Introduction
The reduction of ultimate and fatigue loads plays an important role in
today's wind energy research. In the background of economic efficiency, load
alleviation systems bare potential to reduce rotor weight and costs, to
increase the turbine reliability or allow a further enlargement of the rotor
radius and thus power output. One promising concept to reduce dynamic load
fluctuations are trailing edge flaps applied to the outer part of the rotor
blade. As flaps are able to increase or decrease the local lift by adapting
the deflection angle, it is possible to partly compensate for load variations due
to variations of the effective inflow angle and velocity.
Over the last years, several investigations showed the potential of the flap
concept, such as a test on a full-scale turbine performed by
. In aero-elastic simulations, fatigue load reductions
up to approximately 30 % have been found for a trailing edge flap
covering up to 25 % of the blade span of a 5 MW
turbine . In most of the numerical studies the aerodynamic
loading was computed by blade element momentum (BEM) codes
e.g.,, which have been
extended with different engineering models to account for the unsteady flow
e.g.,. As viscous and unsteady aerodynamics have a
great influence on dynamically deflected flaps , it is,
however, important to also apply higher-fidelity models and gain knowledge of
the flow physics. In this respect, a lot of studies have been performed on
2-D airfoils, for example by and
using computational fluid dynamics (CFD). Comparisons of simulation methods with different aerodynamic fidelities were performed by in 2-D. For the 3-D wind turbine
rotor, only few publications based on higher-fidelity aerodynamic models are
available. In 2012 compared CFD to BEM predictions for a
rotor with trailing edge flap in an artificial half-wake scenario and found a
reasonably good agreement with regard to the complexity of the test case.
investigated trailing edge flaps on the 3-D rotor as part
of the European AVATAR project and proved the load alleviation potential
using a CFD approach. Several comparisons of codes with different aerodynamic
fidelities can also be found in the AVATAR project reports
. A benchmarking within the European
INNWIND.EU project showed, however, that
there are still differences between the results of CFD simulations and BEM
methods which need to be analyzed. While a previous investigation focused on
the analysis of static flap deflection angles by means of
CFD, the main objective of the present work is to study the influence of
unsteady 3-D effects on the example of harmonically oscillating morphing
flaps.
Different deflection frequencies ranging from 1 to 6 p are analyzed on the
DTU 10 MW rotor at rated operational conditions. These
frequencies are considered a realistic operational range for active load
alleviation. The investigated flap layout consists of a single morphing flap
ranging from 70 to 80 % blade radius with 10 % local chord extent.
This limited dimension along the blade span was chosen to obtain a high
impact of 3-D effects. In all cases the flap oscillates with an amplitude of
10∘. It should be noted that the present work does not aim towards an
assessment of the flap concept. The objective is to investigate unsteady 3-D
aerodynamic effects caused by trailing edge flaps and to obtain deeper
knowledge about the dominant phenomena as fundamental basis for an
enhancement of engineering tools commonly used for load calculations. In
this respect aero-elasticity is not considered since, on a flexible blade,
pitching and plunging movements are superimposed onto the flap oscillation and
a distinction of the isolated effects would be difficult.
Aerodynamic effects of trailing edge flaps
Two-dimensional airfoil
Trailing edge flaps are able to increase or decrease the airfoil lift for
respectively positive (downwards) or negative (upwards) deflections due the
change in the airfoil camber. As exemplarily displayed in
Fig. , this leads to a vertical shift of the lift
coefficient cl over angle of attack (AoA) α curve. This
possible lift increase is, however, mostly connected to an increase in drag as
depicted in the drag coefficient's cd–α plot.
Additionally, the moment coefficient around the quarter chord point
cm is also significantly influenced by the change in the airfoil
shape. In general the flap concept aims towards reducing the overall load
fluctuation, but in particular the dominant out-of-plane forces and blade
root bending moment. They are primarily influenced by the lift coefficient.
Example of flap deflection on cl, cd and
cm (FFA-w3-241 airfoil, Reynolds number Re=15.57 × 106, Mach number
M=0.2).
Three-dimensional rotor blade
The increase or decrease in lift in a blade section with trailing edge flap
influences the aerodynamic phenomena in most parts of the rotor blade. A
qualitative illustration of the vortex development around a rotor blade with
deflected flap can be given on the basis of potential flow theory as
illustrated in Fig. . It shows the vortex system with positive
flap deflection in spatial (panel a) and temporal (panel b) consideration.
Sketch of the bound circulation along a wind turbine blade with
trailing (a) and shed (b) vorticity part
a:.
Due to the spatial gradient of bound circulation along the blade radius, a
vortex sheet trails the rotor blade. In the flap section the bound
circulation increases locally due to the change in camber. This leads to
higher gradients at the flap edges and hence stronger trailing vortices at
these locations. Outboard at the blade tip, the tip vortex is shown. Wake
vorticity caused by radially changing bound circulation is commonly referred
to as trailed vorticity.
Generally, the efficiency of the flap with regard to local lift increase or
decrease is reduced by trailed vorticity in the 3-D case. The flap deflection
causes an additional downwash or upwash in the flap section. This leads to a
respectively lower or higher effective AoA in the 3-D case and consequently
to induced drag in relation to the baseline AoA. It is worth noting that, with
respect to the case without flap deflection (β=0∘), the induced
drag is increased in the case of positive deflections and decreased in the case of
negative deflections.
The adverse effect of trailing vortices in the flap section is, however,
countered by a positive effect in the blade parts adjacent to the flap
section. Caused by the sign change in induced velocities over the flap edge,
the described effects for the flap section are experienced in the opposite way at
these blade parts. With regard to integral loads such as power and thrust,
this effect opposes the negative impact of trailing vortices in the flap
section.
The temporal consideration (Fig. b) displays an increase in
bound circulation caused by an increase in the flap angle. This causes shed
vorticity with opposed sense of rotation. Shed vortex structures re-induce
velocities at the blade location and lead to a change in the effective AoA
which in turn influences blade loads. Wake vorticity linked to temporal
changes in bound circulation is called shed vorticity.
Theodorsen theory
Shed vorticity has been analyzed by as well as by
for the 2-D case of an airfoil with flap.
derived an analytical solution for the unsteady
airfoil response caused by sinusoidal flap actuation based on his theory from
for thin airfoils . This
solution is dependent on the reduced frequency (Eq. ), one of
the most important characteristic parameters when it comes to unsteady
aerodynamics . It is a measure of the unsteadiness of a
problem as relation between frequency f and chord length c to the inflow
velocity vinf.
k=π⋅f⋅cvinf
According to Theodorsen's method the lift is given by
cl(t)=2πC(k)F10βπ+F11β˙c4πvinf︸1+c2vinf2-vinfF4β˙-c2F1β¨︸2.
The derived function consists of two terms: a first term which represents the
circulatory forces connected to the bound circulation, and a second term
which accounts for added mass effects. This second term is mostly referred to
as non-circulatory part and depicts the influences of the inertia of the
fluid. In this function, β represents the instantaneous flap angle and
its time derivatives. The coefficients F1 to F11 are geometric terms
solely dependent on the relative flap length to chord. For their definition
it shall be referred to . The function C(k) is the
complex Theodorsen function which accounts for the effects of the shed wake.
Lift amplitude in relation to flap amplitude Δcl/Δβ and phase shift according to
Theodorsen.
The instantaneous lift coefficient cl(t) can be analyzed with
regard to the amplitude Δcl and phase shift ϕ of the
lift response with respect to the input flap signal. Δcl
can also be evaluated in relation to the amplitude of flap deflection
(Δcl/Δβ). Figure shows
the solution for a flap length of 10 % chord as function of the reduced
frequency k.
In the diagram two curves are plotted: a solid curve which represents the
solution of the complete function cl(t), and a dashed curve which
shows the solution if only the circulatory components are regarded and
apparent mass effects are neglected. No major difference between both curves
is apparent in Δcl/Δβ. It is continuously
decreasing with k in the displayed range which is applicable for this work
as the investigated flap frequencies correspond to reduced frequencies of
k=0.024–0.147 at mid-flap position. An increasing lag can be observed in
phase shift, which is more pronounced if only circulatory components are
included. While below a value of roughly 0.1 the differences between both
curves are small, higher discrepancies are observed at larger k when
apparent mass effects become increasingly dominant. In conclusion it is,
however, found that the non-circulatory contribution is in the investigated
range of reduced frequency of minor importance.
Additionally to the lift equation, Theodorsen developed functions for
cm, the pressure drag cdp and the flap hinge
coefficient cm,fh. The reader is referred to for
their definition. Theodorsen's derivations include the assumptions of thin
airfoils and 2-D flow. Both are not applicable for the aerodynamics of a
modern wind turbine since current developments in blade design tend towards
thicker airfoils for increased stiffness. But since the theory is well known
and commonly used to determine unsteady aerodynamic characteristics, it is
compared to the obtained results.
Numerical approach
Simulation process chain
The process chain for the simulation of wind turbines, which was developed at
the Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart
, is used in the present work. The main part constitutes
the CFD code FLOWer of the German Aerospace Center (DLR) .
FLOWer is a compressible code that solves the 3-D Reynolds-averaged
Navier–Stokes (RANS) equations in integral form. The finite-volume numerical
scheme is formulated for block-structured grids. A second-order central
discretization with artificial damping is used to determine the convective
fluxes, which is also called the Jameson–Schmidt–Turkel (JST) method.
Transient simulations make use of the implicit dual-time-stepping scheme. To
close the RANS equation system, several state-of-the-art turbulence models
can be applied, such as the SST model by used in
this study. FLOWer offers the use of the CHIMERA technique for overlapping
meshes which is applied in the simulation of 3-D wind turbines.
Grid generation is widely automated with scripts for 2-D airfoils and 3-D
rotor blades. The generation of the blade grid, for example, is conducted with
Automesh, a script developed at the IAG for the commercial grid generator
Gridgen by Pointwise. The blade grids are of C type with a tip block and
coning towards the blade root in order to connect to the turbine spinner.
Spinner and nacelle are typically included in the simulations. In the case of
pure rotor simulations as performed in this study, the computational domain
is modeled as a 120∘ model with periodic boundary conditions on each
side to reduce computing efforts. For the present study, this means that the
flaps of all blades are deflected simultaneously.
On the post-processing side, again several scripts are available for the
analysis of the simulations. Loads are calculated by the integration of
pressure and friction distribution over the blade surface. Sectional
distributions along the blade radius are determined similarly by dividing the
blade into different radial sections.
Trailing edge flap model
Trailing edge flaps are modeled based on grid deformation in FLOWer.
Therefore, the deformation module was extended by a
polynomial function to describe the shape of
the deflected flap.
w=φ(x)⋅βφ(x)=00≤x<(c-b)(c-x-b)nbn-1(c-b)≤x≤c
In Eq. () c represents the chord length and b the flap length.
The result w is the vertical change in the y direction, while the movement
in the x direction is neglected for small deflection angles up to 10∘.
Using this function requires the chord to be aligned with the x axis. The
polynomial order n is set to 2 for this investigation. In
Fig. the deformation methodology shown for a 2-D airfoil
section. The un-deformed and deformed airfoil surface are shown serving as
input to the grid deformation algorithm, which computes the new simulation
grid at each time step.
Methodology for 2-D deflection
.
The approach for the blade mesh is displayed in Fig. .
There is no separate grid for the flap part. It is integrated into the blade
grid. The connection between the moving flap part and the remaining rigid
blade surface is computed by the deformation algorithm which generates a
smooth transition. At the location of these transitions the blade grid is
radially refined to capture gradients in the flow field which are expected to
occur.
Methodology for 3-D deflection.
Code-to-code validation of the simulations
A baseline simulation setup for the DTU 10 MW turbine without flap has been
examined and validated by code-to-code comparison within the AVATAR project.
For this purpose a simulation of the power curve on the basis of the stiff
straight blade without precone was conducted in steady mode. The comparisons
between the different codes of the project partners are presented by
. The FLOWer results showed good agreement with results
obtained with EllipSys3D by DTU and MapFLow by NTUA. A detailed analysis of
the FLOWer results with a special focus on the comparison of steady and
unsteady simulations is performed in . A comparison of the
simulations with flaps to the AVATAR project partners can be found in
and .
Grid generation
For the 2-D airfoil simulations, the 75 % blade cut of the DTU 10 MW
turbine (FFA-w3-241 airfoil), which is the mid-flap position in the chosen
trailing edge flap configuration, was extracted from the geometry. The
airfoil grid was generated using a script for the commercial grid generator
Pointwise. Approximately 180 000 grid cells have been used with 417 surface
nodes and 205 nodes in the wall-normal direction. Boundary layer resolution was
chosen for ymax+∼1 and the farfield boundary is located at a distance of 150
chords.
For the rotor simulations, the setup used in the code-to-code validation was
modified in order to simulate the rotor with trailing edge flap. Flap edge
refinements were included into the blade grid and a higher resolved
background grid was chosen to accurately capture wake effects. A separate
grid convergence study of the blade grid was performed with a steady flap
deflection ±10∘ and the results are presented in
. The final setup used in the present study amounts to
approximately 21.65 million cells. The distribution between the different
grids is shown in Table .
Grid cells in the 120∘ model.
Blade
Spinner
Nacelle
Background
Total
8.16 × 106
1.39 × 106
1.45 × 106
10.65 × 106
21.65 × 106
Temporal discretization of the simulation setup with flaps
Another critical issue with regard to the unsteady simulations is the
temporal resolution meaning the choice of time step. Unsteady simulations in
FLOWer make use of the dual-time-stepping method as an implicit scheme. In this
approach a pseudo-time is introduced into the equation system at each time
step for which a steady solution is obtained. The method allows the choice of
significantly larger time steps than those dictated by the Courant–Friedrichs–Lewy condition in
explicit schemes. However, the actual eligible size is problem dependent. In
most cases the time step is a trade-off between simulation accuracy and
computational time and it is necessary to determine the largest possible time
step that can still resolve the unsteady flow effects sufficiently. To
analyze the influence of the temporal discretization within this study, a
sensitivity study has been performed based on 2-D and 3-D simulations
First, the 2-D airfoil case (FFA-w3-241, 75 % blade radius) is
presented. The simulations have been performed at a realistic inflow
extracted from the 3-D rotor case at rated operational conditions. These
conditions are specified in Table .
At 75 % radius, the Reynolds number was determined to 15.4 millions, Mach
number to 0.2 and the AoA to 6.5∘. For the determination of the time
step influence, the high flap frequency corresponding to 6 times the
rotational frequency at rated operational point (6 p) was chosen, which
corresponds to a reduced frequency of 0.147. Results for cl and
cd are shown in Fig. . As similar findings were
made for cm, it is not separately shown here but will be regarded
in 3-D. Four different time steps have been selected for the study and 100
inner iterations are performed in the dual-time-stepping scheme for all
investigated time steps except the small step of 0.028∘, for which 30
inner iterations are regarded sufficient. As the results are transferred to
the 3-D case later on, the different time steps are designated by the
corresponding azimuth step in a rotor simulation at rated rotational speed. A
time step of 0.028∘ is for example equivalent to
4.8 × 10-4 s. This very small
step correlates with 100 steps per convective time unit, which is in this case
the chord length. With regard to the computational effort this time step is
not realizable for the 3-D case, but serves as a reference in this 2-D study.
All other discretizations are applicable for pure rotor simulations. In both
aerodynamic coefficients the influence of the time step size is apparent, but
the effect on drag is slightly more distinct. In general, lift and drag agree
well for all resolutions except for 1∘, where major differences are
observed.
Influence of time step in 2-D, lift (no symbol) and drag
(symbol △).
Another parameter of influence is the amount of inner iterations in the
dual-time-stepping scheme, which is analyzed for the time step of 0.5∘
with three different amounts of inner iterations, 50, 100 and 200. The
results are displayed in Fig. . Again the results for the
small time step of 0.028∘ is shown as reference.
Generally, it is observed that the temporal accuracy is more dependent on the
total number of iterations per convective unit than the choice of time step
or inner iterations. By comparing the plots in Figs. and
, it can be seen that, for example, the hysteresis for
1∘ and 100 inner iterations is very similar to the curve with
0.5∘ and 50 inner iterations. This conclusion can, however, not be
transferred to separated flows, for which a small time step is needed to
resolve effects correctly.
Influence of inner iterations in 2-D, lift (no symbol) and drag
(symbol △).
DTU 10 MW turbine, rated operational conditions.
Wind speed
Rotational speed
Blade pitch
Tip speed ratio
11.4 m s-1
9.6 rpm
0∘
7.86
Based on these outcomes, simulations of the rotor model have been conducted
in order to get an impression of the 3-D case. The flap is again oscillating
with 6 p frequency at rated operational condition. Similar time step sizes
to those in the 2-D case have been chosen, replacing 0.028∘ with
0.125∘ as further halving. A total of 100 inner iterations are used.
Figure shows the resulting driving force, thrust and torsion
moment variations at mid-flap position.
The torsion moment is evaluated relative to the blade pitch axis and positive
reducing the blade pitch. The forces and moment are normalized with the total
mean value to allow an easier assessment of the differences. While thrust and
torsion moment show good agreement for nearly all time step sizes, higher
deviations are observed in the driving component in which the drag
differences have a stronger impact. However, a convergence of the curve
progressions with decreasing time step size can be observed, leading to small
differences between 0.125 and 0.25∘.
To conclude the temporal discretization study, a time step size of
0.25∘ with 100 inner iterations for a flap frequency 6 p shows
sufficient accuracy in 3-D simulations as a trade-off to computational time.
This corresponds to 240 steps per flap oscillation. In 2-D simulations
smaller time steps correlated with the convective unit are feasible and
consequently used.
Influence of time step in 3-D, 6 p, 75 % blade cut (driving force, a; thrust, b; and local
torsion moment, c).
Results
Three-dimensional rotor simulations with oscillating flap
First, results of the different flap frequencies are compared for the 3-D
rotor simulations. All simulations have been conducted at rated conditions
(see Table ). As ambient conditions an air density of
1.225 kg m-3 and a temperature of 288.15 K are used. The simulations
were started as a steady-state computation on two multi-grid levels with 8000
iterations each and a flap angle of 0∘. This steady solution
is then restarted in unsteady mode and simulated for the amount of
revolutions required for converged loads. With regard to the Menter SST
turbulence model, it should be mentioned that the required wall distances of
each cell were computed only once at the simulation start and not updated in
every time step. This was done since only a minor influence was found in 2-D
airfoil simulations and to save computation time as in 3-D the wall distance
calculation is very time consuming. Please note that while in the previous
time step study a cosine function is used as the deflection signal, now a sinus
function is applied (Eq. ).
β(t)=10∘⋅sin2πtNTRotorN=1,2,3,6,TRotor=6.25s
The flap frequencies correspond to reduced frequencies of
k=0.024(1p)-0.147(6p) at mid-flap position.
Figure shows the results of integral power and thrust
plotted over one rotor revolution. The effect of the flap can be seen clearly
in both diagrams. Power and thrust are oscillating with the respective
frequency. A higher frequency fluctuation is also apparent in the graphs,
which results from unsteady flow separation at the cylindrical blade root. As
illustrated in the streamlines plot on the blade surface shown in
Fig. , flow separation is dominant there. Due to
this superposition of effects in the integral forces, it is necessary to
regard sectional forces at a blade cut belonging to the flap part in order to
investigate the flap effects.
Integral rotor power (a) and
thrust (b).
In the following, like in the time step study, the 75 % cut as mid-flap
position was extracted from the simulations. Figure shows
the results of the local driving force, thrust and torsion moment at this
location over one flap period for all simulated frequencies. Thrust shows the
expected amplitude decrease and phase shifts with increasing frequency. A
significant reduction of the amplitude is observed between the 2 and 3 p
case, with 3 and 6 p showing a very similar curve progression. These
phenomena will be analyzed in more detail in Sect. . Larger
differences are observed in the driving component. For the 3 and 6 p case, a
second superimposed oscillation is visible from t/TFlap≈0-0.2 and t/TFlap≈0.8-1. This oscillation results from
the overlay of lift and drag forces in the rotor plane. At higher frequencies
drag shows a significant amplitude increase as seen in Fig.
and, additionally, cl and cd are oscillating with
different phases. The superposition of both force components leads to the
curve progression seen in the 3 and 6 p case. This phenomenon in driving
force is especially present at operational conditions with a pitch angle of
0∘, for which the impact of drag is high. Please note that in the
plot of the local torsion moment (Fig. c), the β axis
is reversed in order to be able to better judge the phase shift. The moment
shows an increase in amplitude with increasing flap frequency, mainly between
the 2 and 3 p case like it was observed vice versa for thrust. The high
frequencies 3 and 6 p are again similar. All curves show a lead with respect
to the flap signal. A more detailed elaboration of the torsion moment
relative to the quarter chord point is also performed in
Sect. .
Blade root flow separation – suction
side.
Variation in flap frequency, 75 % blade cut
(driving force, a; thrust, b; and local
torsion moment, c).
In Figs. and sectional
distributions of driving force, thrust and torsion moment over the blade
radius are shown for the 1 and 6 p case respectively. Four instantaneous
solutions are plotted for maximum, minimum and 0∘ flap deflection.
Thrust shows the expected increase and decrease in the flap section with a
smooth load distribution over the flap edges. This smoothing is a consequence
of the positive effect of the flap deflection on neighboring blade sections
as described in Sect. . While trailing vorticity
reduces the effect of the flap in the flap section compared to 2-D, the
sections next to the flap part produce higher/lower lift due to the induced
upwash/downwash for respectively positive/negative flap angles. The change in
sign in induced velocity caused by the flap edge vortices is also apparent in
the driving force as significant steps are appearing at the transition
between flap and rigid rotor part. An opposite behavior of sectional driving
force in relation to thrust can be noticed by comparing the diagrams. When
thrust increases locally in the flap area, the driving force decreases in
relation to neighboring sections. This results again from the strong
influence of drag on the driving component at rated wind turbine conditions
and will be explained on the basis of the 1 p case as follows. Due to the low
reduced frequency in this case (k=0.024), the influence of shed vorticity
is still weak. For maximum positive deflection (t/TFlap=0.25) the
increase in trailing vorticity causes a downwash in the flap section. This
reduces the effective AoA and leads to a rise in induced drag in addition to
the drag augmentation caused by the flap deflection itself. The overall drag
increase is compensated for by the lift increase resulting from the flap
deflection and relative to 0∘ flap deflection an increase in driving
force is achieved. The neighboring sections to the flap experience an
additional upwash in the case of positive deflections. Consequently, the induced
drag reduces associated with the lift increase and these sections produce in
total a higher sectional driving force. Similar observations are made vice
versa for maximum negative deflections, but the driving force increase in the
flap section is less pronounced compared to the decrease in the case of positive
deflection. Further elaborations in this respect can be found in
Sect. , in which lift and drag forces are extracted and
compared. With regard to the torsion moment around the pitch axis, a strong
oscillation is seen in the flap section with steep gradients at the flap edges.
This torsion moment or cm oscillation is typical for trailing
edge flaps and its effect on the overall performance of the
flap concept needs to be investigated separately in an aero-elastic
simulation when the blade is able to twist.
1 p sectional forces (driving force, a; thrust,
b; and local torsion moment, c).
6 p sectional forces (driving force, a; thrust,
b;
and local torsion moment, c).
The differences caused by unsteady effects can also be observed in the plots.
Larger variations of the forces are seen in the 1 p case compared to the
6 p case over the whole blade part influenced by the flap. In contrast the
variation of the moment is slightly increasing. The hysteresis can especially
be noticed at the time instance when the flap is positioned at 0∘
deflection. While in the 1 p case the sectional loads at increasing or
decreasing flap angle are close together, in the 6 p case larger
differences are seen. For the decreasing flap angle at
t/TFlap=0.5 the loads are higher than for the increasing flap
angle at t/TFlap=1.
Comparison to steady flap deflections
In a first step to analyze the influence of unsteady effects in 3-D, the
results are compared to the simulations of steady flap deflection for
±10∘. Like for the oscillating flap cases, the simulations were
initiated in steady state with 16 000 iterations and then restarted in
unsteady mode for three turbine revolutions with a time step corresponding to
2∘ azimuth. This approach is plausible because the flap area is
characterized by a steady flow situation as shown in similar studies
. Table shows the mean
integral loads of the third rotor revolution normalized with the respective
value for β=0∘ for a relative comparison.
By comparing to the results of the oscillating flap
(Fig. ), it can be noticed that in the oscillating
cases an increase in power is possible, while for steady deflections this is
not the case. A negative flap angle even leads to a higher power output than
a positive deflection. This phenomenon is caused by the differences in axial
induction. For positive deflections cl increases, more energy is
extracted from the wind and consequently the axial inductions also rises.
This leads to a lower AoA at the rotor blades, which reverses the effect of
the flap with regard to power. The opposite is observed in the case of negative
flap angles. But since less energy is extracted from the wind, a lower
power output compared to the neutral flap case is still observed. In this
background thrust also shows reasonable values. The magnitudes in the steady
cases are lower compared to the 1 p oscillating case, for which the axial
induction is not able to fully adjust to the changed load situation and
consequently higher oscillations are possible.
Integral loads for steady deflection normalized with value for
β=0∘.
β=10∘
β=-10∘
Normalized power (%)
96.2
98.3
Normalized thrust (%)
103
95.2
To verify this hypothesis an extraction of the local AoA along the blade
radius has been performed according to the reduced axial velocity
method . This method has proven to produce reasonable
results , but it is only applicable for steady
inflow conditions. But since in the 1 p case the reduced frequency is still
very low with a value of 0.024 at mid-flap position, a quasi-steady approach
is appropriate . The method requires annular elements at
different radii in front of and behind the rotor plane as input. These elements
are placed at an axial distance of one local chord to the rotor blade for the
present evaluation. The choice of this axial distance has, however, shown to
have an influence on the results with maximum discrepancies of about
0.16∘ when the axial distance is reduced for example to 0.2 local
chords. Nevertheless, the results of the different cases can be compared to
each other and give a qualitative and within this tolerance quantitative
analysis. The plot displayed in Fig. highlights the
differences for steady and oscillating flap deflections and underlines that
in unsteady cases the axial induction is only slowly adjusting. Consequently,
dynamic inflow effects play an important role on trailing edge flaps,
especially at lower flap frequencies.
Extracted AoA according
to .
Figures and show
extracts from the flow field at an axial distance of one local chord in front
of
and behind the rotor blade for respectively maximum positive and flap
negative angle.
The contour of the axial velocity u is displayed from a front view to the
turbine, which means that the rotational direction is clockwise. The rotor
blade is positioned upright. In each figure three cases are shown: steady
flap deflection, 1 p oscillation and 6 p oscillation. Clearly, the flow
acceleration towards the blade and the reduction of the wind speed in the
blade wake can be seen in all cases. In the flap section the opposing
deflection can mainly be identified in the blade wake, where less reduction
is observed for a negative angle and a higher reduction for a positive angle.
By comparing steady deflections to the 1 p frequency, the different axial
induction can be seen. When comparing the 1 and 6 p frequency, the different
flap frequencies can also be noticed in the blade wake. In the 6 p case at,
for example, β=10∘, an area with increased velocity is apparent.
Flow field extracted at one local chord distance in front of and behind
the rotor blade for β=10∘.
Influence of varying AoA in 3-D
As observed in the previous section, in the 3-D rotor case the local AoA is
oscillating over a flap period as a result of dynamic inflow. This means from
an aerodynamic point of view that two unsteady mechanisms are superimposed:
pitch and flap oscillation. The objective of the present work is, however, to
characterize and quantify unsteady 3-D effects solely due to flap deflection,
and consequently some preliminary considerations have to be made. For this
purpose the 1 p frequency is regarded in the following, for which
quasi-steady assumptions are eligible and the reduced axial velocity method
can be applied. The variation of the local inflow velocity and the AoA is
shown in Fig. for the mid-flap position. While
the inflow velocity shows no major variations, the AoA oscillates with an
amplitude of 0.6∘.
When the instantaneous AoA is extracted from the 3-D simulation it includes
the oscillations caused by both mechanisms, dynamic inflow and flap
oscillation. The dynamic inflow oscillation represents an oscillation of the
baseline AoA as a result of the variation of the axial induction of the
turbine. In contrast the oscillations caused by the flap originate from the
downwash of 3-D trailing vorticity, which changes the effective AoA. As the
objective is to quantify 3-D trailing vorticity, the flap-caused AoA
oscillation should be mimicked to the aerodynamic coefficients, while in
theory the influence of the oscillation caused by dynamic inflow should be
eliminated. A clear distinction between both oscillations is, however, not
possible and requires further aerodynamic modeling. Nevertheless, the
influence of the overall AoA oscillation on the 3-D extracted aerodynamic
coefficients can be assessed.
Flow field extracted at one local chord distance in front of and behind
the rotor blade for β=-10∘.
Figure presents the resulting cl and
cd variations in addition to the resulting variations for an
averaged AoA of 6.5∘ and local inflow velocity of 68 m s-1
(cl,mean, cd,mean). The moment coefficient
cm is not dependent on the inflow direction, so that the
evaluation of the AoA uncertainty for the cm behavior can be
excluded in this section. It can be seen that the AoA oscillations have only
a minor influence on the value of cl but a strong impact on
cd. This is reasonable as for the determination of cl
and cd in the 3-D case, the forces are integrated from the
surface solution as driving force and thrust components at first and then
transferred to the local inflow, also called the aerodynamic coordinate system.
The procedure is shown in Fig. for both components. To
determine total lift and drag forces, both shares by driving force and thrust
are summed up. As the value of cl is roughly 100 times larger
compared to cd, a projection difference of 0.6∘ as
observed in the 1 p case has only a negligible impact on cl.
Consequently, the results for cl and cl,mean are very
similar. However, cd and cd,mean differ strongly.
Based on the previous considerations this difference consists of two parts:
induced drag which originates from trailed vorticity and the drag resulting
from the AoA oscillation caused by dynamic inflow.
1p instantaneous inflow conditions 3-D, 75 %
radius.
1p comparison of lift and drag, 3-D instantaneous/3-D mean, 75 %
radius.
The plot in Fig. can be directly linked to the curves
for driving force and thrust in the 1 p case displayed
Figs. and . Thrust is dominated
by cl and consequently the progressions over a flap period are
very similar. The driving component is a superposition of both forces which
are oscillating at a different phase. This can be noticed, for example, at the
time instance when the flap is deployed to maximum deflection
(t/TFlap=0.25). cl but also cd,mean is
maximal and as a result the progression of driving force flattens.
The phenomenon is observed in the opposite way at minimum deflection (t/TFlap=0.75).
With regard to the objective of this section, determining the influence of
the AoA oscillation on the 3-D extracted aerodynamic coefficients, it can be
concluded that the extracted cl is only marginally affected and it is
appropriate to use cl,mean for the comparison to 2-D simulations and
the evaluation of the impact of trailing vortices. With respect to
cd it is difficult to clearly distinguish between the part caused
by the AoA oscillation and the induced drag from trailed vorticity. In order
to judge the impact of trailing vortices on cd, this
differentiation is, however, necessary and the part caused by the AoA
oscillation needs to be eliminated. The emphasis of the comparison to 2-D
simulations is hence on lift and moment coefficient.
Transformation of driving force and thrust to local aerodynamic
coordinate system (driving force, a; thrust, b).
In order to quantify the limitations of the average approach with respect to
the phase shift, a fast Fourier
transformation (FFT) analysis of the results has been performed for the
1 p frequency and the main peak in frequency was analyzed. The averaged and
instantaneous solution differ by only 1 % in lift amplitude and
0.26∘ in phase shift, which is below the time step resolution of
1.5∘. Due to the limitations caused by the time step, size hysteresis
effects are only qualitatively judged by comparing the curves; lift
amplitudes, however, can be opposed quantitatively.
Comparison to 2-D simulations
To study the unsteady phenomena in more detail and to analyze the main
effects in the 3-D case, the instantaneous cl,mean and
cm results of the 3-D extraction are compared to 2-D airfoil
simulations of the mid-flap position at mean inflow conditions. The mean
inflow conditions used as input for all flap frequencies in 2-D are again
AoA=6.5∘ and vinf=68 m s-1. This
leads to a Mach number of 0.2 and a Reynolds number of 15.4 million.
Results of the cl comparison between 2-D static deflection, 2-D
sinusoidal deflection and 3-D sinusoidal deflection are shown in
Fig. . In order to get an impression of the influence of AoA
oscillation in the 1 p case, additionally an AoA-corrected version of the
2-D sinusoidal oscillation case is plotted which is computed by
Eq. ().
cl,2-Dsinus,AoAcorr(t)=cl,2-Dsinus(t)+2πα3-Dsinus(t)-αmean
Comparison of 2-D/3-D lift for different flap frequencies:
1 p (a), 2 p (b), 3 p (c) and
6 p (d).
As for the higher frequencies the reduced axial velocity method is not
applicable, and as the AoA variations are smaller compared to the 1 p case, no AoA-corrected
curve is plotted. For these cases the results obtained
with the 2-D theory of Theodorsen have been added for comparison. The plots
show the expected decrease in lift amplitude from 2-D static over 2-D
sinusoidal to 3-D sinusoidal. For the 1 p oscillation the comparison of 2-D
static and 2-D sinusoidal results shows the minor influence of unsteady
effects. Even though hysteresis begins to develop in the 2-D sinusoidal
results, the cl amplitude reduction is still small. This result
corresponds well to the low reduced frequency in this case of 0.024. Larger
differences are seen by comparing 2-D and 3-D results which show the decrease
in amplitude caused by trailing and shed vorticity. The reduced amplitude
leads to less shed vorticity and thus less hysteresis is apparent compared to
the 2-D solution.
cl amplitude, 75 % blade cut, 2-D and 3-D results.
1 p
2 p
3 p
6 p
2-D
0.42
0.40
0.38
0.33
3-D
0.30
0.29
0.25
0.24
Δcl,3-D/Δcl,2-D
71 %
73 %
66 %
73 %
2-D, Theodorsen
0.42
0.40
0.38
0.35
The AoA-corrected 2-D curve shows the approximate result for a 2-D simulation
including an AoA variation in the inflow. The curve demonstrates less
hysteresis and a smaller amplitude compared to the baseline progression,
which is reasonable since the AoA progression is a feedback of the
aerodynamic forcing in the 3-D case. Like it was noticed in
Sect. , for low flap frequencies the axial induction is
able to react to the instantaneous load and mimics the effects. The smaller
slope which is seen in the 3-D curve can be explained by the decrease in the
gradient dcl/dβ caused by trailing vortices in 3-D.
In 2-D unsteady effects constantly increase with the flap frequency for the
regarded cases, which corresponds well to the results with Theodorsen's 2-D
theory. The amplitude of lift oscillation is continuously reducing and more
pronounced hysteresis is seen. The results obtained by the Theodorsen's
theory are in fair agreement with the CFD results. A more symmetrical
hysteresis is apparent compared to the slightly bent CFD curves. Generally,
the hysteresis direction is in agreement with the observations made by
. In 3-D, the amplitude is also continuously decreasing
but hysteresis effects show no clear trend. A slightly bigger hysteresis is
seen for the 2 p frequency than for the 3 p frequency. A reason for this
phenomenon could be a different phase lag in the AoA oscillation resulting
from the flap deflection. For clarification, an AoA extraction for unsteady
cases would be required.
cl phase shift, 75 % blade cut, 2-D results.
1 p
2 p
3 p
6 p
2-D
-6.3
-10.2
-12.9
-17.7
2-D, Theodorsen
-4.9
-7.5
-9.2
-11.7
cl amplitude, 75 % blade cut, 15 m s-1,
steady deflections .
β=10∘
β=-10∘
10 % chord
30 % chord
10 % chord
30 % chord
Δcl,3-D,10%span/Δcl,2-D
70 %
69 %
65 %
65 %
Δcl,3-D,20%span/Δcl,2-D
80 %
79 %
75 %
77 %
Table lists amplitudes and Table phase lags of
the lift coefficient for the different cases in order to quantify the
effects. The values were again obtained by applying an FFT on the unsteady lift progression and analyzing the main peak
in the results. Additionally, results of the 2-D theory by Theodorsen
described in Sect. are shown in the table. The
results of this simplified theory are in very good agreement with the results
of 2-D simulations with regard to the cl amplitude. Only at 6 p
is a slight difference noticeable. Larger differences are observed in the
phase shift, for which higher values are determined in the 2-D CFD
simulation. These differences can be caused by the assumption of thin
airfoils in Theodorsen's theory as similar observations were found by
for pitching airfoils.
The comparison of the 2-D and 3-D lift amplitudes shows the expected
reduction due to the influence of trailing vortices. The 3-D lift amplitude
alleviates 66–73 % of the result of 2-D simulations at the same flap
frequency based on the round values listed in Table 4. As it was noticed
before in Sect. with regard to the thrust oscillation,
the lift amplitude of the 3 and 6 p case is similar. This is also seen in the
relative amplitude reduction, as 3 p shows a significantly lower value than
the other cases. Unfortunately, the reason for this phenomenon could not
finally be clarified since this would require a method to extract the AoA in
transient cases, too. This would allow for judging and comparing the AoA
oscillation between all 3-D cases, which provides further insight. Further
research is required in this respect. Nevertheless, this relative reduction
is roughly constant throughout all flap frequencies. An earlier
investigation focused on the impact of steady flap
deflections on the blade performance. Various flap extensions along the blade
radius (10 and 20 %) and chord (10 and 30 %) were analyzed in a
parametric study at 15 m s-1 wind speed. The analysis also included a
comparison to 2-D simulations and an extract of the results is shown in
Table for positive and negative deflections.
A correlation to the results of the present study can be noticed. The
relative amplitude reduction (Δcl,3-D,10%span/Δcl,2-D) is very similar
for both chord extents and in a comparable range as observed in the
oscillating cases. Please note that the slightly lower values for steady
deflections can be explained by the adjusting axial induction as described in
Sect. . The value for 20 % radial extent (Δcl,3-D,20%span/Δcl,2-D) is in contrast
about 10 % higher. Consequently, the relative lift amplitude reduction at
mid-flap position (Δcl,3-D/Δcl,2-D)mid,flap can serve as a rough characteristic value
for a certain flap layout. It also allows a decoupled consideration of 3-D
and unsteady effects at this location. This means that as a first estimation
(Δcl,3-D/Δcl,2-D)mid,flap can be
determined based on a 3-D simulation with static maximum flap deflection. The
amplitude reduction by shed vorticity can be investigated separately in 2-D
and can then be superimposed with the 3-D result. This simplified or quasi-2-D
approach with regard to shed vorticity is, however, only valid at the mid-flap
area. Closer to the flaps edges the unsteady effects will also become
three-dimensional.
Comparison of 2-D/3-D moment coefficient for different flap
frequencies: 1 p (a) and 6 p (b).
With respect to a more general aerodynamic modeling in lower fidelity tools,
the present investigation suggests a simplified consideration of near-wake
effects. The dominant phenomena could be assigned to trailing and shed
vorticity, which can be captured by such approaches. First comparisons of the
present simulations to the near-wake model by , an
implementation in the BEM code HAWC2 by DTU (Technical University of
Denmark), can be found in . have also
published favorable results following a similar approach in the BEM code
FAST, which was developed by NREL (National Renewable Energy Laboratory).
Further benchmarks were performed within the AVATAR project
.
The results of the moment coefficient cm are depicted in
Fig. for the 1 and 6 p frequency. It is noted that while
the previous plots of the 3-D torsion moment were evaluated relative to the
pitch axis, the results have been transferred to the quarter chord point for
the 2-D comparison. In general, only a minor influence of unsteady
aerodynamic effects is observed. A slight reduction of the amplitude and
small hysteresis is seen in the 3-D case. The 2-D results are in good
agreement with , for which similar simulations have been
performed for the same airfoil.