2016_Ramanujam_G.__Ozdemir_H.__Hoeijmakers_H.W.M.-AIAA_2016-0748.pdf

(586 KB) Pobierz
AIAA 2016-0748
AIAA SciTech
4-8 January 2016, San Diego, California, USA
34th Wind Energy Symposium
Improving Airfoil Drag Prediction
Giridhar Ramanujam
†
Energy Research Centre of the Netherlands(ECN), 1755LE, Petten, The Netherlands
‡
University of Twente, 7500AE, Enschede, The Netherlands
¨
H¨seyin Ozdemir
§
u
Energy Research Centre of the Netherlands(ECN), 1755LE, Petten, The Netherlands
Downloaded by Giridhar Ramanujam on January 13, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-0748
H.W.M.Hoeijmakers
University of Twente, 7500AE, Enschede, The Netherlands
An improved formulation of drag estimation for thick airfoils is presented. Drag under-
prediction in XFOIL like viscous-inviscid interaction methods can be quite significant for
thick airfoils used in wind turbine applications (up to
30%
as seen in the present study).
The improved drag formulation predicts the drag accurately for airfoils with reasonably
small trailing edge thickness. The derivation of drag correction is based on the difference
between the actual momentum loss thickness based on free stream velocity and the one
based on the velocity at the edge of the boundary layer. The improved formulation is
implemented in the most recent version of XFOIL and RFOIL (an aerodynamic design
and analysis method based on XFOIL, developed by a consortium of ECN, NLR and TU
Delft after ECN acquired the XFOIL code. After 1996, ECN maintained and improved the
tool.) and the results are compared with experimental data, results from commercial CFD
methods like ANSYS CFX and other methods like DTU-AED EllipSys2D and CENER
WMB. The improved version of RFOIL shows good agreement with experimental data.
Nomenclature
α
∆θ
δ
δ
∗
∞
ρ
θ
ξ, η
A, B
airf oil
c
C
τ
EQ
c
d
c
l
D
e
†
Researcher,
‡
Former
Angle of attack
Error in
θ
Boundary layer thickness
Boundary layer displacement thickness
Subscript for incident free stream condition
Density of fluid
Boundary layer momentum thickness
Streamline space coordinates
G
−
β
equilibrium locus coefficients
Subscript for airfoil parameters
Airfoil chord length
Equilibrium maximum shear stress coefficient
Sectional drag coefficient
Sectional lift coefficient
Drag
Subscript for boundary layer edge condition
Wind Energy Unit, Westerduinweg 3, ramanujam@ecn.nl, Member AIAA
M.Sc. Student, Sustainable Energy Technology, giridhar.ramanujam@gmail.com
§
Researcher, Wind Energy Unit, Westerduinweg 3, h.ozdemir@ecn.nl, Member AIAA
Professor,
Department
of
Mechanical
Engineering,
Engineering
Fluid
Dynamics,
h.w.m.hoeijmakers@utwente.nl, Senior Member AIAA
PO
Box
217,
1 of 12
American Institute of Aeronautics and Astronautics
Copyright © 2015 by Giridhar Ramanujam, Huseyin Ozdemir and H.W.M. Hoeijmakers. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
Downloaded by Giridhar Ramanujam on January 13, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-0748
ef f ective
fp
g
H
H
∗
H
1
H
k
h
T E
L
M
∞
N
crit
Re
θ
Re
c
RF OIL
u
U
∞
U
e
wake
x, y
x
=
∞
x
C
τEQ
XF OIL
Subscript for overall domain parameters
Subscript for flat plate
Correction factor for mass flow shape factor
Shape factor
Kinetic energy shape factor
Mass flow shape factor
Kinematic shape factor
Airfoil trailing edge thickness
Length scale reference parameter
Free stream Mach number
Critical amplification factor
Momentum thickness Reynolds number
Chord length based Reynolds number
Subscript for RFOIL parameters
Velocity
Incident free stream velocity
Boundary layer edge velocity
Subscript for wake parameters
Cartesian space coordinates
Subscript for infinity downstream location
Equilibrium shear stress coefficient multiplier in wake
Subscript for XFOIL parameters
I.
Introduction
ethods
for accurate estimation of drag for airfoils is an important criterion for the analysis and design
of airfoils. Since for an airfoil, the drag is usually two orders of magnitude smaller than lift, even small
errors in drag values can cause a significant change in airfoil performance (lift to drag ratio). Thick airfoils
are commonly used in wind turbine blades with the thickness varying from 15% of chord at the tip section
to about 50% of chord at the root section. It is thus important to have an accurate prediction for drag in
order to determine the performance of the wind turbine. With the trend of size of wind turbines increasing,
thicker airfoils are becoming more and more important as the structural requirements need to be satisfied
while maintaining optimal aerodynamic performance. Present day drag estimation models appear to under-
predict the drag by a significant margin ranging from 10% for thin airfoils to as high as 30% or more for
thick airfoils with negligible trailing edge thickness (h
T E
<
3% of chord) as can be seen in the results of the
present study. In this paper, the method currently used for the calculation of drag for airfoils is studied in
detail and the cause of inaccuracy is analysed. The limitations of the present method are also investigated
in order to explain why it gives acceptable results for a certain range of airfoil types. Finally, a correction is
proposed to improve the prediction of drag and the results based on the correction are discussed.
M
II.
Aerodynamic Methods
For the analysis presented in this paper, XFOIL
1
and RFOIL
2
have been used. XFOIL
a
is a viscous-
inviscid interaction method for predicting flow about airfoils developed by Mark Drela at MIT. It utilizes a
linear-vorticity panel method with Karman-Tsien compressibility correction for analysis in direct and mixed-
inverse modes. Source distributions superimposed on the airfoil and wake permit modelling of viscous layer
effects on potential flow results. A two-equation lagged dissipation integral method is used to represent
the viscous layers. Both laminar and turbulent flows are treated with an
e
N
-type amplification formulation
determining the transition point. The boundary layer and transition equations are solved simultaneously with
the inviscid flow field by a global Newton method. The procedure is especially suitable for rapid analysis
of
low Reynolds number
flows around airfoils with transitional separation bubbles. RFOIL is a modified
version of XFOIL featuring an improved prediction for the maximum lift coefficient and includes a method
for predicting the effect of rotation on airfoil characteristics. Regarding the maximum lift in particular,
a
Refers to XFOIL 6.99 unless stated otherwise
2 of 12
American Institute of Aeronautics and Astronautics
numerical stability improvements were obtained by using the Schlichting velocity profiles for the turbulent
boundary layer, instead of Swafford’s velocity profiles incorporated in XFOIL. Furthermore, the shear lag
coefficient in Green’s lag entrainment equation of the turbulent boundary layer model was adjusted based
on the shape factor of the boundary layer for deviation from the equilibrium flow observed at high values of
the shape factor. From the results of the validation, 10% under-prediction has been found for the drag.
2
III.
Momentum Thickness and Drag - Current Formulation
The currently used formulation for estimation of the drag on an airfoil is based on flat plate boundary
layer theory. The momentum conservation in
x-direction
(direction of incident free stream) yields the drag
on a body immersed in the flow. The expression for drag of a flat plate
3, 4
is given in terms of momentum
thickness as follows,
2
D
f p
=
ρU
∞
[θ
∞
]
x=∞
,
(1)
Downloaded by Giridhar Ramanujam on January 13, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-0748
where the subscript
x
=
∞
indicates value of the parameter at the far wake of the airfoil and
θ
∞
is the
momentum thickness (for incompressible flow) based on free-stream velocity (indicated by subscript
∞)
given by,
δ
θ
∞
=
0
u
U
∞
1
−
u
U
∞
dy.
(2)
The above expression of drag involves the following assumptions,
•
Incident free stream velocity (U
∞
) is in
x-direction
•
U
∞
is a constant in space and time coordinates
•
The velocity outside the boundary layer (the edge velocity) equals
U
∞
. i.e. zero pressure gradient
along the flow
•
The flow is in steady state
•
The flow is incompressible
In dimensionless form, the drag coefficient is given by,
2 [θ
∞
]
x=∞
,
(3)
L
where
L
is a typical length scale reference parameter. For airfoils, the chosen length scale parameter is the
chord length.
c
d
=
In RFOIL (as in XFOIL), the drag coefficient is calculated as given in Eq.(3) but the definition of
momentum thickness is different. The momentum thickness is given in terms of stream-wise coordinates
instead of Cartesian coordinates as defined in Eq.(2) and the reference velocity is the velocity at the edge of
the boundary layer (U
e
) instead of the free stream velocity (U
∞
). The edge velocity (U
e
) on the airfoil surface
is obtained by coupling the boundary layer equations to the potential flow equations describing inviscid,
irrotational flow in the outer region using a strong interaction scheme. The expression for momentum
thickness in RFOIL is as follows,
δ
θ
e
=
0
u
U
e
1
−
u
U
e
dη,
(4)
and the drag coefficient is given by,
2 [θ
e
]
x=∞
,
c
where the subscript
e
denotes the momentum thickness calculated by using
U
e
as reference velocity.
c
d
RF OIL
=
(5)
3 of 12
American Institute of Aeronautics and Astronautics
The two definitions of the drag coefficient are not the same. Eq.(3) gives the force in
x-direction,
which is
the definition of drag, whereas Eq.(5) gives the force along
ξ-direction
(streamwise direction) which may not
necessarily be equal to the actual drag force on the airfoil. But this assumption is reasonable at a location
far downstream of the airfoil, where the flow is almost in the free-stream direction. Thus, in the far wake of
an airfoil, the streamwise and cartesian coordinates are almost coincident.
ξ
∼
x
,
η
∼
y.
(6)
However, the use of
U
e
instead of
U
∞
as the reference velocity in the momentum thickness calculation
would cause a significant difference in the estimation of the drag. In the next section, this difference is
analysed in detail and a correction is proposed.
IV.
Drag Correction
Downloaded by Giridhar Ramanujam on January 13, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-0748
The relation for drag follows from momentum conservation along incident flow direction (x-direction).
However, there is a significant difference between a flat plate and an airfoil in terms of the pressure distribu-
tion. There exists a pressure gradient along and normal to the surface of the airfoil which is a result of the
airfoil being of finite thickness as opposed to a flat plate which is considered to have zero thickness. There
is also an important effect of the angle of attack which is zero for the flat plate theory. As a consequence,
the outer region (outside the boundary layer) which is governed by the potential flow equations has a sig-
nificant pressure gradient in the
η-direction
(normal to streamline direction). This pressure gradient causes
the velocity outside the boundary layer to vary in the
η-direction.
For the analysis, we assume that the
velocity outside the boundary layer is constant in
η-direction
and hence we can neglect the viscous effects
in that region. This assumption is only valid when
U
e
=
U
∞
, which is the flat plate case. For an airfoil,
the assumption that velocity outside the boundary layer is
U
e
at all points in the normal direction results
in an under prediction of the integral quantities displacement thickness (δ
∗
) and momentum thickness (θ).
This also leads to an over prediction of
U
e
. In effect, the under prediction of
θ
reduces the predicted drag
and over prediction of
U
e
increases lift. This behaviour is very commonly seen in viscous-inviscid interaction
methods like XFOIL and RFOIL.
To calculate the error in drag, we need
to estimate the error in the momentum
thickness (∆θ) at the end of wake (taken
to be one chord length downstream of
the airfoil in XFOIL and RFOIL). ∆θ is
the difference between the actual momen-
tum loss thickness
θ
∞
which is based on
free stream velocity (U
∞
) and the pre-
dicted momentum loss thickness
θ
e
which
is based on calculated value of velocity at
the edge of the boundary layer (U
e
). It
is important to note that this error ∆θ is Figure 1: Cartesian (x,
y)
and Streamwise (ξ,
η)
directions on an
airfoil
a deviation of the physical model used
by viscous-inviscid interaction methods
from the real physical process, not an error of a numerical origin. For the following part of this section, all
boundary layer variables are taken to be at the end of wake.
∆θ =
θ
∞
−
θ
e
.
(7)
As mentioned earlier in Eq.(6),
η
∼
y
at the end of wake. This allows us to write the expression of
θ
e
as
a integration in
y-direction.
Also the limits of integration can be taken as the boundary layer thickness (δ),
from 0 to
δ
(the lower limit is taken zero in XFOIL and RFOIL since the wake is considered as symmetric
and the initial values of the boundary layer variables are taken as the sum of upper and lower airfoil surface
trailing edge values). Based on this, we have
4 of 12
American Institute of Aeronautics and Astronautics
δ
∆θ
=
0
δ
u
U
∞
u
1
−
U
∞
δ
dy
−
0
u
U
e
1
−
u
U
e
dy
=
0
u
u
−
U
∞
U
e
δ
0
−
u
−
U
e
u
2
u
2
−
2
2
U
∞
U
e
U
e
+1
U
∞
δ
0
δ
0
dy
u
2
2
U
e
dy
=
=
U
e
−
1
U
∞
U
e
−
1
U
∞
U
e
−
1
U
∞
U
e
−
1
U
∞
U
e
θ
e
−
U
∞
U
e
θ
e
+
U
∞
θ
e
+
u
2
dy
2
U
e
u
2
1
−
2
−
1
dy
U
e
=
=
Downloaded by Giridhar Ramanujam on January 13, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-0748
∗
where
δ
e
≡
δ
0
U
e
∗
(δ +
θ
e
−
δ) ,
U
∞
e
1
−
u
U
e
dy.
Since
H
1
≡
∗
δ
−
δ
e
in XFOIL and RFOIL, we have
θ
e
U
e
U
e
∆θ =
θ
e
1
−
(H
1
−
1)
−
1
,
U
∞
U
∞
(8)
where the value of
H
1
is obtained using the correlation from Green
5
simplified by Drela.
6
H
1
= 3.15 +
1.72
.
H
k
−
1
(9)
This expression is based on experimental results and cor-
∗
δ
−
δ
∞
∗
∗
relates
δ
to
δ
∞
and
θ
∞
as
H
1
≡
. Since
δ
∞
and
θ
∞
∗
θ
∞
are larger than
δ
e
and
θ
e
respectively (as evidenced
from drag under-prediction and lift over-prediction in XFOIL
and RFOIL), using the edge velocity based values will re-
sult in an over prediction of
H
1
. As seen from Eq.(9),
the value of
H
1
depends only on the value of kinematic
shape factor (H
k
), which for incompressible flow is the same
as the shape factor
H
(≡
δ
∗
/θ).
The rate of change of
H
1
with respect to
H
becomes extremely high for values
of
H
tending to unity (
H
→
1, see figure 2). Thus
in the far wake, where the value of
H
is close to unity,
the value of
H
1
is very sensitive to the prediction of
H
and even small changes in the boundary layer model can
cause a significant difference in the results. This differ-
ence is seen in the results of RFOIL compared to XFOIL.
In the next section, this difference in prediction is explored Figure 2:
H
1
−
H
k
correlation from Green
5
in order to obtain a generalised expression for the
θ
correc-
tion.
In order to correct the value of
H
1
, we introduce a constant
g
as a multiplier to
H
1
with the condition
that
g <
1. The value of
g
depends on the turbulent boundary layer method used in the solver and will be
determined later by trial and error. Using this in Eq.(8) and rearranging the terms, we have
∆θ =
θ
e
1
−
From Eqs.(7) and (10), we have,
θ
∞
=
θ
e
+ ∆θ =
θ
e
1 + 1
−
U
e
U
∞
U
e
(g
H
1
−
1)
−
1
U
∞
.
(11)
U
e
U
∞
U
e
(g
H
1
−
1)
−
1
.
U
∞
(10)
5 of 12
American Institute of Aeronautics and Astronautics

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin