WESWind Energy ScienceWESWind Energ. Sci.2366-7451Copernicus PublicationsGöttingen, Germany10.5194/wes-2-15-2017Comparison of a coupled near- and far-wake model with a free-wake vortex codePirrungGeorggepir@dtu.dkhttps://orcid.org/0000-0001-9260-1791RiziotisVasilisMadsenHelgehttps://orcid.org/0000-0002-4647-3706HansenMortenhttps://orcid.org/0000-0002-8868-6152KimTaeseongWind Energy Department, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, DenmarkSchool of Mechanical Engineering, National Technical University of Athens, 9 Heroon Polytechneiou Str., 15780, Athens, GreeceGeorg Pirrung (gepir@dtu.dk)20January201721153312February201610May201616September2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://wes.copernicus.org/articles/2/15/2017/wes-2-15-2017.htmlThe full text article is available as a PDF file from https://wes.copernicus.org/articles/2/15/2017/wes-2-15-2017.pdf
This paper presents the integration of a near-wake model for trailing
vorticity, which is based on a prescribed-wake lifting-line model proposed by
, with a blade element momentum (BEM)-based far-wake model and
a 2-D shed vorticity model. The resulting coupled aerodynamics model is
validated against lifting-surface computations performed using a free-wake
panel code. The focus of the description of the aerodynamics model is on the
numerical stability, the computation speed and the accuracy of unsteady
simulations. To stabilize the near-wake model, it has to be iterated to
convergence, using a relaxation factor that has to be updated during the
computation. Further, the effect of simplifying the exponential function
approximation of the near-wake model to increase the computation speed is
investigated in this work. A modification of the dynamic inflow weighting
factors of the far-wake model is presented that ensures good induction
modeling at slow timescales. Finally, the unsteady airfoil aerodynamics model
is extended to provide the unsteady bound circulation for the near-wake model
and to improve the modeling of the unsteady behavior of cambered airfoils.
The model comparison with results from a free-wake panel code and a BEM model
is centered around the NREL 5 MW reference turbine. The response to pitch
steps at different pitching speeds is compared. By means of prescribed
vibration cases, the effect of the aerodynamic model on the predictions of
the aerodynamic work is investigated. The validation shows that a BEM model
can be improved by adding near-wake trailed vorticity computation. For all
prescribed vibration cases with high aerodynamic damping, results similar to
those obtained by the free-wake model can be achieved in a small fraction of
computation time with the proposed model. In the cases with low aerodynamic
damping, the addition of trailed vorticity modeling shifts the results closer
to those obtained by using the free-wake code, but differences remain.
Introduction
This work is based on a coupled aerodynamics model, where the
trailed vorticity effects in the near wake are computed based on a model
proposed by , and the far-wake contribution is computed using
the well-known blade element momentum (BEM) theory. The near-wake model (NWM)
is a simplified prescribed-wake lifting-line model which efficiently computes
the induction due to the vorticity trailed during a quarter of a rotor
revolution. The coupled model can be seen as a hybrid code between a
traditional BEM model and the more complex vortex codes. Because a BEM model
is based on an actuator disk assumption, it can not model the detailed
dynamic induction response at the individual blades. Therefore, the NWM is
introduced to model these unsteady induction characteristics due to load
changes by pitch, eigenmotion of the blades, turbulent inflow and shear. The
accuracy of the computations is improved due to the added aerodynamic
coupling between airfoil sections through the trailed vorticity, alleviating
the limitations of the BEM strip theory. Especially in cases with large
radial load gradients, for example close to trailing edge flaps or other
aerodynamic devices or close to the blade root and tip, the cross-sectional
coupling will lead to an improved prediction of the steady and dynamic
induction. The addition of the near-wake model in an aeroservoelastic code
has an acceptable effect on the total computation speed. An aeroelastic
simulation with the wind turbine code HAWC2
, of the DTU 10 MW turbine
in normal operation with turbulent
inflow, takes roughly
10 % (30 aerodynamic sections) to 40 % (55 aerodynamic sections)
longer if the near-wake model is enabled than if a pure BEM model is used.
The coupled model using the modified BEM approach for the far wake has been
proposed by and extended by . Further,
improvement has been presented by , where an
iterative procedure was used to ensure convergence and avoid numerical
instabilities of the NWM. An application of the coupled model to estimate the
critical flutter speeds of the NREL 5 MW turbine , also
including blades with modified stiffness, has been described by
, where the coupled aerodynamics model has predicted
4–10 % higher critical flutter speeds than the unsteady BEM model in the
aeroservoelastic wind turbine code HAWC2.
In the present paper, the iteration procedure of the NWM used by
is presented in more detail, as well as a method
to compute the necessary relaxation factor during a simulation. The presented
approach removes the need for additional input or very conservative
relaxation factors that are independent of spatial and temporal
discretization and increase the computation time. Further, the NWM is
simplified to accelerate the computations with small loss of accuracy of the
unsteady results.
The dynamic responses to pitch steps and prescribed blade vibrations are
validated by comparing them to results from the more complex free-wake code
GENUVP . The focus in the pitch step cases is the dynamic
induction response, while the prescribed vibration cases are evaluated based
on aerodynamic work during a period of oscillation. It is found that the
coupled aerodynamic model is capable of producing results that agree much
better with results obtained from the free-wake code than the unsteady BEM
model in most cases, without a dramatic increase in computation time. The
more accurate computation of aerodynamic work can have a considerable impact
on the aeroelastic response in the case where the total damping is close to
zero, such as for edgewise vibrations.
This paper is structured as follows: in the next section a short description
of the NWM and a previous implementation of the coupling to a far-wake model
and shed vorticity model are presented. An overview of the current implementation is given in Sect. 3. In Sect. ,
modifications to far wake and shed vorticity model are proposed to improve
the interaction of these models with the near-wake model and to increase the
accuracy of the dynamic lift computation for cambered airfoils. This is
followed by a description of the iterative procedure to stabilize the
near-wake model in Sect. . A way of simplifying the NWM
to accelerate the computation is presented in Sect. . In
Sect. the free-wake panel code used for validation of
the coupled near- and far-wake model is briefly described. The effects of the
model modifications and results from the code comparison are shown and
discussed in Sect. . An overview of the symbols used in the
equations and figures is included as Appendix A.
Original model description
The structure of the previous implementation of the
model is shown in Fig. . From the velocity triangle, denoted as
VT, follows a geometric angle of attack (AOA) αQS and a
relative velocity vr. An effective AOA αeff is
obtained through a 2-D modeling of the shed vorticity effects, which is
briefly described in Sect. . This effective AOA is
used to determine the aerodynamic forces and the thrust coefficient
CT and torque coefficient CQ. These coefficients lead
to a far-wake induction factor aFW, requiring a coupling
factor kFW as input. Section contains the dynamic
inflow model, using the weighting factors A1 and A2, which is used to
determine the unsteady far-wake induction uFW,dyn.
Using this far-wake induction, and the near-wake induction from the previous
time step, a new intermediate velocity triangle VTi is
determined, with a new quasi-steady AOA and relative velocity. These lead to
the bound circulation ΓQS. The difference in
ΓQS between adjacent blade sections, denoted as ΔΓ in the following, determines the trailed vorticity. In the next
section it is shown how the induced velocity W due to the near wake,
which is added to uFW to obtain the total induced velocity
utot at each blade section, follows from the trailed
vortices. The total induced velocity will then, in addition to the relative
velocity due to blade motion and turbulence in the incoming wind, determine
the velocity triangle after the time step Δt.
The previous implementation of the coupled near- and far-wake model,
as described by and . The numbers in
parentheses refer to the equations in the following sections, [B] to the
original model by , and [A] to . The vector
notation on the induction factors and induced velocity indicates that the
model can be used to compute both axial and tangential
induction.
Near-wake model
The NWM enables a fast computation of the induction due to the trailed
vorticity behind a rotor blade. The trailed wake can be discretized into
trailed vortex arcs from several positions on the blade, where each arc
consists of a number of vortex elements. The induction at a blade section due
to each vortex element can be computed using the Biot–Savart law, but this
computation is numerically expensive as the influence of each vortex element
on the induction at each blade section has to be determined.
proposed to avoid these expensive computations by assuming that the trailed
vorticity follows circular vortex arcs in the rotor plane and limiting the
computation to a quarter rotation. In this quarter rotation, the axial
induction dw from a vortex element at a blade position is
decreasing as the vortex element moves away from the blade, starting with a
value dw0. This decreasing induction, following from the
Biot–Savart law, is approximated by exponential functions:
dwdw0≈1.359e-β/Φ-0.359e-4β/Φ,
where Φ is a geometric factor depending on the radius from which the
vortex is trailed and the distance between the vortex trailing point and the
blade section where the induction is computed. The angle β determines
how much the blade has rotated away from the vortex element. The numerically
efficient trailing wake algorithm gives the axial induction W due to the
trailed vorticity at time step i at a blade section s as
Wsi=∑v=1NvWs,vi,
where Nv is the number of vortex arcs trailed from the blade and Ws,v
is the induction due to a single vortex arc v at the blade section.
The induction consists of components
Xs,vi and Ys,vi, corresponding to both of the exponential
terms in Eq. ():
Ws,vi=Xs,vi+Ys,vi,Xs,vi=Xs,vi-1e-Δβv/Φs,v+DX,s,vΔΓv(1-e-Δβv/Φs,v),Ys,vi=Ys,vi-1e-4Δβv/Φs,v+DY,s,vΔΓv(1-e-4Δβv/Φs,v),
where ΔΓv is the trailed vortex strength, which depends on the
radial difference in bound circulation between the blade sections adjacent to
the vortex trailing point. The relative movement of the blade in the rotor
plane during the time step at the vortex trailing point is denoted as Δβv=(vr, in-plane/r)Δt. The in-plane velocity component
perpendicular to the lifting line is denoted as vr, in-plane.
Equations () and () show that the induction consists of
a decreasing part of the induction at the previous time step, due to the
previously trailed wake moving away from the blade, and the contributions
from the newest element, given by the DX,s,v and DY,s,v terms.
These equations are less time-step-sensitive and computationally faster than
the original equations by . They have been proposed by
, as well as a modification of Φ to account for the helix
angle of the trailed vorticity. Wake expansion could in principle be included
by another modification of Φ, but the effect of wake expansion on the
near-wake induction of the individual blades is expected to be negligible.
also describe how the tangential induction is computed based on
the same approach as the axial induction. In Sect. of the present
article a reduction of the exponential function approximation,
Eq. (), to a single term is proposed. This simplified and
computationally cheaper approach removes the Y component in
Eqs. () and ().
Coupling to far-wake model
The NWM, which only computes a fraction of the total rotor induction, is
complemented by a modified BEM model for the far wake. The total induced
velocity at a blade section is computed as
utot=uFW+W,
where uFW is the far-wake component of the induced velocity
and W is the near-wake component (see Eq. ).
The far-wake component uFW is computed based on the BEM model
implementation in HAWC2 that uses a polynomial to relate the local thrust
coefficient with the axial induction factor at each annular element
. Because a part of the total induction is computed by the
near-wake model, the BEM implementation used to determine the far-wake
induction factor aFW has been modified in two ways compared
to the original BEM implementation in HAWC2. First, the tip loss correction is
removed, because the near-wake model accounts for the increasing induced
velocity towards the blade tip . Also, prior to computing the
induction factors, the thrust coefficient and torque coefficient are reduced
by multiplying by a coupling factor kFW<1 to avoid predicting
exaggerated induction levels on the whole rotor disk. The coupling factor is
automatically adjusted during the computation. The goal of the adjustment is
for the coupled near- and far-wake model to closely match the thrust of a
reference BEM model. The reference axial induction factor aref is
computed as in the regular HAWC2 BEM implementation: including tip loss
effects and without reducing the thrust and torque coefficient .
To account for the far-wake dynamics, this work uses the dynamic inflow model
implemented in HAWC2. The model is applied in the same way for axial and
tangential induction, so in the following a scalar notation is used for
simplicity. Two parallel first-order low-pass filters are applied on the
quasi-steady induced velocities uFW, QS=aFWu∞ from
the BEM model:
uFW, dyni=A1u1i+A2u2i,u1i=u1i-1e-Δt/τ1+uFW,QSi1-e-Δt/τ1,u2i=u2i-1e-Δt/τ2+uFW,QSi1-e-Δt/τ2.
In a pure BEM computation and the previous far-wake model implementation, the
factors Ai are A1=0.6 and A2=0.4. They are used to divide the
induction into a faster and slower reacting part, corresponding to a faster
time constant τ1 and the slower time constant τ2. Both time
constants are a function of radius and mean loading. The constants Ai and
τi have been tuned to actuator disk simulations of step changes in
uniform loading .
Cambered airfoil in parallel inflow to the chord line. The shed wake
corresponds to the time history of the bound circulation.
Unsteady airfoil aerodynamics model
The sketch in Fig. illustrates how the shed vorticity due to the
time variation in the bound circulation induces a downwash w3/4 at the
three-quarter chord of an airfoil. This downwash will change the angle of
attack and thus the lift, drag and moment coefficients according to the
airfoil polars, as well as the directions of the aerodynamic forces. The
inviscid part of the unsteady airfoil aerodynamics model by
treats the shed vorticity effects as a time lag on the angle of attack
according to Jones' function for a flat plate. The effective angle of attack
αeff, which determines the magnitude and direction of the
unsteady aerodynamic forces, is computed as
T0i=c2vri,x1i=x1i-1e-0.0455ΔtT0i+12αQSi+αQSi-10.165vri1-e-0.0455ΔtT0i,x2i=x2i-1e-0.3ΔtT0i+12αQSi+αQSi-10.335vri1-e-0.3ΔtT0i,αeffi=12αQSi+x1i+x2i/vri,
where the superscript i denotes the time step and c the chord length.
Further, αQS is the quasi-steady angle of attack resulting
from the velocity triangle at the blade section and vr denotes
the corresponding relative velocity.
Model overview
The structure of the current implementation of the coupled near- and far-wake
model is shown in Fig. . The changes to the previous
implementation (see Fig. ) are
The weighting factors Ai of the far-wake dynamic inflow are adjusted
during the computation to account for the induction computed by the near-wake
model, which is explained in Sect. .
The trailed vorticity is no longer based on the quasi-steady bound
circulation ΓQS but instead on a dynamic bound circulation
Γdyn. The computation of the dynamic bound circulation is
shown in Sect. .
The near-wake induction is computed in an iteration loop, which is
detailed in Sect. .
The coupling factor is no longer needed as input but instead continually
updated during the computation, as described by .
The trailed vorticity is assumed to follow helix arcs to account for the
downwind convection of the trailed vorticity. To achieve this, Φ is
multiplied by a correction function f, depending on the blade section and
vortex trailing point, as well as the helix angle at which the vortex is
trailed .
The computation of αeff according to shed vorticity effects
is improved for cambered airfoils, which is explained in
Sect. .
Overview of one time step in the coupled near- and far-wake model
used in this work. Relevant equation numbers are included. [P] refers to
.
Modifications to far-wake and shed vorticity modelAdapting the weighting of the dynamic inflow time filters
The dynamic inflow model described in Sect. , which has been
tuned for BEM computations, has to be modified if a part of the induction is
covered by the NWM. The objective is to obtain a similar slow induction
response with the coupled near- and far-wake model as with an unsteady BEM
model.
This requires a modification of the constants A1 and A2 in
Eq. (). The new constants Ai are computed based on
the far-wake axial induction factor aFW and a reference axial
induction factor obtained from a BEM model with tip loss correction
(see Sect. ). The
weighting constants for the far-wake model are determined such that roughly
40 % of the total axial induction is considered to be reacting slowly, as
in the original dynamic inflow model for BEM computation,
Eq. ():
A1,FW=0.4arefaFW,A2,FW=1-A1,FW.
The factors are continuously updated during the computations. A first-order
low-pass filter with the far-wake time constant τ2 of the dynamic
inflow model is applied on A1,FW to make sure this model does
not introduce unphysical rapid induction variations due to instantaneous
changes in the weighting factors.
Extensions of the unsteady airfoil aerodynamics modelUnsteady circulation computation
The influence of shed vorticity on the bound circulation buildup has to be
considered when determining the strength of the trailed vortices of the NWM.
Joukowski's relation between quasi-steady lift LQS and circulation
ΓQS,
ΓQS=LQSρvr=12vrcCL,
which has been used by and to determine the
bound vorticity, is not valid for unsteady conditions. The error of
calculating the circulation based on the unsteady lift at an airfoil section
depends on the reduced frequency k=ωc/(2vr), where
ω is the angular velocity, c is the chord length, and vr
is the relative flow speed. For an airfoil pitching harmonically about the
three-quarter chord point, the error has been estimated by
to be 10 % at k=0.1 and 100 % at k=0.8,
which for the NREL 5 MW reference turbine at rated wind and rotor speed
corresponds to frequencies of about 1.2 and 9.8 Hz at 60 m rotor
radius with a chord of 2 m. Except for the first flapwise and
edgewise bending frequencies, most relevant modal frequencies for modern
blades are between these values, which shows that it is important to include
a modeling of the unsteady circulation.
In this paper, the step response of the circulation is approximated by the
three-term indicial function used by .
Γdyn/ΓQS=1-AΓ,1e-bΓ,1τ-AΓ,2e-bΓ,2τ-AΓ,3e-bΓ,3τ,whereτ=Δt2vrc,AΓ,1=0.5547,AΓ,2=0.1828,AΓ,3=0.2656,bΓ,1=0.3064,bΓ,2=0.0439,bΓ,3=3.227.
In the same way as in the shed vorticity model, the dimensionless time τ
depends on the relative velocity vr, which is updated in each
time step. The algorithm is implemented analogue to the computation for the
effective angle of attack in Eqs. ()–():
xΓ,ji=xΓ,ji-1e-bΓ,jΔtT0i+AΓ,j2ΓQSi+ΓQSi-11-e-bΓ,jΔtT0i,Γdyni=xΓ,1i+xΓ,2i+xΓ,3i,
where the quasi-steady circulation is computed from the quasi-steady lift
coefficient using Eq. ().
Unsteady aerodynamics of cambered airfoils
Any change in bound circulation Γ, which is a function of
vrCL (see Eq. ) should lead to the corresponding shed vorticity. The
implementation of the shed vorticity model according to
(see Eqs. –) is based on the term
αQSvr. The camber of the airfoil is neglected in
this computation of the shed vorticity effects. We propose in this work to
replace αQS in Eqs. ()–() by
αQS, camber, with
αQS, camber=αQS-α0,
where α0 is the zero lift angle of the airfoil.
Effect of including camber in the unsteady aerodynamics model on
effective angle of attack during a step in relative velocity.
Comparison of viscous drag and induced drag during oscillations of a
cambered airfoil parallel to the inflow at 1 Hz. The effect of camber
is included as proposed in Eq. (). The mean drag has been
subtracted.
The impact of this modification is shown for basic cases of relative velocity
changes in Figs. and , where an airfoil with a 2π
lift gradient, a 2 m chord length, a zero lift angle of -3∘ and a
drag coefficient of CD=0.005 has been simulated. The airfoil
characteristics and chord length have been chosen to be similar to the
outboard region of the NREL 5 MW reference turbine, and the geometric angle
of attack has been chosen as zero to show the isolated effect of airfoil
camber. In Fig. , the variation in effective angle of attack due
to a step change in relative speed from 70 to 71 ms-1 within a
time step of 0.01 s is shown. Without the effect of camber, the
change in relative speed has no influence on the angle of attack, because
αQS is a constant zero. The effect of camber leads to a lower
angle of attack due to the shed vorticity caused by the increase in bound
circulation. The camber effect is small, and the angle of attack changes only
by less than 0.02∘ immediately after the relative speed step. In
Fig. the induced drag due to angle of attack changes is compared
to the viscous drag in the case of a vibration of the airfoil section
parallel to the inflow. There would be no induced drag in this example if
camber was excluded from the effective angle of attack computation. The
amplitude of the vibration is 1 m and the frequency 1 Hz. The
effect of induced drag is of the same order of magnitude as the airfoil drag,
which indicates the importance of including the airfoil camber in the
unsteady airfoil aerodynamics model. The camber effect is included in all
further computations presented in this paper except in the left plot of
Fig. , where it is excluded to investigate its importance for
in-plane blade vibrations.
In the unsteady circulation computation described in the previous section,
the camber is accounted for through the quasi-steady circulation
ΓQS, which is based on the lift coefficient
(see Eq. ).
Iterative near-wake and shed vorticity modelIteration scheme
The NWM can become numerically unstable depending on the time step, operating
point of the turbine, blade geometry and radial calculation point
distribution . Figure shows the maximum
time step where a stable computation is possible for a fine- and
coarse-geometry definition, shown in Fig. , of the NREL 5 MW
blade. The coarse-geometry definition is a blade geometry typically
distributed for BEM computations and the fine distribution is more suitable
for computations with higher-fidelity codes. The aerodynamic calculation
points and vortex trailing points follow a cosine distribution, which means
they are placed at equi-angle increments. The time steps have been determined
in a numerical experiment, where the time step has been decreased until large
oscillations of the induction disappear. The results are accurate to the
first significant digit. It can be seen that the finer blade geometry leads
to a more stable computation. This can be explained by the smoother blade
tip, where the blade chord is approaching zero. Thus, the radial circulation
gradient at the very blade tip is smaller and the vortex strength of the tip
vortex is distributed to several weaker trailed vortices in the tip region
that are less likely to cause numerical instabilities. In a coupled
aeroelastic simulation, the small stable time steps for resolutions of 30–60
points would lead to a very slow computation, especially in the case of the
coarser blade geometry.
Coarse and fine blade geometry for the NREL 5 MW reference turbine.
The coarse definition is a typical geometry definition for BEM-based
computations. The finer geometry is smoothed for use in computational fluid
dynamics and free-wake codes.
Maximum stable time step depending on the number of points for the
coarse and fine blade geometries of the NREL 5 MW reference turbine. The
points are distributed using a full cosine distribution
. The results are obtained through a numerical
experiment.
The numerical instability which occurs at larger time steps can be explained
as follows: the axial induction due to trailed vortices typically reduces the
angle of attack at a blade section, which in attached flow leads to a reduced
lift. In the original implementation of the NWM the constant circulation
trailed during a time step is only depending on the flow conditions at the
blade at the beginning of a time step. Therefore, a longer time step will lead to a
larger induction and thus a further reduction in lift in the next time step.
If the time step is too large, the induction can become large enough to create
a negative lift in the next time step that is larger in absolute value than
the previous positive lift. This in turn leads to stronger trailed vortices
of opposite sign, which will cause even larger induced velocities in the
opposite direction, which again leads to stronger vortices.
To stabilize the NWM the balance between trailed vortex strength based on the
sectional circulation and the induced velocities are iterated to equilibrium
in each time step, which removes the need for small time steps to stabilize
the aerodynamics model. The iteration is structured as follows:1.
The quasi-steady circulation is computed according to Joukowski's law
using the velocity triangle at the airfoil section based on the induction
from the last iteration.
2.
The unsteady circulation is computed including shed vorticity effects
(see Sect. ).
3.
This unsteady circulation defines the constant vortex strengths trailed
during a time step.
4.
These constant vortex strengths lead to an induction at all airfoil sections.
5.
The new induction is combined from the inductions from step 1 and 4
by applying a relaxation factor: Wi=Wi-1fr+Wi(1-fr), where the subscript i indicates the iteration number. If
Wi is sufficiently close to Wi-1, it is the desired converged
induction.
The BEM model for the far-wake is excluded from this iteration procedure. The
AOA and relative velocity used to compute the far-wake induction are the
values from the converged iteration in the previous time step. This is
accelerating the computation and is feasible because the near-wake effects
are on a much faster timescale than the dynamic inflow effects in the BEM
model.
Estimation of the necessary relaxation factor
In the following, an estimation of the relaxation factor for a blade section
is described. A conservative estimation is based on the least stable case,
which is characterized by the following properties:
One single blade section with one vortex trailing from each side.
Adjacent sections would tend to have similar circulations and therefore
reduce the vortex strengths and the corresponding induction at the blade
section. The trailed vortices on both sides of the section depend only on the
bound circulation Γ of that section.
The lift coefficient is linearly dependent on the angle of attack,
CL=2πα. A reduced but still positive gradient due to
stall would stabilize the model. Note that a 2π lift gradient is only
assumed in the relaxation factor estimation proposed in this section. In all
other parts of the model the lift coefficients and lift gradients according
to the airfoil polars are used.
No prior trailed vorticity is present. It would stabilize the model,
because the induction would be determined not only by the momentary
circulation at the section but also by the decaying influence of the wake
trailed before. If the model converges in the very first time step, with a
given induction at the section from the previous iteration, then the
iterations will also converge with prior trailed vorticity.
The helix angle at which the vortices are trailed is assumed to be small.
Thus, all the induction due to trailed vorticity is assumed to be axial
induction.
With these assumptions, the downwash after a time step Δt can be
determined by summing up the contributions of the newest element
(see the right terms
in Eqs. and ), for both adjacent vortices:
Wi=∑v=12(-1)vΓDX,v1-e-Δβ/Φv+DY,v1-e-4Δβ/Φv,
where the subscript v denotes the vortex further inboard (v=1) and
outboard (v=2) of the section with the bound circulation Γ. The
subscript i denotes the iteration. Because the tangential induction is
neglected, Δβ is only a function of the rotation speed of the
turbine and the time step. Thus, Γ is the only variable in
Eq. () that depends on the induction at the section:
Γ=ΓdynΓQS12cCLvr,=ΓdynΓQScπvrα,=ΓdynΓQScπv∞-Wi-12+(Ωr)2arctanv∞-Wi-1Ωr,
where v∞ is the free wind speed. The step response function from
Eq. () evaluated at half a time step approximates
Γdyn/ΓQS because we consider a buildup of the
circulation from zero.
Equation () is plotted for different time steps as a function of
Wi-1, the induction from the previous iteration, for the blade tip
section of the NREL 5 MW reference turbine at 8 ms-1 wind speed
and 9.2 rpm rotor speed, corresponding to a tip speed ratio of 7.6, in
Fig. . The airfoil camber is neglected.
Downwash after an iteration as a function of the downwash from the
preceding iteration in the case of a single section with trailed vortices.
The intersections of the curves with the blue curve (Wi=Wi-1) are the
converged solutions, where a new iteration would lead to exactly the same
induction as the previous one. The calculations for different time steps have
different converged inductions, because the length of the trailed vortex
filaments is proportional to the time step. However, not only the converged
solution but also the gradient of the curves changes, which leads to a condition
for convergence: if the distance from the converged solution decreases during
a time step,
Wi-Wconv<Wi-1-Wconv,
the iterative process converges. As seen in Fig. , the gradient of
the curves is almost independent of Wi-1. The gradients are negative
because induction reduces the angle of attack. Therefore, an approximation of
condition () can be used:
dWidWi-1>-1.
This gradient can be derived from Eqs. () and () as
dWidWi-1=ΓdynΓQSπcB1-B2α(v∞-Wi-1)vr+vrΩrv∞-Wi-1Ωr2+1,whereBv=(-1)v(DX,v(1-e-Δβ/Φv)+DY,v1-e-4Δβ/Φv).
The gradient is mainly depending on the time step and point density (through
B1 and B2) and the rotational speed.
Instead of reducing time step and point density until a simulation is stable,
which can lead to time steps orders of magnitude smaller than commonly used
in aeroelastic codes and low spatial resolution, a relaxation factor
fr can be introduced, so that
Wi,r=Wi(1-fr)+Wi-1fr.
The derivative of this downwash with regard to the old downwash is
dWi,rdWi-1=dWidWi-1(1-fr)+fr.
For the minimum relaxation factor fr, which allows for a stable
computation (dWi,r/dWi-1=-1), follows
fr=-1+dWidWi-11-dWidWi-1,
which can be determined depending on the time step Δt, the point
distribution, and the number of points on the blade.
In the initial phase of the simulation, the maximum relaxation factor for all
blade sections can be quickly determined by setting Wi-1=0 in
Eq. () and looping through the sections. The highest
necessary relaxation factor for one section that has been found is then used
for the axial and tangential induction on the whole blade. As the simulation
continues, the relaxation factor can be updated whenever there are large
changes in rotational speed, induction, or blade pitch. If the relaxation
factor is updated every several time steps, then determining the relaxation
factor takes negligible computation time. Choosing a slightly more
conservative relaxation factor than what has been estimated will also ensure
stability in different conditions than the ones the factor was based on.
Accelerating the NWM
In this section, an approach to accelerate the model is presented. The number
of exponential terms used to approximate the decreasing induction with
increasing distance from the blade in Eq. () is reduced to one.
Using only one exponential term removes the Yw component in the
near-wake algorithm, Eq. () and thus halves the computation time.
The reduced approximation function is defined as
dwdw0≈1.359e-β/Φ-0.359e-4β/Φ≈A∗e-β/Φ∗.
The values of A∗ and Φ∗ are found by solving the following equations:
W(β=∞)=∫0∞1.359e-β/Φ-0.359e-4β/Φdβ=∫0∞A∗e-β/Φ∗dβ,∫0∞W(β=∞)-W(β)dβ=∫0∞Φ1.359e-β/Φ-0.3594e-4β/Φdβ=∫0∞Φ∗A∗e-β/Φ∗dβ.
Equation () ensures that the quasi-steady induction W(β=∞) of the reduced model is equal to the one computed by the
original model for a trailed vortex with constant strength.
Equation () ensures a good dynamic behavior by requiring
the time integral of the difference between dynamic and quasi-steady
induction to be identical to the original model. The solution to these
equations is
A∗=(1.359-0.359/4)21.359-0.359/16,Φ∗=Φ1.359-0.359/161.359-0.359/4.
A comparison of the buildup of induction in time, corresponding to the
integral of the exponential functions, is shown in Fig. . The
largest deviations of the reduced model from the original model are below
2.5 % of the quasi-steady induction W(β=∞).
Comparison of induction buildup between full NWM and reduced NWM,
depending on the length of a trailed vortex filament with constant
circulation.
Free-wake code
GENUVP is a potential flow solver combining a panel representation of the
solid boundaries (blades) with a vortex particle representation of the wake.
In the present work, the blades are considered as thin lifting surfaces
carrying piecewise constant dipole distribution (equivalent to horseshoe-type
vortex filaments). Blades shed vorticity in the wake along their trailing
edges and their tips (vorticity emission line). In the model a hybrid wake
approach is followed, which refers to the mixed formulation used in the
representation of the wake. In this formulation, the dipole representation is
retained for the near part (equivalent to horseshoe filaments), while the far
part is modeled by free vortex particles. The near-wake part, consisting of
the newly shed vorticity trailed within the current time step, is modeled as
a vortex sheet also carrying piecewise constant dipole distribution. Within
every time step, a strip of wake panels is released that are in contact with
the emission line. By applying the no-penetration boundary condition at the
center of each solid panel and the Kutta condition along the emission line,
the unknown dipole intensities are determined. Then, at the end of each time
step, the newly shed vorticity is transformed into vortex particles and
all vortex particles are convected downstream with the local flow velocity
(free-wake representation) into their new positions. The layout of the
modeling is shown in Fig. . Details of the model can be found in
.
Layout of the free-wake modeling of a blade: black lines define the
blade surface panels, red lines define the wake generated within a time step,
and symbols represent freely moving particles.
Since GENUVP is defined as a potential flow solver, the loads need correction
in order to account for viscous effects. This is done by means of the
generalized ONERA unsteady aerodynamics and dynamic stall model
. The potential load is calculated by integrating pressures
(pressure differences between pressure and suction side) over the lifting
surfaces. Then, viscous corrections are applied to the potential sectional
loads that require the local flow velocity and angle of attack at
every section as input. In particular, the ONERA model splits the aerodynamic forces
into a potential and a separated flow component. This is done through the
introduction of two equivalent circulation parameters defined both for the
lift and the drag force. In GENUVP and in the case of attached flow conditions, no
correction is applied on the unsteady lift force computed by the free-wake
code. The only correction applied is the inclusion of the viscous drag
contribution to the loads. In the case of separated flow conditions the separated
flow component of the ONERA model is superimposed to the potential loads
provided by the free-wake model .
In the case of a flexible blade, flow equations are solved for the deformed blade
geometry, while deformation velocities are accounted for in formulating the
non-penetration boundary condition.
The GENUVP free-wake code has been thoroughly validated over the past years
against measured data both on wind turbines and helicopter rotors in the
framework of numerous EU-funded projects. Blade loads and wake velocities
comparisons against measurements have been performed on the MEXICO rotor in
the context of Innwind.eu project . Moreover, detailed blade
load calculations have been performed for the NREL test rotor and results
have been compared to experimental data (NREL experiment) and computational fluid dynamics
computations . Extensive validation of the code has been also
performed in the framework of the HeliNovi project, where aerodynamic and
structural loads, wake velocities and elastic deflections have been compared
to tunnel measurements on a BO105 helicopter model .
Results
In the following section, the effectiveness of the iteration procedure and
the estimation of the relaxation factor are demonstrated for a horseshoe
vortex. Then, in Sect. , the unsteady induction predicted by
the coupled near- and far-wake model is compared with results from an unsteady
BEM model and the free-wake code described in Sect. .
Pitch steps and prescribed vibrations of the blades of the NREL 5 MW
reference turbine are investigated.
Iteration procedure
To illustrate the efficiency of the iterative implementation, induction
buildups for a simplified case are shown in Fig. . The simple test
case is a wing with a span of 0.3 m and a constant bound circulation,
so that only two vortices with opposite vortex strength are trailed at the
edges. To use the NWM and ensure parallel flow, the wing is modeled as the
only aerodynamic section at the end of a 10 km long blade. The free stream
velocity is 70 ms-1. At t=1s, the geometric AOA of
the wing with a symmetrical profile is increased from 0 to 5∘ within
0.02 s. The lift coefficient is 2πα and the chord 1 m.
The left side of Fig. shows the induction buildup for different
time steps without iterating, while the right side shows the effect of the
iteration procedure. Both the overshoot of the induction for a time step of
0.002 s and the oscillations for a time step of 0.02 s are
reduced by the iteration procedure.
The relaxation factors estimated as proposed in Sect. are
compared with the lowest stable relaxation factors obtained by
trial and error in Fig. for the NREL 5 MW reference turbine
operating at 8 and 25 ms-1 wind speed in uniform inflow. The
comparison in Fig. shows that the estimated relaxation factor is
conservative, but the safety margin towards unstable computation is smaller
in the 25 ms-1 case.
Buildup of the downwash for a horseshoe vortex depending on the time
step. The NWM tends to be unstable (left) but can be stabilized by iterating
to convergence of the downwash (right).
Estimated relaxation factor compared with the lowest stable
relaxation factor from trial and error depending on the number of aerodynamic
sections. The time step is 0.02 s. The estimated relaxation factors,
Eq. (), are conservative and the influence of the refined
blade geometry is captured.
Comparison of the coupled model with a BEM model and a free-wake panel code
In the following Sect. , the main differences between
the coupled near- and far-wake model and the free-wake code are described. It
is also highlighted how the validation cases in the following sections
(predicted force responses to pitch steps, Sect. ; blade
vibrations, Sect. ) are chosen to investigate the effects of the
various model differences. All computations use the refined blade model shown
in Fig. . The blade has been discretized using 40 radial
aerodynamic stations in the BEM-based codes, with corresponding 41 vortices
in the coupled model trailed from root, tip and in between stations. For the
lifting-surface free-wake simulations, the blade has been discretized using
35 spanwise and 11 chordwise grid lines. Compared to the faster models,
the resolution was mainly reduced close to the blade root.
Inherent differences between coupled near- and far-wake model and free-wake code
This paragraph contains an overview of the inherent modeling differences
between the different models. It is also detailed how these differences are
investigated in the following comparisons of aerodynamic response to pitch
steps and prescribed blade vibrations.
The free-wake code uses a lifting-surface approach, while the near-wake model
uses a lifting-line approach. The free-wake code models the airfoils as the
camber line of the airfoil sections (thin airfoil approach) using potential
theory. Thereby, the different sections of the blade have a lift gradient of
2π and zero drag. Drag and the lift gradient deviation from 2π is
then added afterwards as part of the ONERA unsteady airfoil aerodynamics
model. The faster models, on the other hand, all use the airfoil data
directly. Thus, the measured lift gradients are used for both the shed
vorticity model and the near-wake model. The influence of this difference is
evaluated in the vibration comparisons section by setting the lift gradient
to 2π and drag to zero in the case of an edgewise vibration, where drag
contributes significantly to the aerodynamic work.
The time constants for the shed vorticity model in the BEM-based codes do not
contain a correction for airfoil thickness but are instead the
approximations for a flat plate originally obtained by Jones. The flat plate
approximation agrees with the thin lifting surface in the free-wake
simulations. The comparisons of the aerodynamic response to prescribed blade
vibrations contain also results from a BEM model with deactivated shed
vorticity model. These results are included to evaluate the isolated
influence of the shed vorticity modeling and distinguish the dynamic effects
of shed and trailed vorticity. In the free-wake code, the shed vorticity is
inherently modeled and can not be turned off. In contrast to the fast dynamic
effects due to trailed and shed vorticity close to the blade, there are also
slow dynamic inflow effects. This term is used here to describe effects that
would also be visible in actuator disk simulations where the individual
blades are not modeled at all. The slow dynamic effects are modeled directly
in the free-wake code and by means of a dynamic inflow model in the other
codes. The influence of these effects is compared for pitch steps where the
free-wake code results directly model the influence of wake expansion. In the
case of the blade vibrations, on the other hand, the main effects occur in
the direct wake close to the blade. The influence of dynamic inflow in these
cases is very small, which is reflected in the large time constants in the
modeling. Wake expansion is also expected to be of minor importance in these
cases.
Scaled axial force at different radial positions during and after a
pitch step by 5∘ in 1 s.
Scaled axial force at different radial positions during and after a
pitch step by 5∘ in 4 s.
Another difference between the free-wake code and the BEM-based models is
that the dynamic interaction between a blade and the wake of the other blades
is only modeled in the free-wake code. This introduces dynamic variations in
the pitch step cases that are missing in the BEM-based codes. If this
blade–wake interaction were to play an important role in the vibration cases,
the agreement between the codes should be better in the high wind speed cases
with a larger helix angle than in the low wind cases, where the wake of
previous blades is closer to the rotor plane when a blade is passing.
In order to avoid additional uncertainties due to dynamic stall modeling, all
cases have been chosen such that stall is mostly avoided. To obtain this, the
pitch steps are conducted from 5∘ to feather towards normal
operation, and the amplitudes in the vibration cases are generally small.
There may be stall at the very root in the prescribed vibration cases, but
the amplitude there is almost zero. Thus, there is very little contribution
from the root section to the aerodynamic work and differences in dynamic
stall modeling do not visibly change the results.
Pitch steps
Pitch steps with stiff blades have been performed where the NREL 5 MW
reference turbine is operating at a wind speed of 8 ms-1 and a
rotation speed of 9.2 rpm. The turbine starts with blades that are
pitched by 5∘ to feather. The time steps are 0.054 s
(120 steps per revolution) for GENUVP and 0.05 s for the other
models. After 60 s simulated time, the blades are pitched to 0∘ at
constant pitch rate in 1 or 4 s. The forces are normalized to compare
the dynamics of the pitch response, such that the force before the pitch step
is 0 and the force 45 s after the pitch step is 1.
Left plot: force distribution before the partial pitch step and
50 s after. Right plot: time history of the axial force comparing
coupled model and GENUVP at 21.6 m radius (34.3 % rotor radius). The BEM
gives a constant force.
Figure show the axial force response at a position at mid-blade
and close to the blade tip for the fast pitch step. The free-wake code
predicts a slower force response during the pitch step than the BEM model.
The results of the coupled model during the pitch step lie in between the
other codes. In the free-wake code results, some oscillations are present
after the pitch step, especially at the mid-blade section. The oscillations
occur at a 3P frequency and are likely due to the individual blades passing
by the trailed vortex wakes of the other blades. These oscillations make it
difficult to judge whether the BEM model or the coupled model is predicting the
overshoot closer to the free-wake code, the results of which are in between
the two.
The results of the slower pitch step in Fig. show less
oscillations of the free-wake code results. In this case, the predicted
results from the coupled model clearly agree better with the free-wake code
than the BEM results, both on the slope during the pitching motion and on the
predicted overshoot.
Axial force distributions for a partial pitch comparison at
8 ms-1 are shown in Fig. . In this case, only the
outer half of the blade is pitched from 5 to 0∘ over 1 s.
As shown in the left plot of Fig. , the trailed vorticity effect
at the mid-blade is predicted by both the coupled aerodynamics model and the
free-wake code to a similar degree. The effect looks like a radial smoothing
of the loading, because the trailed vorticity increases the angle of attack
inboard of the pitched blade half and decreases the angle of attack at the
beginning of the higher loaded outer half of the blade. In the right plot of
Fig. , the time history of the axial force is compared at
34.3 % radius. The constant force predicted by the BEM model on this
non-pitching part of the blade is not included in this comparison. The
overshoot is underpredicted by the coupled model by around 40 %. The
induction predicted by the coupled model stops increasing after
1.6 s, corresponding to a quarter revolution at 9.2 rpm,
while the force predicted by GENUVP continues to increase. Figure
thus illustrates that the coupled model can predict a change in loading that
can not be computed based on BEM theory, but the restriction to a quarter
revolution is limiting in this case. The difference in the overshoot
prediction at 21.6 m radius amounts to roughly 70 Nm-1.
Prescribed vibrations
The aerodynamic response to blade vibrations is investigated for normal
operation at 8 and 25 ms-1. The corresponding rotor speeds are
9.2 and 12.1 rpm and the pitch angles 0 and 23.2∘,
respectively. The force response is compared in terms of radial distributions
of aerodynamic work during one oscillation, where a positive aerodynamic work
corresponds to a positive aerodynamic damping of the vibration. The mode
shapes are chosen as the first and second structural mode shapes of the NREL
5 MW reference turbine blade at standstill (see Fig. ). To simplify the comparison, the
vibrations have been prescribed as collective in-plane or out-of-plane
vibrations. The frequencies, amplitudes and time steps used for the
computations are shown in Table , as well as the modal
masses that are used for damping estimations. For the first modes only
results at the larger amplitudes are shown in the following. Investigating
the results at the smaller amplitudes leads to the same conclusions.
Mode shapes used in the work computations, which are simplified to
be purely in-plane or out-of-plane deflections.
Modal parameters and time steps prescribed in the work comparison.
The time steps for the first flap were chosen as 180 steps per revolution and
thus depend on the rotor speed.
In the BEM and coupled model, the blade section velocities due to the
vibrations are added to the relative wind speed. The deflection of the blade
and the resulting change in the section positions have been neglected because
the amplitudes are small compared to the blade radius. In the free-wake code,
not only deflection velocities but also deformation of the blade shape is
considered.
Aerodynamic work per oscillation of first flap motion at
8 ms-1 (left) and 25 ms-1 (right) wind speed with
an amplitude of 0.5 m.
Aerodynamic work per oscillation of first edge motion at
8 ms-1 at an amplitude of 1 m. Drag has been excluded,
and CL=2πα. In the left plot, αQS is used
in the unsteady airfoil aerodynamics model in both coupled model and BEM
(see
Eqs. –). In the right plot, the unsteady
airfoil aerodynamics model uses αQS-α0
(see Eq. ),
improving the agreement with the free-wake code results.
To distinguish the effects of shed and trailed vorticity, the following
comparisons include a BEM computation with a disabled shed vorticity model.
The aerodynamic work during out-of-plane motion according to the first flap
mode shape is shown in Fig. for 8 and 25 ms-1. The
work integrated over the blade is overpredicted by the BEM model (dashed
black line) by about 10 % compared to the free-wake code in both cases.
If the three BEM-based results are compared, it shows that the trailed
vorticity (dotted blue line versus dashed black line) has an influence of the
same order of magnitude as the shed vorticity (solid purple versus dashed
black line). The trailed vorticity effects are more important close to the
tip vortex, while the influence of the shed vorticity extends across the
whole blade. The results of the coupled model are very close to the free-wake
code results but deviate slightly towards higher work. The influence of the
trailed vorticity behind the other two rotor blades, which is not included in
the NWM, on the vibration response is found to be small compared to the
influence of the wake of the blade itself in normal operation.
The in-plane vibrations at 8 ms-1 are almost parallel to the
inflow and the drag forces contribute much more to the work than in the other
cases. To simplify the problem, drag has been excluded from the aerodynamic
work computations presented in Fig. . Further, the lift gradient
has been assumed as 2π. In the left plot of Fig. , the
quasi-steady angle of attack is used in the unsteady airfoil aerodynamics
model (see
Eqs. –). The agreement in this case is
poor. In the right plot, the zero lift angle due to camber is included in the
quasi-steady angle of attack (see
Eq. ). This approach leads to a much improved result for the
BEM-based codes and good agreement of the coupled model and GENUVP.
With the airfoil polars of the NREL 5 MW reference turbine the agreement
between the codes is not as good in the 8 ms-1 case; see the
left plot of Fig. .
Aerodynamic work per oscillation of first edge motion at
8 ms-1 (left) and 25 ms-1 (right) wind speed with
an amplitude of 1.0 m.
However, the coupled model produces results much closer to the free-wake code
close to the blade tip than the BEM model. At 25 ms-1, where the
work is predominantly due to the vibration component perpendicular to the
inflow as a result of blade pitch, the coupled near- and far-wake model agrees
similarly well with the free-wake code as in the cases with out-of-plane
vibrations discussed above. The shed vorticity effects on the in-plane
vibrations are larger than on the out-of-plane vibrations due to the higher
frequency and the larger relative velocity variations.
Figure shows results for the second modes.
Aerodynamic work per oscillation of second flap (top) and edge
(bottom) motion at 0.25 m amplitude at 8 ms-1 (left) and
25 ms-1 (right).
The BEM model results compare similarly well with the GENUVP results as for
the first modes. Because the frequencies of the second modes are higher the
shed vorticity model is more important in these cases. The importance of the
trailed vorticity model at higher frequencies does not increase by the same
amount, because the higher reduced frequencies affect the buildup of the
unsteady bound circulation (see
Eq. ). The coupled model results are closer to the free-wake
results than the BEM results in all cases, but as opposed to the comparisons
above, the coupled model is underestimating the effects of the dynamics of
the tip vortex. Further inboard the free-wake code predicts slightly lower
aerodynamic work in the 25 ms-1 case than the BEM model, which
can not be seen in the coupled model results. Also, the agreement of the
coupled model and GENUVP is worse in the edgewise case than in the flapwise
case at 25 ms-1, which has not been seen to that extend for the
first edgewise cases (see
Figs. and ). A reason for this might be that the
second edgewise case is computed with fewer time steps per period of
oscillation to limit the computational cost.
For easier evaluation of the force response differences, the aerodynamic work
can be expressed in terms of a damping ratio of a respective blade mode.
Because the computations have been based on prescribed purely in-plane and
out-of-plane structural mode shapes, these dampings do not correspond to any
aeroelastic blade modes. For a system with a single degree of freedom with the
modal mass m and frequency f, given in Table , the
damping ratio ξ and logarithmic decrement δ are
ξ=Waero8π3A2f2m=11+2πδ2,
where A is the amplitude and Waero the aerodynamic work per
oscillation period. The estimated logarithmic decrements according to
Eq. () corresponding to the first flap motion at
8 ms-1 with an amplitude of 0.5 m
(see Fig. ) are
334 % for the BEM results, 300 % for the coupled model and 292 %
for the free-wake code results. Flapwise modes are highly damped and thus
these changes in the damping will not significantly alter the blade fatigue
loads. On the other hand, lower aerodynamic damping of flapwise blade motion
will correspond to a lower aerodynamic damping of tower fore–aft motion and
might thus lead to increased tower fatigue loads. It is expected that this
lower aerodynamic damping is balanced to some degree because the near-wake
effects reduce the aerodynamic excitation due to atmospheric turbulence.
Aerodynamic damping estimations for the first in-plane vibrations at
8 ms-1 at 1 m amplitude are shown in Table .
The damping has been estimated in four cases, which differ in the airfoil
polars and the modeling of the camber effect on the unsteady airfoil
aerodynamics. Comparison of the first two cases of Table
shows that the induced drag caused by airfoil camber in the shed vorticity
modeling results in a damping of roughly 0.7 % logarithmic decrement.
According to the BEM and coupled model results in cases (3) and (4), the
airfoil drag increases the logarithmic decrement by about 0.3 %. The
trailed vorticity decreases the absolute value of the damping by roughly
0.14 % log decrement. Further, comparing columns (2) and (4), the
combined influence of airfoil polars with lift coefficients other than 2π and non-zero drag is close to 3 times larger in the free-wake code
computations, which is caused by the different unsteady drag modeling. The
small differences in estimated logarithmic decrement can have an impact on
loads and stability computations for edgewise modes with a very low
aeroelastic damping.
In the out-of-plane prescribed vibration cases investigated, the trailed
vorticity reduces the aerodynamic work. Further, a previous study by
showed that the trailed vorticity effects will delay the
onset of flutter towards higher rotor speeds. This is in agreement with
findings on the influence of shed vorticity, which leads to both a decrease
of the flapwise damping and increased flutter speeds of a vibrating 2-D blade
section .
Conclusions
In this paper, several modifications of a coupled model
consisting of a trailed vorticity model for the near-wake and a BEM-based
model for the far wake have been presented and validated. Results from the
coupled model are compared to free-wake panel code and a BEM model to
evaluate the benefits and limitations of the added trailed vorticity
modeling.
Estimated logarithmic decrements [%] corresponding to the
aerodynamic work of first in-plane vibrations at 8 ms-1, based
on the results at an amplitude of 1 m.
It has been shown that the acceleration of the model by reducing the number
of exponential functions in the trailing wake approximation from two to one
is possible with negligible effect on the results. The approach presented
here does not change the steady results predicted by the NWM.
An iteration scheme to stabilize the model has been presented. It applies a
relaxation factor that is computed dynamically based on the blade
discretization and the operating point of the turbine. To evaluate the
computed relaxation factors, minimum necessary relaxation factors have been
determined by trial and error and the estimated factors are found to be
conservative. The iterative process enables stable computations without the
need for very small time steps and reduces oscillations of the near-wake
induction.
The 2-D shed vorticity modeling, based on thin airfoil theory, has been
extended by including the unsteady effects on the bound circulation. Further,
it has been found that it is necessary to include airfoil camber in the
modeling of the influence of varying inflow velocity on the dynamic angle of
attack to obtain good results if the direction of vibration is close to
parallel to the inflow direction.
A comparison of pitch step responses of the NREL 5 MW reference turbine using
the coupled near- and far-wake model, a BEM model based on the aerodynamics
model in HAWC2 and the free-wake panel code GENUVP has been presented. The
trailed vorticity modeling in the coupled model gives results closer to the
free-wake code than the BEM model during the pitching motion, and for a slow
pitching rate a clear improvement is seen in the computation of the
overshoot. Fast pitch rates resulted in oscillations due to the motion of the
wake in the free-wake code, which could not be achieved in the coupled model
due to the prescribed wake assumption. The response to a partial pitch of the
outer half of the blade demonstrated the cross-sectional aerodynamic
coupling, which will have an influence on the load distribution in the
presence of trailing edge flaps.
The coupled model agreed better than the BEM model with the free-wake code in
all prescribed vibration cases investigated. The main improvement due to the
trailed vorticity is found close to the tip of the blade, even in the case of the
higher modes investigated. The work response to the edgewise vibrations has
been found to be difficult to model if the direction of vibration is close to
parallel to the inflow direction. The results in this case compare much
better if no drag forces are computed. If drag is included, the coupled model
still compares well with the free-wake code close to the blade tip, but there
are larger deviations in the results of all models further inboard. In
general, the simulations agreed better for out-of-plane vibrations than
in-plane vibrations.
The implementation of the coupled near- and far-wake model presented here
delivers promising results and will be further investigated and validated
against computational fluid dynamics results and measurements in future work.
In particular, the more accurate prediction of aerodynamic work for edgewise
vibrations is considered to be important for stability analyses and load
predictions due to the low aeroelastic damping typically associated with
these vibrations.
Data availability
A repository contains the source code of the aerodynamics
program used for the BEM and coupled wake model computations. The relevant
input data for the NREL 5MW reference turbine are also
included.
NomenclatureSymbolUnitsDescriptionA[m]Amplitude of vibrationA1, A2[–]Dynamic inflow weighting factorsAΓ,1, AΓ,2, AΓ,3[–]Coefficients in bound circulation step responseA*[–]Coefficient of accelerated near-wake modelaFW[–]Far-wake induction factoraref[–]Induction factor according to BEM polynomialbΓ,1, bΓ,2, bΓ,3[–]Dimensionless time constants in bound circulation step responseCD[–]Drag coefficientCL[–]Lift coefficientCL′[–]Gradient of lift coefficient with respect to angle of attackCT[–]Thrust coefficientc[m]Chord lengthDX,s,v[m-1]Slowly decaying component of induced velocity due to infinitely long vortex arc v with vortex strength 1 at section sDY,s,v[m-1]Fast decaying component of induced velocity due to infinitely long vortex arc v with vortex strength 1 at section sdw[ms-1]Induced velocity due to infinitesimal vortex elementdw0[ms-1]Induced velocity due to infinitesimal vortex element starting at the bladeFaero[Nm-1]Aerodynamic forces per unit radiusFip[Nm-1]In-plane component of the aerodynamic forces per unit radiusFoop[Nm-1]Out-of-plane component of the aerodynamic forces per unit radiusF[–]Tip loss factorf[Hz]Frequency of vibrationfr[–]Relaxation factork[–]reduced frequencykFW[–]Coupling factorL[Nm-1]Lift force per unit radiusm[kg]Modal massNv[–]Number of vortex arcs trailed from a bladeu1, u2[ms-1]Components of time-filtered far-wake-induced velocityuFW,dyn[ms-1]Time filtered far-wake-induced velocityuFW,QS[ms-1]Quasi-steady far-wake-induced velocityutot[ms-1]Total induced velocity due to the combined near and far wakeu∞[ms-1]Free wind speedT0[s]Time constant for unsteady airfoil aerodynamics modelT[s]Period of oscillationΔt[s]Time stepvip[ms-1]In-plane component of the blade section velocityvoop[ms-1]Out-of-plane component of the blade section velocityvr, in-plane[ms-1]In-plane component of relative velocityvr[ms-1]Relative velocityWaero[Nmm-1]Aerodynamic work during one period of oscillation per unit radiusWs,v[ms-1]Induced velocity due to vortex arc v at section sWs[ms-1]Induced velocity due to all vortex arcs at section sXs,v[ms-1]Slowly decaying component of induced velocity due to vortex arc v at section sx1, x2[ms-1]Components of effective angle of attackxΓ,1, xΓ,2, xΓ,3[m2s-1]Components of bound circulationYs,v[ms-1]Fast decaying component of induced velocity due to vortex arc v at section sαQS[rad]Geometric angle of attackαQS, camber[rad]Geometric angle of attack relative to zero lift angleα0[rad]Zero lift angleαeff[rad]Effective angle of attackβ[rad]Angle a blade has rotated since a vortex has been trailedΔβ[rad]Angle a blade rotates in one time stepδ[–]Logarithmic decrementΓQS[m2s-1]Bound circulationΔΓ[m2s-1]Trailed vortex strengthΩ[rads-1]Rotor speedω[rads-1]Angular velocityΦ[–]Geometric parameter determining how fast the influence of a trailed vortex element decaysΦ*[–]Geometric parameter of accelerated near-wake modelτ1, τ2[s]Dynamic inflow time constantsτ[–]Dimensionless time in bound circulation step responseξ[–]Damping ratio
The authors declare that they have no conflict of
interest.
References
Andersen, P. B.: Advanced Load Alleviation for Wind Turbines using Adaptive
Trailing Edge Flaps: Sensoring and Control, PhD thesis, The Technical
University of Denmark, Roskilde, Denmark, 133 pp., 2010.
Bak, C., Bitsche, R., Yde, A., Kim, T., Hansen, M., Zahle, F., Gaunaa, M.,
Blasques, J., Døssing, M., Wedel Heinen, J., and Behrens, T.: Light Rotor:
The 10-MW reference wind turbine, in: Proceedings of EWEA 2012 – European
Wind Energy Conference & Exhibition, European Wind Energy Association
(EWEA), 10 pp., 2012.
Beddoes, T. S.: A near wake dynamic model, Aerodynamics and Aeroacoustics
National Specialist Meeting, 25–27 February 1987, Arlington, TX, USA,
1987.
Chassapoyiannis, P. and Voutsinas, S. G.: Aerodynamic Simulations of the flow
around a Horizontal Axis Wind Turbine using the GAST software,
Deliverable 2.2 of THALIS – NTUA: Development of know-how for the
aeroelastic analysis and the design, optimization of wind turbines MIS:
379421 project, 2013.
Dieterich, O., Langer, H., Sneider, O., Imbert, G., Hounjet, M., Riziotis,
V., Cafarelli, I., Calvo Alonso, R., Clerc, C., and Pengel, K.: HeliNOVI:
Current vibration research activities, 31st European Rotorcraft forum,
Florence, Italy, 13–15 September 2005, 83.1–83.14, 2005.Hansen, M. H.: Aeroelastic instability problems for wind turbines, Wind
Energy, 10, 551–577, 10.1002/we.242, 2007.
Hansen, M. H., Gaunaa, M., and Madsen, H. A.: A Beddoes-Leishman type dynamic
stall model in state-space and indicial formulations, Risø-R-1354,
Roskilde, Denmark, 2004.
Jonkman, J., Butterfield, S., Musial, W., and Scott, G.: Definition of a 5-MW
Reference Wind Turbine for Offshore System Development, National Renewable
Energy Laboratory, Tech. rep., NREL/TP-500-38060, 75 pp., 2009.
Kim, T., Hansen, A. M., and Branner, K.: Development of an anisotropic beam
finite element for composite wind turbine blades in multibody system, Renew.
Energ., 59, 172–183, 2013.Larsen, T. J. and Hansen, A. M.: How 2 HAWC2, the user's manual, Denmark,
Forskningscenter Risoe, Risoe-R-1597, 2007.
Larsen, T. J., Madsen, H. A., Larsen, G. C., and Hansen, K. S.: Validation of
the dynamic wake meander model for loads and power production in the Egmond
aan Zee wind farm, Wind Energy, 16, 605–624, 10.1002/we.1563, 2013.
Madsen, H. A. and Gaunaa, M.: Udvikling af model for 3D induktions- og
stallmodellering, Risø-R-1509(DA), Roskilde, Denmark, 2004.Madsen, H. A. and Rasmussen, F.: A near wake model for trailing vorticity
compared with the blade element momentum theory, Wind Energy, 7, 325–341,
10.1002/we.131, 2004.Madsen, H. A., Bak, C., Døssing, M., Mikkelsen, R., and Øye, S.:
Validation and modification of the Blade Element Momentum theory based on
comparisons with actuator disc simulations, Wind Energy, 13, 373–389,
10.1002/we.359, 2010.
Madsen, H. A. (Ed.), Prospathopoulos, J., Voutsinas, S., Riziotis, V.,
Diakakis, K., Chassapoyiannis, P., Gomez-Iradi, S., Echarri, X. M., Ruiz,
A. I., Shen, W. Z., and Sørensen, N. N.: Validation of high rotational
speed aerodynamics by wind tunnel tests, Deliverable 2.13, InnWind project,
2015.
Petot, D.: Differential equation modeling of dynamic stall, La Recherche
Aerospatiale, English Edn., 59–72, 1989.Pirrung, G. R.: Aerodynamics code used in Wind Energy Science paper
“Comparison of a coupled near- and far-wake model with a free-wake vortex
code”, Data set, Zenodo, 10.5281/zenodo.242498, 2017.Pirrung, G. R., Hansen, M. H., and Madsen, H. A.: Improvement of a near wake
model for trailing vorticity, J. Phys. Conf. Ser., 555, 012083,
10.1088/1742-6596/555/1/012083, 2012.Pirrung, G. R., Madsen, H. A., and Kim, T.: The influence of trailed
vorticity on flutter speed estimations, J. Phys. Conf. Ser., 524, 012048,
10.1088/1742-6596/524/1/012048, 2014.Pirrung, G. R., Madsen, H. A., Kim, T., and Heinz, J.: A coupled near and far
wake model for wind turbine aerodynamics, Wind Energy, 19, 2053–2069,
10.1002/we.1969, 2016.
Riziotis, V. A. and Voutsinas, S. G.: Dynamic Stall on Wind Turbine Rotors:
Comparative Evaluation Study of Different Models, in: Proceedings of the 1997
European Wind Energy Conference and Exhibition, Dublin, Ireland, October
1997.
Sørensen, N. N. and Madsen, H. A.: Modelling of transient wind turbine
loads during pitch motion, European Wind Energy Association (EWEA), 786–795,
2006.
Voutsinas, S. G.: Vortex Methods in Aeronautics: How to make things work,
Int. J. Comput. Fluid D., 20, 3–18, 2006.