Robust and accurate dynamic stall modeling remains one of the most
difficult tasks in wind turbine load calculations despite its long research
effort in the past. In the present paper, a new second-order dynamic stall
model is developed with the main aim to model the higher harmonics of the
vortex shedding while retaining its robustness for various flow conditions
and airfoils. Comprehensive investigations and tests are performed at
various flow conditions. The occurring physical characteristics for each
case are discussed and evaluated in the present studies. The improved model
is also tested on four different airfoils with different relative
thicknesses. The validation against measurement data demonstrates that the
improved model is able to reproduce the dynamic polar accurately without
airfoil-specific parameter calibration for each investigated flow condition
and airfoil. This can deliver further benefits to industrial applications
where experimental/reference data for calibrating the model are not always
available.
Introduction
An accurate prediction of wind turbine blade loads is influenced by many
parameters including 3D and unsteady effects. The first mainly occurs in the
root and tip areas of the blade due to radial flow and induced velocity
influences, respectively . The latter can occur due to
variation in the inflow conditions caused by yaw misalignment, wind
turbulence, shear and gusts, tower shadow, and aeroelastic effects of the
blade. The abovementioned phenomena may result in dynamic stall (DS).
Experimental studies showed that the aerodynamic forces can differ
significantly in comparison to the static condition. DS is often initiated by
the generation of a leading edge vortex (LEV), which increases the positive
circulation effect on the airfoil suction side, causing delayed stall
. This intense leading edge vortex is convected
downstream along the airfoil towards the trailing edge. At the same time, the
lift force increases significantly, and the pitching moment becomes more
negative compared to the static values. A significant drag increase is
observed at large angles of attack. An example is shown in
Fig. . Afterwards, a trailing edge vortex (TEV) with opposite
rotational direction than LEV is formed, which pushes the leading edge vortex
towards the wake area. This onset may result in a significant drop of the
lift coefficient (CL) and can be dangerous for the blade
structure itself, although dynamic stall also generally enhances aerodynamic
damping.
Typical dynamic stall behavior of S801 airfoil. Data obtained from
.
To model the behavior of the airfoil under these situations, semiempirical
models can be used. The models are known to produce reasonable results
without any notable increase in computational effort. Despite that, these
models usually cannot reproduce higher harmonics of the load fluctuations.
Furthermore, the applied constants shall be adjusted according to the flow
conditions and airfoils. Leishman and Beddoes (LB)
have developed a model for dynamic stall combining the flow delay effects of
attached flow with an approximate representation of the development and
effect of separation . This model was developed for
helicopter applications and therefore includes a fairly elaborate
representation of the nonstationary attached flow depending on the Mach
number and a rather complex structure of the equations representing the time
delays . simplified the
model for wind turbine applications by removing the consideration of
compressibility effects and the leading edge separation. The latter was
argued because the relative thickness of wind turbine airfoil is typically no
less than 15 %. This model was called the Risø model in
. Examples of the other models are given by Øye
, Tran and Petot (ONERA model),
and Tarzanin (Boeing Vertol model) . To better
model the vortex shedding characteristics at large angles of attack,
second-order dynamic stall models were introduced. An example of this model
was given by which makes use of the difference between the
inviscid and viscous static polar data as a main forcing term for the dynamic
polar reconstruction, in contrast to the LB model that uses the changes of
the angle of attack over the time. An improved version of the Snel model was
proposed recently by to cover for the increased shedding
effects in the downstroke phase. All abovementioned models employ the static
polar data and dynamic flow parameters as the input needed for the dynamic
polar reconstruction. Then, the models compute the dynamic force difference
required for the reconstruction process.
Although many studies have been dedicated to dynamic stall modeling
,
engineering calculations in the industry are still relying on the very basic
classical dynamic stall models such as the Leishman–Beddoes and Snel models.
The reason is the simplicity to tune in the models for different airfoils and
for different flow conditions. Therefore, one major key for a model to be
used in industrial applications is robustness of the model itself; i.e., the
model is easy to apply with a small number of well-defined user parameters.
One of the purposes of this paper is to document widely used dynamic stall
models in research and industries. These include the first-order LB model and
the second-order Snel model. A very recently improved Snel model according to
will also be evaluated. The mathematical formulations of
these models will be presented in this report. Weaknesses of existing dynamic
stall modeling shall be identified, and possible corrections to those
limitations will be described. Finally, a new second-order dynamic stall
modeling will be proposed that is able to model not only the second-order
lift and drag forces, but also the pitching moment along with calculation
examples in comparison to experimental data for different airfoils and flow
conditions.
The paper is organized as follows. Section describes
the mathematical formulation of existing dynamic stall models and the new
model developed in this work. Then, in Sect. assessments
are carried out on the performance of the second-order dynamic stall models
in comparison with measurement data. The new model is further tested at
various flow conditions and to examine its robustness on four different
airfoils without further calibrating the constants. Finally, all results will
be concluded in Sect. .
Mathematical formulations
In this section the mathematical formulations of each model are described in
detail. The reasons are mainly to provide information on how each model was
employed and to gain deeper insights for further developing the new model.
Note that each existing model was developed by different authors; thus
different symbols and formulation methods were adopted in those publications
. In this paper, all models are described in a consistent
way for clarity and for an easier interpretation and implementation process.
Leishman–Beddoes model
The original Leishman–Beddoes model is composed of three main contributions
representing various flow regimes: (1) unsteady attached flow, (2) unsteady
separated flow and (3) dynamic stall. The present section will elaborate the
mathematical description and its physical interpretation of each module.
Figure illustrates several main parameters needed for
modeling the dynamic stall characteristics.
Illustration of the main aerodynamic parameters needed for modeling
the dynamic stall characteristics.
Unsteady attached flow
In this module, the unsteady aerodynamic response of the loads is represented
by the time delay effects. The indical
formulas were constructed based on the work of
and have been refined by . The loads are
assumed to originate from two main sources: one for the initial
noncirculatory loading from the piston theory and another for the circulatory
loading which builds up quickly to the steady-state value
. In the formulation, the relative distance traveled
by the airfoil in terms of semi-chords is represented by s=2Vt/c, which can
also be used to describe the nondimensional time. Note that V, t and
c are free-stream wind speed, time and chord length, respectively. For a
continuously changing angle of attack αn, the effective angle of
attack (αen) can be represented as
αen=αn-Xn-Yn,
where n is the current sample time. The last two terms describe the
deficiency functions that are given by
2Xn=Xn-1exp-b1β2Δs+A1Δαnexp-b1β2Δs/2,3Yn=Yn-1exp-b2β2Δs+A2Δαnexp-b2β2Δs/2,
where
4Δαn=αn+1-αn,5Δs=sn-sn-1.
In these equations, b1, b2, A1 and A2 are constants. The
variable β represents the compressibility effects and is formulated as
β=1-M2. Because information about the previous cycle is
needed in the formulations, initializations are required. The solution needs
to develop for a certain time until convergence of the resulting unsteady
loads is obtained.
The circulatory normal force due to an accumulating series of step inputs in
angle of attack can be obtained using
CNnC=dCNdααen-α0INV.
The variable α0INV is the angle of attack for zero
inviscid normal force. The original formulation of the model disregarded the
use of α0INV. However, this term is important when the
airfoil has a finite camber. This has been pointed out as well by
.
The noncirculatory (impulsive) normal force is obtained by
CNnI=4KαTIMΔαnΔt-Dn,
where TI is given by TI=Mc/V. The deficiency
function Dn is given by
Dn=Dn-1exp-ΔtKαTI+Δαn-Δαn-1Δtexp-Δt2KαTI
and Δt=tn-tn-1.
The total normal force coefficient under attached flow conditions is given by
the sum of circulatory and noncirculatory components as
CNnP=CNnC+CNnI.
Unsteady separated flow
stated that the onset of leading edge separation is
the most important aspect in dynamic stall modeling. The condition under
which leading edge stall occurs is controlled by a critical leading edge
pressure coefficient that is linked into the formulation by defining a lagged
normal force coefficient CNnP1 as
CNnP1=CNnP-Dpn,
where Dpn is given by
Dpn=Dpn-1exp-ΔsTp+CNnP-CNn-1Pexp-Δs2Tp.
It has been investigated by that the calibration
time constant Tp is largely independent of the airfoil shape. The
substitute value of the effective angle of attack incorporating the leading
edge pressure lag response may be obtained using
αfn=α0INV+CNnP1dCN/dα.
In most airfoil shapes, the progressive trailing edge separation causes loss
of circulation and introduces nonlinear effects on the lift, drag and
pitching moment, especially on cambered airfoils. This is even more important
for wind turbine airfoils because the relative thickness is large. To derive
a correlation between the normal force coefficient and the separation
location (fn), the relation based on the flat plate from Kirchhoff and
Helmholtz can be used, which reads
CNnVISC=dCNdα1+fn22αn-α0VISC.
The location of the separation point is usually obtained by a curve-fitting
procedure in literature. For example, proposed the
following correlation:
fn=1-0.3expαn-α1S1;αfn≤α10.04+0.66expα1-αnS2;αfn>α1.
The coefficients S1 and S2 define the static stall characteristic
while α1 defines the static stall angle. The derivation was based on
the NACA 0012, HH-02 and SC-1095 airfoils that have a single break point of
the static lift force coefficient. proposed the
formulation for the S809 airfoil as
fn=c1+a1expS1αn;αfn≤α1c2+a2expS2αn;α1<αfn<α2c3+a3expS3αn:αfn≥α2
that has two break points (α1 and α2) of the static lift
force coefficient, where c1, c2, c3, a1, a2 and
a3 are constants.
The additional effects of the unsteady boundary layer response may be
represented by application of a first-order lag to the value of fn to
produce the final value for the unsteady trailing edge separation
point f2n. This can be represented as
f2n=fn-Dfn,
where Dfn is given by
Dfn=Dfn-1exp-ΔsTf+fn-fn-1exp-Δs2Tf,
and Tf is a constant. Then, the unsteady viscous normal force
coefficient for each sample time can be obtained using
CNnf=dCNdα1+f2n22αen-α0VISC+CNnI.
The tangential component of the force can be obtained by
CTnf=-ηdCNdααen2f2n.
Note that positive CTnf is defined in the
direction of the trailing edge while η is a constant.
According to and , a general
expression for the pitching moment behavior cannot be obtained from Kirchhoff
theory, and an alternative empirical relation must be formulated.
proposed the following formulation for the S809
airfoil
CMf=CM0+K0+K11-f2n+K2sinπf2nm;αn≤α2CM0+K0+K3expK4f2nm;αn>α2.
where CM0 defines the moment coefficient at zero normal
force and K0 is the mean offset of the aerodynamic center from the
quarter chord position. K1, K2, K3, K4 and m are
constants.
Dynamic stall
The third part of the model describes the post-stall characteristics where
the vortical disturbances near the leading edge become stronger. The effect
of vortex shedding is given by defining the vortex lift as the difference
between the linearized value of the unsteady circulatory normal force and the
unsteady nonlinear normal force obtained from the Kirchhoff approximation,
which reads
CVn=CNnC1-Kn,
where Kn is given by
Kn=141+f2n2.
The normal force is allowed to decay, but it is updated with a new increment
in the normal force based on prior forcing conditions, which can be defined
as
CNnV=CNn-1Vexp-ΔsTv+CVn-CVn-1exp-Δs2Tv;if0<τvn<TvlCNn-1Vexp-ΔsTv;otherwise,
where Tv and Tvl are the vortex decay and center of
pressure travel time constants, respectively. The nondimensional vortex time
is given by τvn=τvn-1+0.45ΔtcV;ifCNnP1>CNCRIT0;ifCNnP1<CNCRITandΔαn>0,
with CNCRIT being the inviscid critical static normal
force, usually indicated by the break of the (viscous) moment polar at the
critical angle of attack αnCRIT. This can be formulated
as
CNCRIT=dCNdααnCRIT-α0INV.
The idealized variation in the center of pressure with the convection of the
leading edge vortex can be modeled by
CPvn=Kv1-cosπτvTvl.
The dynamic moment coefficient can be formulated as
CMnV=-CPvnCNnV.
Therefore, the total dynamic loading on the airfoil from all modules can be
written as
28CNnD=CNnf+CNnV,29CTnD=CTnf,30CMnD=CMnf+CMnV,
and by converting these forces into lift and drag, one obtains
31CLnD=CNnDcosαn-CTnDsinαn,32CDnD=CNnDsinαn+CTnDcosαn.
Note to present implementation
In Eqs. () and (), a curve-fitting
procedure is usually adopted in literature. In this sense, the parameters or
even the formulation need to be adjusted when the airfoil is different.
Therefore, in the present implementation, the separation point is derived
directly from the static polar data using the inversion of
Eq. () as
fn=2CNnVISCdCNdααfn-α0VISC-1.02.
The same approach was used for example by . This
way, the user can avoid dealing with curve fitting adjustment (which requires
changes on the constants for different airfoils and flow conditions) as long
as the static polar data are available.
In the original formulation, the pitching moment is also obtained by a
curve-fitting procedure in Eq. (). Again, this kind of
approach is not straightforward as the user needs to perform curve fitting of
the polar data. In the present implementation, the moment coefficient is
easily obtained from the static viscous polar data by interpolating the value
at the effective angle of attack, incorporating the leading edge pressure
time lag αfn, which reads
CMnf=CMVISCαfn.
In this sense, the moment coefficient can be reconstructed easily without the
need to adjust the parameters in advance, minimizing the user error.
Furthermore, to avoid discontinuity in the downstroke phase for
Eq. (), an additional condition is applied in the
present implementation as
τvn=τvn-1+0.45ΔtcV;ifCNnP1>CNCRIT0;ifCNnP1<CNCRITandΔαn≥0τvn-1;otherwise.
Snel second-order model
The history of the Snel second-order model dates back to 1993
based on Truong's observation on dynamic lift coefficient characteristics
. Truong proposed that the difference between the static
and dynamic lift can be divided into two terms: the forcing frequency
response and the higher-frequency dynamics of a self-excited nature. The
total dynamic response of the airfoil is formulated as
36CLnD=CLnVISC+ΔCLnD1+ΔCLnD2,37CDnD=CDnVISC+,38CMnD=CMnVISC+,
with D1 and D2 being the first- and second-order
corrections, respectively. The first correction is modeled using an ordinary
differential equation (ODE) by applying a spring-damping like function as
τΔC˙LnD1+Kf10nΔCLnD1=F1n.
The frequency of the first-order corrected lift follows the frequency of the
forcing term F1. This term is based on the time derivative of the
difference between the steady inviscid
coefficient CLnINV and viscous lift
coefficient CLnVISC of an
airfoil (ΔCLnINV) as
40F1n=τΔC˙LnINV,41ΔCLnINV=CLnINV-CLnVISC=dCLdααn-α0INV-CLnVISC,
with n and dCL/dα as the current sample
time and inviscid lift gradient, respectively. The time constant τ in
the above equation represents the time required for the flow to travel half a
chord distance as
τ=c2V.
The “stiffness” coefficient of the first-order term Kf10n
can be expressed as
Kf10n=1+0.5ΔCLnINV81+60ταn˙;ifαn˙CLnINV≤01+0.5ΔCLnINV81+80ταn˙;ifαn˙CLnINV>0.
As shown in , the above equation becomes
numerically unstable if αn˙ is large (increasing reduced
frequency above 0.1) for
αn˙CLnINV≤0. The reason is that
the denominator goes to zero and then negative, causing numerical integration
instability. Thus, based on pure intuition the denominator value was set to a
minimum of 2.0 in . In the present implementation,
a similar approach is adopted but the limit differs. Instead, the minimum
denominator value is limited to 1×10-5, because it yields more
physical results for several cases tested by the authors.
To incorporate the higher-order frequency dynamics, a second-order ODE is
used to describe the second-order correction term. The general form may be
written as
τ2ΔC¨LnD2+Kf21nΔC˙LnD2+Kf20nΔCLnD2=F2n
similar to the first-order correction. The frequency of the higher-order
dynamics is determined by the forcing term F2n, defined as
F2n=0.1ks-0.15ΔCLnINV+0.05ΔC˙LnINV.
It is noted that the value 0.1 as a constant was chosen according to
. This is not a fixed value and can be adjusted based on the
evaluated cases as seen in literature .
Variable ks is a constant with a typical value of 0.2. The spring
coefficient is given by
Kf20n=ks21+3ΔCLnD221+3αn˙2
and the damping coefficient as
Kf21n=60τks-0.01ΔCLnINV-0.5+2ΔCLnD22;ifαn˙>02τks;ifαn˙≤0.
Adema–Snel second-order model
The recently developed model of improves the original Snel
model in several aspects. Instead of using the lift
coefficient (CL), the normal force coefficient (CN)
is used, similar to the LB model . The total dynamic
response of the airfoil is formulated as follows.
48CNnD=CNnVISC+ΔCNnD1+ΔCNnD249CTnD=CTnVISC+50CMnD=CMnVISC+
The model introduces some modifications of the original model in terms of
(1) projected ks, (2) the first-order coefficient and (3) the
second-order coefficient. The mathematical formulation of the first-order
term of the model is listed as
51τΔC˙NnD1+Kf10nΔCNnD1=F1n,52F1n=τΔC˙NnINV,53ΔCNnINV=CNnINV-CNnVISC=dCNdααn-α0INV-CNnVISC,54Kf10n=1+0.2ΔCNnINV81+60ταn˙;ifαn˙CNnINV≤01+0.2ΔCNnINV81+80ταn˙;ifαn˙CNnINV>0,
and for the second-order correction term as
55τ2ΔC¨NnD2+Kf21nΔC˙NnD2+Kf20nΔCNnD2=F2n,56F2n=0.01ks-0.04ΔCNnINV+1.5τΔC˙NnINV,57Kf20n=10kssinαn21+3ΔCNnD221+2802τ2αn˙2,58Kf21n=60τks-0.01ΔCNnINV-0.5+2ΔCNnD22;ifαn˙>060τks-0.01ΔCNnINV-0.5+14ΔCNnD22;ifαn˙≤0.
One may notice that Eq. () contains τ in the second
term of the right-hand side (RHS). This is intended to remove the dependency
of the model on the velocity as the input parameter. The other main
difference with the original model is also observed in
Eq. (), where ks is projected
by sinαn. At last, the downstroke motion of the second-order term
of Eq. () is modified to enable vortex shedding
effects.
To sum up the characteristics of the above-discussed state-of-the-art dynamic
stall models, Table lists the properties of each
model and in which aspects the model can be improved further.
Properties of the discussed dynamic stall
models.
Model nameFirst or secondHigherModelModelModelorderharmonicsCLCDCMLeishman–Beddoesfirst order–xxxSnelsecond orderxx––Adema–Snelsecond orderxxx–New second-order IAG model
The proposed IAG model is developed based on knowledge gained from four
different models: Leishman–Beddoes, Snel, Adema–Snel and ONERA
models with
modifications. Similar to the modern models like those from Snel (and ONERA)
and its derivatives, the present model is constructed by two main terms: the
first-order and second-order corrections. The total dynamic response of the
airfoil is formulated as
59CLnD=CLnD1+ΔCLnD2,60CDnD=CDnD1+ΔCDnD2,61CMnD=CMnD1+ΔCMnD2,
with D1 and D2 being the first- and second-order
corrections, respectively. Below the description of the modifications made
for the new model will be discussed in detail.
First-order correction
Based on the Hopf bifurcation model of that used the LB
model as the starting point of the first-order correction, the present model
operates similarly. Despite that, the LB model is not transferred into the
state-space formulation, but it is retained as the indical formulation. The
model applies the superposition of the solution using a finite-difference
approximation to Duhamel's integral to construct the cumulative effect on an
arbitrary time history of angle of attack. The LB model described in
Sect. to will be used with the following
modifications.
In the above LB model, predictions for drag are not accurate as will be shown
in Sect. . This inaccuracy lies in the
determination of η in Eq. () for the tangential force
component because drag is more sensitive to tangential force than the lift
force is. Also, to maintain simplicity, parameter η is removed, and the
tangential force is obtained from the static data at the time-lagged angle of
attack αfn by
CTnf=CTVISCαfn.
If one uses this formulation directly, at some point drag still becomes lower
than the static drag value by a significant amount. By evaluating the
experimental data for several airfoils and various flow conditions, this is
not physical at small angles of attack, especially in the downstroke regime,
where it usually just returns to the static value. In fact, those
experimental data infer that strong drag hysteresis occurs only at high
angles of attack beyond stall. Similarly, in the upstroke regime the drag
value increases only slightly (approximately only 20 %). In
Fig. , one can see that drag hysteresis occurs when
ζn≲ζv, with
ζn=1πdCNdα1+fn22,
and ζv is a constant with a value of 0.76. Based on these
observations, a simple drag limiting factor is adopted when
ζn≥0.76 as
CDnD=1.2CDnVISC;ifCDnD>1.2CDnVISCandCNnP-CNn-1P≥0.0CDnVISC;ifCNnP-CNn-1P<0.0CDnD;otherwise.
Note that for the purpose of numerical implementation, it is always
recommended in practice to adopt relaxation to avoid any discontinuity which
may present in the above formulation. Furthermore, the value
of ζv may be chosen differently for different airfoils
depending on the vortex lift influence on drag. Further studies on this
aspect are encouraged. The effects of these modifications are displayed in
Fig. .
Relation between drag hysteresis in the stall regime with weighted
separation parameter ζ for four airfoils. From (a)
to (d): S801 (13.5 %), NACA4415 (15 %), S809 (21 %) and S814
(24 %).
It will also be shown in Sect. that predicting
moment coefficient directly from the static polar data by means of the
time-lagged angle of attack has its drawback in the correct damping effect
calculation. One may obtain better results by using the “fitting function”
as in Eq. () instead of using Eq. ().
However, this limits the usability for different airfoils, since the fitting
has to be done again for each individual airfoil. For wind turbine
simulations, this is fairly impractical because a wind turbine blade is
usually constructed by several different airfoils, not to mention the
interpolated shapes in between each airfoil position. To solve for this
issue, a relatively simple approach is introduced by applying a time delay to
the circulatory moment response as
CMnC=CMn-1Cexp-ΔsTMU-CPfnCVn-CVn-1exp-Δs2TMU;ifτvn<TvlandΔαn≥0CMn-1Cexp-ΔsTMD-CPfnCVn-CVn-1exp-Δs2TMD;ifΔαn<0CMn-1C;otherwise
where
CPfn=KfCdCNdααnCRIT-α0INV,
with KfC, TMU and
TMD being constants relatively insensitive to airfoils.
Furthermore, the second condition of Eq. () is
modified to avoid discontinuity which occurs at a large reduced frequency
(e.g., k=0.2) for any LB-based models without recalibration of the time
constant as
τvn=τvn-1+0.45ΔtcV;ifCNnP1>CNCRITτvn-1exp(-Δs);ifCNnP1<CNCRITandΔαn≥0τvn-1;otherwise.
The effects of these modifications are displayed in
Fig. .
Drag reconstruction in comparison with the experimental data for the
S801 airfoil applying
(a) Eq. (), (b) Eq. (), and
(c) Eqs. ()
and ().
Moment reconstruction in comparison with the experimental data for
the S801 airfoil applying
(a) Eq. () and
(b) Eq. ().
The total first-order dynamic response of the airfoil is formulated as
68CNnD1=CNnf+CNnV,69CTnD1=CTnf,70CMnD1=CMnf+CMnV+CMnC,
where the definition and description of each variable were given in
Sect. for the LB model. Thus the first-order lift
and drag responses can be obtained by
71CLnD1=CNnD1cosαn-CTnD1sinαn,72CDnD1=CNnD1sinαn+CTnD1cosαn.
Second-order correction
The second-order correction takes the form of the non-linear ordinary
differential equation according to the second-order correction of the Snel
and Adema–Snel models. Generally, the basis
of implementation of the present model mostly uses the Adema–Snel
model where the normal force is used instead of the lift
force as for the original Snel model as
ΔC¨NnD2+Kf21nΔC˙NnD2+Kf20nΔCNnD2=F2n.
This way, shedding effects apply not only to the lift force but also to the
drag force. Note that τ is not present in Eq. () in
contrast to the original formulation in Eqs. ()
and (). The equation is changed because the time
derivatives in the above equation are no longer with respect to time but to
s=2Vt/c, similar to the ONERA model . This is done to
nondimensionalize the equations.
In Eq. (), the ks was projected as a
function of the angle of attack by sinαn. This modification causes
problems when the hysteresis effect takes place in both positive and negative
angles because Eq. () will be zero and then
negative, causing instability of the ODE. Thus, the original form of the Snel
model is retained in the present model, but the constant is
modified as
Kf20n=20ks21+3ΔCNnD221+3αn˙2.
The idea for the downstroke damping as in Eq. () is
adopted in the present model; the following form and constants are used:
Kf21n=150ks-0.01ΔCNnINV-0.5+2ΔCNnD22;ifαn˙>030ks-0.01ΔCNnINV-0.5+14ΔCNnD22;ifαn˙≤0andαn≥αnCRIT0.2ks;ifαn˙≤0andαn<αnCRIT.
Note again that τ is not present in the above equation. The original
formulation in Eq. () replaces the damped oscillator
when αn˙≤0 for a self-excited oscillator of Van der Pol
type with more damping. This is in contrast with the implementation done in
, where the self-excited oscillator is only
replaced by the damped oscillator, when the flow is reattached on the return
cycle. Under such circumstances, the oscillatory behavior still subsists in
the return cycle, albeit with smaller amplitude. To accommodate this aspect,
the last term of Eq. () is applied when the angle is
smaller than αnCRIT. As for the forcing term, the
original form of the Snel model is adopted as
F2n=0.5ks-0.15ΔCNnINV+0.05ΔC˙NnINV.
To facilitate the inclusion of the higher harmonic effects for the pitching
moment, the idealized center of pressure obtained in the first-order
correction given in Eq. () is passed into the second-order
model. Thus, the dynamic moment reaction takes the form
ΔCMnD2=-CPvnΔCNnD2.
Regarding the tangential force, a similar assumption is made as in
Eq. () where the influence of
ΔCTnD2 is neglected in the formulation, but
ΔCTnD1 is considered in the first-order
term correction described above. Finally, the second-order term of the lift
(ΔCLnD2) and drag
(ΔCDnD2) can be calculated easily. The
effects of the second-order term are displayed in
Fig. .
Airfoil response reconstruction in comparison with the experimental
data for the S801 airfoil applying only the
first-order correction and with inclusion of the second-order term.
(a) Lift, (b) drag and (c) pitching
moment.
Constants applied for the investigated dynamic stall models
The following constants are applied in the implemented dynamic stall models.
These values are kept constant throughout the paper. The constants for the
Leishman–Beddoes model and for the proposed IAG model are given in
Tables and , respectively. For
any model developed based on the Leishman–Beddoes type, the critical angle
of attack plays a major role. This can be obtained as the angle where the
viscous pitching moment breaks or when the drag increases significantly (or
the stall angle). The applied critical angles are given in
Table . The validation is done by comparing
the calculations with experimental data obtained at The Ohio State University
for the S801 airfoil (13.5 % relative thickness) ,
NACA4415 airfoil (15 % relative thickness) , S809
airfoil (21 % relative thickness) and S814 airfoil
(24 % relative thickness) . All selected test cases for
the airfoils are employed with a leading edge grit (turbulator) to enable the
“soiled” effects on a wind turbine blade at a Reynolds number of around
750 K. Note that these polar data are different from those used for example
by where the natural transition cases were
taken. Therefore, the critical angles of attack are also different. The
results of the studies are presented in the following sections.
Critical angle of attack (αnCRIT) applied for
the Leishman–Beddoes and IAG models.
S801NACA4415S809S81415.1∘10∘14.1∘10∘Results and discussion
The three state-of-the-art dynamic stall models reviewed above
(Leishman–Beddoes, Snel, Adema–Snel) have been used as a basis for
examining the dynamic loads of four different pitching airfoils at various
flow conditions. Experience gained from those models is used to formulate a
new second-order dynamic stall model, namely the IAG model, by evaluating the
weakness and strength of each model. The presented second-order models need
to solve a set of differential equations. For this purpose, the Euler–Heun
forward integration method is used.
Comparison against experimental data
This section compares the predicted dynamic forces and the measurement data.
For a fair comparison, all models are assessed with the same time step size
of Δt=T/1440, with T being the pitching period. The evaluations
are performed on the S801 airfoil at k=0.073. The comparison of each model
is shown in Figs. to for
the Snel, Adema and IAG models, respectively. Note that the constants of the
other existing dynamic stall models are taken directly from literature
without further calibration for the S801 airfoil. Therefore, it is already
expected that their performance will not be optimal. The main purpose of the
comparison is not to study their accuracy but to analyze the robustness of
each model for a different airfoil without tuning the constants. On the other
hand, the constants for the IAG model in Table were
obtained using the S801 airfoil. To enable a fair assessment of the model
robustness, the IAG model will also be used to reconstruct the dynamic polar
data of four airfoils with different relative thicknesses without changing
the constants in Sect. .
Dynamic force reconstruction using the Snel model in comparison with
the measurement data for Δt=T/1440. S801
airfoil, k=0.073, α‾=20∘ and
Δα=10∘.
Dynamic force reconstruction using the Adema model in comparison
with the measurement data for Δt=T/1440.
S801 airfoil, k=0.073, α‾=20∘ and
Δα=10∘.
Dynamic force reconstruction using the IAG model in comparison with
the measurement data for Δt=T/1440. S801
airfoil, k=0.073, α‾=20∘ and
Δα=10∘.
The original Snel models cannot predict the drag and moment coefficients in
the original formulations. Thus, only the static polar data are shown. The
Snel model actually shows an acceptable accuracy even though the constants
are taken as found in literature. The higher harmonic effects are
unfortunately not captured by this model. This is further refined by the
Adema model which was developed as an improvement for the Snel model. The
model performs fairly well for the lift and drag predictions, though the drag
value at small angles of attack is a bit off. The pitching moment prediction
is also not included in its formulations. These disadvantages are better
treated in the proposed IAG model. Not only the prediction of the lift
coefficient but also the accuracy of drag prediction are improved
significantly. The modifications described in Sect.
result in the improvement at low and high angle-of-attack regimes. The model
is also able to reconstruct the pitching moment polar accurately, which is
important for aeroelastic calculations of wind turbine blades.
For the following sections, the proposed IAG model will be tested under
various flow conditions and for several airfoils at various relative
thicknesses in comparison with measurement data. Note that these calculations
are performed without changing the constants to assess the robustness of the
model at different flow conditions.
Effects of time signal deviation
The actual pitching motion within The Ohio State University (OSU)
measurement differs slightly from the intended motion. The actual time series
of the angle of attack is included in the experimental data
. To assess
the effects of this time signal deviation on the aerodynamic response, the
calculations using these time signal data were performed applying the IAG
model. The results are compared with the experimental data and the results of
the IAG model presented in Sect. . Note that
these time signal data are fairly coarse and can cause problems for
second-order dynamic stall models because the gradient of α change can
be extremely large. To cover for this issue, the time signal is interpolated
in between each available point using a third-order cubic-spline
interpolation. The time signals are discretized by Δt=T/1440 over a
single pitching period. The first period of oscillation is used for
initialization of the time integration; thus a constant angle of attack is
applied as shown in Fig. .
Comparison of the time series of the idealized sinusoidal angle of
attack to the exact signals in the experimental campaign for the S801
airfoil, k=0.073, α‾=20∘ and
Δα=10∘.
Figure presents the influence of the time signal
variation on the aerodynamic performance in terms of CL,
CD and CM. TS labels the exact time signals in the
experimental campaign. Although the time signal difference has almost no
influence on the global prediction characteristics, some deviation from the
idealized sinusoidal motion can be noticed clearly. For example, one can see
the increased lift buildup in the upstroke regime before stall and the
location of the lift stall. Some deviations in the drag and pitching moment
coefficients are observed as well. For the rest of the paper, the prediction
procedure using the actual time signal from the experimental data is used for
the best consistency with the experimental campaign.
Dynamic force reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the
actual angle of attack in the experimental campaign. TS labels the exact time
signals in the experimental campaign. S801 airfoil, k=0.073,
α‾=20∘ and
Δα=10∘.
Performance of the model for different mean angles of incidence
In this section, the effects of the mean angle of attack are evaluated. Three
different angles of attack at the same inflow conditions are selected for
this purpose. These are α‾=8, 14 and 20∘. Note that
these mean angles of attack are only approximations since the actual time
signal data from the experimental campaign are used. The selected mean angles
represent the regime of attached flow, partly separated flow and fully
separated flow conditions. These are helpful to assess the model performance
under various flow situations.
Figure presents the results for the lift
coefficient under these three investigated mean angles of attack. The model
performs very well for these different cases. The maximum lift is a bit
overestimated in the model for the lowest α‾, but in general
all unsteady lift characteristics in the measurement data are reproduced in a
sound agreement with the experimental data. A similar behavior is shown for
the drag prediction depicted in Fig. . The
proposed model captures the increased drag effect and its shedding
characteristics well. The simple modifications applied in
Sect. result in a good prediction of the drag
coefficient compared with the experimental data. In
Fig. , the prediction for pitching moment
is shown. Here the predicted moment coefficient is in a good agreement with
the measured values.
Lift reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the
actual angle of attack in the experimental campaign at various
α‾ values. From (a) to (c):
α‾=8, α‾=14 and
α‾=20∘. S801 airfoil, k=0.073 and
Δα=10∘.
Drag reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the
actual angle of attack in the experimental campaign at
various α‾ values. From (a) to (c):
α‾=8, α‾=14 and
α‾=20∘. S801 airfoil, k=0.073 and
Δα=10∘.
Pitching moment reconstruction by the IAG model in comparison with
the measurement data for Δt=T/1440 using
the actual angle of attack in the experimental campaign at various
α‾ values. From (a) to (c):
α‾=8, α‾=14 and
α‾=20∘. S801 airfoil, k=0.073 and
Δα=10∘.
Performance of the model for different reduced frequencies
The effects of pitching frequency on the aerodynamic response will be
discussed in this section. Three different reduced frequencies are examined,
namely k=0.036, 0.073 and 0.111. The stall regime is shown here, where the
prediction is the most challenging. The actual time signals as of the
measurement campaign are used, following the procedure described in
Sect. .
Figure displays the results for the
dynamic lift coefficient response. The lowest reduced frequency of 0.036 is
dominated by the viscous effects. It represents the case where the
“delayed” angle-of-attack response is the weakest. It can be seen that the
maximum attained lift coefficient increases with increasing k. The gradient
of the lift polar in the upstroke and downstroke phases also increases as
well. These characteristics are present in both experimental data and
predictions delivered by the IAG model. A similar behavior is also displayed
in drag and pitching moments in Figs.
and , respectively. It is obvious that
stall occurs much earlier for a smaller k value. One can see that the
maximum amplitude of all three force components increases with increasing
k. This can be dangerous for the structural stability, since the amplitude
determines the fatigue loads.
Lift reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the
actual angle of attack in the experimental campaign at various k values.
From (a) to (c): k=0.036, k=0.073 and k=0.111. S801
airfoil, α‾=20 and
Δα=10∘.
Drag reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the
actual angle of attack in the experimental campaign at various k values.
From (a) to (c): k=0.036, k=0.073 and k=0.111. S801
airfoil, α‾=20 and
Δα=10∘.
Pitching moment reconstruction by the IAG model in comparison with
the measurement data for Δt=T/1440 using
the actual angle of attack in the experimental campaign at various k
values. From (a) to (c): k=0.036, k=0.073 and
k=0.111. S801 airfoil, α‾=20 and
Δα=10∘.
To better investigate the effects of k, the IAG model is used to
reconstruct the dynamic polar data at various k values by applying an
idealized sinusoidal motion as presented in
Fig. . Only the last DS cycle is shown for
clarity of the observation. While the maximum amplitude of all three force
components at low-frequency domains increases with increasing k (blue and
green markers), the amplitudes for all three forces at high-frequency domains
show different characteristics as shown in the Fourier transformation in
Fig. , albeit with much smaller values. The
higher harmonic amplitudes are attributed to flow separation effects, while
for low-frequency domains are driven by the pitching motion (i.e., external
unsteadiness or inflow).
Effects of k on the aerodynamic response by the IAG model for
Δt=T/1440. S801 airfoil, α‾=20 and
Δα=10∘. (a–c) Polar; (d–f) time
series.
Fourier transformation of the predicted forces presented in
Fig. . fs=f/f0 with
f0 being the pitching frequency.
Performance of the model for different pitching amplitudes
In this section, the effects of pitching amplitude on the aerodynamic
response of a pitching airfoil are investigated. The mean angle of attack is
fixed at α‾=20∘. Note again that
α‾ is only an approximation because the actual time signal
data from the measurement campaign are applied. This large mean angle of
attack is purposely selected because the post-stall characteristic is of
interest and is well known for its violent vibration, even for the static
condition. The small amplitude in this case means that the whole pitch
oscillation occurs within the stall regime.
Figures
to display the dynamic force
responses due to pitching motion of the airfoil predicted by the IAG model in
comparison with the experimental data. The model accurately reconstructs the
dynamic forces despite the predicted case being challenging within the
post-stall regime. Interesting to note is that the small-pitching-amplitude
case induces stronger shedding effects for lift than the larger-amplitude
case. This can be explained as follows. As described by Leishman in his
papers , the airfoil sees a lagged force response compared to the
imposed disturbance. Therefore, in his model, a time-lagged angle of attack
is introduced as the “effective” angle actually seen by the airfoil
section. When the pitching motion takes place partly within the fully
separated region (in the static case) and partly in the attached/partly
separated flow region, the airfoil still sees the lower angle (where the flow
is still attached) even though the pitching motion already reaches the
post-stall regime. This effect stops/decreases when the effective angle is
larger than the critical angle defined in
Table . As the critical angle for the S801
airfoil is defined at 15.1∘, the lower-amplitude case is fully
operating within the stall regime, where the attached flow effect is not
present.
Lift reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the
actual angle of attack in the experimental campaign at various Δα
values. (a)Δα=5.5∘;
(b)Δα=10∘. S801 airfoil, k=0.073 and
α‾=20∘.
Drag reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the
actual angle of attack in the experimental campaign at various Δα
values. (a)Δα=5.5∘;
(b)Δα=10∘. S801 airfoil, k=0.073 and
α‾=20∘.
Pitching moment reconstruction by the IAG model in comparison with
the measurement data for Δt=T/1440 using
the actual angle of attack in the experimental campaign at various
Δα values. (a)Δα=5.5∘;
(b)Δα=10∘. S801 airfoil, k=0.073 and
α‾=20∘.
Performance of the model for different airfoils
In this section, the performance and robustness of the proposed IAG model are
assessed for airfoils with different relative thickness. All model constants
in Table remain the same for all calculations. The
difference from one airfoil calculation to the other lies only in the
critical angle of attack value as shown in
Table . The value was obtained simply by
looking at the static polar data where the viscous pitching moment breaks or
when the drag increases significantly.
Despite the increased airfoil thickness from 13.5 % to 24 %,
Figs.
to demonstrate that the
reconstructed dynamic forces are in a good agreement with the experimental
data, not only for the general trend but also the higher harmonic effects.
As is also the case for the Leishman–Beddoes model, it is important to
select the appropriate value for the critical angle of attack. The simple
approach used in the present paper has shown its usefulness and potentially
reduces the complexity of parameter tuning for industrial applications.
for example defined two different critical angles
of attack, one for CN and the other for CT that were
shown to improve the prediction accuracy. Although their attempt might be
beneficial, this is not followed in this work because one main aim of the
studies is to reduce parameter tuning required for one case to the other
cases.
Lift reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the actual angle of attack in the
experimental campaign for different airfoils. From (a)
to (d): S801 (13.5 %), NACA4415 (15 %), S809 (21 %) and S814
(24 %). k=0.073, α‾=20∘ and
Δα=10∘.
Drag reconstruction by the IAG model in comparison with the
measurement data for Δt=T/1440 using the actual angle of attack in the
experimental campaign for different airfoils. From (a)
to (d): S801 (13.5 %), NACA4415 (15 %), S809 (21 %) and S814
(24 %). k=0.073, α‾=20∘ and
Δα=10∘.
Pitching moment reconstruction by the IAG model in comparison with
the measurement data for Δt=T/1440 using the actual angle of attack in the
experimental campaign for different airfoils. From (a)
to (d): S801 (13.5 %), NACA4415 (15 %), S809 (21 %) and S814
(24 %). k=0.073, α‾=20∘ and
Δα=10∘.
Predictions of the center of pressure
To further complement the analyses conducted in
Sect. , the location of the actual pressure
center is calculated in this section as
Xp=-CMCL,
which indicates the distance of the pressure point to the quarter chord
position where CM is defined.
A correct location of the pressure point is important for determining the
stability on aeroelastic simulations of wind turbine blades. The results of
the calculations both for the experimental data and for the proposed IAG
model are presented in Fig. for all
four investigated airfoils both as time series and as the polar plot. It can
be seen clearly that the agreement between the experimental data and the
present predictions is excellent for all investigated airfoils.
Center of pressure reconstruction in comparison with the measurement
data by the IAG model for Δt=T/1440 using the actual angle of attack
in the experimental campaign for different airfoils. From top to bottom: S801
(13.5 %), NACA4415 (15 %), S809 (21 %) and S814 (24 %). k=0.073,
α‾=2∘0 and
Δα=10∘.
L2 norm of error analyses
introduced a way of quantifying the absolute
error between the experimental data and modeled lift coefficient. The general
formulation reads
L2ϕ=1N∑iNϕimod-ϕiexp2,
with ϕ being the variable of interest, i the current sample and
N the total number of samples. In their paper, however, only lift was
considered. Here all three force components will be shown for four different
airfoils. Figure displays the quantified error for two
different flow categories, attached and deep stall. The time series of the
angle of attack was obtained from the measured data by applying a third-order
cubic-spline interpolation in between each available point. One can see that
generally the attached flow case is predicted very well, while the error
increases as the flow condition becomes more complicated. Interestingly,
especially for lift, it seems that the error decreases with increasing
airfoil thickness. The reason for the larger error obtained for the thinner
airfoil is attributed to the complex characteristics of the leading edge
stall, causing severe load variations, especially with increasing angle of
attack. Thus, it makes the prediction more challenging. Furthermore, the
quantification of the error was also performed on two other dynamic stall
models, Snel and Adema–Snel. The same approach for the angle of attack
signal was applied. One can see that the IAG model shows its improved
prediction in particular for the deep-stall case for all three force
components.
Quantified L2 norm of error with respect to the measurement
data for four airfoils. (a–c) Attached flow case (k=0.073,
α‾=8∘, Δα=5.5∘);
(d–f) deep-stall case (k=0.073, α‾=20∘,
Δα=10∘).
Conclusions
Comprehensive studies on the accuracy of several state-of-the-art dynamic
models to predict the aerodynamic loads of a pitching airfoil have been
conducted. From the studies, the strength and weaknesses of each model were
highlighted. This information was then transferred to develop a new
second-order dynamic stall model proposed in this paper. The new model
improves the prediction for the aerodynamic forces and their higher-harmonic
effects due to vortex shedding, developed for robustness to improve its
usability in practical wind turbine calculations. Details on the model
characteristics, modifications and treatment for numerical implementation
were summarized in the present paper. The studies were conducted by examining
the influence of the time step size, time signal deviation, mean angle of
attack, reduced frequency, pitching amplitude and variation in the airfoil
thickness. Several main conclusions can be drawn from the work.
The general characteristics of the polar data can be predicted by all
investigated dynamic stall models. Despite that, only the Adema model
and the present IAG model are able to demonstrate the higher harmonic
effects among the three investigated models.
The exact time signal imposed based on the measurement campaign
improves the prediction accuracy of the IAG model in comparison with
the idealized sinusoidal motion.
The dynamic forces reconstructed by the IAG model are in a sound
agreement with the experimental data under various flow conditions by
variation in α‾, k and Δα and for four
different airfoils by changing only the values of the critical angle of
attack.
The amplitudes at low-frequency domains increase with increasing k
and can be attributed to the effects of inflow/external unsteadiness.
The amplitudes at high-frequency domains decrease with increasing k
values, which are driven by flow separation effects.
When the airfoil operates at a high α‾ within the
stall regime, a small Δα leads to increased vibrations for
lift. The opposite is true for the pitching moment.
Recommendations for future work
The present paper evaluates the newly developed IAG model under various flow
conditions for four different airfoils. The following aspects are encouraged
for future work.
In the present studies, the assessment was mainly carried out for the
S801 airfoil having a relative thickness of 13.5 %. This airfoil is
mainly characterized by leading edge separation, which is very
challenging for validating the accuracy of a dynamic stall model.
However, typical modern wind turbine blades usually employ airfoils
with no less than 18 % relative thickness and a much higher Reynolds
number. Therefore, future investigations shall be done for thicker
airfoils at various flow conditions as well.
The above statement is also true for the current available
experimental data. Therefore, experiments on dynamic stall for thick
airfoils at a much higher Reynolds number are encouraged.
Three-dimensional effects (Himmelskamp or tip loss effects) for a
rotating blade can alter the loads significantly even under a steady
inflow condition. Further consideration and examination of the model
under this condition shall be carried out.
Further tests and recalibration of the model for deep-stall
conditions at extremely large angles of attack are encouraged, which
can be relevant for a turbine in standstill.
List of symbols
VariablessNondimensional time (s)VIncoming wind speed (m s-1)tTime (s)cChord (m)CNNormal force coefficient (–)CTTangential force coefficient (–)CLLift force coefficient (–)CDDrag force coefficient (–)CMPitching moment coefficient (–)CNPTotal inviscid CN(–)CNP1Time-lagged total inviscid CN(–)CNIImpulsive inviscid CN(–)CNCCirculatory inviscid CN(–)CNfViscous CN(–)CTfViscous tangential force coefficient (–)CMfViscous pitching moment coefficient (–)CMCCirculatory pitching moment coefficient (–)CNVVortex lift CN(–)CNCRITCritical CN(–)CPfStepping parameter moment (–)fFrequency (Hz)f0Pitching frequency (Hz)fn, f1, f2Separation factor (–)F1, F2First- and second-order forcing term (–)kReduced frequency (k=πf0c/V) (–)KfStiffness coefficient (–)ksConstant close to Strouhal number (–)MMach number (–)X, Y, DDeficiency functions (–)c1, c2, c3Curve-fitting constants (–)a1, a2, a3Curve-fitting constants (–)S1, S2, S3, α1Curve-fitting constants (–)A1, A2, b1, b2Model constants (–)Kα, Tp, Tf, Tv, TvlModel constants (–)Kv, η, KfC, TmU, TmDModel constants (–)Greek lettersαAngle of attack (rad (unless stated otherwise))α0Zero lift α(rad (unless stated otherwise))αeEffective α(rad (unless stated otherwise))αfTime-lagged αe(rad (unless stated otherwise))αCRITCritical α(rad (unless stated otherwise))βMach-number-dependent parameter (–)τvNondimensional vortex time (–)τTime constant (–)ζvVortex lift drag limiting factor (–)SubscriptsnPresent sampling time (–)fViscous lagged value (–)vVortex lift affected value (–)
The raw data of the simulation results can be shared by
contacting the corresponding author of the paper.
Author contributions
GB developed the new model, designed the studies and
conducted the analyses. TL and MA supported the research and provided
suggestions and discussion about the manuscript.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors gratefully acknowledge Wobben Research and Development GmbH for
providing the research funding through the collaborative joint work DSWind.
The measurement data provided from The Ohio State University are highly
appreciated.
Financial support
This research has been supported by the Wobben Research and
Development GmbH (grant no. DSWind). This
open-access publication was funded by the University of
Stuttgart.
Review statement
This paper was edited by Alessandro Bianchini and reviewed
by Khiem V. Truong and Gerard Schepers.
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