Design Of The Wind Turbine Airfoil Family Riso-A-Xx 1998

30 Pages·1998·0.44 MB·English

Ris��R�0(EN)

Design
of
the
Wind
T
urbine

Airfoil
F
amily
RIS��A�XX

Kristian
S.
Dahl,
P
eter
F
uglsang

Ris�
National
Lab
oratory
,
Roskilde,
Denmark

Decem
b
er

Abstract
A
metho
d
for
design
of
wind
turbine
airfoils
is
presen
ted.
The
design

metho
d
is
based
on
direct
n
umerical
optimization
of
a
B-spline
represen
tation

of
the
airfoil
shap
e.
F
or
�exibilit
y
,
the
optimization
algorithm
relies
on
separate,

stand
alone
to
ols
for
the
analysis
of
aero
dynamic
and
structural
prop
erties.
The

panel
metho
d
based
XF
OIL
is
used
during
the
optimization
whereas
the
Na
vier-

Stok
es
solv
er
EllipSysD
is
used
in
the
ev
aluation
of
the
results.

The
metho
d
is
demonstrated
b
y
the
design
of
an
airfoil
family
comp
osed
of

airfoils
ranging
in
thic
kness
from
%
to
0%.
The
design
is
based
on
Reynolds
and

Mac
h
n
um
b
ers
represen
tativ
e
of
a
00
k
W
wind
turbine.
The
airfoils
are
designed

to
ha
v
e
maxim
um
lift-drag
ratio
un
til
just
b
elo
w
stall,
a
design
lift
co
e�cien
t
of

ab
out
.
at
an
angle
of
attac
k
of
0

�

and
a
maxim
um
lift
co
e�cien
t
of
..
The

airfoils
are
made
insensitiv
e
to
leading
edge
roughness
b
y
securing
that
transition

from
laminar
to
turbulen
t
�o
w
on
the
suction
side
o
ccurs
close
to
the
leading
edge

for
p
ost
stall
angles
of
attac
k.

The
design
metho
d
and
the
airfoil
family
pro
vides
a
sound
basis
for
further

enhancing
the
c
haracteristics
of
airfoils
for
wind
turbines
and
to
tailor
airfoils
for

sp
eci�c
rotor
sizes
and
p
o
w
er
regulation
principles.

The
Danish
Energy
Agency
funded
the
presen
t
w
ork
under
the
con
tract
ENS

/ -000.

Ris��R�0

ISBN
�0��

ISBN
�0��
(In
ternet)

ISSN
00�0

Information
Service
Departmen
t,
Ris�
�

Con
ten
ts

In
tro
duction

Wind
turbine
airfoil
c
haracteristics

Design
metho
d

.
Design
algorithm

.
Geometry
description

.
Optimization
algorithm

.
Sensitivit
y
analysis

.
Flo
w
analysis

.
Structural
analysis

Design
strategy
0

Airfoil
family

.
V
alidit
y
of
geometry
description

.
Design
criteria

.
Geometric
prop
erties

.
A
ero
dynamic
prop
erties

.
Comparison
of
XF
OIL
and
EllipSysD
predictions
0

.
Comparison
of
clean
and
dirt
y
p
erformance

Conclusion

References

Ris��R�0(EN)

In
tro
duction

Design
of
tailored
airfoils
for
wind
turbine
rotor
blades
is
imp
ortan
t
for
the
con
tin-

uing
dev
elopmen
t
of
wind
turbines.
Optimization
studies
sho
w
that
airfoils
with

suitable
c
haracteristics
are
imp
ortan
t
to
further
reduce
the
cost
of
the
pro
duced

energy
,
F
uglsang
and
Madsen
[
].
The
airfoils
that
are
curren
tly
used
range
from

rather
old
NA
CA
airfoil
series
originally
dev
elop
ed
for
airplanes,
Abb
ot
and
Do
en-

ho�
[
]
to
dedicated
wind
turbine
airfoils.
Wind
turbine
airfoils
should
di�er
from

traditional
a
viation
airfoils
in
c
hoice
of
design
p
oin
t,
o�-design
capabilities
and

structural
prop
erties.

The
dev
elopmen
t
of
wind
turbine
airfoils
has
b
een
ongoing
since
the
mid
0's

and
a
large
e�ort
w
as
done
b
y
T
angler
and
Somers
[
],
who
dev
elop
ed
sev
eral

airfoil
families.
Other
airfoil
designs
for
wind
turbines
can
b
e
found
in
Bj�rk
[],

Timmer
and
v
an
Ro
o
y
[0
],
Hill
and
Garrad
[]
and
Cha
viarop
oulos
et
al.
[].

Most
of
these
airfoil
designs
w
ere
dev
elop
ed
b
y
use
of
in
v
erse
design
metho
ds.

Numerous
metho
ds
for
airfoil
design
are
a
v
ailable
and
a
surv
ey
of
suc
h
meth-

o
ds
and
a
v
ailable
references
can
b
e
found
in
Henne
[
]
and
Dulikra
vic
h
[0
].
In

traditional
in
v
erse
design,
the
airfoil
surface
�o
w
is
prescrib
ed
at
sp
eci�ed
op
era-

tional
conditions
and
a
shap
e
is
found
that
will
generate
these
surface
conditions.

F
ull-in
v
erse
metho
ds
determine
the
o
v
erall
airfoil
geometry
from
the
o
v
erall
sur-

face
pressure
distribution
whereas
mixed-in
v
erse
metho
ds
determine
parts
of
the

airfoil
con
tour
while
holding
the
rest
unc
hanged.

A
full-in
v
erse
approac
h
for
incompressible
�o
ws
is
the
complex
mapping
metho
d

originally
form
ulated
b
y
Mangler
[
]
and
Ligh
thill
[0
].
A
metho
d
that
includes

a
b
oundary
la
y
er
form
ulation
is
later
dev
elop
ed
b
y
Lieb
ec
k
[
].
On
basis
of
these

metho
ds,
Eppler
and
Somers
dev
elop
ed
their
computer
co
de
[
],
whic
h
has
b
een

used
for
dev
elopmen
t
of
n
umerous
wind
turbine
airfoils,
e.g.,
[
].
A
p
opular

mixed-in
v
erse
metho
d
is
the
XF
OIL
co
de
b
y
Drela
[ ]
that
uses
a
global
Newton

metho
d.
XF
OIL
w
as
used
for
design
of
wind
turbine
airfoils
b
y
Bj�rk
[
]
among

others.

T
raditional
in
v
erse
design
metho
ds
in
general
ha
v
e
limited
capabilities
for
m
ul-

tiple
design
p
oin
ts,
since
there
is
only
a
single
target
pressure
distribution
at

a
single
design
p
oin
t.
Ho
w
ev
er,
a
metho
d
for
m
ulti-p
oin
t
design
using
an
in
v
erse

metho
d
w
as
dev
elop
ed
b
y
Selig
and
Maughmer
[
].
They
allo
w
di�eren
t
segmen
ts

of
the
airfoil
shap
e
to
b
e
determined
b
y
di�eren
t
�o
w
constrain
ts.
In
v
erse
design

metho
ds
can
not
treat
m
ultidisciplinary
design
problems
and
allo
w
only
limited

o�-design
considerations.
These
matters
are
most
often
tak
en
care
of
man
ually
b
y

the
designer
in
a
cut-and-try
pro
cess.

Direct
design
metho
ds
based
on
n
umerical
optimization
pro
vide
basically
a
ra-

tional
m
ultidisciplinary
design
pro
cedure
where
sev
eral
design
parameters
can
b
e

impro
v
ed
and
m
ultiple
constrain
ts
can
b
e
imp
osed.
A
general
�o
w
solv
er
and

ev
en
tually
a
structural
co
de
are
coupled
with
a
n
umerical
optimization
algorithm.

The
optimization
algorithm
generates
an
optim
um
airfoil
shap
e
that
has
desir-

able
c
haracteristics,
as
sp
eci�ed
b
y
the
designer,
while
satisfying
aero
dynamic

and
structural
constrain
ts.

Most
direct
design
metho
ds
use
gradien
t-based
algorithms.
Hic
ks
et
al.
[
]

used
a
simple
feasible
direction
algorithm
with
a
panel
metho
d
in
their
design
of

transonic
airfoils.
When
a
more
complex
�o
w
solv
er
is
used,
suc
h
as
in
Eyi
and
Lee,

[
],
the
computational
costs
increase
b
ecause
of
the
sensitivit
y
analysis,
whic
h

requires
a
large
n
um
b
er
of
analysis
runs.
In
the
case
of
Na
vier-Stok
es
or
Euler

solv
ers,
computational
costs
can
b
e
reduced
b
y
the
use
of
adjoin
t
op
erator/con
trol

theory
,
Jameson
[
].
Another
category
of
metho
ds
are
ev
olutionary
algorithms

suc
h
as
in
Oba
y
ashi
and
T
ak
anashi
[
]
and
sto
c
hastic
approac
hes,
Aly
et
al.
[].

Ris��R�0(EN)

They
are
less
sensitiv
e
to
lo
cal
minima
but
ha
v
e
v
ery
high
computational
costs.

Airfoil
design
is
a
m
ultidisciplinary
�eld,
in
v
olving
aero
dynamics,
structural

dynamics,
stabilit
y
and
con
trol,
man
ufacturing
and
main
tenance
considerations.

Despite
a
v
ailable
design
metho
ds,
airfoil
design
remains
to
a
great
exten
t
a
cut-

and-try
pro
cedure
where
adv
anced
design
metho
ds
assist
the
designer.
The
pur-

p
ose
of
the
presen
t
w
ork
w
as
to
further
automate
the
airfoil
design
pro
cess
b
y

dev
eloping
an
in
terdisciplinary
optimization
metho
d
for
airfoil
design,
whic
h
used

a
n
umerical
optimization
algorithm.
The
metho
d
relied
on
a
state
of
the
art
to
ol

for
analysis
of
the
�o
w
�eld
and
included
simple
structural
calculations.
A
tten
tion

w
as
paid
to
the
abilit
y
to
design
airfoils
from
scratc
h
and
a
strategy
for
tailoring

of
wind
turbine
airfoils
w
as
dev
elop
ed.
The
design
metho
d
w
as
demonstrated
b
y

the
design
of
an
airfoil
family
for
pitc
h-
or
stall-regulated
wind
turbines
with
a

rated
p
o
w
er
around
00
k
W.

Ris��R�0(EN)

Wind
turbine
airfoil
c
haracteris-

tics

The
c
haracteristics
of
an
ideal
wind
turbine
airfoil
dep
end
in
principle
on
the

sp
eci�c
rotor
the
airfoil
is
in
tended
for.
But,
in
general,
some
prop
erties
can
b
e

lab
eled
as
desirable
for
most
wind
turbine
airfoils.

F
or
maxim
um
p
o
w
er
pro
duction,
the
lift-drag
ratio
should
b
e
high
for
airfoils

used
on
the
outer
part
of
the
blades.
In
case
of
pitc
h
regulation
and
activ
e
stall

regulation,
the
lift-drag
ratio
should
b
e
high
at
and
near
the
op
erational
p
oin
t.
F
or

stall
regulation,
the
lift-drag
ratio
should
b
e
high
in
the
en
tire
op
erational
range,

i.e.,
angle
of
attac
k
b
elo
w
the
maxim
um
lift
co
e�cien
t.
On
the
in
b
oard
part
of

the
blades,
the
lift-drag
ratio
is
of
less
imp
ortance,
but
the
maxim
um
lift
should

b
e
high
to
reduce
the
blade
area.

The
op
erational
p
oin
t
should
b
e
close
to
maxim
um
lift.
This
ensures
high
lift-

drag
b
elo
w
stall
for
stall
regulation
and
in
case
of
wind
gusts
for
pitc
h
regulation

an
autonomous
stall
con
trol
is
build
in
to
reduce
p
o
w
er
p
eaks.

Go
o
d
o�-design
c
haracteristics
are
imp
ortan
t
b
ecause
of
the
wide
v
ariation
in

the
angle
of
attac
k
during
normal
op
eration
(this
is
in
con
trast
to
a
viation
op
er-

ating
conditions).
F
or
stall
regulation,
the
�o
w
at
maxim
um
lift
should
separate

from
the
trailing
edge
to
ha
v
e
a
smo
oth
lift
curv
e
in
stall
whic
h
reduces
the
risk

of
stall
induced
vibrations
in
con
trast
to
massiv
e
leading
edge
separation.
The

transition
from
the
linear
part
of
the
lift
curv
e
to
the
p
ost
stall
area
should
b
e

w
ell-de�ned
and
smo
oth.
F
urthermore,
the
airfoil
should
b
e
insensitiv
e
to
double

stall,
Bak
[].

In
natural
conditions,
bugs
and
dirt
often
soil
wind
turbine
blades
at
the
leading

edge.
Roughness
at
the
leading
edge
will
cause
early
transition
from
laminar
to

turbulen
t
�o
w
and
an
ev
en
tual
jump
in
the
b
oundary
la
y
er
momen
tum
thic
kness.

This
reduces
maxim
um
lift,
lo
w
er
the
lift
curv
e
slop
e
and
increase
the
skin
friction

resulting
in
loss
of
p
o
w
er
pro
duction.
Esp
ecially
for
stall
regulation,
the
maxim
um

lift
co
e�cien
t
should
b
e
insensitiv
e
to
leading
edge
roughness.

On
the
in
b
oard
blade
section,
the
airfoils
should
ha
v
e
high
cross
section
sti�ness,

to
limit
blade
w
eigh
t
and
tip
de�ection.
This
is
most
easily
obtained
b
y
increasing

the
airfoil
maxim
um
thic
kness
at
the
exp
ense
of
aero
dynamic
p
erformance,
e.g.,

reduced
lift-drag
ratio.

The
desirable
airfoil
c
haracteristics
constitute
b
oth
aero
dynamic
and
structural

prop
erties
and
m
ultiple
con�icting
c
haracteristics
are
in
v
olv
ed.
High
lift-drag
is
in

con
trast
to
high
airfoil
thic
kness.
High
maxim
um
lift
is
in
con
trast
to
insensitivit
y

to
leading
edge
roughness.
High
lift-drag
ratio
at
the
design
p
oin
t
is
di�cult

to
obtain
together
with
extensiv
e
o�-design
requiremen
ts.
But,
this
is
exactly

where
n
umerical
optimization
is
useful,
b
ecause
it
can
searc
h
the
design
space

in
a
systematic
manner
and
�nd
the
b
est
compromise
b
et
w
een
these
con�icting

requiremen
ts.
The
designer
of
course
still
has
to
mak
e
quali�ed
decisions
on
the

relativ
e
w
eigh
ting
of
the
di�eren
t
desirable
prop
erties.

Ris��R�0(EN)

Design
metho
d

The
design
metho
d
is
based
on
n
umerical
optimization.
The
general
form
ulation

of
an
optimization
problem
is,
e.g.,
[
]:

Minimize:
F
(x)

Sub
ject
to:
G

j

(x)
�
0,
j
=
0;
m

where
m
+

is
the
n
um
b
er
of
constrain
ts.
The
ob
jectiv
e
function,
F
(x),
is
mini-

mized
b
y
c
hanging
the
design
v
ariables
that
comp
ose
the
design
v
ector,
x
.
Here,

the
design
v
ariables
are
the
co
ordinate
p
oin
ts
that
describ
e
the
airfoil
shap
e.
The

inequalit
y
constrain
ts,
G

j

(x),
are
side
v
alues
for
the
design
v
ariables
and
b
ounds

on
resp
onse
parameters.
Equalit
y
constrain
ts
can
b
e
replaced
with
t
w
o
inequalit
y

constrain
ts
with
opp
osing
signs.

.
Design
algorithm

The
com
bination
of
n
umerical
optimization
and
di�eren
t
to
ols
for
�o
w
and
struc-

tural
calculations
are
sho
wn
in
Figure
.

'

&

$

%

Initial
airfoil
shap
e

?

'

&

$

%

Ob
jectiv
e
function

Constrain
ts

Design
v
ariables

-

�

Optimization

algorithm

�

-

In
terface

-

-

Flo
w
solv
er

Structural
calc.

T
arget
curv
e

?

�

�

�

�

Optim
um
airfoil
shap
e

Figur
e
.
Flow
chart
of
the
design
metho
d.

An
airfoil
shap
e
(in
principle,
an
y
airfoil-lik
e
shap
e)
is
input
together
with
a

de�nition
of
the
ob
jectiv
e
function,
the
design
v
ariables
and
the
constrain
ts.
The

optimization
pro
cess
is
iterativ
e
and
the
iteration
lo
op
in
v
olv
es
sev
eral
calcula-

tions
of
�o
w
and
structural
prop
erties.
Di�eren
t
to
ols
carry
out
these
tasks.
An

in
terface
handles
the
necessary
b
o
ok-k
eeping
of
design
v
ariables
and
constrain
ts

and
the
calculation
of
sensitivit
y
information.
The
in
terface
con
v
erts
the
actual

design
v
ector
in
to
an
airfoil
shap
e.
The
�o
w
and
structural
calculations
are
used
to

estimate
the
v
alue
of
the
ob
jectiv
e
function
and
the
constrain
ts.
Multiple
angles

of
attac
k
can
b
e
calculated
to
allo
w
o�-design
optimizations
and
the
com
bina-

tion
of
�o
w
and
structural
resp
onses
allo
ws
in
terdisciplinary
optimization.
When

a
v
ailable,
other
calculation
to
ols,
suc
h
as
calculation
of
aero
dynamic
self
noise

can
easily
b
e
incorp
orated.
T
raditional
in
v
erse
airfoil
design
is
made
p
ossible
b
y

comparing
the
actual
�o
w
resp
onse
with
prescrib
ed
target
v
alues.

Ris��R�0(EN)

.
Geometry
description

A
smo
oth
airfoil
shap
e
is
imp
ortan
t
for
the
optimization
results.
In
principle,
an
y

ph
ysically
realistic
shap
e
should
b
e
p
ossible
to
allo
w
design
from
scratc
h.
The

shap
e
description
should
ha
v
e
as
m
uc
h
geometric
�exibilit
y
as
p
ossible
with
as

few
design
v
ariables
as
p
ossible
to
secure
an
e�ectiv
e
and
represen
tativ
e
searc
h

of
the
design
space
with
acceptable
computational
costs.
It
is
imp
ortan
t
that
the

geometric
description
do
es
not
limit
the
design
space
to
o
m
uc
h
a
priori.
Di�eren
t

approac
hes
can
b
e
used.
Hic
ks
et
al.,

[
]
describ
e
the
airfoil
thic
kness
b
y
a

p
olynomial
where
the
co
e�cien
ts
are
design
v
ariables.
Others
suc
h
as
[]
represen
t

the
airfoil
surface
b
y
p
olynomials.
An
initial
airfoil
shap
e
can
b
e
mo
di�ed
b
y

adding
smo
oth
p
erturbations
as
in
[
]
where
a
linear
com
bination
of
a
set
of
base

functions
is
used
with
w
eigh
ting
co
e�cien
ts
as
design
v
ariables.
Ho
w
ev
er,
these

metho
ds
need
a
large
n
um
b
er
of
design
v
ariables
to
ha
v
e
su�cien
tly
geometric

degrees
of
freedom
and
this
increases
computational
costs
and
migh
t
cause
scatter

in
the
airfoil
geometry
.

In
the
presen
t
case,
the
airfoil
shap
e
is
represen
ted
b
y
a
single
B-spline
curv
e

de�ned
b
y
a
set
of
con
trol
p
oin
ts
[]:

p(u)
=

n

X

i=0

P

i

N

i;k

(u)

where
0
<
u
<
n
�
k
+
,
k
is
the
order
of
con
tin
uit
y
,
P

i

(�

i

;
�

i

)
are
the
co
ordinate

p
oin
ts,
n
+

is
the
n
um
b
er
of
co
ordinate
p
oin
ts,
N

i;k

(u)
are
in�uence
functions.

The
B-spline
curv
e
w
as
de�ned
clo
c
kwise
from
the
airfoil
trailing
edge
and

the
airfoil
shap
e
w
as
transformed
in
to
a
standard
x
�
y
co
ordinate
system
with

the
c
hord
along
the
x-axis.
The
B-spline
curv
e
is
con
tin
uous
of
the
k
'th
order

and
no
sp
ecial
considerations
are
necessary
for
the
airfoil
nose
region.
B-splines,

furthermore,
ha
v
e
the
adv
an
tage
that
k
determines
ho
w
large
a
part
of
the
en
tire

curv
e
that
is
altered
when
a
single
con
trol
p
oin
t
is
mo
v
ed.
High
v
alues
of
k
result

in
a
smo
oth
curv
e,
whereas
small
v
alues
of
k
create
a
more
liv
ely
curv
e.
Figure

sho
ws
an
example
with
n
+

=
,
k
=
,
whic
h
w
ere
common
v
alues
for
the

presen
t
study
.
Most
of
the
con
trol
p
oin
ts
w
ere
only
allo
w
ed
to
mo
v
e
in
the
y

direction,
whic
h
limits
the
n
um
b
er
of
design
v
ariables
to
b
e
close
to
n
+
.

�

�

y

x

p(u)

Figur
e
.
B-spline
r
epr
esenting
the
airfoil
shap
e,
n
+

=
,
k
=
.
the
dots
ar
e

the
c
ontr
ol
p
oints/design
variables.

Ris��R�0(EN)

.
Optimization
algorithm

The
c
hoice
of
optimization
algorithm
is
basically
a
c
hoice
b
et
w
een
gradien
t
based

metho
ds
and
global
metho
ds
suc
h
as
ev
olutionary
t
yp
e
algorithms.
Ev
olutionary

algorithms
are
less
sensitiv
e
to
lo
cal
minima.
Ho
w
ev
er
they
are
time
consuming

and
constrain
ts
ha
v
e
to
b
e
included
as
a
p
enalt
y
term
on
the
ob
jectiv
e
function.

Gradien
t
based
metho
ds
on
the
other
hand
allo
w
m
ultiple
constrain
ts
but
lac
k

global
optimalit
y
.

W
e
c
hose
a
traditional
simplex
optimization
algorithm
based
on
sequen
tial
linear

programming
with
mo
v
e
limits
in
a
standard
b
ound
form
ulation
[
].
Simplex

metho
ds
are
searc
h
metho
d
that
are
simple,
robust
and
reasonably
fast.
They

require
the
gradien
ts
of
the
ob
jectiv
e
function
and
of
the
constrain
ts
whic
h
are

pro
vided
b
y
a
sensitivit
y
analysis.

.
Sensitivit
y
analysis

A
djoin
t
op
erator/con
trol
theory
metho
ds
ha
v
e
recen
tly
b
een
applied
to
�uid
�o
w

equations
[
].
This
approac
h
requires
the
additional
solving
of
adjoin
t
equations.

Compared
to
traditional
n
umerical
�nite
di�erences,
these
metho
ds
are
time
sa
v-

ing
when
the
n
um
b
er
of
design
v
ariables
is
large.
Ho
w
ev
er,
the
adjoin
t
equations

ha
v
e
to
b
e
deriv
ed
for
eac
h
of
the
go
v
erning
�o
w
equation.

W
e
based
the
sensitivit
y
analysis
on
n
umerical
�nite
di�erences.
This
w
as
more

time
consuming,
but
ensured
�exibilit
y
in
the
c
hoice
of
�o
w
solv
er
and
structural

calculations.

.
Flo
w
analysis

In
principle,
there
are
no
restrictions
on
the
c
hoice
of
�o
w
solv
er.
Since
the
op-

timization
pro
cess
requires
man
y
ev
aluations
of
the
ob
jectiv
e
function
and
the

constrain
ts
b
efore
an
optim
um
design
is
obtained,
computational
costs
are
high

when
a
Na
vier-Stok
es
solv
er
is
used
for
eac
h
�o
w
calculation
as
in
[
].
In
stead,

w
e
c
hose
XF
OIL[ ]
for
the
�o
w
calculations.
XF
OIL
is
an
in
viscid
linear-v
orticit
y

panel
metho
d
with
source
distributions
sup
erimp
osed
on
the
airfoil
and
its
w
ak
e

allo
wing
mo
deling
of
viscous
la
y
er
in�uence
on
the
p
oten
tial
�o
w.
A
t
w
o-equation

in
tegral
b
oundary
la
y
er
metho
d
is
used
to
represen
t
the
viscous
la
y
er
[].
XF
OIL

is
dev
elop
ed
for
transonic
and
lo
w
Reynolds
n
um
b
er
�o
ws
and
is
w
ell
suited
for

optimization
b
ecause
of
the
relativ
e
fast
and
robust
viscous/in
viscid
in
teraction

sc
heme.

F
or
giv
en
angle
of
attac
k,
Reynolds
n
um
b
er
and
Mac
h
n
um
b
er,
XF
OIL
pro-

vides
pressure
distribution,
lift
and
drag
co
e�cien
ts.
In
addition,
n
umerous
b
ound-

ary
la
y
er
parameters
are
calculated,
e.g.,
displacemen
t
and
momen
tum
thic
kness,

shap
e
factor,
skin
friction,
transition
p
oin
t
lo
cation,
etc.
In
XF
OIL,
transition
is

mo
deled
b
y
the
e

n

metho
d
with
n
=

as
default
v
alue.

.
Structural
analysis

Simple
structural
calculations
w
ere
carried
out
on
the
airfoil
cross
section
suc
h

as
the
airfoil
thic
kness
and
mean
line
distributions,
the
airfoil
maxim
um
relativ
e

thic
kness,
area,
and
area
momen
ts
of
inertia.

Ris��R�0(EN)

Design
strategy

Before
turning
to
the
sp
eci�c
design
of
the
airfoil
family
,
w
e
describ
e
in
general

terms
the
design
strategy
follo
w
ed.
The
design
task
or
rather
the
optimization

problem
is
de�ned
b
y
the
design
v
ariables,
the
op
erating
conditions,
the
design

ob
jectiv
es
and
the
constrain
ts.

Design
v
ariables

The
design
v
ariables
are
c
hosen
among
the
con
trol
p
oin
ts
of
the
B-spline
describing

the
airfoil
shap
e.
The
con
trol
p
oin
ts
at
the
trailing
edge
are
t
ypically
�xed
in
b
oth

the
x
and
y
directions
to
pro
vide
the
desired
trailing
edge
thic
kness.
F
or
most
of

the
con
trol
p
oin
ts
only
the
y
-co
ordinate
is
a
design
v
ariable
to
limit
the
n
um
b
er

of
design
v
ariables
and
to
ensure
a
uniform
spacing
b
et
w
een
the
con
trol
p
oin
ts.

Op
erational
conditions

The
o
v
erall
op
erational
conditions
are
de�ned
b
y
the
Reynolds
n
um
b
er
based
on

c
hord
and
the
Mac
h
n
um
b
er.
The
Reynolds
n
um
b
er
for
an
airfoil
section
on
a

wind
turbine
blade
dep
ends
on
the
span-wise
lo
cation
and
on
the
size
of
the
wind

turbine.
Since
the
maxim
um
Mac
h
n
um
b
er
is
usually
around
0.,
the
�o
w
can
b
e

considered
incompressible
with
go
o
d
appro
ximation.

Design
ob
jectiv
es

T
o
allo
w
b
oth
aero
dynamic
and
structural
ob
jectiv
es
and
o�-design
ob
jectiv
es,
the

ob
jectiv
e
function
is
de�ned
as
a
linear
com
bination
of
ob
jectiv
es,
F
=

P

n

i=

a

i

f

i

,

where
a

i

are
w
eigh
t
factors
and
f

i

are
the
di�eren
t
ob
jectiv
es.

The
ob
jectiv
es
can
b
e
b
oth
aero
dynamic
(e.g.,
lift-drag
ratio
for
one
or
more

angles
of
attac
k)
and
structural
(e.g.,
momen
t
of
inertia
of
thic
kness
at
a
certain

c
hord-wise
p
osition).

The
w
eigh
ting
of
the
di�eren
t
ob
jectiv
es
is
the
resp
onsibilit
y
of
the
designer
and

this
has
ob
viously
great
in�uence
on
the
�nal
design.

The
ob
jectiv
e
at
the
design
angle
of
attac
k
is
usually
giv
en
a
high
w
eigh
t
factorto

secure
go
o
d
p
erformance
at
the
design
p
oin
t.

Design
constrain
ts

T
o
conclude
the
de�nition
of
the
optimization
problem,
constrain
ts
are
imp
osed

on
the
design.
T
o
obtain
the
desired
maxim
um
lift
co
e�cien
t
and
lift
curv
e,
upp
er

and
lo
w
er
limits
are
imp
osed
on
the
lift
co
e�cien
t
at
the
design
angle
of
attac
k

and
other
angles
of
attac
k,
e.g.,
the
C

Lmax

-angle
of
attac
k
and
in
the
p
ost-stall

region,
Figure
.

The
design
angle
of
attac
k,
�

d

should
b
e
c
hosen
-
degrees
b
elo
w
C

Lmax

to

ensure
a
linear
C

L

(�)
and
lo
w
drag
at
angles
un
til
C

Lmax

.
In
principle,
�

d

can
b
e

an
ywhere
on
the
linear
part
of
the
lift
curv
e,
�

d

can
ev
en
b
e
a
design
v
ariable.

Dep
ending
on
the
desired
p
ost
stall
c
haracteristics,
constrain
ts
can
also
b
e
added

to
the
suction
side
separation
p
oin
t,
S

sep

,
that
should
b
e
at
the
trailing
edge
at

�

d

and
then
mo
v
e
to
w
ards
the
leading
edge
just
b
efore
C

Lmax

.
T
o
ensure
a
w
ell

de�ned
stall,
there
should
b
e
a
sudden
mo
v
emen
t
in
S

sep

at
C

Lmax

.
A
smo
oth

trailing
edge
stall
can
b
e
sp
eci�ed
with
a
lo
w
negativ
e
slop
e
for
S

sep

(�)
in
stall,

whereas
an
abrupt
stall
can
b
e
ac
hiev
ed
with
a
signi�can
t
drop
in
S

sep

to
w
ards

the
leading
edge
at
stall.

0
Ris��R�0(EN)

�

C

L

Figur
e
.
Constr
aints
on
the
lift
curve

Insensitivit
y
to
leading
edge
roughness
is
obtained
b
y
con
trolling
the
lo
cation
of

the
transition
p
oin
t
on
the
suction
side,
S

tr

(�
)
b
efore
and
after
C

Lmax

.
T
o
increase

the
lift-drag
ratios
at
the
angles
of
attac
k
corresp
onding
to
the
design
ob
jectiv
es,

S

tr

should
in
general
b
e
as
far
do
wnstream
as
p
ossible
at
�

d

and
other
angles
of

attac
k
b
elo
w
stall.
A
t
C

Lmax

,
S

tr

should
b
e
close
to
the
leading
edge.
The
�o
w
on

most
of
the
suction
side
w
ould
then
b
e
turbulen
t
b
ecause
of
early
transition
and

the
transition
p
oin
ts
w
ould
b
e
equally
lo
cated
for
b
oth
smo
oth
and
rough
leading

edges
securing
minimal
di�erence
in
C

Lmax

and
lift
curv
e
slop
es.
The
transition

p
oin
t
should
remain
close
to
the
leading
edge
throughout
the
p
ost
stall
region.

The
remaining
e�ect
from
leading
edge
roughness
w
ould
b
e
an
increase
in
drag.

As
a
structural
constrain
t,
the
airfoil
thic
kness
as
a
function
of
c
hord-wise
p
osi-

tion
is
constrained
to
giv
e
the
desired
relativ
e
thic
kness,
but
also
to
a
v
oid
negativ
e

thic
kness.

Other
constrain
ts
can
b
e
added
to
the
airfoil
shap
e
or
the
v
elo
cit
y
distribu-

tion,
the
maxim
um
suction
side
v
elo
cities,
structural
requiremen
ts
or
aero
dynamic

requiremen
ts
at
other
angles
of
attac
k.
F
or
dev
elopmen
t
of
airfoil
families,
con-

strain
ts
can
ensure
compatibilit
y
of
b
oth
the
aero
dynamic
c
haracteristics
and
of

the
airfoil
shap
es.

T
o
run
an
optimization,
an
initial
airfoil
shap
e
is
generated.
This
can
in
principle

b
e
an
arbitrary
shap
e
that
migh
t
b
e
v
ery
di�eren
t
from
the
optim
um
shap
e.

Ho
w
ev
er,
computational
costs
are
reduced
when
the
initial
design
is
close
to
the

optim
um
design.
Side
constrain
ts
are
added
to
the
design
v
ariables
to
ensure

that
they
mo
v
e
within
reasonable
limits.
During
the
optimization,
the
�o
w
solv
er

calculates
the
�o
w
for
all
angles
of
attac
k
where
ob
jectiv
es
and
constrain
ts
are

de�ned.
T
ypically
the
�o
w
is
solv
ed
at
a
few
angles
of
attac
k
b
efore
stall
and
at

sev
eral
angles
of
attac
k
in
stall.
F
or
a
reliable
optimization
pro
cess,
con
v
ergence

problems
in
the
�o
w
predictions
should
b
e
a
v
oided.

Ris��R�0(EN)

Airfoil
family

In
this
section,
w
e
presen
t
the
basis
for
and
result
of
the
design
of
the
airfoil
family
.

.
V
alidit
y
of
geometry
description

Before
presen
ting
the
results
of
the
airfoil
design,
w
e
c
hec
k
that
the
geometry

description
is
acceptable,
in
the
sense
that
it
should
b
e
able
to
represen
t
man
y

di�eren
t
airfoils
with
a
limited
n
um
b
er
of
design
v
ariables.
This
is
done
b
y
letting

the
design
to
ol
minimize
the
geometric
di�erence
b
et
w
een
a
new
design
and
t
ypical

wind
turbine
airfoils,
i.e.,
NA
CA
-,
FF
A-W-,
and
DU
-W-0.
That

is,
the
ro
ot
mean
square
sum
of
di�erences
in
y-co
ordinates
is
minimized.
The

results
are
giv
en
in
Figure
,
and
they
sho
w
that
the
geometry
description
can

repro
duce
the
v
arious
shap
es
reasonably
w
ell.

NACA 63418

Initial design variables

Initial design

NACA 63418

Final design variables

Final design

(a)
NA
CA
-

FFAW3241

Initial design variables

Initial design

FFAW3241

Final design variables

Final design

(b)
FF
A-W-

DU 91-W2-250

Initial design variables

Initial design

DU 91-W2-250

Final design variables

Final design

(c)
DU
-W-0

Figur
e
.
Ge
ometric
r
epr
esentation
of
wind
turbine
airfoils

On
the
basis
of
this
exercise,
w
e
assume
that
the
geometry
description
based
on

a
B-spline
is
capable
of
generating
a
large
part
of
the
in�nite
n
um
b
ers
of
p
ossible

airfoil
shap
es.

Ris��R�0(EN)

.
Design
criteria

In
the
follo
wing,
the
design
of

airfoils
is
describ
ed.
The
relativ
e
airfoil
thic
knesses

range
from
%
to
0%.
The
design
angle
of
attac
k
is
0

�

.
The
ob
jectiv
e
function

is
the
sum
of
lift-drag
ratios
at
angle
of
attac
ks
of

�

,

�

,

�

,

�

,
and
0

�

.
The

w
eigh
t
factors
are
all
the
same,
but
in
a
sense
the
lift-drag
ratio
at
0

�

has
the

largest
w
eigh
t
factor
since
it
is
the
largest
and
the
optimization
algorithm
w
ould

tend
to
optimize
here.
F
urthermore,
a
high
lift-drag
ratio
at
0

�

leads
to
high

lift-drag
ratios
also
at
lo
w
er
angles
of
attac
k.
The
constrain
ts
for
the
three
thin

airfoils
are
giv
en
in
T
able

and
in
T
able

for
the
four
thic
k
airfoils.

F
or
all
sev
en
airfoils,
the
upp
er
and
lo
w
er
limits
on
the
lift
curv
e
are
iden
tical.

The
design
lift
at
0

�

is
b
et
w
een
.
and
.
and
the
C

Lmax

of
.
should
b
e

reac
hed
at
ab
out

�

.

The
separation
p
oin
t,
S

sep

on
the
suction
side
is
�xed
to
the
trailing
edge
un
til

C

Lmax

is
reac
hed.
Separation
for
a
turbulen
t
b
oundary
la
y
er
w
as
estimated
from

H<
.
as
separation
criterion
as
in
[
],
where
H
is
the
b
oundary
la
y
er
shap
e

factor.

The
constrain
ts
on
the
suction
side
transition
p
oin
t
di�er
for
the
thin
and
thic
k

airfoils.
F
or
RIS��A�,
RIS��A�,
and
RIS��A�,
the
transition
p
oin
t,
S

tr

is
lo
cated
on
the
�rst
%
of
the
c
hord
for
angles
of
attac
k
ab
o
v
e
the
C

Lmax

-angle.

F
or
the
remaining
thic
k
er
airfoils,
the
transition
p
oin
t
is
on
the
�rst
0%
of
the

c
hord.

F
or
RIS��A�,
RIS��A�,
and
RIS��A�0,
an
additional
constrain
t
is
that

the
�o
w
on
the
suction
side
decelerates
from
0:
�
x=c
�
0:
for
�
=
0

�

.

T
able
.
Constr
aints
for
RIS��A�,
RIS��A�
and
RIS��A�.

�
0.0

�

0.

�

.0

�

.

�

.0

�

.

�

.0

�

.

�

C

Lmin

.
.
.
.
.0
.
.

C

Lmax

.
.
.
.
.
.
.

S

sep;min

0.
0.
0.

S

tr;max

0.0
0.0
0.0
0.0
0.0

T
able
.
Constr
aints
for
RIS��A�,
RIS��A�y,
RIS��A�yand
RIS��A�

0y.

�
0.0

�

0.

�

.0

�

.

�

.0

�

.

�

.0

�

.

�

C

Lmin

.
.
.
.
.0
.
.

C

Lmax

.
.
.
.
.
.
.

S

sep;min

0.
0.
0.

S

tr;max

0.0
0.0
0.0
0.0
0.0

ydv
i

s

�
0
for
�
=
0

�

and
0:
�
x=c
�
0:

In
T
able
,
the
op
erational
conditions
are
giv
en
together
with
selected
prop
er-

ties
of
the
resulting
airfoil
design.
The
op
erational
conditions
are
the
Reynolds

n
um
b
ers
and
the
Mac
h
n
um
b
ers
corresp
onding
to
a
t
ypical
00k
W
wind
turbine.

The
Reynolds
and
Mac
h
n
um
b
ers
are
relativ
ely
high
for
the
thinner
airfoils
in
the

tip
region
and
on
the
mid
section
but
lo
w
er
for
the
thic
k
er
airfoil
used
in
the
ro
ot

region.

The
maxim
um
lift
co
e�cien
ts
(according
to
XF
OIL)
are
also
giv
en
for
b
oth
clean

and
dirt
y
conditions
(i.e.,
rough
leading
edge).
In
calculations
with
rough
leading

Ris��R�0(EN)

edge,
the
transition
p
oin
ts
for
the
suction
and
pressure
sides
w
ere
�xed
to
%
and

0%,
resp
ectiv
ely
as
in
[].
W
e
see
that
going
from
clean
to
dirt
y
conditions
C

Lmax

drops
ab
out
0%
for
RIS��A�
to
RIS��A�
and
ab
out
%
for
RIS��A�

and
RIS��A�0.

T
able
.
Op
er
ational
c
onditions
and
sele
cte
d
pr
op
erties
the
airfoil
design

t=c
Re
�
0

�

M
a
C

Lmax

(�)y
C

Lmax

(�
)z

RIS��A�
%
:00
0:0
.(.

�

)
.(0.0

�

)

RIS��A�
%
:00
0:
.(.

�

)
.(.0

�

)

RIS��A�
%
:00
0:
.(.0

�

)
.(.

�

)

RIS��A�
%
:00
0:0
.(.0

�

)
.0(.0

�

)

RIS��A�
%
:
0:0
.(.0

�

)
.(0.

�

)

RIS��A�
%
:
0:0
.(.0

�

)
. (.0

�

)

RIS��A�0
0%
:0
0:0
.(.0

�

)
.(.0

�

)

yfree
transition,
z�xed
transition

.
Geometric
prop
erties

The
airfoil
shap
es
are
giv
en
in
Figures

and
.
Geometrically
,
RIS��A�
to

RIS��A�0
are
clearly
a
family
,
whereas
RIS��A�
and
RIS��A�
do
not

lo
ok
lik
e
their
thic
k
er
relativ
es.
The
en
tire
family
is
c
haracterized
b
y
a
sharp
nose.

F
or
RIS��A�
and
RIS��A�0
the
rear
part
of
the
suction
side
is
sligh
tly

w
a
vy
,
whic
h
it
migh
t
b
e
p
ossible
to
remo
v
e
if
not
for
an
ything
else
as
for
aesthetic

reasons
with
out
compromising
aero
dynamic
p
erformance.
This
has
not
b
een
tried

in
this
w
ork
but
it
is
an
ob
vious
p
ossibilit
y
for
future
impro
v
emen
t
of
the
design.

RISØ-A-XX

Figur
e
.
A
irfoil
shap
es

Ris��R�0(EN)

RISØ-A-12

RISØ-A-15

RISØ-A-18

RISØ-A-21

RISØ-A-24

RISØ-A-27

RISØ-A-30

Figur
e
.
A
irfoil
shap
es,
r
evisite
d

Ris��R�0(EN)