WESWind Energy ScienceWESWind Energ. Sci.2366-7451Copernicus PublicationsGöttingen, Germany10.5194/wes-2-521-2017Trailed vorticity modeling for aeroelastic wind turbine simulations in standstillPirrungGeorg R.gepir@dtu.dkhttps://orcid.org/0000-0001-9260-1791MadsenHelge A.https://orcid.org/0000-0002-4647-3706SchreckScottWind Energy Department, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, DenmarkNational Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, CO 80401, USAGeorg R. Pirrung (gepir@dtu.dk)20November2017225215327January20176March20176September201710October2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://wes.copernicus.org/articles/2/521/2017/wes-2-521-2017.htmlThe full text article is available as a PDF file from https://wes.copernicus.org/articles/2/521/2017/wes-2-521-2017.pdf
Current fast
aeroelastic wind turbine codes suitable for certification lack an induction
model for standstill conditions. A trailed vorticity model previously used as
an addition to a blade element momentum theory based aerodynamic model in
normal operation has been extended to allow computing the induced velocities
in standstill. The model is validated against analytical results for an
elliptical wing in constant inflow and against standstill measurements from
the NREL/NASA Phase VI unsteady experiment. The extended model obtains good
results in the case of the elliptical wing but underpredicts the steady
loading for the Phase VI blade in attached flow. The prediction of the
dynamic force coefficient loops from the Phase VI experiment is improved by
the trailed vorticity modeling in both attached flow and stall in most cases.
The exception is the tangential force coefficient in stall, where the codes
and measurements deviate and no clear improvement is visible. This article
also contains aeroelastic simulations of the DTU
10 MW reference turbine in standstill at turbulent inflow with a fixed and
idling rotor. The influence of the trailed vorticity modeling on the extreme
flapwise blade root bending moment is found to be small.
Introduction
State-of-the-art aeroelastic wind turbine codes that are suitable for
simulating the many time series needed for certification typically use an
aerodynamics model based on blade element momentum (BEM) theory. These BEM-based models can be extended by tip loss corrections and so-called dynamic
inflow models that take the wake inertia into account. With this extension,
they are suitable for predicting the varying induced velocities in an unsteady
aeroelastic simulation. In addition to the dynamic induced velocities, there
are also dynamic effects due to shed vorticity and dynamic stall, which occur
on faster timescales than the dynamic inflow and are typically taken into
account by 2-D unsteady airfoil aerodynamics models, in this work the one
described in .
Thus, both the larger-scale wake effects and the smaller-scale unsteady
airfoil aerodynamics are taken into account if the turbine is in operation.
In standstill, however, BEM theory cannot be used because the basic
assumption in BEM, i.e., that the rotor can be approximated by a disc, is violated. Therefore, the induced velocities due to the vortices trailed from the blades
are not modeled, which results in both a wrong steady-state load distribution
and missing dynamics.
Wind turbine blades are twisted to ensure a reasonable angle of attack
distribution along the blade in operation. In standstill, on the other hand,
the blade twist leads to large load variations along the blade and thus
strong trailed vorticity that is not modeled in the aeroelastic codes used
for wind turbine certification. Further, the inflow turbulence, which in
normal operation only affects a part of the relative flow velocity at the
airfoils (the other part being due to rotor rotation), causes very large
dynamic variations in the angle of attack (AOA) along the blade in standstill.
In idling conditions a yaw error, as well as nacelle tilt and wind
inclination, is directly translated into AOA variations as the blades rotate
slowly.
In this work, a trailed vorticity model, which was originally designed for
normal operation and implemented as part of a BEM-based model in the aeroelastic code HAWC2,
, has been extended so that it
can be used in standstill conditions. Results from this extended model are
compared to the analytical constant downwash at an elliptical wing and
measurements from the NREL/NASA Phase VI unsteady experiment,
. Aeroelastic computations in standstill with
turbulent inflow are performed to evaluate the influence of the model on
radial load distributions and extreme flapwise blade root bending moment.
Near-wake model description
The near-wake model (NWM) for trailed vorticity was originally developed for
use in helicopter aerodynamics. It was assumed in the original model that the
trailed vorticity stays in the rotor plane. The induced velocity at a blade
section due to a trailed vortex element decreases as that vortex element
moves away from the blade. This decreasing induction is approximated by
exponential functions. This approximation makes it possible to use an
indicial function algorithm to avoid the time-consuming numerical integration
of vortex arcs based on the Biot–Savart law. The model has since been
modified to enable the computation of the induction due to trailed helical
vortex arcs , which is important in normal operation at high
wind speed. Further, it has been shown by that using one
exponential function instead of two is possible with negligible accuracy
loss.
A sketch of the near-wake geometry is shown in Fig. . The
induction W at a blade section s at a time step i is found as the sum
of the induced velocities due to all vortex arcs v trailed from a blade:
Wsi=∑v=1NvWs,vi.
The induced velocity due to an individual vortex arc is
Ws,vi=Ws,vi-1e-Δβ/Φs,v+Dw,s,vΔΓ(1-e-Δβ/Φs,v),
where Δβ is the angle the blade rotates during a time step and
Φ is a geometric parameter depending on the positions of vortex trailing
point and blade section, as well as the helix angle of the trailed vortex
arc. The trailed vortex strength, which depends on the radial gradient of the
bound circulation, is ΔΓ, and Dw,s,v describes the induced
velocity at section s due to a trailed vortex arc v with circulation 1
that starts directly at the blade. The advantage of using exponential
functions is apparent: to obtain the induction at a new time step, the
induction due to the newly trailed vortex element, the right term in
Eq. (), is added to the exponentially decreasing induced velocity
due to all previously trailed elements contained in Ws,vi-1.
Sketch of the geometry in the near wake. The vortex arc v, trailed
at radius r, induces axial (out-of-plane) and tangential (in-plane)
velocities at the section s. The radial distance between vortex trailing
point and section position is denoted h and h>0 if the vortex is trailed
outboard of the section. The angle β describes how far a vortex element
with length ds has moved away from the blade. The figure is
adapted from . In practice, the blade is discretized
into many sections and vortices are trailed from the root and tip and in
between sections.
Model extension
In order to enable the computation of standstill cases, a new definition of the
angle β is necessary. The previous implementation used the projection
of the trailed vortex filament in the rotor plane, which is not possible in
standstill conditions. Thus, the angle β is redefined:
β*=vrelΔtr,
where vrel is the relative flow velocity at the radial station
from which the vortex filament is trailed. If the trailed vorticity stays in
the rotor plane, the old and new definitions are identical: β=β*.
The new definition of β* accounts for the differing element length
trailed in one time step due to the downwind convection velocity. The axial
and tangential components of the induced velocities due to the newest
element (cf. Eq. ), can be determined based on the helix angle
φ:
Dw,s,v,axial=Dw,s,vcosφ,Dw,s,v,tangential=Dw,s,vsinφ.
Because the near-wake model is mainly meant to capture trailed vorticity
effects close to the blade, the local inflow angle is used as helix angle
φ. This inflow angle is computed based on the velocity triangle at
the vortex trailing point and is affected by the free wind speed including
turbulence, the movement of the blade and the induced velocities due to near
and far wake. This way the near-wake flow situation depends only on the
velocities at the blade section, which is similar to how the 2-D unsteady
aerodynamics effects are computed; see . The time simulation
of axial and tangential induction is then computed independently, so
Eqs. () and () are evaluated twice for each
section–vortex-arc combination.
If the downwind convection velocity increases, the paths of the trailed
vorticity change from circular (at zero convection speed) over helical (at
moderate convection speed) to straight (at standstill). This influences both
the steady-state value of the induction from trailed vorticity and the
dynamic behavior. Both of these can be modeled by changing the parameter
Φ in Eq. ().
As described in , an optimal value of Φ can be computed.
With this Φopt, the indicial function approximation computes the
same steady-state induced velocity as the Biot–Savart law:
∫0π/2hr21-1-hrcosβ1+1-hr2-21-hrcosβ+βtanφ23/2dβ=∫0π/21.359e-β/Φopt-0.359e-4β/Φoptdβ.
Here, r denotes the radial position where the vortex is trailed and h
the distance from the vortex trailing position to the radial position of the
blade section where the induction is to be determined (positive if the
section is inboard the vortex); cf. Fig. . For straight vortices,
Φ varies linearly with h/r:
Φs=0.788hrfor0<h/r<1-0.788hrforh/r<0.
To ensure that the model can be used for straight vortices in standstill
conditions and helical vortices in normal operation, a new
Φ* is
computed, that is a linear interpolation between Φs for
straight vortices Eq. () and the expression of Wang and
Coton for circular vortices, :
Φ*=kΦΦs+(1-kΦ)Φ,
where the interpolation kΦ is a function of both h/r and the tangent
of the helix angle. The straight and circular Φ approach each other for
h/r→0, meaning for sections very close to vortex trailing
points, where the influence of the vortex is large and an accurate
computation of Φ is thus very important. Therefore, the interpolation
proposed in Eq. () ensures good results for close positions,
which would be difficult to achieve by direct curve fitting of Φ to the
optimal value according to Eq. ().
For positive values of h/r, kΦ can be approximated as
kΦ=ahr,1+ahr,2φ+ahr,3φ2+ahr,4φ3,ahr,i=pi,1+pi,2hr+pi,3hr2+pi,4hr3.
For negative values of h/rkΦ=ahr,1+ahr,2eahr,3(π2-φ)+ahr,4e-8(π2-φ)-ahr,2-ahr,4,ahr,i=ni,1+ni,2eni,3hr+ni,4eni,5hr-ni,4-ni,2.
The values ni,j and pi,j are collected in the matrices N
and P:
Optimal and approximated values for Φ are shown for different helix
angles ranging from 0 (circular arcs) to 89∘ in Fig. .
The 89∘ have been chosen because for 90∘ the integral in
Eq. () cannot be evaluated. It is shown clearly that the
approximation gives a good representation of Φ across the range of helix
angles. There are some deviations for h/r→1, which represents
the influence of vortices close to the tip on the root sections. Because the
deviations only influence roughly the innermost 5 % of the rotor radius,
where often no aerodynamic profiles are installed, the quality of the
approximation is acceptable.
Approximation of Φ at different helix angles compared to the
optimal Φ value.
Unsteady airfoil aerodynamics model
The 2-D unsteady airfoil aerodynamics model in HAWC2 consists of both an
attached flow model for the 2-D shed vorticity effects and a dynamic stall
model to predict unsteady flow separation, as described in .
The attached flow model uses indicial functions assuming a flat plate. The
dynamic stall model interpolates between a fully attached and fully separated
airfoil polar, based on a time-lagged trailing edge separation point. The
dynamic stall model does not include leading-edge separation.
Interaction between dynamic stall model and trailed vorticity model
The trailed vortex strength
ΔΓ in the near-wake model (see Eq. ) is given by
the difference in bound circulation between the adjacent sections to a vortex
trailing point. The near-wake model needs to be iterated to convergence. The
bound circulation is part of that iteration loop, including attached flow
airfoil aerodynamics effects. Inside that loop, the quasi steady bound
circulation is computed according to the quasi steady lift coefficient
. This accounts for stall in the bound circulation
computation. The converged induced velocity due to the trailed vorticity is
then used to compute the angle of attack, which is the input to the unsteady
airfoil aerodynamics model. The unsteady airfoil aerodynamics model then
computes the effective angle of attack and the influence of dynamic stall on
the aerodynamic forces. The only deviation from the basic structure of the
implementation from the structure outlined in Fig. 3 of the article by
is that no far-wake model is used because the BEM
modeling is not valid in standstill conditions and the near-wake model
computes the full induction due to the semi-infinite trailed vorticity behind
the blades.
ResultsElliptical wing
The case of an elliptical wing with a 10 m span has been used previously to
test the NWM . In these earlier publications,
the wing was placed at the end of a very long, slowly rotating blade to
ensure an almost parallel inflow. In this work, the wing is instead mounted
on a 0.5 m long, nonrotating hub in a uniform inflow of 35 m s-1.
The previous publications prescribed an elliptical circulation distribution,
but in this work the wing is modeled with a geometric AOA to the inflow of
5.45∘ and a maximum chord length of 5.21 m. The geometric AOA is
defined as the angle of the local chord line with respect to the inflow
direction in HAWC2, which corresponds to the wind tunnel center line in the case of the Phase VI measurements discussed later. A lift gradient of 2π is
used, which leads to the analytical result of a constant downwash of
1.5 m s-1 at the wing.
Figure compares downwash at the lifting line computed from
original and extended NWM with the analytical solution. The original model
fails to predict the constant downwash in standstill, while the results from
the extended model are in good agreement with the analytical solution.
Results for an elliptical wing.
Radial distribution of normal force coefficients at 3.5 (a)
and 18.2 (b) degrees geometric AOA at 47 % blade radius. Results
from HAWC2 (denoted H2) and HAWC2 including the extended near-wake model
presented in this paper (H2 NW) are compared to measurements.
NREL Phase VI rotor in standstill
In all following comparisons, “HAWC2” refers to HAWC2 standstill
simulations. The BEM model and dynamic inflow model are disabled because the
BEM model is not valid in standstill and the dynamic inflow model simulates
the unsteady behavior of the BEM induction. The 2-D unsteady aerodynamics
model containing shed vorticity and dynamic stall modeling as introduced in
Sect. is active.
In addition to the 2-D unsteady aerodynamics model, the “HAWC2 NW”
simulations include the trailed vorticity modeling by the extended near-wake
model.
Constant pitch angle
Besides measurements at operation,
the Phase VI experiment also contained measurements in standstill, some of
which have been compared to computational fluid dynamics (CFD) results by
and . Here, some steady comparisons with
measurements published in are shown together with comparisons
at a lower geometric angle of attack.
The inflow speed in the cases presented here is 20 m s-1, which
results in a Reynolds number of 0.86 million at 47 % blade radius. The
Reynolds number varies along the blade with the chord length (ignoring
induced velocity effects on the Reynolds number), and the aerodynamic code
interpolates accordingly between different airfoil polars.
A comparison of the radial distribution of the normal force coefficient is
shown in Fig. at 3.5 and 18.2∘ geometric AOA at 47 %
blade radius. At 3.5∘ geometric AOA at the 47 % station, most of
the blade is in attached flow. In this case, the near-wake model predicts a
radial distribution of the normal coefficient that agrees well with the
measurements in terms of the radial load gradients, but there is an offset to
the measurements. No explanation for this offset has been found. The results
at the higher geometric AOA, where most of the blade is in stall, are shown
in Fig. b. In this case the near-wake model can predict the root
vortex well, but the agreement with the measurements becomes worse toward the
tip, where the blade is in deep stall.
The steady-state comparison of the tangential force coefficients in these
cases in Fig. leads to the same conclusions. Again there appears
to be an offset between near-wake computations and measurements in the
3.5∘ case. In the 18.2∘ case the prediction of the root
vortex by the near-wake model is clear, but the agreement gets worse towards
the stalled tip.
Radial distribution of tangential force coefficients at
3.5 (a) and 18.2 (b) degrees geometric AOA at 47 %
blade radius. Results from HAWC2 (denoted H2) and HAWC2 including the
extended near-wake model presented in this paper (H2 NW) are compared to
measurements.
Varying pitch angle
Two cases of a pitching blade are presented here: case O47010 with a mean
geometric AOA at the 47 % station of 3∘ and a pitching amplitude
of 2∘ at a frequency of 0.739 Hz (reduced frequency K=0.0625 at
47 %) and case O47320 with a mean geometric AOA at the 47 % station
of 14∘ and a pitching amplitude of 5.5∘ at a frequency of
1.183 Hz (reduced frequency K=0.1 at 47 %). The free-stream velocity
in both cases is 23.3 m s-1.
Case O47010. Variation about mean cn.
The normal force coefficient variation for the O47010 case is shown in
Fig. . The mean geometric AOA is only half a degree different than
in the steady case in Fig. , and almost the full blade is in
attached flow. The unsteady simulation agrees with the steady simulation in
an offset, where the HAWC2 NW results are below the measurements at every
station but the blade tip. A comparison of the mean values would thus not
lead to new conclusions. To make the comparison of the dynamic behavior
easier, the mean values of cn have been subtracted in Fig. .
The near-wake modeling leads to improved agreement with the measurements
everywhere except at the 63 % station, where the differences between the
predicted and measured loops are small. At the other radial stations, HAWC2
NW predicts the loop openings and gradients much better than HAWC2. At the
80 % station, for example, HAWC2 NW predicts the slight loop opening due
to beginning separation that is seen in the measurements. The HAWC2
computations show a cn gradient and loop opening that is characteristic of too
large a mean AOA. This observation is in conflict with the comparison of the
steady-state cn in Fig. . There, the larger values predicted by
HAWC2 results are in better agreement with the experimental data at the
80 % section than the HAWC2 NW results.
Case O47010. Variation about mean ct.
The ct variations in Fig. show improved simulation results due
to the NWM at most blade stations except 47 %. Similar as in the cn
comparison above, HAWC2 NW predicts gradients and loop openings that are very
close to the measurements. The normal force coefficient loops for the O47320
case (14∘ mean AOA at the 47 % radial station) are shown in
Fig. . Because the amplitudes are larger and the mean AOA is
higher in this case, the loops are more open and more nonlinear. Therefore,
the unsteady aerodynamics model has a larger influence on the mean values of
cn and ct than in case O47010, and it has been chosen not to subtract
the mean values in the O47320 results. The flow is only attached at the
30 % radial station (cf. Fig. ), and there HAWC2 NW predicts
the cn gradient more accurately than HAWC2. At the 47 % radial
station, HAWC2 NW predicts a slightly higher range of cn that is closer
to the measurements. Also, the shape of the loop predicted by HAWC2 NW agrees
better with the measurements than that predicted by HAWC2, but both models do
not reach as high maximum cn values as the measurements. At the 63 %
station HAWC2 NW predicts a slightly more open loop than HAWC2 up to
17∘ geometric AOA, which is in better agreement with the
measurements. Also in agreement is the increasing normal force coefficient
towards higher AOA, which is not predicted by HAWC2. As at the 47 %
station, and also further outboard, the models underpredict the maximum
measured cn values. However, the local increase in cn at high AOA in
the measurements at the 63 % section appears to be due to leading-edge
vortex formation. This effect is not included in the dynamic stall modeling; therefore, this overshoot of cn cannot be predicted by HAWC2. To a lesser
extend, the same effect can be seen at the 47 and 80 % stations. At both
the 80 and 95 % stations, the loops predicted by HAWC2 are narrowing
towards the high angles of attack. This is because the dynamic stall model
interpolates between a fully attached and fully separated curve; cf.
Sect. . At high angles of attack, where the flow is fully
separated, the dynamic stall model becomes steady because the separation
point does not move any more, and accordingly the loops close. Due to the
trailed vorticity in the HAWC2 NW computations, the local angles of attack at
the radial stations close to the tip are lower than the geometric angles of
attack, and therefore the flow is not yet considered fully separated. Thus, the loops predicted by HAWC2 NW do not become more narrow towards high
geometric AOA at the 80 and 95 % stations. However, the narrowing dynamic
stall loops are just delayed towards higher geometric AOA.The underlying
issue, that the dynamic stall model is not suited for deep stall conditions,
remains. Even though the loop opening predicted by HAWC2 NW at the 95 %
radial station is closer to the measurements, the gradient of the cn loop
cannot be predicted.
The loops of the tangential force coefficient in the O47320 case are shown in
Fig. . At the 30 % station in attached flow, HAWC2 NW clearly
predicts a loop opening and gradient that agrees better with the measurements
than the results from HAWC2. At the 47 % radial station, HAWC2 NW
predicts the form of the loop slightly better, but the opening in the
measured loop is considerably larger. At the further outboard stalled
stations, there is generally a large disagreement between both codes and the
measurements and it is difficult to state which codes' predictions agree
better with the measurements.
Case O47320; cn.
Case O47320; ct.
DTU 10 MW in standstill with turbulent inflow
Aeroelastic simulations on the DTU 10 MW reference turbine
have been performed to investigate the effect of the trailed vorticity model
in standstill. The mean wind speed in these simulations is 50 m s-1,
the shear exponent is 0.11, and the turbulence intensity is 11 %. These
values are based on design load case (DLC) 6.2 in the design load basis for onshore wind turbines by
. All blades are pitched to 82∘, and the simulated time is
700 s, where the first 100 s are removed to avoid transients. The blades
are discretized into 30 equidistantly spaced aerodynamic sections.
Distribution of induction, AOA, and loads for the upward-pointing
blade. The mean values are shown as solid lines, and the dashed lines
indicate the standard deviations; IP indicates in-plane.
Distribution of induction, AOA, and loads for the upward-pointing
blade. The mean values are shown as solid lines, and the dashed lines
indicate the standard deviations.
The compared codes are HAWC2 with dynamic stall enabled but without induction
model and HAWC2 NW, where both dynamic stall and induction due to the trailed
vorticity are modeled. In order to enable direct comparisons between the
different aerodynamic models a few computations have been performed with a
locked rotor and only a single turbulence seed. The radial distributions of
AOA, induced velocities, and aerodynamic forces on the blade pointing
vertically upward are discussed in Sect. at wind directions
of 0, -15, and 15∘. Wind direction misalignments larger than
15∘ are not included in the locked rotor analysis because they can
lead to standstill vibrations of the DTU 10 MW . These
vibrations make it difficult to compare radial load distributions, and their
analysis is outside the scope of the present work.
To evaluate the extreme blade root flapwise bending moments, simulations with
an idling rotor in the wind direction range of -30 to 30∘
(5∘ resolution) are presented in Sect. . A number of
36 turbulence seeds has been used at each wind direction and for each
aerodynamic model. It is important to note that yaw angles outside of the
±15∘ range are at the boundaries of the near-wake model validity,
which is at its core a simplified lifting line model. The yaw errors up to
±30∘ are included in the idling analysis to investigate how the
model behaves in these difficult conditions.
Distribution of induction, AOA, and loads for the upward-pointing
blade. The mean values are shown as solid lines, and the dashed lines
indicate the standard deviations.
Locked rotor
The radial distributions of in-plane induced velocity, AOA, and edgewise and
flapwise aerodynamic forces are shown in Fig. for a wind
direction of -15∘. The solid lines show the mean values, and the
standard deviations are indicated by dashed lines. At this wind direction,
most of the upward-pointing blade is in negative stall, except for the area
around the tip where the flow is attached. Because the near-wake model is
implemented such that a bound circulation is computed based on the lift
coefficient, a slight reduction in the mean aerodynamic forces can be seen
when the near-wake model is enabled. The AOA and induction distributions
clearly show the prediction of a root vortex, while the influence of the
trailed vorticity on the mean loading toward the blade tip is limited.
Because the lift gradient is larger in attached flow, the standard deviation
of the induced velocity increases towards the tip. These increased induced
velocity variations counteract the force variations due to inflow turbulence,
which causes slightly smaller standard deviations of the angle of attack and
flapwise force if the trailed vorticity modeling is active. There is no clear
tip vortex visible in the mean induction distribution because the loading
approaches zero towards the tip even without an induction model. Further the
equidistant point spacing means that the small drop in the loading to zero at
the very last blade section is not so finely resolved. At 0∘ wind
direction, the flow is attached on the whole blade; see Fig. b.
Therefore, the lift gradients are large and the trailed vorticity modeling
has a larger influence on the loading. Both the mean aerodynamic forces as
well as the standard deviations of these forces are reduced along almost the
whole blade. The root vortex is clearly visible in the plot of the induced
velocity in Fig. a.
Maximum flapwise (a) edgewise and (b) blade root
bending moment for wind directions between -30 and 30∘ yaw error.
Dashed lines show the mean of the maximum values for the 36 different seeds,
the solid lines the maximum of the maxima.
The flow is stalled along the whole blade at 15∘ wind direction.
Therefore, the quite large induced velocities if the near-wake model is
active have only a minor effect on the aerodynamic forces, even though using
the near-wake model changes the mean local AOA by up to 5∘; see
Fig. . The forces at the blade root, with around 25 m radius, are
clearly smaller in the HAWC2 NW than in the HAWC2 computations, but because
of the small lift gradient around stall, this reduction does not affect the
standard deviations of the aerodynamic forces as much.
Idling rotor
Computations with 36 different turbulence seeds per wind direction and for
each of the two aerodynamic models have been performed to investigate the
influence of the aerodynamic model on the extreme flapwise and edgewise blade
root bending moment. The result of these simulations is shown in
Fig. .
The dashed lines represent the mean value of the maximum absolute flapwise
and edgewise blade root bending moment in the 36 simulations. The absolute
maximum of the maxima encountered in the simulations is shown as solid lines.
It can be seen that including the near-wake model in the simulations reduces
the mean maximum value by roughly 0.5 to 1.5 %, depending on the wind
direction. An exception for this reduction is the edgewise moment at
25∘ yaw error, where vibrations are present and distort the analysis.
The influence of the aerodynamic model on the absolute maximum encountered in
the 36 seeds (see the solid lines in Fig. ) is less clear. An
explanation for this might be that in the idling cases with yaw error, the
blades see different AOA distributions as a function of azimuth position. A
very high flapwise blade root bending moment will occur if a high wind speed
hits a large part of the blade at an angle of attack corresponding to the
maximum lift coefficient. Because the lift gradient at the maximum lift
coefficient is small, the influence of the near-wake model in this extreme
case is small as well; therefore, the highest flapwise blade root bending
moments at yaw error are very similar in HAWC2 and HAWC2NW. This extreme case
of high wind speed at maximum lift coefficient does not occur with each
turbulence seed though. If the extreme loading does not occur at the maximum
lift coefficient in a 600 s HAWC2 simulation, then the trailed vorticity
leads to a reduced maximum loading. Therefore, the average extreme loading of
the simulations with 36 turbulence seeds is decreased when the near-wake
model is active. At 25 and 30∘ yaw error, the maximum edgewise blade
root moments indicate edgewise vibrations. The near-wake model seems to
reduce the absolute maximum of edgewise blade root bending moment encountered
in the 36 seeds when vibrations are present, but the effect on the mean
maximum values is smaller.
The mean values of the mean and standard deviation of the idling speed for
the 36 seeds are shown in Fig. . If the near-wake model is active,
HAWC2 predicts both slightly lower idling speeds and lower idling speed
variations. This is consistent with the generally slightly lower loading
observed in Figs. to .
Mean and standard deviation of the idling rotor speed. The near-wake
model slightly reduces mean idling rotor speed as well as its standard
deviation, which is consistent with the generally lower flapwise loading
predicted by HAWC2 NW.
The load comparison in idling conditions shows that adding the near-wake
model might appear to reduce the extreme loading if a small number of
turbulence seeds is used. A high number of turbulence seeds, on the other
hand, is expected to lead to the same extreme loading independent of trailed
vorticity model. Another conclusion is that the maximum extreme loading is
much higher than the average extremes of the 10 min time series with
different turbulence seeds. A large number of seeds might be necessary to
achieve realistic extreme values in an aeroelastic load analysis in
standstill conditions.
Conclusions
The near-wake model has been extended to compute the induction due to trailed
vorticity in standstill and idling conditions. Due to the twist distribution
of a wind turbine blade and the larger effect of turbulence in standstill
when compared to operational conditions, strong vortices can be trailed from
any position along the span of the blade. In idling conditions yaw errors,
tilt angle, and wind inclinations directly translate into AOA variations on
the slowly rotating blades. Comparison with the analytical solution of a
constant downwash for an elliptical wing shows good agreement with results
from the extended near-wake model, with the original model wrongly predicting
large radial variations in the downwash.
Comparison with measurements from the NREL/NASA Ames Phase VI experiment in
attached flow conditions shows an unexplained offset between the steady-state
normal and tangential force coefficients measured and predicted by HAWC2 NW.
However, the HAWC2 NW code predicts the effect of the trailed vorticity on
the radial load gradients in steady state.
A comparison of the dynamic variation in the force coefficients for a
sinusoidally pitching blade in attached flow shows that HAWC2 NW can predict
dynamic loops that agree much better with the measurements than those
predicted by HAWC2 on the major part of the blade. The agreement is improved
both in terms of cn–AOA curve gradients and openings of the loops.
In a steady-state comparison at high mean AOA, where the flow is separated at
most of the blade, the near-wake model can predict the root vortex at the
inner part of the blade in attached flow. At the rest of the blade, no clear
improvement due to the added trailed vorticity modeling is visible. At the
tip, which is in deep stall, the predicted normal force coefficient agrees
less well with the measurements.
The unsteady comparison at high AOA shows a clear improvement at the inner
part of the blade, which is in attached flow. Also on the outer part the
openings of the cn loops are predicted better by HAWC2 NW than HAWC2,
mainly because the flow in the HAWC2 simulations is close to fully separated,
where the dynamic stall model cannot predict the dynamic behavior. The HAWC2
NW simulations predict lower AOAs close to the blade tip, and thus the
dynamic stall loops stay open. Even though the trailed vorticity modeling
leads to improved predictions in this case, the basic weakness of the
Beddoes–Leishman-type dynamic stall model in deep stall should be addressed
in future research. Further, the outboard ct loops in stall are found to
be difficult to model. However, the Unsteady Aerodynamics Experiment (UAE) Phase VI ct
measurements, and ct measurements in general, are highly sensitive to
pressure tap distribution, which can lead to increased uncertainties in this
measurement.
As expected, the aeroelastic computations in standstill with turbulent
inflow and a fixed rotor show that the near-wake model reduces the mean blade
loading mainly at radial positions in attached flow compared to the standard
standstill aerodynamic model without induction. The standard deviations of
the force variations are reduced accordingly. Because the relative velocity
in standstill is similar at all radial positions of the blade and the chord
gets smaller towards the tip, no large tip loss effects have been observed
and the main induction is clearly due to the root vortex.
Also computations with idling rotor in a yaw error range of -30 to
+30∘ have been performed with 36 turbulence seeds per wind
direction. The absolute maximum of flapwise blade root bending moment shows
only a very small influence of the aerodynamic model. The mean maximum
encountered with the different turbulence seeds shows a small but clear
reduction due to the trailed vorticity modeling.
Future work
For a different turbine and more flexible blade design, standstill
vibrations in attached flow can be possible. The impact of the trailed
vorticity modeling on these vibrations could be addressed in future research.
The damping of vibrations in parked or idling conditions can also be highly
dependent on the dynamic stall model parameters. The attached flow parameters
used in HAWC2 are based on the analytical solution for the dynamic lift and
drag of a flat plate, and the airfoil thickness could be taken into account
here. Also the time constants for the flow separation are currently assumed
to be independent of the airfoil, as well as identical in positive and
negative stall. This assumption is certainly wrong for cambered airfoils, and
a better approach could be identified in the future.
Leading-edge separation, which is not part of the current dynamic stall model
implementation, could become very important at extreme yaw errors, where the
AOA is around 180∘. Then, the trailing edge of the airfoil acts as a
sharp leading edge and leading-edge separation is much more likely than in
the cases investigated in the present article.
The measurement data are available by request from NREL
using the relevant case numbers found in the article. The aerodynamic
computations have been executed with HAWC2. Commercial and research licenses
for HAWC2 can be purchased from DTU.
The authors declare that they have no conflict of
interest.
This article is part of the special issue “The Science of
Making Torque from Wind (TORQUE) 2016”. It is a result of the The Science of
Making Torque from Wind (TORQUE 2016) conference, Munich, Germany, 5–7
October 2016.
Acknowledgements
The work has been conducted within the project “Research and development of
optimal wind turbine rotors under offshore wind conditions in China
(OffWindChina)”, funded by “Det Strategiske Forskningsråd ved
Programkomiteen for Bæredygtig Energi og Miljø”, contract
12-130590. Edited by: Gerard J. W. van
Bussel Reviewed by: Xabier Munduate and Vasilis A. Riziotis
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