An aeroelastic model for horizontal axis wind turbines

8/10/2019 An aeroelastic model for horizontal axis wind turbines

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MATHEMATICS IN ENGINEERING, SCIENCE AND AEROSPACE

MESA - www.journalmesa.com

Vol. 4, No. 3, pp. 249-264, 2013

c CSP - Cambridge, UK; I&S - Florida, USA, 2013

An aeroelastic model for horizontal axis wind turbines

Florin FRUNZULICA1,, Alexandru DUMITRACHE2, Horia DUMITRESCU2

1 POLITEHNICA University of Bucharest, Faculty of Aerospace Engineering, Polizu 1-6, RO-

011061, Bucharest, Romania,

2 Gheorghe Mihoc - Caius Iacob Institute of Mathematical Statistics and Applied Mathematics,

P.O. Box 1-24,RO-010145, Bucharest, Romania.

Corresponding Author.E-mail: [email protected]

Dedicated to the memory of late Professor Mihai Popescu.

Abstract. In this paper an advanced aeroelastic numerical tool for horizontal axis wind turbines

(HAWT) is presented. The tool is created by coupling an unsteady aerodynamic model based on the

vortex method (using the vortex ring model for the lifting surface coupled with an unsteady free-wake

vortex particle model) with an elastodynamic model based on the beam approximation. The coupling

is non-linear in the sense that at every time step the two models interact through data transfer from

the one to the other. The aeroelastic model is evaluated through comparisons of its predictions with

experimental data as well as with predictions obtained by simpler models.

1 Introduction

The design problem of horizontal axis wind turbines is related to a number of physical processes of

varying complexity. Due to the non-linear as well as stochastic (in some aspects), nature of these

processes, a number of simplifications have been introduced up to now. Most of them lead to the

decoupling of the various processes in order to treat them separately. In this connection, there are two

dominant model problems: the aerodynamic problem and the elastodynamic one. Their combinationleads to the aeroelastic problem of a horizontal axis wind turbine (HAWT)[1]. Input to this problem is

the wind inflow conditions. The solution of the aerodynamic and aeroelastic problems can be used to

provide useful information to a number of design problems. Thus, the post-processing of the results

can lead to a fatique and reability analysis, the analysis of wake effects for the design of wind parks,

the prediction of power fluctuations, etc.

In this work, the authors present a non-linear advanced aeroelastic model based on a vortex method

as regards the aerodynamics, and a beam structural model adapted to this problem.

2010 Mathematics Subject Classification: 76G25, 74B20, 74S05

Keywords: vortex lattice method, beam theory, finite elements method, aeroelasticity, HAWT.


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250 F. Frunzulica, Al. Dumitrache, H. Dumitrescu

2 The numerical method

The key point of the approach adopted herein is based on the formulation of the aeroelastic problemas an appropriate coupling of two different problems: the aerodynamic and the elastodynamic. In

brief, the aerodynamic part aims at the calculation of the loads exercised on the structure due to the

dynamics of the surrounding fluid.

The loads are then introduced as forcing to the elastodynamic equations where from the defor-

mations, the rates of deformations and the accelerations are calculated along the blade. The rates of

deformations are feed back into the aerodynamic part as excess body velocities of the blade (in addi-

tion to the rigid body motions). This results in a modification of the non-entry boundary condition for

the aerodynamic part that represents the effects of flexibility on the fluid dynamics.

Regarding the wind turbine and its surrounding fluid as one mechanical system, it is clear that

the procedure just outlined defines a two-way channel of communication between the two parts of

the system. This communication is achieved by means of two interfaces: one for the transfer of theaerodynamic loads from the aerodynamics to the elastodynamic part, and one for the transfer of the

rats of deformations from the elastodynamics to the aerodynamic part. Note that in both cases the data

that are transferred concern solely the solid surface of the blades, i.e. the domain of contact between

and the solid part of the system. This is always the case regardless the type of approximation used

for the either the fluid or the solid. In what follows, the flow problem is approximated by a free-wake

aerodynamic model, whereas the elastic part is modeled by the beam approximation.

2.1 The aerodynamic model

For rotor computations, the blades element momentum methods are easily understandable and appli-

cable, using minimum computation requirements [1,2]. Anyhow, there are cases when these methodsdo not provide the desired precision. The design interest cases include asymmetric aerodynamic un-

steady flow (especially the dynamic effects of incident flow). An evident alternative is the detailed

computation of the induced velocities field, based on the wake vortex distribution [3,4,5,6]. The

vortex methods are interesting, even while requiring significant computation resources, due to their

possibility to observe the vortex systems main structure. In the following sections, we will present

a calculation algorithm where the near-wake strip elements are transformed into vortex particles and

become part of the far-wake. Integrating the vorticity of each near-wake dipole element produces a

vortex particle. The new vortex particles became part of the far wake which evolves prior to the next

time step using a Lagrangean description of the flow .

Hypothesis.The working fluid is a continuous barotropic ideal fluid which fills an unlimited,

simple, contiguous domain

. In the flow is adiabatic and non-rotational, where =

R.The domainR is the rotational flow domain (i.e. for a blade this is represented by the solid bladesurface and the wake flow- field domain).

The blades have a rotational motion related to the reference system attached to the airflow speed

direction. In the calculation points the velocity is given by sum of three components: the cinematic

velocity corresponding to blade motion, the induced velocities and the wake induced velocities. The

generated free wake is attached to the leading edge of the blade.

The time steps are chosen so that vortexes generated by the trailing/leading edge should not trans-

ported on distance greater than the smallest panel dimension.

The Vortex Lattice Method (VLM).When the blades lifting surface has a relatively reduced thick-

ness (valid for most free rotor blades) it might be selected a scientific method requiring the vortex ring


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An Aeroelastic Model for HAWT 251

distribution on a surface built by means of the generatrix camber line. Therefore the numeric calcula-

tion can determine the pressure jump p= pi pe between upper and lower surfaces.

Figure 1 shows the blade decomposing into panels. The vortex element on the panel at front edge islocated at of the panels chord, while the collocation point is located at of panels chord. Numerically,

the vortex rings are stored in a matrix with the indexes i and j. Each vortex ring is perfectly sitting on

the mean wing surface, to enable a correct representation from the mathematical point of view. The

induced velocity in a control pointP is calculated with the Biot-Savart formula [3].

Fig. 1 Discretization of the blade into panels (a) and vortex rings distribution (b).

To determine the vortex rings intensities, i,j, it will be imposed that the resulting speed in the

calculation point shall be tangent to the mean blade surface:

(M

i=1

N

j=1

(Vind,P(t,i,j))i,j+Mw,T E

i=1

N

j=1

(Vw,P(t,i,j))i,j+

Mw,T E

j=1

Vw,P(t,j)) nP=Vk,P(t) nP (2.1)

whereMand Nare the number of panels along the chord, respectively along the wing span, Mw,T Erepresent the number of free vortexes in each closed wake strip generated by the trailing edge and

Mw,T Erepresent the number of particles generated in the wake by the trailing edge (the formula (1)

includes the wake separation at leading edge for high angle of attack). Numerically, we will consider

a finite wake (about 3 rotor diameters). Vortexes generated at the leading and trailing edges for each

nearby adjacent panel will change their intensity in time and will move in space with local airflow

speed. Equation (2.1) provides (at each time step) a linear algebraic system A = b. Solving thissystem by an exact or iterative method enables to determine the vortex rings intensity i,j,i=1,...,M,

j=1...N. MatrixAis the influence coefficients matrix, while vector bis the right side of equation (2.1).

Leading Edge Wake Model.To take into account the leading edge stall there is necessary to know

the wake vortex near the leading edge at any time. Physically the phenomenon is a consequence of

high incidence viscous flow and to introduce it in the numerical algorithm the ONERA method applies

[1,7]. This gives the changes of the s circulation, while on blade elements (element strips along the

chord), under stall conditions, by solving a second order differential equation:

s+

s+

r

2s=

r

2VtCL

E

Vn (2.2)

where =2Vt/c,c is the blade element chord, Vtis the velocity component along the chord,Vnis thechord normal velocity component, CL is the lift coefficient difference between potential flow and

stationary flow. Dumitrescu et al. present in detail the empiric coefficients r,E,[1].To determine CLit is necessary to find out each blade elements local angle of incidence. Usually,

this is the global angle of incidence for a flow completely attached to the blade.


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The differential equation (2.2) applied for each blade element enables determination of the first

vortex ring in the leading edge generated wake, representing the support of the wake particles carrying

vortices.T ELE= (at+s) (2.3)

where T E is the trailing edge circulation, while at is circulation corresponding to the flow fully

attached along the local blade strip.

The Trailing Edge Wake Model.Expressing the (far) wake with particles means that each particle

is defined by its position xi, its vortices i and its kernel radius i [4,7]. Thus, the local vorticity in

wake can be determined with the formula:

w(x, t) =i

i(t) g (xxi(t) ,i(t)) (2.4)

where g represents the vortices distribution function for the particle i (i= 1...Nw,p), while i is the

length defining the particle kernels support domain. In three-dimensional space, the function g hasthe expression [5]:

g (r,) = 3

43exp

(|r|/)3

(2.5)

Thus, the speed induced in a point by a set of particles will be:

Vw(x, t) = 1

4

w

(xx)w(x, t)

|xx|3 d =

1

4

Nw,p

i=1

i(t) (xxi)

|xxi|3

f(xxi,i) (2.6)

where: f(r,) =1 exp(|r|/)3

.

Therefore, the evolution of a vortex lattice is reduced to the calculation of a particles assembly, j:

Dxj(t)

Dt=V (xj, t) ,

Dj(t)

Dt= (j(t) ) V (xj, t) (2.7)

whereV is the total fluid velocity in the point xj.

For a time period t, the Adams-Bashforth formula is approximating the trajectory of a fluid

particle:

xj(t+t) =xj(t) + [1.5V (xj(t) , t)0.5 V (xj(tt) , tt)] t (2.8)

Changing a particles kernel radius due to viscous diffusion is similarly to the viscous diffusion of

a vortex filament in a plan perpendicularly to this filament:

D

Dt =

c2

2t , c=2,242 (2.9)

with the kernel radius at time,t+tdetermined by the relation:

|t+t|=|t| (t/t+t)2

(2.10)

In applications, the near wake is including a single vortex ring line, usually the one generated by

the trailing edge TE(a similar approach is available also for the leading edge ,LE, wake). Figure 2

shows the particles transforming algorithm.

At the time momentt,calculation to solving the systemA = b, provides the circulation intensityof the vortex ring tm1,jand of the neighbor rings, the location of the cross points 1 and 2 being from

the previous time step:


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Fig. 2 Generation of aTEvortex ring (a), convection and transformation in an equivalent particle (b).

x1=xm,j+ Vttm,j t , x2=xm,j+1+ V

ttm,j+1t (2.11)

On the edges 14 and 23, the effective circulation is equally distributed between the panels adja-cent to these edges, while on edge 12 the circulation is provided by the difference between current

intensities and those of the previous time moment. The new positions of the cross points defining the

panel 1234 is determined for each time step with the equation:

x i=xi+ Vtit, i=1,2,3,4 (2.12)

IfSis the panelSvorticity (in its new position), S(x) =Sdx, then the particlesjvorticity

associated to panel,S, is determined with the relation:

j=

SSd (2.13)

while the particles position in time will be determined with the relation:

xjj=

SxSd (2.14)

The transport of the new particlejis further done according to the equations (2.7).

Blade pressure calculation.To determine the pressure difference on the blade we will decompose

the local velocity V on the blade surface, panel (i,j), related to two directions: i (tangent direction

oriented along the chord) and tauj (tangent direction along the span). Defining the characteristic

length for a panel (i,j) along the chord and the span, ci,j andbi,j, the pressure difference will be

provided by the equation [1,3]:

pi,j= (Vp)i,j ii,ji1,j

ci,j+ (VP)i,j j

i,ji,j1

bi,j+i,j

t (2.15)

while the force on the panel (i,j) will be provided by:

Fi,j=pi,jni,jSi,j= Di,ji +Fyi,jj +Li,jk (2.16)

The next phase of the aerodynamic computation consists of calculations of blade forces. The first

is the tangential force per unit blade length Ftacting along the chord line of a blade element in the

direction of motion. The second is the normal force per unit blade lengthFnacting in the direction of

the unit vector normal to the chord line. A complete set of two dimensional aerodynamic forces would

also include a pitching moment about the spanwise axis. In general this moment is small and is thus

neglected. From (2.16) we can compute in spanwise direction, lift coefficient CL and drag coefficient

CD(including given airfoil drag and locally induced drag).


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2.2 The elastodynamic model

The aspect ratio of wind turbine blades is, usually large and, therefore, beam theory can be used

to describe, the elastodynamic behaviour of the blade. Let O [Xe,Ye,Ze]denote the beam coordinatesystem.and it is assumed that the elastic axis is straight and coincides with axisYe.

The beam theory. In this model three types of deformations are introduced: x(y)- the bendingdeformation along Ye direction (flapwise bending),z(y)- the bending deformation alongZe direction(leadwise bending) andy(y)- the spanwise torsional deformation [8,9].

According to the linear beam theory, the equations for the equilibrium of forces and moments in

the(X Z)eplane are as follows:

2

y2

EIz

2xy2

+

2xt2

zcm2yt2

+ fxa+fxg+fxc= 0 (2.17)

2

y2

EIx

2

zy2

+

2

zt2 xcm

2

yt2 + fza+fzg+fzc=0 (2.18)

y

GIt

yy

Ip2yt2

+xcm2zt2

zcm2xt2

+ mya+ myg+ myc= 0 (2.19)

where:E(y)- the Young modulus (N/m2),G (y)- the shear modulus (N/m2), (y)- the linear density(kg/m), xcm- the distance of the mass center from Ze axis (m), zcm- the distance of the mass centerfromXe axis (m),Ix(y)- the moment of inertia about Xe axis (m

4),Iz(y)- the moment of inertia aboutZe axis (m

4), Ip(y)- the polar moment of inertia about the elastic center (m4),It(y)- St. Venant tor-

sional moment of inertia (m4), f(y)the bending distributed loads (N/m),m(y)the distributed torque(Nm/m), and subscripts a, g, c denotes the aerodynamic loads, gravitational loads and centrifugalloads respectively.

The first step in structural computation is to calculate beam cross-sectional properties of thin-

walled beam, multi-cell, nonhomogeneous, closed sections, within the framework of Bernoullis

bending theory and St. Venant torsion theory [8,9]. The key idea is the approximation of the airfoils

shape byne straight, homogeneous elements. The thickness of every element is considered constant

and is evenly distributed across the two sides of its midline. For each element, the following data

are necessary:Ee,Ge moduli (N/m2), densitye (kg/m

3), thicknesste(m), coordinates of element in

plane(X1e,Z1e)and(X2e,Z2e)(figure 3).

Fig. 3 The discretized airfoil.

At the beginning, the coordinates(X1e,Z1e) and (X2e,Z2e)can be given with respect to any co-ordinate system (figure 2), but after the calculation of the elastic center coordinates, we switch to the


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elastic coordinate system. Following the notations of figure 2, the section characteristics are obtained

as follows:

Le=

(X2e X1e)2

+ (Z2e Z1e)2

, Ae= Le te, A=

ne

e=1Ae (2.20)

E=ne

e=1

EeAe/A, G=ne

e=1

GeAe/A (2.21)

=ne

e=1

eAe (2.22)

xel=ne

e=1

EeAe(X1e+X2e)

2

1

EA, zel=

ne

e=1

EeAe(Z1e+Z2e)

2

1

EA(2.23)

where Ae is the elements area, Le is the elements length, and ((xel,zel )) are the coordinates of the

elastic center. Now, the whole analysis is referred to the elastic coordinate system. The remainingsectional properties are defined as follows:

Xcm=ne

e=1

eAe(X1e+X2e)

2

1

A, xcm=Xcm xsc

Zcm=ne

e=1

eAe(Z1e+Z2e)

2

1

A, zcm=Zcm zsc (2.24)

where(Xcm,Zcm)are the coordinates of the mass center in the elastic coordinate system, and (xsc,zsc)are the coordinates of the shear center in the elastic coordinate system. The calculation of the shear

center will be described later.

St. Venant torsional constant It. The St. Venant torsional constant of a section is defined as:

It= Mt/(G) (2.25)

whereMtis the torque that acts on the plane of the section, and is the rate of twist due to this torque.

In the case of a thin-walled, multi-cell section, the calculation of necessitates the calculation of

the shear flows in the skin of the section. For three cells section (figure 4), we apply the following

equations:

-the equation of the equilibrium of moments (Bredt-Batho formula):

2q11+ 2q22+ 2q33= Mt (2.26)

-the equations for the shear flow balance at the junctions:

q1= q4+ q2 , q2= q3+ q5 (2.27)

-and the equations for the consistency of torsional deformations for every cell:

1= 2= 3 (2.28)

or in extended form

1

2G11(q11+ q44) =

1

2G22(q2(2+6)q44+ q55) =

1

2G33(q33 q55) (2.29)


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where1,2,3 are the areas of cells,Gj the tangential modulus, averaged over cell j, and

j=

Sj

ds

t =iSj

Li

ti (2.30)

The segment Sj represents the curvilinear segment where acting the shear flow qj, j= 1,2...6.Equations (2.26)-(2.29) form a 5x5 system from which the values ofqj, j= 1,2...6, can be computed.The torsional specific deformation of the section is equal to the torsional specific deformation of

each one of the cells,

= 1

2G11(q11+ q44) (2.31)

and now,Itcan be obtained by equation (2.25).

Fig. 4 The shear flow for the thin-walled, three cell sections, under twisting moment.

The shear center (SC).The shear center of the section represents the point through which the plane

of the resultant loads must pass to prevent the development of twisting moments on the section. Tocalculate the coordinates of the SC we assume that a shearing force Tacts through the SC at distance

dfrom the arbitrary reference point (the elastic center). Our section is three-celled and is three times

statically indeterminate. We perform a cut in each cell and as consequences the structure is reduced

to the thin-walled open section. To restore continuity at every cut, a supplementary shear flow is

introduced in celli:

qj= qj0+ qj (2.32)

whereq j0 is the shear flow at any point of cell j assuming cuts at all cells, and qj is the constant

shear flow which is developed if we close the cut of cellj. The shear flowqj0 is defined as [9]:

qj0=IxTx IxzTz

IxIz I2

xz

Qz IzTz IxzTx

IxIz I2

xz

Qx (2.33)

where

Ixz= 1

E

ne

e=1

EeAeX1e+X2e

2

Z1e+Z2e2

(2.34)

and

Qz= 1

E

s0

E(s)x t(s)ds, Qx= 1

E

s0

E(s)z t(s)ds (2.35)

In equation (2.33),Qxand Qzare the only quantities which change from point to point.

We evaluate the shear flowqj0 from the cut whereqj0=0 (the inferior limit of integrals (2.35))and we continue moving along the correspondent line until we meet a junction. In junction we apply

a balance of shear flow (figure 5); for exemplification, in junction 1 we have


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Fig. 5 The shear flow for the thin-walled, three cell sections, under shear load.

q14=q11b q

12a (2.36)

and now, the shear flowq40 in spar web 1-4 is

q40=q14+

IxTx IxzTzIxIz I2xz

Qz IzTz IxzTx

IxIz I2xzQx

14

(2.37)

The final shear flow can be interpreted as algebraic sum of the shear flows qj0 in the opened

section, and the corrective shear flows qj applied independently in each cell. The total relative warping

at the three cuts is obviously zero if all cells are closed. Thus, if the rate of twist is set equal to zero,

for cellj we have the following equation:

1

Gj

Sj

qj ds

t=

1

Gj

Sj

qjds

t

Sjk

qkds

t

Sji

qi ds

t+

Sj

qj0 ds

t

=0 (2.38)

whereSjkis the web common to cells j and k. If we denote

j0= 1

Gj

Sj

qj0 ds

t, j j=

1

Gj

Sj

ds

t, ji=

1

Gj

Sji

ds

t, jk=

1

Gj

Sjk

ds

t(2.39)

then the equation (2.38) becomes

jiqi +j jq

j+jkq

k=j0 (2.40)

and represents a 3x3 system which can be solved with respect to qi . Now we can obtain the shear

flow distribution at any point of the section using eq. (2.32). The moment M0 produced by the shear

flows about the arbitrary reference point (the elastic center) is

M0=ne

e=1

qe(X1eZ2e X2eZ1e) (2.41)

whereqeis the shear flow that correspond to thee-th element.

For the x-coordinate of the shear center, an external load in the z direction must be imposed. In

this case:

xsc=M0/Tz (2.42)

For thez-coordinate of the shear center, the load is imposed in the x direction:

zsc=M0/Tx (2.43)


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The finite element technique. By using Lagrange equations the following linear equations of

motion are obtained [10,11]:

MDn+ C

Dn+ KDn= R

ext

n (2.44)where

-the matrixM=

VNTNdVis the mass matrix

-the matrixC=

VNTCNddV is the structural damping

-the matrixK=

VNTdE NddVis the stiffness matrix

-the vectorRextis the load vector

-the vector D is the displacement vector, which contains the degrees of freedom: x - edgewise

bending, x - the bending slope at XY plane, z - flapwise bending, z - the bending slope at ZY

plane,y - the torsional deformation at X Zplane, and

-Nd- the derivative matrix of shape function

-N- the matrix of shape function (the shape functions most commonly used are the third-degree

polynomials and the first degree in the case of torsion).

The time advancementof the equation (2.44) with the appropriate initial conditions is performed

with the specialized algorithm (Crank-Nicolson) method [10]. This is an unconditionally stable im-

plicit one-step method, which is second order accurate in time and relates the displacements, veloci-

ties, and acceleration as

Dn+1=Dn+t

2

Dn+ Dn+1

, Dn+1= Dn+

t

2

Dn+Dn+1

(2.45)

Subscript n corresponds to the old time and n+1 is the new time. t is the time step. Rearranging

equations (2.45) gives

Dn+1= 2t

(Dn+1 Dn) Dn, Dn+1= 4t2

(Dn+1 Dn) 4t

Dn Dn (2.46)

By combining equations (2.46) with the equations of motion (2.44) at timet= tn+1, we obtain:

Ke f f Dn+1=Re f fn+1 (2.47)

where the effective stiffness matrix, Ke f f, and the effective load vector,Re f fn+1, are, respectively,

Ke f f = 4

t2M +

2

tC + K (2.48)

and

Re f fn+1= R

extn+1+ M

4

t2Dn+

4

tDn+ Dn

+ C

2

tDn+ Dn

(2.49)

2.3 Coupling models

The solution of the aeroelastic problem requires the coupling of an aerodynamic and an elastodynamic

model. In previous paragraphs a brief description of each part was done separately. In this paragraph

the basic principles of the communication between the two parts will be discussed.

As regards the elastodynamic part, the load vector must be input. This vector is calculated by

superimposing the gravitational forces on the aerodynamic loads. The quantities that have to be trans-

ferred from the aerodynamic part are, therefore, the aerodynamic forces that act on the blade.


11/16


The solution of equation (2.44) yields the vector D of the deformations, the vector D of the de-

formation rates and the vector D of the accelerations at the nodes of the beam that simulates the

blade.The feedback to the aerodynamic part are the deformations rates, which, in the case of non-linear

coupling, are included in the aerodynamic problem along with the body motion velocities. This results

in a change of the effective velocity of the blade seen by the fluid. Consequently, the angles of attack

and, therefore, the lift and drag coefficients change.

Within the context of linear elasticity, deformations are assumed small. On the other hand, it is

not expected to encounter considerable deformations on a blade of a wind turbine. Therefore, it is

approximately correct to retain the undeformed geometry of the blade throughout the whole analysis.

The main modules can be summarized in the following flowchart (figure 6):

Fig. 6 The flowchart of the coupling between aerodynamic and elastodynamic models.

The flow chart of the aeroelastic code has the following steps: (a) initialize code; (b) perform same

pure aerodynamic steps for the calculation of the circulation distribution. For every time step: (c.1)

start time step; (c.2) calculate the aerodynamic forces distribution; (c.3) calculate the force and the

velocity distribution on the blades; (c.4) perform elastodynamic steps for a period of time equal to

aerodynamic time step; (c.5) circulation calculation step; (c.6) go to next time step.

For a non-linear coupling, in the step c.2 the aerodynamic forces is fulfilled including the rates of

elastodynamic deformations, which have been calculated during the previous time step. In the case of

a linear coupling, there is no feedback from the elastodynamic part to the aerodynamic part.

The only communication between the two parts is in step c.4 where the aerodynamic forces areimposed on the beam and the elastodynamic problem is solved.

3 Results

3.1 Test case 1: Modal analysis ( =25rev/min)

In this numerical test we consider a reference blade [10] with complete structure. The blade is dis-

cretized in finite elements (3094 nodes which define 2985 shells), metallic structures (figure 7). Using

the complete model and equivalent model of the blade structure, we computed the natural frequencies

and modal forms for both models. The firsts five modes are show comparatively in the figure 8, and


12/16


the correspondent frequencies are summarized in the table 1.

Table 1. The natural frequencies (Hz): beam model vs. complete model.

Mode Completestructure

Complete structurewith centrifugal ef-

fect

Beammodel

Beam model withcentrifugal effect

Flapwise I 1,208 1,321 1,199 1,318

Flapwise II 2,985 3,23 2,921 3,221

Edgewise I 5,521 5,66 5,486 5,601

Edgewise III 9,128 9,257 9,055 9,193

Torsion I 13,951 14,877 13,804 14,698

Fig. 7 The discretized structure of the blade in finite elements.

3.2 Test case 2: Dynamic response

In this test, we computed the dynamic response of HAWT (the same beam model as the test case 1)

at impulsive wind speedVw= 15 m/s (nominal regime 12 m/s, = 25 rev/min) [10]. In the figure9 we show the wake development at nominal regime. The figure 10 presents the variation of the

displacement and pitch angle at the blade tip, and flapwise and edgewise bending moment at the

blade root, as a function of azimuth angle. The oscillations are quickly damping, after 210 degrees.

3.3 Test case 3.

The results presented in the sequel concern the two cases of double pitch steps for the Tjaereborg

HAWT, for which experimental data are available [12] (figure 11). The parameters used for each case

are (table 2): the inflow velocity - U (m/s), the starting time of first pitch step t1,st(s), - the ending

time of first pitch step t1,end (s), the starting time of second pitch step t2,st(s), the ending time of

second pitch stept2,end(s), the initial pitch angle 1 (deg), the pitch angle after the first pitch step2(deg).


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Fig. 8 The comparison of the modal forms: beam model (b) vs. complete model (a).

Fig. 9 The free-wake of the rotor.


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Fig. 10 Test case 2: the dynamic response at impulsive wind.1. stationary nominal regime; 2. induced oscilation regime

(transient regime); a. flap displacement of tip blade, b. flapwise bending moment at root blade, c. pitch angle variation, d.

edgewise bending moment at root blade.


15/16


Table 2. Comparison between experiment and numerical results.

Case 3.1 Case 3.2Experiment Aeroelastic

code

Experiment Aeroelastic

code

U (m/s) 12.5 12.5 8.7 8.7

t1,st (s) 4.70 14.0 2.0 21.0

t1,end(s) 6.00 15.3 2.5 21.5

t2,st (s) 34.58 24.0 32.0 34.0

t2,end(s) 35.7 25.12 32.7 34.7

1 (deg) -1.164 -1.164 -0.07 -0.07

2 (deg) -3.19 -3.19 -3.716 -3.716

Fig. 11 Comparison between experiment and aeroelastic code.


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4 Conclusions

A complete aeroelastic tool has been presented together with its self consistency tests and someresults. In this stage, we cannot conclude on its accuracy. However, the experience suggests that this

could be expected.

There are three points that must be underlined: (1) in some tests it appeared necessary to introduce

artificial damping, (2) the coupling, within the context of approach described, must be non-linear and

(3) the computational effort required to couple the aerodynamics with the structural part, is small

compared to the whole.

Prospective work: we will made a most elaborate model based on the coupling of the aerodynamic

model with a structural code based on the complete structural model and composites materials.

Acknowledgement

Supports for this work by PCCA-PARTNERSHIP Program, with the support of ANCS, CNDI-

UEFISCDI, project no. PN-II-PT-PCCA-2011-3.2-1670, are gratefully acknowledged.

References

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[2] Dumitrescu, H., Cardos, V., Prediction of unsteady HAWT aerodynamics by lifting line theory, International Journal

of Applied Mechanics and Engineering, Vol. 9, No. 4, p.675686, 2004.

[3] Katz, J., Plotkin, A., Low-Speed Aerodynamics, Cambridge University Press 2001.

[4] Leonard, A., Computing three-dimensional incompressible flows with vortex filaments, Ann. Rev. Fluid Mech., Vol.

17, p.523559, 1985.

[5] Voutsinas, S.G., Belessis, M.A., Rados, K.G., Investigation of the yawed operation of wind turbines by means of a vor-

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[7] Riziotis, V.A., Voutsinas, S.G., Zervos, A., Investigation of the yaw induced stall and its impact to the design of wind

turbines, European Union Wind Energy Conference, 20-24 May 1996, Gteborg, Sweden.

[8] Timoshenko S., Goodier J.N.: Theory of elasticity, Mc Graw-Hill, 1951 (1992, Ed.).

[9] Megson T.H.G.: Aircraft structures for engineering students. Edward Arnold (Publishers) Ltd, 1977.[10] Zienkiewiecs O. C.: The Finite Element in Engineering Science, London-New York, McGraw-Hill, 1971.

[11] Bisplinghoff R.L., Ashley H., Halfman R.: Aeroelasticity. Dover Publications, (1996 Ed.).

[12] Oye S.: Tjaereborg wind turbine: Fifth dynamic inflow measurement, AFM VK-233, Department of Fluid Mechanics,

Technical University of Denmark, 1992.

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An aeroelastic model for horizontal axis wind turbines