A Comprehensive Numerical Model for Horizontal Axis Wind Turbines Aeroelasticity

1 RUZGEM 2013

Proceedings of the Conference on Wind Energy Science and Technology RUZGEM 2013

October 3-4, 2013, Ankara, TURKEY

RZGM2013-24

A COMPREHENSIVE NUMERICAL MODEL FOR

HORIZONTAL AXIS WIND TURBINES AEROELASTICITY

Angelo Calabretta1

Roma Tre University Rome, Italy

Marco Molica Colella2


Luca Greco3

CNR-INSEAN Rome, Italy

Giulio Dubbioso4


Claudio Testa5


Massimo Gennaretti6


ABSTRACT

This paper deals with a computational aeroelastic tool aimed at the analysis of performance,

response and stability of horizontal axis wind turbines. It couples a nonlinear beam model

for blades structural dynamics with an unsteady state-space sectional aerodynamic load

model taking into account dynamic stall and inflow effects induced by rotor wake. An

extension of 2D static coefficients for high angles of attack is provided to characterize operations in deep stall regime. The Galerkin method is applied to the aeroelastic differential

system, with the introduction of a novel approach for the spatial integration of the additional

aerodynamic states related to wake vorticity and dynamic stall. Periodic blade responses are

determined by a harmonic balance approach and a standard eigenproblem is solved for the

stability analysis. Validation of the applied unsteady, sectional aerodynamics model is

performed through comparisons with experimental data concerning NACA0012 and S809

airfoil undergoing oscillatory pitch motion. Further, results obtained by the aeroelastic code

including dynamic stall modeling applied to the NREL/NASA Ames Phase VI two-bladed

rotor in axial flow are presented, with comparisons to available experimental and numerical

data.

NOMENCLATURE

C

DOF f

=

= =

Chord length [m]

Degrees Of Freedom Point of separation’s function

k

M

=

= , reduced frequency

Mach number

q = , non-dimensional pitch rate

s = , non dimensional time

1 PhD Student, Department of Engineering, [email protected]

2 Fellow Researcher, PhD, Department of Engineering, [email protected]

3 Researcher, CNR-INSEAN Italian Ship Model Basin, [email protected]



6 Professor, Department of Engineering, [email protected]

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t = Time [s]

VW = Free stream velocity [m/s]

x = Aerodynamic states

x’ = , aerodynamic states derivatives with respect to non-dimensional time

α = Angle of attack [rad]

= Angle of attack at ¾ chord point [rad]

ω = Motion pulsation [rad/s]

INTRODUCTION

This paper presents a comprehensive computational tool for the aeroelastic analysis of horizontal axis

wind turbines. The effectiveness of a wind turbine relies on the capacity of rotor blades to extract energy from

the incoming wind with high levels of reliability, sustainability and safety standards. Hence, multidisciplinary

tools able to examine rotor aerodynamics, aeroelasticity and control strategies with good levels of accuracy at

reduced computational costs are of great interest for wind turbines designers. This issue is particularly crucial for

very large wind turbines and is the subject of many research projects worldwide (see e.g. the EU-FP6 project

UPWIND in [1]).

Key issues for the successful analysis and design of wind turbines are rotor aerodynamics and structural

dynamics modeling, along with the strategies adopted for the spatial/temporal integration of the aeroelastic

system [2]. In order to predict rotor operative equilibrium conditions and stability boundaries, tools able to take

into account rotor unsteady operating conditions induced by both onset wind variations and structural dynamics, are mandatory. The structure of currently applied wind turbine blades is quite stiff, and thus almost all blade

models are based on linear beam theories. However, such an approach cannot take into account blade

geometrical non-linearities. Moreover, as wind turbines size increases and blades get more flexible, the use of

non-linear structural models should be considered.

Concerning aerodynamic simulation, state-of-the-art aeroelastic codes for wind turbine design relies on

the well established engineering methods based on the Blade Element Momentum Theory (BEMT) technique

[3]. It is enhanced through tuned-up corrections such as blade tip flow corrections [4], dynamic inflow [5] and

dynamic stall models [6,7] to take into account three-dimensional flow, spatial/temporal variability of wind

speed and direction, transient loads induced by unsteady wake effects, flow separation and centrifugal effects.

More recently, CFD tools based on RANSE, DES and LES aerodynamics solvers have been proposed to achieve

a physically consistent description of viscosity and turbulence effects on wind turbines blades. Nevertheless the

high computational costs limit their use to the prediction of isolated wind turbines performance and to the

detailed analysis of blades local flow. Thus their application to the aeroelastic analysis for practical preliminary

design use is still a future scenario [8]. Alternative to BEMT models, three-dimensional inviscid-flow methods

allow to describe wind turbines flow at still limited computational costs [9,10]. In this framework, solvers based

on a Boundary Element Method (BEM) may efficiently predict rotor loads and blades wake induced inflow. This

kind of prediction tools have been extensively validated in the past by some of the authors for the analysis of rotorcraft [11] and marine propellers [12], whereas applications to wind and tidal turbines are more recent [13]. A

drawback of these methods is that viscosity effects on blade loads can be only approximately described, thus for

aeroelastic and performance analysis concerning off-design operations, the inclusion of further modeling to

account for unsteady flow separation is necessary. From the structural point of view, state-of-the-art aeroelastic

tools often relies on Finite Element Method (FEM) approaches, which allow the description of complex blades

deformation states. Nevertheless, in the framework of design and optimization applications, modal (spectral)

approaches provide a computationally efficient alternative way to predict the structural dynamics of turbines

within aeroelastic solution tools.

The present work proposes a rotor aeroelastic model where generalized aerodynamic loads are

evaluated through a sectional formulation taking into account unsteady effects due to both shed vorticity and

dynamic stall onset [6,7]. Three-dimensional wake inflow correction is provided by a BEM solver for

incompressible, potential, attached flows. An extension of 2D static coefficients at high blade angles of attack is

presented to characterize operations in deep stall regime. Centrifugal effects are considered by Snel’s correction

[14]. This unsteady aerodynamic model is coupled with a nonlinear, integro-differential formulation for the

structural dynamics analysis of bending-torsional blades subject to moderate deformations. A Galerkin approach

for the spatial integration of the aeroelastic system is applied, using a novel technique for the spectral description

of the additional states appearing in the sectional aerodynamics model to describe wake vorticity and dynamic stall. The time response may be evaluated either by a harmonic-balance technique (suitable for steady operative

condition) or by a time-marching solution algorithm.

In the following, first, structural and aerodynamics models are briefly outlined, then the techniques

applied for numerical integration of the fully coupled aeroelastic system are illustrated. Furthermore, some

results concerning validation performed for the unsteady sectional dynamic stall model with experimental data

(carried out on NACA0012 and S809 airfoils) are presented. Finally, a detailed aeroelastic analysis of the

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NREL/NASA Ames Phase VI two-bladed rotor in axial flow [15] is shown, along with comparisons with

available experimental and numerical data.

ROTOR AEROELASTIC MODELING

Rotor Structural Model Beam-like model are applied to describe the structural dynamics of rotor blades; these are based on the

nonlinear bending-torsion formulation described in [16], that is valid for straight, slender, homogenous,

isotropic, nonuniform, twisted blades, undergoing moderate displacements. The resulting structural operator

consists of a set of coupled nonlinear differential equations governing the bending of the elastic axis and the

blade torsion [17]. If present, effects of blade pre-cone angle, hinge offset, torque offset and mass offset are

included in the model.

Combining this structural dynamics model with a suitable description of the distributed aerodynamic

loads yields the aeroelastic formulation that is herein used for rotor performance, response and stability analyses.

Unsteady Rotor Aerodynamics Model The rotor aerodynamics model is based on the state-space formulation proposed by Beddoes-Leishman

(B-L model in the following) [6, 7, 18] to predict unsteady compressible aerodynamic loads generated on an

airfoil undergoing simple oscillatory pitch and plunge motion. This numerical model (used for industrial

applications) is based on experimental data obtained at low frequencies, so turbulent phenomena (having higher

frequencies) are not considered. This model consists of a set of 12 aerodynamic states governed by ordinary

differential equations (ODE) and describing, at an engineering level of approximation, the different physical

aspects involved in 2D unsteady aerodynamics. Eight states provide the solution for linear unsteady attached

potential flows, three states the nonlinear effects of trailing edge separation and a single state the leading edge

flow separation characterizing dynamic stall events.

Denoting with , vector state-space B-L model can be written in the form

and the resulting aerodynamic forces are

(1)

where CN and CC are normal and chord force coefficients, CM is the pitching moment coefficient, is the angle

of attack and q is the pitch rate.

Dealing with attached flow conditions, and take into account the effects of shed vorticity from trailing edge, thus being fully equivalent to the circulatory terms of the Theodorsen theory [19], as shown in

Fig.1 where the hystheresis loop of normal force coefficient is presented. The first order ODEs governing these

states are forced by the effective angle of attack at airfoil three-quarters chord point ( ) which can be

expressed as a function of the downwash ( ) as follows

(2)

where VW is the free-stream velocity.

Figure 1. Normal force coefficient on a NACA0012 airfoil undergoing pitch motion (k=0.099):

comparison of B-L and Theodorsen formulations for unsteady attached flow

The next six states reproduce non circulatory loads accounting for flow compressibility. This aspect is

crucial: at very low values of free stream velocity this formulation tends to provide singular values for

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aerodynamic loads, due to Mach number appearing at denominator of some terms of Eq. (1) [18]. Under these

conditions, the present formulation may be substituted by a simplified one for incompressible flows where

unsteady loads are described by two additional aerodynamic states providing circulatory loads added to the non

circulatory terms described by the Theodorsen theory. The analysis of normal force coefficient hysteresis loop

for a NACA0012 airfoil undergoing purely harmonic pitch motion between 0° and 20° (Fig.2) shows that for

the original B-L model converges, whereas for lower Mach it is unstable.

(a) M = 0.3 (b) M = 0.0015 (c) M = 0.001

Figure 2. Comparison between B-L model for compressible and incompressible flow

As already mentioned, separated flow conditions are considered through the remaining four aerodynamic

states of the B-L model. Three aerodynamic states are introduced to simulate the effects of trailing edge

separation. Specifically, for an airfoil undergoing unsteady motion such that the angle of attack is close to or

greater than the static stall angle, flow separation onset is modeled through a time lag between separated and

fully attached force coefficients (state . This leads to the definition of a quasi-steady separation point characterized by a time lag with respect to the instantaneous angle of incidence. Progressive flow separation at

trailing edge is governed by the dynamics of the boundary layer which leads to a time lag of the unsteady

separation point (states and ). The last aerodynamic state represents leading edge separation (state )

and models the dynamic stall effect on airfoil unsteady loads [6,18]; this phenomenon occurs when the leading

edge pressure reaches a critical value causing the shedding of a vortex travelling over the airfoil. The lift increase

effect of the travelling vortex on airfoil loads is shown in Fig.3 where the normal force coefficient for a

NACA0012 airfoil undergoing harmonic pitch motion with amplitude 10° around a mean value of 10° is

predicted by B-L model with and without the activation of the state .

Figure 3. Comparison between B-L model and experimental data (k=0.099)

Dealing with rotating blades aerodynamics, it is necessary to extend 2D static aerodynamic coefficients

between the range -180° < α < +180° for operations at high angles of attack that occur in deep stall regimes,

where no experimental data are available [20]; thus, Viterna-Corrigan model [20], that predicts lift and drag

coefficients beyond 2D static polars, is used:

(3)

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where

(4)

is the angle of attack where the maximum of lift coefficient is observed, and denote lift and

drag coefficients at stall angle , respectively and is the maximum drag coefficient (at α = 90°). Within

the framework of Viterna-Corrigan stall model, depends on the aspect ratio λ (defined as the ratio

between blade span and mean chord) as follows:

(5)

The case λ > 50 may be interpreted as an infinite blade whose aerodynamic properties are fully described

by 2D aerodynamic coefficients, whereas the case λ 50 also considers 3D (finite span) effects.

It has been observed that, for rotating blades, stall regime starts at higher angles of attack as compared

with wind tunnel measurements on static aerofoils [20]. This has been attributed to centrifugal pumping effects,

reducing the adverse pressure gradient that leads to aerofoil stall. This effect is more evident at the blade root than blade tip. A widely used empirical description of this effects based on the laminar boundary layer theory is

given by Snel [14].

The B-L model is implicitly two-dimensional. Dealing with rotating blades aerodynamics, an

enhancement of this approach is herein proposed: to overcome the limitations in the description of unsteady

attached flow and to take into account flow three-dimensional effects, a modification of the boundary condition

at the airfoil three-quarters chord (downwash) point is introduced. In detail, the perturbation velocity induced by

the rotor wake is evaluated at the rear neutral point of each blade section by a Boundary Element Method. This

approach is based on a boundary integral formulation for potential flows around lifting bodies which is suitable

to describe strong three-dimensional and unsteady flow effects and blade wake shape evolution [11,12,13]. Note

that the shed vorticity generated by blade sections is already taken into account by the unsteady airfoil theory,

thus its contribution to downwash shall be neglected. Following [12], once the solution in terms of velocity

potential on the blade surface is evaluated by the BEM approach, wake induced inflow ( ) can be determined. Finally, the downwash induced by the wake on blade sections is combined with blade motion to obtain the

effective angle of attack as follows

(6)

where , , is the rotor angular velocity and is the blade local twist (Fig.4).

Figure 4. Effective angle of attack at three-quarter chord point

Integration of the Turbine Aeroelastic System Coupling the equations governing blade structural dynamics with the unsteady aerodynamic loads model

yields the aeroelastic integro-partial differential equations to be integrated. The space integration is performed

through the Galerkin approach [21].

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For this purpose there is the need of knowing forcing terms distribution in the added states equations. The

method here applied consists of projecting them on the respective trial functions, that are the same as those used

for the structure. Applying the principle of superposition, added states equation ( - ) are decomposed into

bending (in-plane, v and out-of-plane, w since they have same trial function), torsion () and rigid motion contributions, each forced by the corresponding distribution of the forcing terms:

(7)

where

(8)

In this way, all terms are presents and appear only once in the equations and coupling of the aerodynamic

Jacobian matrix with the structural one is straightforward [21]. This approach allows to express dynamic stall

variables through the same set of functions used for the structure.

Remaining states equations ( - ), piecewise continuous due to the action of the separation point

function, f (equal to 1 for attached flow sections and 0 for fully separated flow sections), are solved differently.

Along the rotor blade span a discrete number of characteristic sections are defined, and on each of them the

equations for - are independently integrated; an interpolation between the different sections yields the

radial distribution of states - .

The aeroelastic system can be solved by using the harmonic balance approach as described in [21].

Alternatively, a time marching solution can be obtained by a standard Newmark-β algorithm.

RESULTS

The proposed aeroelastic model is applied to the analysis of the NREL Phase VI wind turbine tested at

NASA Ames wind tunnel. This experimental dataset is described in [15] and, despite the design constraints due

to scientific needs, it is representative of modern commercial wind turbines. In detail, a 10.58 m diameter two-

bladed turbine rotor in axial uniform inflow is considered (upwind S configuration). Blade shape is characterized

by a S809 airfoil with linear chord and non-linear twist spanwise distributions (Fig.5). Moreover, a cylindrical

and a linear transition zones characterize the blade up to 25% of the radius. A constant pitch value of 3° is

considered and the blade cone angle is set to 0°. Further details on the rotor mechanical and geometrical

characteristics can be found in [15] and [22].

Figure 5. CAD rendering of NREL Wind Turbine blade

In the following, validation of the B-L unsteady aerodynamics model is firstly addressed by comparing

present computations with available experimental and numerical data. Specifically, a NACA0012 and the S809

airfoils are considered. Then, a thorough analysis of NREL Phase VI wind turbine is performed.

Validation of B-L model applied to 2D airfoil The B-L model is herein validated through the analysis of a 2D airfoil undergoing harmonic pitch motion

about the quarter-chord point. Numerical results are compared to experimental data [18,23]. Figure 6 presents the

aerodynamic normal force and moment coefficients hysteresis loops for the NACA0012 airfoil [18] for the

motion law where k=0.099. Figure 7 shows similar results for the S809 airfoil [23]

oscillating about a mean angle of 7°, with amplitude of 10° and k=0.077. For both test cases, numerical

predictions show that the proposed implementation of the B-L model reproduces CN and CM magnitude with

reasonable accuracy in the considered range of α. Slight overestimation on CN is observed in the reattachment

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phase during the downstroke for the normal force coefficient, whereas major discrepancies appear on CM, even if

the general quality of the numerical data is still reasonable. Greater differences in the hystheresis loop of the

moment coefficient are evidenced on the S809 aerofoil. A devoted tailoring of the semi-empirical coefficients

appearing in the aerodynamic formulation should improve the numerical predictions: the present analysis has

been performed by using the same set of parameters provided by [18] for the NACA airfoil.

Figure 6. Normal force and moment coefficients for the NACA0012 airfoil

Figure 7. Normal force and moment coefficients for the S809 airfoil

3D BEM wake inflow In this section, the wake inflow determined by the BEM solver to evaluate blade effective angle of attack

is examined. Specifically, two different operating conditions are considered = 5 m/s and 8 m/s. For these conditions, Table 1 summarizes BEM predicted turbine thrust and torque along with the corresponding measured

data and the relative percentage errors.

Table 1. Comparison between numerical and experimental data for turbine thrust and torque

VW [m/s]

Numerical data (BEM) Experimental data ε Thrust

ε Torque Thrust [N] Torque [N•m] Thrust [N] Torque [N•m]

5 654 275 681 292 4 % 6 %

8 1475 1137 1352 1103 9 % 3 %

The influence of three-dimensional effects on blade loading can be evaluated by the analysis of blade

circulation, which is directly correlated to the intensity of wake vortices. To this aim, Figure 8 shows the

spanwise distribution of circulation for the considered operating conditions. Excluding the irregular trend in the

cylindrical and transition zone of the blade (r/R < 0. 25), blade circulation tends to increase smoothly along the

span highlighting 3D flow effects that rapidly increase near the tip. Moreover, the variation of circulation along the blade increases with wind velocity (and hence with blade loading).

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Figure 8. Spanwise distribution of circulation for two different operating conditions

This kind of blade behavior is responsible for the distribution of wake induced axial velocity shown in

Fig. 9: as expected, at higher blade loading greater values of are predicted. Finally, the effective angle of

attack, evaluated by Eq. 6, is shown in Fig. 10.

Figure 9. Axial component of rotor wake induced velocity Figure 10. Blade effective angle of attack

Turbine performance and stability analysis The proposed aerodynamic model is herein used to predict rotor performance. First, a rigid blade

configuration is considered. A crucial issue when wake induced velocity coming from a BEM approach is used

to enhance a 2D aerodynamic solver, is the range of wind speed (and thus of blade sections angle of attack) for

which values can be considered reliable. As a matter of fact, as the wind speed increases, larger portions of the

blades start to face flow separation and the BEM approach provides overestimated airloads (Fig.11) resulting in

unphysical high values of induced velocity. The analysis of BEM predicted rotor performance (and of blade

sections angle of attack, Fig.10) shows that fully attached conditions are achieved only when the wind speed is

less than 8 m/s.

In the present approach, for blade sections experiencing an angle of attack lower than 20°, the sectional aerodynamics formulation is corrected by the wake inflow evaluated through a BEM approach; for blade

sections where the angle of attack exceeds 20°, the 2D aerodynamics coefficients are given by Viterna-Corrigan

formulas for a flat-plate, with the addition of a centrifugal pumping variation according to Snel’s model.

Figure 11. Turbine thrust (left) and torque (right): predictions by BEM compared to experiments

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Once the above described tuning process is performed, the proposed model is then used to evaluate rotor

performance. Figure 12 shows the comparison of the proposed approach with measured data and numerical

computations by the code PROPID©

[24], which is based on a Blade Element Momentum Theory approach and

includes a 3D post-stall correction to airfoil behavior. This figure concerns thrust and torque generated by the

rotor as function of the wind velocity. Compared to BEM solution, for wind speeds above 8 m/s, the present

approach provides reasonable results and closer to experimental ones. In particular, regarding thrust, solution deviates from experimental data in the same way as that obtained from code PROPID, while torque prediction

reproduces experimental data in a better way.

Figure 12. Turbine thrust (left) and torque (right): predictions by present model compared to

experiments and numerical data by PROPID© code

Next, blade elasticity effects have been introduced in order to assess their influence on predicted rotor

performance. The corresponding computed thrust and torque are compared with those obtained under rigid-blade

rotor assumption in Fig.13. It demonstrates that, under the examined conditions, the predicted rotor performance

is barely affected by blade deformation effects. This is a consequence of blade stiffness characteristics provided

in [15].

Figure 13. Turbine thrust (left) and torque (right): influence of blade elasticity

Figure 14. Rotor aeroelastic eigenvalues (zoom of the low-damping region is shown in the right picture)

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Then, the aeroelastic stability of the turbine has been analyzed. To this aim, first the linearized time-

invariant state-space approximation of the small-perturbation rotor aeroelastic equations (about a reference trim

configuration identified through the harmonic balance approach) has been derived, and then the corresponding

eigenproblem has been solved, for a given set of wind speeds. Figure 14 presents the turbine aeroelastic

eigenvalues predicted in the most critical condition, that corresponds to the maximum wind speed (Vw=25m/s). A

stable response to perturbation is observed (as detailed by the zoom in the right hand side picture), with high modal frequencies due to blade stiffness (indeed, small trim blade tip deformations have been evaluated for

5m/s≤ Vw ≤25m/s, as shown in Fig.15). In Fig.14 the zero-frequency eigenvalues are related to the additional

aerodynamic states, whereas the remaining ones concern the first flap, lag and torsion modes (from lower to

higher frequency).

Figure 15. Blade tip deflections for the reference trim configuration: influence of wind speed

Finally, the time marching response to a perturbation of the complete, nonlinear aeroelastic system has

been carried out. In order to have a comprehensive tool for turbines aeroelastic analysis, such a capability is of

great importance indeed, in that operating conditions might be such to generate non periodic rotor response,

strongly affected by the contribution from nonlinear terms (as it might happen because of gust occurrence or non

uniform wind distribution). Moreover, the parameters appearing in the equations governing some of the

additional aerodynamic states of the B-L model, are characterized by discrete variations. Thus, the transition

from attached to separated conditions may be simulated only through a time marching solution. Figure 16

presents the turbine response to an arbitrary perturbation in terms of the bending and rigid components of the

first additional state. The rigid component, uncoupled with respect to the structural dynamics, shows a first-order

behavior, whereas the bending component shows a damped oscillatory response. The application of the

logarithmic decrement rule to the time signals allows the identification of the damping coefficient that may be

compared with the one determined through the eigenvalue analysis. Table 2 compares the perturbed response

dynamics characteristics as determined from the eigenanalysis and from the time marching solution for state x1. Although not coincident, these appears to be in good agreement. However, note that for the examined test case,

only trailing edge separation occurs, while leading-edge one does not appear: the presence of leading edge

separation would increase the difference between the results from the two stability prediction strategies, thus

making the application of the time marching approach necessary for system stability assessment.

Figure 16. Bending and rigid components time marching solution of added state x1

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Table 2. Frequency and damping results of the fully coupled system

Eigenvalues analysis Post processing of time marching solution

Frequency Damping Frequency Damping

=9.13

-2 +11

CONCLUSIONS

A fully coupled aeroelastic tool for the analysis of horizontal axis wind turbines in axial flow is

presented. Specific focus is posed to the rotor aerodynamics model which combines the B-L dynamic stall model

with the induced inflow velocity provided by a three-dimensional Boundary Element Method. A Galerkin

approach is applied for the spatial integration of the aeroelastic system, using a novel approach for the spectral

description of the additional states appearing in the sectional aerodynamic model to describe wake vorticity and

dynamic stall. The numerical analysis conducted has led to the following results:

- The B-L aerodynamic model, as expected, is fully adequate in predicting the unsteady airloads acting on the

two pitching airfoils considered for validation;

- The correction of the 2D aerodynamic solver through a 3D BEM approach to evaluate rotor wake induced

velocity due to trailed vorticity yields quantitatively reasonable turbine performance predictions in the

considered wind range, especially in terms of generated torque. Modeling of blades sections transition from

attached to separated flow conditions remains a critical issue when dealing with performance predictions;

- For the present test case, blades elasticity and deformation do not affect turbine performance. Moreover, the

stability analysis, performed both through a time marching solution of the complete non linear aeroelastic system and by a linearized eigenvalues approach, has indicated a stable behavior of the rotor in the

considered wind range.

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A Comprehensive Numerical Model for Horizontal Axis Wind Turbines Aeroelasticity

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