Downloaded from orbit.dtu.dk on: Jul 12, 2017
A Beddoes-Leishman type dynamic stall model in state-space and indicial formulations
Hansen, Morten Hartvig; Gaunaa, Mac; Aagaard Madsen , Helge
Publication date:
2004
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Hansen, M. H., Gaunaa, M., & Aagaard Madsen, H. (2004). A Beddoes-Leishman type dynamic stall model in
state-space and indicial formulations. (Denmark. Forskningscenter Risoe. Risoe-R; No. 1354(EN)).
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
• You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
! "#!
$ %&&
!
" # $
$ % &
!
& '() *+,,-.-*/-# *+,,-.-/-0 '
'(() -1-234*0-
41
44
4.
40
4,
5 6
7 .1 8
.4 5
01 9
04 9
0. )
!
"
:;3:31.,0<)
.
0
:;3:31.,0<)
7
$
<$
$
7 Æ
Æ
5 Æ
6
Æ
)
= : 4
& )
7 4
! ! 7 4 7 : % 5 !
! : 5
'
:;3:31.,0<)
,
2
:;3:31.,0<)
$ =>7 ?1 4@
" (
& $
A " $
?.@ &$ " " B
,-A $
?0@# & +-A %
1/*. ?,@# 9 $
C)<: ?2 +@ # ?* / 1-@
?11@ ! ?14@ ! " D
( " ?1. 10@ ?1,@ 5 ?12@ $
! ?1+@ !
?1*@
" $ +-3*- E ! !
1, F =
& " >
! " ?1/@ ?1. 10@ % ! G '
(
4 :;3:31.,0<)
+
(
. (
0 < )(! *
:;3:31.,0<)
&
%' (
% 6 -. " % < ! " < " " $ ?.@ $ &
$ ! % 1
A ! 5
$
# $ ?1/@
9
> H 5A '
K
I
H 0 I J J J - J 1
I ! " K harmonic wake
U
w3/4
:;3:31.,0<)
/
4 H
4
! & 1 0 % K I I 1 5A ! 4 = 1 H - J
.
5A H -
0
D
4 4 H -
,
¼
4
% ! >
LK
$ H 1 2
?1-@ & (
2 H , 4 Ê 4 H
+
¼
¼
¼
(
+ , H 1 J J *
Ê 4
H /
¼
H 1 4 5
/ 4 H 4 I J 1-
H 1 4 - H - 1-
:;3:31.,0<)
=
C5< 1- ! "
Æ
" = &
! ! B
H H C5< 1- I
4
4
J
11
I J
4 H - H - > !
H 1 J J 14
" 1 H 4 J I 4 1.
! ) $
11 '
! ! < < < !
1, F
% $ < $ ! < $
1J
10
4
" L " ?4-@ " < &
% 4 " H 1 H -
H x/c =
U
f
separation point
wake
!" ! :;3:31.,0<)
11
! <
10
H 4
1
1,
10 % $ < 1 &
H $ 12
! " B ! 12 = 7%5 ! " (
! H H 0 ! $
10 H -
$
Æ
"
H J 1 1+
Æ
" H 1*
1 &
1,
! " H -
) <
1* ! H 1 = 1, 1* 1 1/
4
=
" ! " % . $ 1+ G %
$ 1, &
Æ
" 1*
" ! ! < # ' I J H 4-
14
:;3:31.,0<)
1.5
1
0.5
0
-0.5
Lift coefficient for stationary flow
Lift coefficient for fully attached flow
Lift coefficient for fully separated flow
Position of separation point
-1
-1.5
-40
-20
0
20
40
Angle of attack [deg]
#$ % & 1+' "
Æ
!" &
& 1* &
& 1, %
Æ
" 1. - H -
Æ
H Æ
" Æ
! G H J % ! H <
1,
< I J H 41
- H - B
1+
H Æ
< H J 1 J I
44
1. H
4.
4
) " H " <
44 11 4- 41 ! :;3:31.,0<)
1.
%'% "
=>7 ?1 4@ G (
% '
! 9 !
'
!
' &
$ = B
" ! ! B
! ! ! &
! % 0 Æ
M H 40
14 44 ! " #
!
B
" ! # "
>
< a - aE
CLdyn
CLind @ CLdyn
DCDind @ (a - aE) CLdyn
U
wwake a
aE
wwake
(
& ") "
"
10
:;3:31.,0<)
'
< $
" L " ?4-@ ( $
10 $
4,
H 1 4 ' " 1 0 ?11@
$
4, % %& " $
$ LA '
1 0 ! M
¼¼
H 1
4
1
4
42
Æ
LA $
Æ
H J M J M
4+
& % Æ
4+ Æ
M - M -
¼¼
¼¼
%' #
< = Æ
$ H J M I 0 4*
M < ) &
"
< % , &
¼¼
¼¼
H :;3:31.,0<)
4/
1,
CL
st
CMst - CM0
equivalent pressure center
c/4
a
U
- ast = -
*
CMst - CM0
CLst
+, +%
Æ
( Æ
! ., 4/ $
1, % 2 <
% . ' !
! $
< % " < % &
= ! & < % < Æ
M H .-
Æ
<
44
Æ
¼¼
Arm to pressure center, ast [-]
0
-0.05
-0.1
-0.15
Separation on upper surface
Separation on lower surface
-0.2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Separation point, f [-]
1
- *
+%
#
" 12
:;3:31.,0<)
%' ) ( *
C5< % + " ( =
! I ! $ 11 4-
41 C5< I J J I 4 H I J J I 4 H
I J H J I .1
I J H J - H - ! 14 ) H 4 Æ
H J 1 J I
H J .4
J 4
0
4 I
&
1, 12 1* 4/ <
.1 .4 <
6
= 7 ?1,@ =
(
. H
J external inflow
a
a
a3/4
U
U
Dynamic stall
model
Structural
model
CL
dyn
CD
dyn
CM
dyn
external forces
"
%& !"
:;3:31.,0<)
1+
# $ %' + >
$ $
= & $
H J H J H J ..
! !!
1
.1 H J H 1 4 . 0
.0
'
.. .0 .1 H H H H .,
I
I J H I J H
I J H
I J H
I
J I .2
H H 4 H - J J ! <
14
$
! $
1, 4
H 4
.+
.2 (
- H - $
Æ
.4 $
H J J J I H J J J .*
1*
H
J J 4 I :;3:31.,0<)
Æ
1
H
J
H
H
H H
H
1 0 1 0 ./
Æ
! H
1
J 1 0-
H 1 H - " Æ
./ % H 1 Æ
" H <
1/ % H - Æ
H H - &
& Æ
H - H - Æ
" H - <
.2
:;3:31.,0<)
1/
4-
:;3:31.,0<)
'
$ >
5 Æ
$
$
.1
' & I J H 01
$ Æ
H H
Ê J 04
H J M J M
H
H
J
Ê Ê Ê J Ê J 0.
> H - <
04 0. $ J M H J ! "
00
Ê H 0,
!
H
Ê "
02
Æ
! <
01
'% "+
( :;3:31.,0<)
41
$ B
H 0+
N
1
"
0*
!
H N
$ N H -", J 0/
N H -", J ,-
) $
Æ
$
H 1 H 4 J I J I N H J
,1
J N H J ,4
Æ
H . H 0 N H ,.
N H -", N
N J 4 I
H H -", J 4 J I ,0
,,
,2
J J J %! &
6 $
$ B
H ,+
!
H
N
J
44
1
MN
H
!
H
N
J
J
1 1
N
M
,*
,/
1
4MN
M
4 J 4M N
1 1
N
M
2-
4M J 4 1
N N
:;3:31.,0<)
Æ
$
H 1 H 4
H 4 21
Æ
H . H 0 H
H
J 4 I
J H 1 J J 24
2.
20
2,
4 I
> 0+0* ,+,* ,/2-
C
<
00 Æ
<
.4
H H J M
H
:;3:31.,0<)
J 4.
40
:;3:31.,0<)
'
& !
" ' , ( - #
% " $ (
' ( & " " $ % * " " " : &
!
" $
" <
2 6
! I J K = -4 '
' )7 2.1,
% / ' !
'
! (
4 ! $ ' $ ! % !
$ :;3:31.,0<)
4,
0.75
Indicial
General Solver
Indicial +full VM
2πα
0.7
0.65
CL [−]
0.6
0.55
0.5
0.45
0.4
0.35
3
3.5
4
4.5
5
5.5
6
6.5
7
0.05
0.1
0.15
0.2
o
α [ ]
1
0.9
0.8
0.7
CL [−]
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.2
Indicial
General algorithm
Indicial +full VM
−0.15
−0.1
−0.05
0
Y/c [−]
1.8
1.6
1.4
Indicial, λ=0.8
Indicial, λ=0.4
General algorithm, λ=0.8
General algorithm, λ=0.4
Indicial +full VM, λ=0.8
Indicial +full VM, λ=0.4
Quasisteady value
CL [−]
1.2
1
0.8
0.6
0.4
0
90
180
270
360
ωt [o]
(
% .' H ,Æ 4Æ /'
% .' H -"4 H ,Æ / " ."' H
H -"0 -"*0 H ,Æ / ! 1
&
" % & 2
$ 2 ' &
% +
H 4 H -"4
"
42
:;3:31.,0<)
1.6
1.4
1.2
CL [−]
1
0.8
0.6
0.4
Static
o o
α=3 ±4
α=12o±4o
α=21o±4o
0.2
0
0
5
10
15
20
25
30
25
30
α [o]
0.3
Static
o o
α=3 ±4
o o
α=12 ±4
α=21o±4o
0.25
0.15
C
D
[−]
0.2
0.1
0.05
0
−0.05
0
5
10
15
20
o
α [ ]
Static
α=3o±4o
o o
α=12 ±4
α=21o±4o
−0.06
−0.07
−0.08
−0.1
C
M
[−]
−0.09
−0.11
−0.12
−0.13
−0.14
−0.15
0
5
10
15
20
25
30
o
α [ ]
3
./' ./ ."/ ,
4212 - 3 +
H 4 H -"1'
M H 0Æ :;3:31.,0<)
4+
'% , ( - #
% 1- & M H 0 M H 4 ' C
M H 4 % M H 1 ?41 44@ Æ
Æ
Æ
Æ
1.6
1.6
1.4
1.4
1.2
1.2
CL [−]
1
CL [−]
1
0.8
0.8
0.6
0.6
Static
o
Nonlinear ∆α=4
o
Linear ∆α=4
0.4
0.2
0
0
5
10
15
20
25
Static
o
Nonlinear ∆α=2
o
Linear ∆α=2
0.4
0.2
0
30
0
5
10
α [o]
Static
o
Nonlinear ∆α=4
Linear ∆α=4o
0.3
0.25
0.25
30
25
30
0.15
0.1
0.1
0.05
0.05
0
0
0
5
10
15
20
25
30
−0.05
0
5
10
α [o]
−0.07
15
20
α [o]
Static
o
Nonlinear ∆α=4
o
Linear ∆α=4
−0.06
Static
o
Nonlinear ∆α=2
o
Linear ∆α=2
−0.06
−0.07
−0.08
−0.09
−0.09
CM [−]
−0.08
CM [−]
25
0.2
CD [−]
CD [−]
0.2
−0.1
−0.1
−0.11
−0.11
−0.12
−0.12
−0.13
−0.13
−0.14
−0.15
20
Static
o
Nonlinear ∆α=2
Linear ∆α=2o
0.3
0.15
−0.05
15
α [o]
−0.14
0
5
10
15
o
α [ ]
20
25
30
−0.15
0
5
10
15
20
25
30
o
α [ ]
5 1
3,
./' ./ ."/ 4212 - M H 0Æ ./ M H 4Æ ./0 & " +
H 4 H -"1
4*
:;3:31.,0<)
' .+ ( -! - /0
&
! >
?4.@ 1/4, = &
!
&
?40@ '
!
!
% 11 !
" :; 140 :; 140 40F !
& ?4,@ :; 140 &
" $
O
?42@ $
>
' & ! " 8
?4+@
$
$
>
O
$ &
&
!
$
$
B
:; 140 $ $
$
:6( $ $
$ 1F $
$
<
2 1
Æ
$
$
" :; 140 1
0.9
CL/CL,stat [−]
0.8
0.7
0.6
0.5
0.4
0.3
0
Jones approximate flat plate
Panel code Risoe A1−24
Approximate Risoe A1−24
5
10
15
20
τ [−]
25
30
(
% ) ! 36 2, H 4 % O
?42@ -12, -.., --0,, -.--:; 140 -4/0 -..1 --220 -.422
& 1Æ
$ % :;3:31.,0<)
4/
.E0 &
' & &
!
= )(! ! )(! #* (;
?4*@ 6
?4/ .-@ :
)(! # $ % ! &
)(! ! % >
<
5
:;
% 14 :'(P 140 ' >
$
O
&
)(! & :'(P 140 )(! & $ &
!
$
$
)(!
Æ
M H -"-12 Æ
)(! )(! Æ
0"4 )(! Æ
1.05
1
CL [−]
0.95
0.9
0.85
0.8
Indicial flat plate
Indicial Risoe A1−24
Navier−Stokes Risoe A1−24 Unsteady
Navier−Stokes Risoe A1−24 Steady
0.75
0.7
2.5
3
3.5
4
4.5
5
5.5
6
α [o]
3
H 0"4Æ 1",Æ " +
H 4 H -"-/4 3(7 2, 1
&,
"
&
& 4%,) ,
& !, 3(7 2, 4%,) M H -"-12' " &"
4%,) 8, H ."12Æ
.-
:;3:31.,0<)
% " $
" "
" " $
'
$ $ !
$ % M H 4 =>7( ?41 44@
%
&
!
& Æ
:;3:31.,0<)
.1
.4
:;3:31.,0<)
?1@ D
O Q 7 = 7 3 6 7
R 2 2 1 9
&
1
6 D
9 $ S' '
<
B
5
! 1//2 14/31.,
?4@ D
O 6
= TK! <
D P ( 8
=
>
! 5 QD
5
'
9
(R : :;:1-0,<) :; :! 5
! 6 1//*
?.@ Q8
'
6
%R 4212 3 - 1/., 01.30..
?0@ %
U 7 2
(
2 O
> V
(
'
1/,, 5 1//.
?,@ %
D D Q%
(
:>
( :
DR : 9 + ) 4 1/*. 1-13101
?2@ 7 D 5 Q(< 6 5
(
9 7
:
= % %R - #
3 ;"
< 2 ' ; ; 1/*-
?+@ 5 : 7 Q'
( % (< 6 45 % R : 9 + ) 4 1/*. +.3*2
?*@ ( Q (
B
< '
( =R : 9 1 1/+2 11.314.
?/@ ( Q:
R : 9 + ) 4
1/*. 1*.31/+
?1-@ ( QD 7
B
R : 9 *
) 1 1/*0 ,,3+1
?11@ O 8 ( Q 8
6 B
5
( B
'
6R
;
2
2
= >
57 O
1/*2
?14@ O 8 ( Q (< 6 5
(R >
2
= 9 .0 ) . 1/*/
.31+
?1.@ O 8 7 8 Q (( 6 B
7 % % R 2(22 ; ,
55 1/*/ 1311
?10@ O 8 )
L W Q(( :
B
R 2(22 >
9 4* ) , 6 1//- *.23*00
?1,@ O 8 7 8 Q(( 6 B
5
(R 2(22 , 1/*/ 1.+431.*.
?12@ O 8 Q 5
6 B
5 (R >
2 9 4, ) + O 1/** 22,3222
?1+@ ( Q ) >! 5
6R 2
= 4
? @
2
2 % 1/*+ 13/
:;3:31.,0<)
..
?1*@ O 8 ;
= 2
7 B
D 4---
?1/@ 9
> 8 O 8 QC
'
"
9
% 9 B
R >
2
=,
9 ./ ) 0 C 1//0 4,3.2
?4-@ < (
& 2
7 B
D
1/21
?41@ =
6 = Q R ;
#
9
#
1
6 (
O
4--. D 75
?44@ =
6 = Q R #
3 %
8
(
4--. D 75
?4.@ >
= QBK <
"K
R A " @ @ 9 , 1/4, 1+3.,
?40@ 7 7 U =
6 L Q '
5 '
( %
!
R > @ 9 11* 1/*4 ./.3
0-/
?4,@ %
D Q5
9&
:'(P1 % >
R ; 2 @
:
) B( O
4--1 ' 4--13--4*
?42@ O
: Q B
>
%
:R : 2*1 )7 : 1/0-
?4+@ 8
6 B
2
4212 55 2 =,
@
D5 B
5
!
5L4*-- L 5
! 4--4
?4*@ (;
) ) Q8
D % ( % C =R
: :;:*4+<) :; )
1//,
?4/@ 6
O Q.5 D 5
6! D5<
(R : %6 /4-, 5
% 6
B
5
! 1//4
?.-@ 6
O Q!( 6 (
45 .5 <
D5<AR : %6 /0-2 5
% 6
B
5
! 1//0
?.1@ ' 5
< 9 9
5 D
1/0/
?.4@ P ( Q'
X ! !X Y &R : %6 *1-, ) 6<L B
5
!
5
! ) 1/*1
?..@ 5
> % 2
5 1/0.
.0
:;3:31.,0<)
$ " " 8 ?.1@ " &
!
Z (Z
& ' H J
1
4 ' H Z
4
) ' ! ' % 1. ' $ ( & ' $ ' H
4' ' '
H
14
14 (Z
.
( ! $
& ' ' H
0
4' ' '
# ' H
I J ) I J '
I
,
*
!
I K
I K
C
I K
!
'
*
% %
, ' " " ,
' 8 & * )' ' " % ) ' % C
? %
& ' ' %% ' " ,
+' & ' ' " % " :;3:31.,0<)
.,
)
" $ ' J ' J # ' H 2
'
<
.3, )
2
14
14 (Z J 4&'' ' '
I J ) I J '
I H +
Æ
1 & ' '
H 4) I I I J
*
' 1
4 & ' H 4I J
1& ' '
/
' 1
4
'
' 1
H
& ' '
1-
' 1
&
! H 14
' !
& ' '
11
T ! H 14& ' 14
D )" H ) I J ! ) I J 4) I !
1.
'
M% H
%
%
I & J H )
!
& '"
10
% ,
-
H
H
H
.2
M% H )
I & J !
& ' '
1,
' M% '
'
)
I & J !
& '
'
12
:;3:31.,0<)
'
& <
1 Æ
<
*31 ?.4@
, H 4) I ?) I I -",I @
J)
&
'
1
J ?) I I K K K@
H J -",4) I ?) I I
-
J)
I I '
&
'
1
1+
' -",)
-",I @
I I
J ?) I I K K K@ 1*K
1*
$ ?..@
&
1
H
) I I
I J
'
1/
4
'
1
Æ
&
T
H , 4-
. H , J 41
! L
A " & "
[ H
& '
H
& '"
44
% <
1 [ H
& '
H J -", 4.
'
<
* / L
A <
44 'J1
& ' H 4 ?) I I -",I @ "
40
' 1
! <
40 ! " ' ! ' J 1
'J1
& ' H
& '
4,
' 1
' 1
!
B
! ! ' J 1
1
' J '
42
&
' H
& ' J ' 1
' 1
!
:;3:31.,0<)
.+
7
<
40 42 !
&
H
4 ?)
I I
-",I @ J
' J
'
1
& !! ! ! "
4+
B
<
4+ ! ) M
4*
'
H 1 ' J
% ! <
4+ 4* "
<
1+ 1* 1/ 4- 41
' ,1 ( - -#
! "
" !
&
A " " ! > ?1/@
) &
>
?4.@ 1/4, Æ
H -", H 4 \ &
4 "
H
4/
$ \ <
4/ $ $ $
$ $
: O
?42@ \ H 1 -"12, # -".., # .-
'
% 10 $ $
1F " " *! &
! & ?.@ 1/., % 1, &
'
.*
:;3:31.,0<)
1
0.9
Φ(τ) [−]
0.8
0.7
0.6
General algorithm
R.T.Jones approximate
0.5
0
5
10
15
τ [−]
20
(
% % !" ! 1
$ $
>
D-E' #+
.25/' ! "
%
12 " (
" 0.02
General algorithm
Theodorsen Analytical
0.65
0.02
General algorithm
Theodorsen Analytical
0.015
0.6
[−]
[−]
0.005
C
0.45
0
−0.005
0.35
3.5
4
4.5
5
α [o]
5.5
6
6.5
−0.01
0
−0.005
0.4
0.3
0.005
M
D
0.5
C
CL [−]
0.01
0.01
0.55
General algorithm
Theodorsen Analytical
0.015
−0.01
3.5
4
4.5
5
α [o]
5.5
6
6.5
−0.015
3.5
4
4.5
5
α [o]
5.5
6
6.5
(
% ! 0
$
& " 3 +
H 4 H -". 2
) H ,Æ 1",Æ ' "
F $ ' " ' ' %
1
0.02
0.015
0.8
0.01
0
0.6
0.005
CM [−]
C
L
CD [−]
[−]
−0.02
0.4
−0.04
0.2
−0.06
0
−0.2
0
−0.005
General algorithm
Theodorsen Analytical
−0.1
0
Y/c [−]
−0.01
General algorithm
Theodorsen Analytical
0.1
0.2
−0.08
−0.2
−0.1
0
Y/c [−]
0.1
0.2
−0.015
−0.2
General algorithm
Theodorsen Analytical
−0.1
0
Y/c [−]
0.1
0.2
- (
% %
! 0
$
& " +
H 4 H -"4 ' " ' ' %
:;3:31.,0<)
./
3.5
3
λ=0.0
λ=0.4
λ=0.8
CL/CL0 [−]
2.5
2
1.5
1
0.5
0
90
180
/ H 270
360
? @
Æ
(
% "
! 0 ") %
9 <
DE . ' DE/ " .
/ +
H 4 H -"4' " $ % ) !
! > ?1/@ % 1+
' > &
" " 0-
:;3:31.,0<)
+
! " ,-.,./01
6
= =
6 8
= 6
*+,,-.-*/-# *+,,-.-/-0 '
-1-234*0-
5
>
<
5
O
1* 4--0
0-
1
<)( 1.2.E-----+
<)L27 4-----.41+
..
! "# $%%% !
" # $
$ % &
!
& &
>')5 B:')<(# <:C5U)6'7(# <:C<( '7' U# 5U)6'7(