Gasch R., Twele J. (Eds.) Wind Power Plants: Fundamentals, Design, Construction and Operation
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6.9 Limits of the blade element theory and three-dimensional calculation methods
242
where
x
cB
D.ma1
;
A.s
2
D.maxD.s
A.s
2
sin
cos
1
ccB ;
c
D.max
is around 1.3 for typical rotor blades of wind turbines
α
A.s
, c
D.s
– last measured data point of profile data (s = start of separation)
The corresponding lift coefficient is
A
A
2
2A1
sin
cos
2sin
AAc
L
(6.26)
where
2
1
1
B
A
;
A.s
2
A.s
A.sA.sD.maxL.s2
cos
sin
cossin
ccA
c
L.s
– last measured data point of profile data (s = start of separation)
After testing a prototype these values should be replaced with values determined
from the prototype measurements.
If the rotor blades rotate, further 3D-effects act on the flow. Measurements
indicate higher lift values in the near-hub region of the blade. Coriolis and
centrifugal forces influence the flow. Snel published an estimate to calculate the
additional lift [9]. He found that the local solidity (cf. Fig. 5-15) influences the lift
coefficients.
LL.2DL.3D.rot
ccc
(6.27)
The additional lift coefficient c
L
depends on the local radius r and the local chord
length c, the value c
L.inv
is from equation (6.30):
L.2DL.inv
2
L
3 cc
r
c
c
. (6.28)
Provided that
- the lift does not exceed the lift for attached flow,
- the lift is zero for very high angles of attack (α
A
= 90°)
- it is practical to modify equation (6.27) to
6 Calculation of performance characteristics and partial load behaviour
243
A
2
L.2DL.inv
2
L
cos3tanh
cc
r
c
c
(6.29)
The correction term is, however, limited to positive lift values with this modifica-
tion. And it is only valid for rotating blades. Its influence is limited to the near-hub
region of rotor blades with a large maximum chord length.
Fig. 6-24 Influence of 3D-effects on the lift coefficient c
L
. 2D-measurement, 3D-measurement
and 3D-corrected lift coefficient c
L
for 30% and 70% of the blade length [9]
There are different theories about the drag under these conditions. Usually the
changes are ignored because they only have a small impact on the turbine. Of
course, all this is only a rough approximation to describe the start of flow separa-
tion which, in reality, involves very complex physics.
Nowadays, all these “3D-effects” can be calculated by Computational Fluid
Dynamics (CFD) which also takes into account the viscous effects. The results
depend strongly on the chosen turbulence model and the Navier-Stokes equation
solver [10], cf. section 6.9.5. The main problem is that the calculation with these
programs is very time-consuming. So it seems to be expedient to back-calculate
the 2D-lift curves from these models and then use them in conventional BEM-
programs to calculate the dynamic loads.
6.9 Limits of the blade element theory and three-dimensional calculation methods
244
6.9.2 Dynamic flow separation (Dynamic stall)
A model for the dynamic flow separation is needed to calculate the aerodynamic
forces at elastic blades in turbulent flow. Otherwise the calculated dynamic loads
can be at an unrealistic high level. The real physics is very complex, especially for
leading edge stall. On rotor blades of wind turbines with large Reynolds numbers
and thick leading edges we expect mostly trailing edge stall. The Beddoes-
Leishman model was developed for helicopters, and this was further simplified for
wind turbine rotors by Øye [11]. This model works very well for trailing edge
separation. At first the lift coefficient curve is modelled according to the charac-
teristics of a flat plate at attached flow:
c
L.inv
= 2π(α
A
- α
A.0
) (for thin airfoils) (6.30)
The zero-lift angle α
A.0
is the angle of attack at which the lift coefficient of the
chosen airfoil is zero. The curve of separated flow is then modelled. According to
Hoerner [8] for a flat plate and fully separated flow (index sep) we can say that
c
L.sep
(α
A
)
= c
D.max
sin(α
A
- α
A.0
) cos (α
A
- α
A.0
) (6.31)
The lift coefficients of an existing profile can now be described as a function of
the degree of separation f
A.sepA.inv
A.sepA
A
)(
cc
cc
f
(6.32)
For each angle of attack α
A
the steady-state degree of separation f can now be
determined, and for each degree of separation f there is now a corresponding lift
coefficient c
A
. The degree of separation varies between zero and one and may be
interpreted as the location of the separation point on the blade (cf. Fig. 6-27). If
there is a quick change in the angle of attack, the separation point needs a certain
time to move to the new steady-state point. The time response is described in the
model of Øye by a simple PT1 behavior
fak
stat
ff
dt
df
(6.33)
The time delay constant τ
fak
depends on the chord length c and the relative velocity
w:
w
c
a
fak
(6.34)
6 Calculation of performance characteristics and partial load behaviour
245
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
-10-5 0 5 1015202530
Ca,inv
Ca,stat
Ca,sep
fstat
A
[°]
c
L
[ ]
c
L.inv
c
L.stat
c
L.sep
f
stat
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
-10-5 0 5 1015202530
Ca,inv
Ca,stat
Ca,sep
fstat
A
[°]
c
L
[ ]
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
-10-5 0 5 1015202530
Ca,inv
Ca,stat
Ca,sep
fstat
A
[°]
c
L
[ ]
c
L.inv
c
L.stat
c
L.sep
f
stat
Fig. 6-25 Lift coefficient c
L
composed of auxiliary curves for attached flow c
L.inv
and fully sepa-
rated flow c
L.sep
using the degree of separation f
stat
depending on the angle of attack
A
Measurements to determine the factor a are needed. For one wind turbine it was
found that a = 4 [11, 12]. This means for a simulation in the time domain
fak
1istat1ii
)(
t
ffftf
(6.35)
The lift coefficient can now be obtained by
)()(1)()(),(
AA.sepAA.invAL
ctfctftc
. (6.36)
6.9.3 Method of singularities
An older method based on the Laplace equation which describes the inviscid flow
(potential flow theory) is the method of singularities. Objects in the flow, together,
if possible, with the known fluid dynamic effects at the rotor, are described by
vortices or by sources and sinks which are superimposed on the main flow. It is
difficult to describe the wake flow behind the rotor. This is either assumed to be
static, or its shape is determined by iteration. Despite the Laplace equation being
only valid for inviscid flow, the method of singularities is from time to time still
applied in combination with the airfoil characteristics, although good results are
only achieved in the range of low profile drag.
6.9 Limits of the blade element theory and three-dimensional calculation methods
246
y
z
x
Fig. 6-26 Model for a free wake rotation
6.9.4 Computational fluid dynamics applied to wind turbines
Preliminary remark on computational fluid dynamics
The aim of numerical flow simulation by Computational Fluid Dynamics (CFD) is
the calculation and visualization of flow fields with a spatial and temporal resolution.
It provides a deeper understanding of the flow phenomena and saves cost-
intensive experimental investigations which, for wind turbines and other large
rotordynamic machines, are only possible using significantly smaller turbine models
or the already erected prototype. By choosing an appropriate physical model of
the flow it is hoped that the so-called “numerical wind tunnel” will represent the
occurring flow effects better than the blade element momentum method. The aero-
dynamic simulation of wind turbine components (like blades, spinner and nacelle
housing as well as the ventilation and cooling system) helps to optimize the distri-
bution of forces, to minimize flow losses, to improve acoustics, and so on. CFD is
based on the Euler equations of motion for inviscid fluid, or the Navier-Stokes
equations of motion if viscosity is considered [e.g. 13, 14]. Using a time-step re-
solved simulation, CFD allows the investigation of unsteady processes. Moreover,
6 Calculation of performance characteristics and partial load behaviour
247
it is possible to choose any section of the flow volume for visualisation thus facili-
tating the investigation of areas where access with measuring instruments is
difficult.
If CFD is applied by an experienced expert using reasonable boundary condi-
tions, the results for steady flow obtained in an acceptable period of time are quite
close to reality, e.g. close to the design point of the rotor. The computational
efforts required for unsteady flow, e.g. when stalling, are significant, and there is
still intense research needed to improve the modelling [10].
The basic equations of fluid dynamics applied in CFD can only be solved as
coupled partial differential equations and by iteration. Therefore, there is always a
residual deviation from the exact solution.
w
Stagnation point S
Laminar BL
Transition point U
Separation point A
Turbulent BL
Recirculation area
Separated BL
Viscous sublayer
w
Stagnation point S
Laminar BL
Transition point U
Separation point A
Turbulent BL
Recirculation area
Separated BL
Viscous sublayer
Fig. 6-27 Scheme of the development of the boundary layer (BL) and separation along the pro-
file section
ϕ
y
Outlet
Outlet
Inlet
Inlet
w
x
Fine
O-grid
Coarse
outer grid
ϕ
y
Outlet
Outlet
Inlet
Inlet
w
x
Fine
O-grid
Coarse
outer grid
Fig. 6-28 Scheme of CFD-grid with a block-structured 2D-grid including an O-grid around a
blade profile [15, modified]
6.9 Limits of the blade element theory and three-dimensional calculation methods
248
The choice of flow model, which includes e.g. the approaches for modelling tur-
bulence, has a significant influence on the result. For example, a simple turbulence
model with two equations may be able to perform calculations quickly but can
neither able to represent the development of the boundary layer (and the resulting
force distribution on the blade surface) nor compute the transition from laminar to
turbulent flow, Fig. 6-27. Complex flow models, in contrast, yield more exact
results but with a significant increase in computing time.
Besides displaying the flow properties of interest (visualization as well as inte-
gral values), the evaluation of CFD results always includes a critical assessment of
the chosen flow model and boundary conditions. If the CFD results are validated
with existing measurements, both, the results of simulation and measurement
should be viewed critically. It should be remembered that every measurement on
real wind turbines has an uncertainty. In some research work the comparison with
CFD revealed measuring discrepancies.
The procedure of numerical flow simulation comprises the following steps:
1. Discretisation of the flow region into many small partial volumes (finite
volume method) which form the computational grid, cf. Fig. 6-28 as an
example of the blocks of a 2D grid.
2. Description of the flow phenomena using suitable partial differential
equations (for single-phase flow the equation of continuity, i.e. mass con-
servation and momentum balance) and, for example, the ideal gas law for
consideration of the temperature.
3. Choice of a suitable simplified physical flow model for the solver and
selection of the boundary conditions (e.g. inflow and outflow velocity
and angle, as well as other conditions at the inlet, outlet and the walls, cf.
Fig. 6-28)
4. Solving the equations by iteration for each discrete partial volume until
the chosen residual (remaining difference to exact solution) is reached.
5. Evaluation and visualization of the results including critical assessment.
In the following, examples of flow simulation in wind turbines are presented to
illustrate the application of CFD in this area. Section 6.9.6 of the 4th German
edition of this book includes some deeper discussion of the steps in the CFD pro-
cedure mentioned above.
6.9.5 Examples of CFD application to wind turbines
The motivation for using CFD on wind turbines arises, at least in part, from the
fact that in the real wind turbines the observation of flow phenomena by means of
measurement techniques is difficult. Moreover, there is the constant attempt to
describe ahead of time the flow effects relevant for the efficiency and structure
6 Calculation of performance characteristics and partial load behaviour
249
which result from the interaction of blade, nacelle and tower and the wind farm
effects as well.
A: Separation point
Suction side
A
Pressure side
Line of flow
separation
Rotor axis
Sectional
plane
Exit of the radial flow
through the
separation bubble
Flow sucked in
from the near-hub
region
Attached flow
Deviated outer
flow
Tip vortex
Separated
helical flow
c
A: Separation point
Suction side
A
Pressure side
Line of flow
separation
Rotor axis
Sectional
plane
Exit of the radial flow
through the
separation bubble
Flow sucked in
from the near-hub
region
Attached flow
Deviated outer
flow
Tip vortex
Separated
helical flow
c
Fig. 6-29 Schematic representation of the flow in the separation area on the suction side of a
blade at higher wind speed [15, modified]
Early experimental works on propellers revealed that´, due to 3D effects, there are
higher lift coefficients in the near-hub area compared to the blade tip, and that
there are other lift curves than those measured for a 2D airfoil in the wind tunnel
[16]. It was found by means of experiments [e.g. 17] and numerical simulation
[18] that on the suction side, especially in the near-hub region, there are large
areas of separated flow due to the high blade twist angle, the contour change from
the circular blade root to the profiled blade and the unfavourable inflow. Fig. 6-29
provides the schematic representation of the effects described in the literature.
In the region of separated flow there are reduced flow velocities and therefore
the centrifugal and Coriolis forces have a stronger effect. A helical flow forms in
the separation bubble heading radially towards the blade tip.
In the European project VISCVIND, [10], various aspects of CFD were inves-
tigated and compared to measurement data. The aim was to compare the suit-
ability of the CFD software of different institutions for flow simulations at wind
turbines, to make improvements and to bundle the knowledge for further under-
standing of 3D effects.
6.9 Limits of the blade element theory and three-dimensional calculation methods
250
Fig. 6-30 shows the shear stress lines (near-wall stream lines) on the blade surface
obtained by a 3D simulation of the flow around the rotating blade LM 19.1. The
region of flow separation on the suction side of the profile with the nearly radial
streamlines expands towards the blade tip with increasing wind speed but constant
rotational speed (i.e. approaching stall). At v
1
=15 m/s wind speed backflow is
clearly visible in the region of separated flow. With real blades this is visualized
by attaching filament probes on the blade surface - or accidently by dirt on the
blades (e.g. oil or grease from the pitch mechanism or the blade bearing).
Separation region with nearly radial near-wall flow
RISO, LM19.1, v
1
= 7 m/s, n = 27.1 rpm, mesh = 28x24
3
RISO, LM19.1, v
1
= 10 m/s, n = 27.1 rpm, mesh = 28x24
3
Separation region with nearly radial near-wall flow
Separation region with nearly radial near-wall flow
RISO, LM19.1, v
1
= 15 m/s, n = 27.1 rpm, mesh = 28x24
3
Separation region with nearly radial near-wall flow
RISO, LM19.1, v
1
= 7 m/s, n = 27.1 rpm, mesh = 28x24
3
RISO, LM19.1, v
1
= 10 m/s, n = 27.1 rpm, mesh = 28x24
3
Separation region with nearly radial near-wall flow
Separation region with nearly radial near-wall flow
RISO, LM19.1, v
1
= 15 m/s, n = 27.1 rpm, mesh = 28x24
3
Fig. 6-30 Streamlines (wall shear stress lines) on the suction side surface of an LM-blade, from
a CFD calculation at different wind speeds; rotor blade length 19.1 m [10]
a) 2D calculation
b) Quasi-3D calculation
a) 2D calculation
b) Quasi-3D calculation
Fig. 6-31 Streamlines a) from a 2D-CFD calculation of the flow around an NACA 63
2
-415 air
foil at an angle of attack
A
= 20° and Re = 1.5*10
6
, b) Reduction of the separation area in a
quasi-3D calculation by introduction of radial force influences (centrifugal forces)
6 Calculation of performance characteristics and partial load behaviour
251
It is possible to transform 2D flow simulations around the airfoil into quasi-3D
simulations by introducing additional terms for the radial centrifugal forces. For
the Quasi-3D simulation, which considers centrifugal forces as 3D effect, Fig. 6-31
right, simulation of the flow around a rotating airfoil at an angle of attack of 20°
results in a reduced size of the separation bubble on the suction side compared to
the pure 2D simulation, Fig. 6-31 left. This improves the pressure distribution
which reflects the increased lift values observed in the near-hub area of rotor
blades.
These comparisons helped in the development of new correction equations,
based on existing approaches [9, 19], for transforming wind tunnel measured 2D-
profile characteristics to be then suitable as well for the tip and near-hub region of
the rotating blade. These transformations consider not only the ratio c/r of the
local chord length c to the local blade radius r (e.g. the values of 30 and 70% in
Fig. 6-24), but also the blade twist angle
[10].
If simulations are computed for a fully turbulent flow around airfoils the drag
coefficient is over-estimated in most cases, compared to measured profile charac-
teristics (as there are only a few specially designed full turbulent profiles). The
reasons for this may lie, among other, in the faster and stronger growth of the
simulated fully turbulent boundary layer compared to the real boundary layer
which is in its first section a laminar boundary layer, cf. Fig. 6-27. The effective
profile shape results from the profile shape itself and the boundary layer around it,
cf. Fig. 6-31. Therefore the fully turbulent approach may cause a displacement of
the stagnation point (cf. Fig. 6-27) as well as a “deformation” of the effective pro-
file.
Comparison studies of measured and CFD-simulated wind turbine power
curves are performed at present mostly by simulating a rotor section of 120°
around one blade, followed by a transformation of the result to the real wind tur-
bine with three blades. In most cases the rotor-tower-nacelle interactions as well
as the influence of the wind shear have been ignored so far to save computing
time.
Such a comparison study was performed e.g. for the stall-controlled wind turbine
Nordtank N500/41 [10] , Fig. 6-32. As long as the flow is attached, simpler turbu-
lence models pretty much agree on the measurement as well. However, the SST
model is obviously better for the more complex range with stall effect (strongly
separated flow).