(a) mode 1

(b) mode 2

(c) mode 3

Figure 18: POD modes on the streamwise plane behind the nacelle

(a) mode 1

(b) mode 2

(c) mode 3

Figure 19: POD modes on the streamwise plane behind the tower ($-D/2$)

(a) mode 1

(b) mode 2

(c) mode 3

Figure 20: POD modes on the crosswind (ZX) plane behind nacelle

(a) mode 1

(b) mode 2

(c) mode 3

Figure 21: POD modes on the crosswind plane at the blade’s tip ($+D/2$)

To analyse the wake region behind the turbine a couple of planes were created
in the streamwise and spanwise directions. As illustrated in Fig. 10, multiple
planes were strategically positioned for detailed wake analysis. Along the
streamwise (XY) direction, three planes were constructed: one immediately
behind the nacelle, another at the turbine tip’s highest z-coordinate (+D/2),
and a third at the tip’s lowest z-coordinate (-D/2) to account for tower
effects. Additionally, two planes were created along the crosswind (ZX)
direction: one aligned with the nacelle and another at the farthest point in
the Y direction (+D/2) at the tip intersection. This arrangement facilitated a
comprehensive examination of the wake flow field in the streamwise direction.
For evaluating wake characteristics in the spanwise (YZ) direction, three sets
of planes were established, ranging from $0.5$\mathrm{m}$$ to
$2.5$\mathrm{m}$$. The size info of the mentioned planes is presented in Fig.
10.

The modal accumulative energy of the velocity magnitude at the streamwise
plane is shown in Fig. 11. Wake regions of rigid and flexible blades
demonstrated a similar trend and close values, especially after the second
mode. The highest difference was observed within the first mode which
corresponds to less than 2%. Moreover, the value of the first mode of the
flexible turbine was less than that of the rigid turbine due to the blade
deflection (harmonic nature of flap-wise and torsional deflections) and
corresponding flow alterations. After all, this result indicates the
unnecessity of considering blade deformation for wake analysis behind large-
scale wind turbines.

The modal cumulative energy at different spanwise planes from 0.5D to 2.5D
away from the rotor has been presented in Fig. 12. It is evident that the
farthest plane located 2.5D away from the rotor contains the lowest amount of
cumulative energy in the first mode. As a result, it is expected to observe
more flow structures at the furthest plane. In addition, the modal energy for
the planes from 0.5D to 2D demonstrated a similar trend to each other.
Therefore, the flow structures at the near plates could represent analogous
vortical characteristics. In this regard, in order to have a deeper insight
into the wake, the spatial modes of the streamwise component of velocity at
planes located 0.5D, 1.5D and 2.5D behind the rotor have been presented in
Fig. 15. The first modes as seen, mainly show the area affected by the
presence of the rotor which was also found in the work of (De Cillis  _et al._
, 2021). Additionally, this area is subtly affected by the presence of the
tower and nacelle. To illustrate, in the first mode at 0.5D presented in Fig.
15, the affected area by the energy extraction is distorted towards the lower
part due to the wake shedding of the tower whose trace can be more clearly
seen in the consecutive planes. For instance, in the first mode at 1.5D, the
wake shed from the tower becomes more pronounced indicating that moving away
from the rotor, the effect of energy extraction is outweighed by the wake
shed. Furthermore, the first mode at 1.5D vividly shows the tip vortices
emanating from the blades’ loading. The decay of this trend can be observed at
farther planes at 2.5D. Moreover, the flow structures due to the wake shed
from the tower tend to move towards the center and merge with that of the
nacelle leading to an integral vortical structure 2.5D away from the rotor. In
addition, the symmetrical shape of the first mode starts to fade away with the
increase in the distance from the rotor proving that while the work extraction
is the main characteristic of the planes near the rotor, the effects of wake
shed and tip vortices rise into importance in the farther planes.

The second mode at 0.5D from the rotor illustrates the structures stemming
from the tower wake and the tip vortices which are the most effective patterns
in the wake of the wind turbine. This observation has also been reported in
(Sheidani  _et al._ , 2023b; Iungo  _et al._ , 2015; Bastine  _et al._ , 2014;
Andersen, Sørensen, and Mikkelsen, 2014). It should be noted that with the
increase in the distance from the wind turbine, the decay of the structures at
the plane in 0.5D is observed at the same mode number. To illustrate, the flow
structures shedding behind the tower are observed at the same positions,
however, at smaller dimensions which are also observed for the tip vortices.
Moreover, as also shown for the first modes, the flow structures due to the
flow interaction with the tower begin to move towards the central part forming
an integral structure at 2.5D behind the rotor. However, it should be noted
that the central integral structure at 2.5D at the first and the second modes
decays into separate structures in the next modes. In addition, the structures
emanating from the flow interaction with the tower prove to sustain themselves
for a longer distance and through higher mode numbers. This observation
insinuates that the tower wake along with those stemming from the blades plays
a more important role in the far wake compared to the tip vortices. Also, it
is concluded that the exclusion of the tower leads to missing a major part of
the wake structures, especially at far wake positions which have also been
reported in (Premaratne, Tian, and Hu, 2022; Wang  _et al._ , 2012).

In order to investigate the effect of the tower on the downstream wake, the
velocity magnitude contours at different sections have been presented in Fig.
14. As seen, the wake behind the tower leaves a significantly more sustainable
trace compared to the wake of the blades. This is due to the fact that as also
evident in the POD modes, the blades at the rated TSR undergo a suitable angle
of attack leading to a less intense wake shed compared to the tower. As a
result, it is found that the main elements affecting the wake of the wind
turbine are the tower, tip vortex and to a lesser extent the blades
especially, at the mid and tip sections. As for the comparison of the tower
wake and the blade tip vortices which was also drawn in the spatial modes
section, the vorticity field coloured by the velocity magnitude has been
presented in Fig. 13. It is observed that the tip vortices sustain themselves
for a longer distance compared to the blade wake, while as also previously
explicated the tower causes the most significant effect on the wake region
which could be traced for a much longer distance. In addition, the merging of
the blade wake with that of the tower in the relatively farther planes behind
the rotor could be observed which was explicitly discussed in the spatial mode
section.

In order to gain a deeper understanding of the wake, the accumulative energy
and the POD modes of the streamwise component of velocity in the cross-wind
and streamwise planes both on the blade’s tip have been presented in Fig. 16.
First of all, it is evident in the figure that the accumulative energy of both
planes demonstrates quite similar trends and values proving that with three
modes almost all the dominant flow structures could be extracted. In this
regard, the wake behaviour has been deeply investigated on the streamwise (ZX)
planes at the blade tip, nacelle and the tower depicted in Fig.17, Fig.18 and
Fig.19, respectively.

As seen, the first mode in all the planes corresponds to the area affected by
the presence of the body responsible for the energy extraction as also
reported in Sheidani  _et al._ (2023c). However, as shown in Fig.17(a),
Fig.18(a) and Fig.19(a) that near the blade tip, the wake region is much more
uniform compared to the other two planes located at a lower altitude for the
first mode. This observation proves that despite the highest deflection which
is experienced at the tip in flexible blades, the adversarial effect of the
nacelle and tower is yet significantly more noticeable. In addition, the first
mode for the plane at the blade tip and the tower bear a closer resemblance
with respect to that of the nacelle. As a result, the nacelle is expected to
leave more remarkable traces in the wake compared to other parts of the wind
turbine. Also, the wake region on the blade tip plane appears to be far more
uniform than the planes corresponding to cross sections of lower aerodynamic
efficiency i.e. nacelle and tower are.

This finding could be better understood by moving on to the next modes. For
instance, the trend in the second modes for all the cross sections in spanwise
planes shown in Fig.17(b), Fig.18(b) and Fig.19(b) reveals that the dominant
structures correspond to the entrainment stemming from the wake region
collapse Sheidani  _et al._ (2023d). However, considering the cross sections
of lower aerodynamic efficiencies i.e. at the tower and nacelle, other
structures emanating from the flow separation in congruous with sources of
performance loss are observed. To illustrate, the second mode at the tower
plane demonstrates typical structures in the wake of a cylinder (Noack  _et
al._ , 2016) insinuating the importance of the tower on the wake and its modal
behaviour. Furthermore, regarding the second mode on the nacelle plane, the
structures related to the vortex shedding from the nacelle geometry are
analogous to that of the tower plane, nonetheless, at a lower intensity. In
this regard, the flow structures of the second mode at the blade tip plane are
mainly concentrated around the wake region collapse indicating the higher
performance of the tip section as the structures due to the flow separation
being of the same modal energy of the expected dominant structure i.e. wake
region collapse is not observed. Therefore, considering the effect of
flexibility of the blades does not cause the hindrance against the power
production by the blade to overweight that of the nacelle and tower.

Presented Fig.17(c), Fig.18(c) and Fig.19(c) demonstrate the third POD modes
for the blade tip, nacelle and tower sections respectively. The decay of the
structures observed previously in the second mode could be observed in all
three planes. The third mode for the blade tip and the nacelle show similar
symmetrical structures near the wake region borders which are more pronounced
near the tip. These structures come into existence due to the velocity
difference between the wake region and the area beyond it as explained in
(Sheidani  _et al._ , 2023c, d). However, the third mode of the tower plane,
as seen in Fig.19(c), shows the structures to be mostly concentrated near the
central part of the wake region. Therefore, as expected, the section with the
lowest efficiency cannot bring about a considerable velocity gradient between
the wake and the far-field regions. It should be noted that, as for the
nacelle and tower planes modes, the decay of the structures emanating from the
nacelle is still observed, while the flow structures at the blade tip are more
pronounced near the wake region collapse proving the higher efficiency of this
section.

Fig.20 and Fig.21 display the spatial POD modes on the crosswind plane,
illustrating the potential interactions and phenomena on the wake region. It
is important to highlight that two distinct planes are used for evaluation;
one intersecting with the nacelle, called the near plane, and the other
positioned farther from the nacelle at $+D/2$, referred to as the far plane.
This allows for separate assessments of their effects on the wake.

Beginning with the first mode on the near plane presented in Fig.20(a), it is
evident that the modal behaviour is quite similar to that of the spanwise
plane at the nacelle rather than the tower plane. However, the first POD mode
at the far plane exhibits a greater resemblance with the tower plane. This
proves that the effect of the nacelle on the wake could be regarded as even
more influential than the tower. The significant discrepancies between the
modal behaviour of the spanwise and the streamwise planes begin after the
second mode shown in g.20(b) and Fig.21(b). To illustrate, the second mode of
both near and far planes, despite the presence of similar structures
especially, near the point of wake region collapse, the streamwise plane
suggests more structures emanating mostly from the tower. Therefore, it is
found that the effects of the tower on the wake region could be taken into
consideration after the second mode in the near plane, specifically.

The third mode, while sustaining the main trend, shows the decay of the
structure observed in the previous mode as expected Sheidani  _et al._
(2023c). Nonetheless, the far and near planes reveal a fundamental difference
regarding the behaviour of the third mode. As seen in Fig.21(c), the main
structures on the far plane are mainly concentrated around four to five times
the wind turbine diameter illustrating the effect of wake region collapse. On
the other hand, the third mode on the near plane demonstrates other structures
than those corresponding to the decay of the second mode near the wake region.
First of all, the vortex shedding of the nacelle around the centre line is
evident in Fig.20(c) which is obviously absent on the far plane due to its
distant position from the nacelle.

Moreover, the symmetrical structures near the boundaries of the wake region
are observed on the near which is yet another remarkable difference between
the far and near plane’s third modes. It is observed in Fig.20(c) that the
presence of the symmetrical structures is more pronounced on the lower part of
the part of the wake region. This is attributed to the fact that the lower
part is prone to the flow structures ensuing from the presence of the tower,
while the upper part of the wake region is not exposed to a significant source
of vortex shedding upstream. As a result, the symmetrical structures on the
lower part are intensified resulting in an asymmetrical wake region shape on
the near plane.

##  IV Conclusion

To analyse the wake behaviour behind a full-scale onshore HAWT a fully coupled
two-way FSI simulation has been implemented. The proper orthogonal
decomposition has been applied in this matter to extract spatial and temporal
modes along with modal energy at the wake region. Results indicate a similar
modal behaviour for the rigid and flexible turbines at the wake region which
implies the negligible effect of the blade deformation on the wake
characteristics. Also, the main source of this deviation is the harmonic
nature of the blade deformation through each rotation cycle.

By observing the wake characteristics at streamwise planes behind the turbine,
it was concluded that the value of accumulative energy of the first mode
(convective mode) decreases as getting further from the turbine. As a
consequence, more flow structures in terms of local vortices and fluctuating
velocity fields exist at the furthest plane. The flow structures due to the
wake shed from the tower tend to move towards the center and merge with that
of the nacelle leading to an integral vortical structure 2.5D away from the
rotor. In addition, the symmetrical shape of the first mode starts to fade
away with the increase in the distance from the rotor proving that while the
work extraction is the main characteristic of the planes near the rotor, the
effects of wake shed and tip vortices rise into importance in the farther
planes. Also, it is concluded that the exclusion of the tower leads to missing
a major part of the wake structures, especially at far wake positions.

In summary, the main findings of the paper can be listed as follows:

  * •

A similar modal behaviour was observed for the rigid and flexible wind
turbines indicating the negligible effect of the blades’ flexibility on the
wake characteristics behind the HAWT,

  * •

Value of the first mode (convective mode) decreased as getting farther in the
wake region,

  * •

The effect of tip vortices caused the elimination of the symmetrical behaviour
of the first mode of velocity in locations more than 2D of the rotor,

  * •

The necessity of inclusion of the tower in the simulation especially in case
of examination of the far wake region.

  * •

The flexibility of the blades does not still affect the wake as much as the
tower and nacelle do.

  * •

The modes on the streamwise plane demonstrate different behaviours on the near
and the far planes.

  * •

It was proven that up to the second the mode the near and far planes show a
modal behavior analogous to that of the nacelle and tower on the spanwise
plane respectively.

Nomenclature

C | Damping Coefficient [$\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{s}$]
---|---
d | Structural Displacement [$\mathrm{m}$]
d | Stiffness Coefficient [$\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{s}^{2}$]
F | Force Vector [$\mathrm{k}\mathrm{g}\,\mathrm{m}\mathrm{/}\mathrm{s}^{2}$]
M | Mass Tensor [$\mathrm{k}\mathrm{g}$]
U | Velocity Vector [$\mathrm{m}\mathrm{/}\mathrm{s}$]
P | Pressure [$\mathrm{P}\mathrm{a}$]
S | Strain Rate Tensor [$\mathrm{1}\mathrm{/}\mathrm{s}$]
t | Time [$\mathrm{s}$]
v | Node Velocity of Interface [$\mathrm{m}\mathrm{/}\mathrm{s}$]
$\rho$ | Density [$\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$]
$\mu$ | Dynamic Viscosity [$\mathrm{P}\mathrm{a}\,\mathrm{s}$]
$\hat{n}$ | Normal Vector of Interface
$\sigma$ | Traction Forces [$\mathrm{N}$]
$\Phi_{U}$ | Left Eigenvector
$\Phi_{V}$ | Right Eigenvector
$\Sigma$ | Eigenvectors
$\theta_{p}$ | Pre-cone Angle [$\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$]
$\theta_{A}$ | Azimuth Angle [$\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$]

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