fsi of wt airfoils

Fluid Structure Interaction of Wind Turbine Airfoils

Ouahiba Guerri1*, Aziz Hamdouni2 and Anas Sakout2

1.Centre de Développement des Energies Renouvelables BP 62, Bouzaréah, CP 16 340,Alger,Algeria.Email: [email protected]. LEPTIAB, Départ. Génie Civil et Mécanique, Pôle Sciences et Technologies, Univ. de La Rochelle,Av. Michel Crépeau, 17042, La Rochelle, Cedex 01, France. Email: [email protected] [email protected] *Corresponding author

WIND ENGINEERING VOLUME 32, NO. 6, 2008 PP 539–557 539

ABSTRACTFlow induced vibrations of two airfoils used in wind turbine blades are investigated by a

strong coupled fluid structure interaction approach. The method is based on a general

Computational Fluid Dynamics (CFD) code that solves the Navier-Stokes equations defined

in Arbitrary Lagrangian Eulerian (ALE) coordinates by a finite volume method. A

straightforward technique is implemented in a user subroutine for the coupling of the CFD

code to a structural dynamics program to determine the airfoil displacements due to the

aerodynamics forces and for updating the grid at each time step. Simulations are carried out

for a free pitch oscillating airfoil and for a combined pitch and vertical oscillating airfoil.

Beforehand, the problem of the flow around a forced pitch oscillating airfoil is considered to

check the reliability of the moving mesh technique and the CFD computations.

All computations are performed in 2D, incompressible and low Reynolds number flows.

KKeeyywwoorrddss:: Aeroelasticity. Unsteady aerodynamics. Incompressible Navier-Stokes

equations. ALE formulation. Moving mesh.

1. INTRODUCTIONWind turbines are subjected to a hard environment such as (i) the atmospheric turbulence,

(ii) the ground boundary layer, (iii) the rapid variations in wind speed and direction and (iv)

the tower shadow for downwind turbine. This stochastic inflow, associated to the architecture

of the rotor leads to a 3D unsteady aerodynamics and dynamic stall [1, 2]. The dynamic stall

results in fluctuating blade force and blade oscillations, known as aeroelastic phenomena.

Problems of aeroelastic stability can be encountered on the new large wind turbine blades as

well as on the rotating wind turbine blades and on parked wind turbine blades at high wind

speeds [3].

Most of the wind turbine aeroelastic analyses were performed using engineering methods

where the blade forces were often computed by the BEM theory and the dynamic stall was

modeled with empirical models such as the ONERA or the Beddoes-Leishman models. The

dynamic response of the wind turbine was thereafter determined using structural

computational tools [3-5]. Displacements of the structure were then computed after

convergence (or partial convergence) of the aerodynamic part of the aeroelastic code. The

problems of viscous fluid flow and elastic body deformation were then studied separately.

These weak coupled approaches are limited to small deformations and low non-linearity. In

most aeroelastic problems, the interaction between these two media and the modeling of the

unsteady aerodynamics and dynamic stall are of great importance for the aeroelastic

stability study. We need then to perform aeroelastic computations by means of a strong

coupled approach with Fluid Structure Interaction (FSI) techniques where the aerodynamic

forces are computed from the solution of the time-accurate Navier-Stokes equations and used

for the solution of the rotor dynamic equations to determine the response of the structure at

each time step.

FSI techniques have been widely used in many industrial problems [6, 7], for aerospace

applications where most papers are applied to compressible flows [8]. It has also been shown

that compressibility effect plays a significant role in aeroelastic stability limits. As wind

turbines operate in an incompressible environment, specific studies have to be performed for

wind turbine blades. To the author’s knowledge, the main contribution with an FSI approach

applied to wind turbine blades has been carried out in the frame of the European project

KnowBlade where Ellipsys3D, an in house Computational Fluid Dynamics (CFD) code, and a

structural code have been used to simulate flap-lead/lag vibrations of a wind turbine blade

[9]. Recently, Svacek et al. [10] presented the study of the classical flutter by a FSI approach

where the incompressible fluid equations were solved by the finite element method using an

Arbitrary Lagrangian Eulerian (ALE) formulation of the Navier Stokes equations.

In this paper, the problem of classical flutter phenomenon deals with a strong coupled

method where the dynamic response of the wind turbine blade is determined in time accurate

sequences. The aim of this study is to propose a straightforward technique to be used with

general computer tools. The numerical simulations are performed using StarCD, an industrial

CFD code which is coupled to a computational structural program for the solution of the

dynamic equations of the blade. A technique is implemented in a user subroutine for the

coupling of both codes and for updating the grid. The viscous flow induced vibrations on two

airfoils are then simulated. First of all, the moving mesh technique is applied to a forced

oscillating airfoil to check the reliability of the CFD computations. The hysteresis loops of the

dynamic stall phenomenon are then highlighted. All computations are carried in a 2D,

incompressible and laminar flow. In addition, for one case study, comparison is made between

laminar and turbulent computations carried out with the solution of the Reynolds Averaged

Navier-Stokes equations (RANS).

2. NUMERICAL APPROACHThe fluid flow equations, the dynamic equations of the oscillating airfoil, the coupling

approach and the equations describing the moving mesh technique are summarized in the

following.

2.1. The fluid equationsThe fluid governing equations are described in ALE coordinates where the grid is considered

as a referential frame moving with an arbitrary velocity. A detailed description of the

derivation of the ALE formulation can be found in the literature (see e.g. Refs. 11 or 12). There,

the expression of the Navier-Stokes equations defined in ALE coordinates is given.

Let Ω f ∈ R2 a 2D spatial domain occupied by the fluid. The incompressible fluid equations

in ALE form are written as, in tensor notation:

(1) ∂∂

=u

xi

i

0

540 FLUID STRUCTURE INTERACTION OF WIND TURBINE AIRFOILS

(2)

where t is the time, xi the Cartesian coordinate of a point of Ω f, ui the absolute velocity

component in the direction i, ucj the velocity of the moving grid, p the pressure, ρ the fluid

density, g the determinant of the metric tensor and τ ij the stress tensor components defined

as in the case of laminar flows,

(3)

with µ, the molecular dynamic fluid viscosity. For turbulent flows, ui and p denote their

ensemble averaged values and the stress tensor τ ij is expressed as:

(4)

where u′i denotes the fluctuating part of the instantaneous velocity and the over bar

denotes the ensemble averaging process. The term ρ · u′i u′j is the additional Reynolds stresses

due to turbulence that are described here by the SST k-ω turbulence model of Menter [13]. This

model is a combination of the k-ε and k-ω models which uses the k-ω model near the wall but

switches through a function to a k-ε model when away from the wall, closer to the upper limit

of the boundary layer. Consequently, the k-ε equation is transformed into a k-ω formulation.

The SST k-ω model of Menter has shown to give results for flow with strong adverse pressure

gradients that are far higher than those obtained with either the original k-ε model and its

variants or the k-ω model [14, 15].

When the RANS equations are written in ALE formulation, the terms ∂uik/∂xi and ∂uik/ωxi of

the transport equations for the turbulence energy k and the turbulent specific dissipation ωare expressed as:

and (5)

GGeeoommeettrriicc ccoonnsseerrvvaattiioonn llaaww.. An additional equation called the Space - Conservation Law

(SCL) is enforced to avoid errors that could be induced by the moving grid cells:

(6)

where V is the cell volume, S the surface enclosing the volume V and n is the surface vector.

This equation describes the conservation of space when the control volume changes its shape

and/or position with time [16].

BBoouunnddaarryy ccoonnddiittiioonnss.. An overview of the computational domain is shown in Fig. 1. The

boundary conditions are defined as follows. The inflow condition is applied to the West

boundary having defined the free-stream velocity. The outflow condition is defined at the East

d

dtdV u ndS

V

cj

S

∫ ∫− ⋅ = 0

∂ −( )∂

u u

xi ci

i

ω∂ −( )∂

u u k

xi ci

i

τ µ ρij

i

j

j

ii j

u

x

u

xu u=

∂∂

+∂

∂

− ⋅ ' '

τ µij

i

j

j

i

u

x

u

x=

∂∂

+∂

∂

ρ ρτ1

g

g u

t

u u u

x x

p

x

i i j cj

j

ij

j

∂( )∂

+∂ −( )( )

∂−

∂( )∂

= −∂∂

ii

WIND ENGINEERING VOLUME 32, NO. 6, 2008 541

boundary. A symmetry condition is applied to the South and the North boundaries. At the

airfoil surface, the fluid velocity is equal to the airfoil speed:

(7)

where u is the fluid velocity adjacent to the airfoil face and vb is the velocity of the airfoil.

Figure 1: Block topology

CCoommppuuttaattiioonn ooff tthhee aaeerrooddyynnaammiicc ffoorrcceess.. The resultant of the aerodynamic force is

obtained from the solution of the Navier-Stokes equations by integrating the pressure p and

the shear stress τw over the blade surface S :

(8)

where:

• n→

is the outward-pointing vector normal to the airfoil surface

• N is the number of the airfoil faces

• pi, τwi and δSi are the wall cell pressure, shear stress and elementary area

respectively

• n→

i is the outward-pointing vector normal to δSi

The drag force Fx and the lift force Fy are computed as:

(9)

where X→

and Y→

are the unit vectors in the parallel and vertical directions of the flow,

respectively. The lift and drag coefficients CL and CD are then derived with:

F F Yy i

i

N

= ⋅=

∑

1

F F XX i

i

N

= ⋅=

∑

1

F p n n dS p n n S

wS i i wi i ii

N

= − + ⋅( ) = − + ⋅( )∫=

τ τ δ1

∑∑

u vb

=


and (10)

where Aref is the plan form area of the blade.

AAllggoorriitthhmm aanndd ddiissccrreettiissaattiioonn sscchheemmeess.. The Navier Stokes equations are solved by a finite

volume method. The PISO algorithm is applied for the solution of the coupled pressure

velocity equations. The first order differencing UPWIND scheme is used for the discretisation

of the convection-diffusion terms. Higher order schemes are more accurate; however

instability problems can be encountered when the Peclet number is too high. As it is important

to distinguish the numerical instabilities of the solution from the unsteady flow associated

with the airfoil motions, the computations are performed with the first order UPWIND

scheme, less accurate but more stable. The temporal discretisation is performed with the

implicit θ - scheme with θ = 0.8.

2.2. The dynamic equationsTwo ways are applied for the simulation of the flow around the vibrating airfoils: (i) forced

oscillations and (ii) free oscillations.

FFoorrcceedd oosscciillllaattiinngg aaiirrffooiill:: The instantaneous angle of attack of the airfoil in forced pitch

oscillations is given by the relation:

(11)

where α0 is the mean angle of attack of the oscillating airfoil, αm is the oscillation amplitude

and ω = 2πf with f the frequency of oscillation.

As the airfoil displacements are imposed, the coupling procedure is reduced to the

updating of the computational grid in accordance with the airfoil motion at each time step.

FFrreeee oosscciillllaattiinngg aaiirrffooiill:: It is assumed that the airfoil is an elastic body and the airfoil

elasticity is described by a simple two degrees of freedom model. The dynamic equations of

the airfoil are defined from the Lagrange equations. In the case of an airfoil in pitch oscillations

(or torsion around the elastic axis EO) and in flap-wise oscillations (or vibrations in the

vertical direction) as drawn in Fig. 2, the non linear equations that describe the airfoil

oscillations are written as:

(12a)

(12b)

where α, y, α.

, y., α

..and y

..are the airfoil rotational and vertical displacement, velocity and

acceleration respectively; Fy is the vertical component of the aerodynamic force; M0 the

torsion moment; m is the airfoil mass; Sα = –mrG with rG the distance between the centre of

gravity and the elastic axis EO; Jα the inertia moment around the elastic axis EO, kα and ky are

the torsion and bending stiffness; Cα and Cy are defined as:

(13a)

(13b)C my y y

= 2 ω ζ

C Jα α α αω ζ= 2

J C S y k M α α αα α α+ + + =0α

m y C y S k y Fy y y

+ + + = −α α

α α α ω( ) sin( )t tm

= +0

CF

A VDX

ref

=⋅ ⋅ ∞1 2 2ρ

CF

A VLY

ref

=⋅ ⋅ ∞1 2 2ρ


Figure 2: The airfoil scheme

with ζα and ζy the damping coefficients, and are the airfoil

natural frequencies.

Time integration of the dynamic equations. The algorithm used for the solution of the

dynamic equations is based on a trapezoidal time advancement scheme. Equations (12a) and

(12b) that describe the airfoil oscillations are first re-written in a matricial form:

(14)

where q is a generalized coordinate, q.

and q..

the first and second derivatives with respect

to time. M, C and K are the mass, damping and stiffness matrices respectively and R the

resultant force matrices. Then, given ∆t the time step, the terms q.

and q..

are discretised using

a trapezoidal time advancement scheme:

(15)

(16)

The substitution of these discretised forms of q.

and q..

in the equation (14) leads to the

applied algorithm written as, at each time step:

(17)

(18)

(19)

All computations are started with an initial displacement and zero vibration velocity.

R R Mt

qt

q q Ct

q qen n n n n n n+ = + + +

+ +1

2

4 4 2

∆ ∆ ∆

Kt

MtC K

e= + +

4 22∆ ∆

K q Re

nen⋅ =+ +1 1

qt

q q qn n n n+ += −( )−1 12

∆

qt

q q qn n n n+ += −( )−1 12

∆

M q C q K q R + + =

ωy y

k m=ωα α α= k J


2.3. The coupling schemeAt each time step, the angular and vertical displacements of the airfoil surface are derived

from the solution of the dynamic equations with:

(20)

(21)

where the superscripts n and n+1 refer to times tn and tn+1 respectively. ∆ α and ∆ y are then

used to move the mesh nodes.

This explicit coupling procedure is first order accurate scheme and it does not conserve

energy at the moving fluid - solid interface. It is well known that implicit algorithms are more

suitable [11]. However, implicit schemes are more computationally expensive and heavy to

implement and to integrate in a commercial CFD code. Therefore, they are not used in this

study.

2.4. The moving mesh techniqueThe airfoil is located at the centre of an O-H block structured grid (Fig. 1) that allows a meshing

technique easy to implement. The moving mesh method is based on algebraic interpolations.

Given:

i. Rni and θn

i the cylindrical coordinates of the vertices Vi at time tn;

ii. xni and y

ni the related Cartesian coordinates;

iii. ∆ α and ∆ y the airfoil displacements that are calculated with the solution of the

dynamic equations;

The displacement of the vertices is performed as follow:

• The vertices of the circular sub-domain of radius R1 move at the airfoil velocity:

(22)

(23)

(24)

• For the vertices of the annular sub-domain delimited by the circles of radius R1 and

R2, the applied relations are:

(25)

(26)

where:

(27)

(28)y R yn n n1

11 1

1+ += ( )+sin θ ∆

θ θ αin

in i

R R

R R+ = + −

−−

1 1

2 1

1 ∆

y R y yNi

nin

in n

R

+− −

+ += ( )+ −( )11 1

12 1

1 1sin θ

x Rin

in

in+ += ( )1 1cos θ

y R yin

in

in+ += ( )+1 1sin θ ∆

x Rin

in

in+ += ( )1 1cos θ

θ θ αin

in+ = +1 ∆

∆y y yn n= −+1

∆α α= −+n nα1


(29)

NR is the number of radial cells in the annular sub domain delimited by the radius R2 and R1

(see in Fig. 3 the illustration of the vertices notation).

• The vertices of the outer domain with radius Ri ≥ R2 are stationary.

With this moving mesh algorithm, the mesh distortions are small and the original mesh

quality is preserved. Given an appropriate choice of the radius R1 and R2, this technique can

also be applied even for large displacements of the airfoil. This meshing technique is

implemented in a user subroutine called by the CFD code at the beginning of each time step.

Figure 3: Sketch of the vertices notation

3. APPLICATIONTwo airfoil models have been selected for this study: a symmetrical NACA 0012 airfoil used on

vertical axis wind turbine blades and a cambered NACA 632415 used on large horizontal axis

wind turbine blades.

In all computations, the properties of the fluid are those of air with a density ρ = 1.2 kg/m3

and a dynamic viscosity coefficient µ = 1.8 10-5 Pa⋅s. The stability of the solution for the flow

field is checked with the maximal value of the CUNO parameter, a number identical to the CFL

number but whose calculation is based on the face fluxes. It is recommended to ensure that the

peak values of the CUNO do not exceed 300 (http://www.adapco-online.com/). However, this

criterion proved to be insufficient and the time step has been fixed so as to get CUNO values

lower than 20.

CCoommppuuttaattiioonnaall mmeesshh.. The computational domain extends to a distance equal to –8.1 × C at

upstream and the outlet boundary is located at a distance equal to 15 × C . The North and South

boundaries are located at 10 × C respectively. The grid is an O-H block structured mesh and

consist of 111000 cells with 720 cells at the periphery of the sub-domain of radius R1 , 500 cells

around the airfoil and approximately 10 grid points vertically in the airfoil boundary layer

(Fig. 4).

Vi-1(Ri-1, θ

i-1)

V1(R1, θ

1)

V2(R2, θ

2)

Vi(Ri, θ

i)

y R2 2 2

= ( )sin θ


Figure 4: The mesh near the airfoil

3.1. NACA 0012 in forced pitch oscillationsThese computations are performed for an airfoil with a chord length C = 1 m. The axis of

rotation of the airfoil is located at 25 % chord length from the leading edge. The computations

are performed with the chord Reynolds number Re = ρ • U∞ • C/µ = 104 and a constant

integration time step ∆t = 10-2 s. The instantaneous value of the angle of attack is given by

equation (10) with α0 = 0° and αm = 10° . Two cases are selected for the oscillation frequency ωcorresponding to k* = 0.19 and 0.45 where k* = ωC/2U∞ is the dimensionless reduced frequency

based on half chord and the free-stream velocity. At time t = 0 s, the airfoil incidence is zero.

First, computations are carried out for an airfoil with a fixed angle of attack to get the initial

flow field. The simulations for the airfoil in forced oscillations are initiated when the solution

becomes stable.

All results are depicted as a function of the dimensionless time t* = t/T where T = 2π/ω is

the period of oscillation. The time functions of the force and moment coefficients are shown in

Fig. 5, compared to the oscillations of the airfoil:

• The lift and moment coefficients have the same frequency as that of the airfoil

oscillation. However, the force and moment coefficient time histories are not pure

sinusoidal. Higher harmonics from the fundamental pitching airfoil frequency can

be observed

• For the reduced frequency k* = 0.19, a slight lag has been observed in the phase

difference between the lift force and the angle of attack. The variation of the lift and

moment are in phase with the airfoil oscillations when k* = 0.45. Consequently, an

increase is expected in the phase lead as k* increases. Similar behavior has been

found with the linear theory of Theodorsen.

• The drag frequency is one half the lift frequency. Similarly, the behavior for the drag

history was found by Yang et al. [17] with Euler computations of the flow over an

oscillating airfoil. In Ref. 18 (p. 47), it was reported that this has been shown in

experiments and that it is a consequence of the geometry of the vortex street. These

results remind us that the flow around a circular cylinder when a regular Karman

street is observed (see. eg. [19]).


The hysteresis loops CL (α) and CD (α) are shown in Fig. 6. When the airfoil is oscillating,

both lift and drag coefficients increase compared to the 2D steady state. The mean values of

the drag coefficient corresponding to α = 0° are CD = 0.0775 and 0.0684 with k* = 0.19 and 0.45

respectively. In previous computations of the flow around a fixed NACA 0012 airfoil at zero

incidence, it was found that CD = 0.0374. The lift coefficients vary roughly between 0.37 and

0.12 with k* = 0.19 and 0.45 respectively. For these two case studies, the lift hysteresis loops are

then widen when the reduced frequency is lower. It is found a small decrease of CL max when

the reduced frequency k* increases. Similar results have been found with the linear theory of

Theodorsen.

Figure 5: Time histories of the lift, drag and moment coefficients for the forced oscillating NACA 0012

airfoil


Figure 6: Hysteresis loops of the lift, drag and moment coefficients for the forced oscillating NACA

0012 airfoil

3.2. NACA 0012 in free pitch oscillationsThe model is a NACA 0012 airfoil with a chord length C = 0.12 m and a span of 0.50 m that

oscillates around an axis located at 25% of C from the leading edge. The airfoil natural

frequency is ωα = 57 and the damping ratio isς = 0.05. These computations have been

performed with the chord Reynolds number Re = 76 000 and the initial angle of attack

αinit = 10°. The initial flow field is determined by simulations that have been carried out for an

airfoil at the fixed 10° incidence until time t = 0.60 s (corresponding to the dimensionless time

t* = U∞t/C ≈ 48), when the time wise variations of the force coefficients become periodic. The

airfoil is then released in the fluid and the computations are performed in a strong coupled

Fluid Structure Interaction.


The time-wise variation of the airfoil position is shown on Fig. 7 (a). The curve shows that

the airfoil oscillations are damped and that the pitch angle tends to a zero incidence. This can

be seen on Fig. 7 (b) that shows the variation of the torsion moment as a function of the angle

of attack where counter-clockwise rotation indicates positive aerodynamic damping. The

time history of the lift coefficient, not shown here, shows that the lift coefficient decreases

quickly as soon as the profile is free oscillating, until reaching a zero lift coefficient. This

positive aerodynamic damping was expected as the elastic axis is located at the aerodynamic

centre: the pitching moment always lags the airfoil motion and the aerodynamic damping in

pitch tends to suppress torsional oscillations at all frequencies [20].

Figure 7: Time histories of the pitch oscillations and moment versus the pitch angle for the free

oscillating NACA 0012 airfoil


3.3. NACA 632 415 airfoil in pitch and vertical oscillationsThe model is similar to that of Ref. [10] and has a chord length C = 0.30 m, a span of 0.50 m and

the following elastic parameters:

• m = 8.6622 10-2 kg , Sα = –7.79673 10-4 kgm, Jα = 4.87291 10-4 kgm2

• ky = 105.109 N/m, kα = 3.695582 Nm/rad

• Cy = 105.109 and Cα = 3.695582 10-3

The elastic axis EO is localized at 40 % of C from the leading edge and the centre of gravity

G is at 37 % from the leading edge. Computations have been started with the airfoil initial

position yinit = +0.050 m and θinit = 6°, where yinit and θinit are the initial vertical and angular

positions, respectively. The simulations have been performed with the free stream velocities

U∞ = 2, 26 and 45 m/s. With U∞ = 45 m/s, the simulations have been carried out for laminar and

turbulent flows for comparison purposes.

As previously, the initial flow-field is determined with computations carried out for a fixed

airfoil at the above initial position. The airfoil is then released in the fluid and the computations

have been performed in FSI. The obtained results have been depicted according the speed

index, a dimensionless parameter defined as where and

is the mass ratio. The aim of this was to verify if the flutter point will be

attained or not. With the applied values of the airfoil elastic parameters, the velocities U∞ = 2,

26 and 45 m/s correspond to the speed indices V* = 0.15, 1.96 and 3.40, respectively.

The contours of velocity magnitudes obtained with the different values of V* are shown in

Figs. 8 and 9. Umax is the maximal velocity magnitude obtained over the airfoil surface. At time

t = 0 s, when the airfoil is at rest, the ratio Umax /U∞ increases with the free stream velocity (Fig.

8). When U∞ = 2 m/s, it is found that the vortex streets are larger in the near wake of the airfoil

than with the higher free stream velocities. On Fig. 9, the results are also referenced to the

dimensionless time t* = U∞ t/C. When V* = 0.15 (U∞ = 2 m/s), the ratio Umax /U∞ increases at the

beginning of the oscillations and remains important at time t* = 7.0 with a value Umax /U∞ = 2.10.

However, with V* = 1.96 and 3.40 (U∞ = 26 and 45 m/s, respectively) the ratio Umax /U∞

decreases quickly as soon as the airfoil is released in the fluid. At V* = 3.40 , Umax /U∞ = 1.30 at

time t* = 7.5 and with turbulent computations, lower values of the velocity magnitude have

been found (Figs. 9 (e) and 9 (f)).

The time-wise variations of the force components are shown in Fig. 10. Higher values of the

force coefficients are found whenU∞ = 2 m/s. In addition, laminar and turbulent computations

resulted in different values of the aerodynamic forces with lower lift and drag coefficients for

turbulent computations.

As expected, the CFD results are sensitive to the free stream velocity value. However, it

has been found that the value of the inlet fluid velocity does not influence the airfoil responses.

The phenomenon of divergence has not been observed. On Fig. 11, it can be observed that after

some time, the pitch motion becomes periodic while the vertical displacement shows a

positive damping. In addition, due to the frequency ratio ωy/ωα = 0.4, the vertical motion has a

smaller frequency than the pitch motion. The non-influence of the free stream velocity on the

dynamic response of the airfoil is due to the airfoil elasticity parameters and damping and to

the low values of the speed indices. In addition, due to different initial airfoil positions, our

computed airfoil displacements are different from those of Ref. [10]. Indeed, it was shown in [21]

that the motion of oscillating airfoils is stable or unstable depending on both values of the

speed index V* and the initial position.

µ π ρs

m C= ⋅ ⋅( )4 2

U U C* = ⋅( )∞2 ωαV Us

* *= µ


Other simulations whose results are presented in another paper have been performed for

an airfoil without dampening and with different value of the airfoil natural frequency. The

influence of the free-stream velocity on the airfoil oscillations is then more significant.

Figure 8: Contours of the velocity magnitude around the free oscillating NACA 632415 airfoil at time t = 0 s


Figure 9: Contours of the velocity magnitude around the free oscillating NACA 632415 airfoil at

selected time


Figure 10: Time histories of the lift coefficient for the free oscillating NACA 632415 airfoil


Figure 11: Time histories of the pitch and vertical oscillations for the free oscillating NACA 632415 airfoil

4. SUMMARY AND CONCLUSIONFlow induced vibrations have been investigated for two airfoils used on wind turbine blades.

The method is based on a strong coupled Fluid Structure Interaction approach where the

dynamic blade response due to the fluid forces is determined in time accurate sequence. CFD

computations have been performed for incompressible and low Reynolds number flows, using


a commercial CFD code that solves the Navier Stokes equations defined in ALE coordinates.

A straightforward coupling and meshing technique was implemented in a user subroutine

called by the CFD code at each time step.

The coupling method was successfully applied for the computation of the viscous laminar

flows around (i) a forced pitch oscillating airfoil where the hysteresis loops of the dynamic

stall were highlighted, (ii) a free pitch oscillating airfoil and (iii) an airfoil in combined pitch

and vertical oscillations (classic flutter). As wind turbines operate in turbulent environment,

computations were also performed with the resolution of the incompressible Reynolds

averaged Navier-Stokes equations for one case of study. Laminar and turbulent flow

computations resulted in different values of the aerodynamic force and similar airfoil

response.

ACKNOWLEDGMENTS The first author is grateful for the support provided by the La Rochelle University and the

Région Poitou - Charentes (France) in form of grants.

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