HAL Id: tel-00685850 https://tel.archives-ouvertes.fr/tel-00685850 Submitted on 6 Apr 2012 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Analyse numérique des hydroliennes à axe vertical munies d’un carénage Ane Menchaca Roa To cite this version: Ane Menchaca Roa. Analyse numérique des hydroliennes à axe vertical munies d’un carénage. Autre. Université de Grenoble, 2011. Français. <NNT : 2011GRENI050>. <tel-00685850>
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HAL Id: tel-00685850https://tel.archives-ouvertes.fr/tel-00685850
Submitted on 6 Apr 2012
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Analyse numérique des hydroliennes à axe verticalmunies d’un carénage
Ane Menchaca Roa
To cite this version:Ane Menchaca Roa. Analyse numérique des hydroliennes à axe vertical munies d’un carénage. Autre.Université de Grenoble, 2011. Français. <NNT : 2011GRENI050>. <tel-00685850>
THÈSEPour obtenir le grade de
DOCTEUR DE L’UNIVERSITÉ DE GRENOBLESpécialité : Mécanique des Fluides, Procédés, Energétique
Arrêté ministériel : 7 août 2006
Présentée par
Ane MENCHACA ROA
Thèse dirigée par Thierry MAITREet codirigée par Christian PELLONE
préparée au sein duLaboratoire des Ecoulements Géophysiques et Industrielsdans l’Ecole Doctorale Ingénierie - Matériaux, Mécanique,Environnement, Energétique, Procédés, Production
Analyse numérique deshydroliennes à axe vertical munies
d’un carénage
Thèse soutenue publiquement le 30 septembre 2011,devant le jury composé de :
M. Gérard BOISProfesseur des Universités, ENSAM de Lille - Président
M. Jacques-André ASTOLFIMaître de Conférences ENSAM HDR, IRENav de Brest - Rapporteur
M. Andrei-Mugur GEORGESCUMaître de Conférences, UTCB, Bucarest, Roumanie - Rapporteur
M. Christophe CORREProfesseur des Universités Grenoble-INP, Grenoble-INP - Examinateur
M. Thierry MAITREMaître de Conférences HDR, Grenoble-INP - Directeur de thèse
M. Christian PELLONEChargé de Recherche CR1, CNRS - Co-Directeur de thèse
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M#; 8-A.1) 30, ,+('-+'(0, J&(+!-!+)!(0, ,'( 4) 9)40$ )' -&'(, 3='*0 (&+)+!&*#
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C4@ ;<%+&'#(/ &52#-'<%*/ ,#%- &-+'&*- $*/ 56)-#$'*((*/ *( .#(E3%-+&'#( .#(E(2*4 S9
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#005!9
P =1
2ρSV 3
∞
Protor = P ηhydrolienne ηm canique η lectrique
ηhydrolienne
CP
CP 16 27
16 25
η$ =
CP = 0.32 CP = 0.56
CP
16/27 59%
CS
CS
FS
CS
V0/Va = 1/3
(pp1 − pp2)
(1 + CS)
!"#$%&' () '%"% *' +,"&% !
!"#$%&'
(%)*
+#* '
"#$
Ap(p
p2−
p p1)
=ρ
ApV
p(V
0−
Va)
Ap(p
p2−
p p1)+
FS
=ρ
ApV
p(V
0−
Va)
%&'()*+,*-)(*'./*0)
FS
=0
FS
=1 2C
S[ρ
Ap(V
2 0−
V2 a)]
1)'/&2--,
(pp2−
p p1)
=1 2ρ
(V2 0−
V2 a)
32,44*/()'.(25.'.)
P=
ApV
p(p
p2−
p p1)
=1 2ρ
ApV
p(V
2 0−
V2 a)
6,7)44)*2(&-
Vp
=1 2
(V0+
Va)
Vp
=1 2
(1+
CS)(V
0+
Va)
8&)9:7'*;/.)'&7&'
CT≡
FT
1 2ρA
pV
2 a
=(p
p1−
p p2)A
p
1 2ρA
pV
2 a
8&)9:52,44*/()
CP
=(p
p1−
p p2)V
P
1 2ρV
3 a
=1 2C
T
(
1+√
1−
CT
)
CP
=(p
p1−
p p2)V
P
1 2ρV
3 a
=1 2C
T(1
+C
S)(
1+√
1−
CT
)
8&)9:52,44*/()<*+:
CP
max
(
V0/V
a=
1 3
)
=16 27
CP
max
(
V0/V
a=
1 3
)
=16 27
(1+
CS)
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<-$-=%>&- +$-#,2? #$A B) R! E #? RRQ' J.03=6 `=60O7'
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(06=/>:6? c+,/>06 R$R'
Chapitre 2
H R = 2R N N = 3C
V∞
ωH R C N V∞ ω
λω R
λ λopt
λ =ω R
V∞
Re
Re =UL
ν
U [m/s] L[m]ν[m2/s]
U L
Re
Re =V∞
ν
Re
ReC =(ω R) C
ν=
(V∞λ) C
ν
105 Re ReC
σ
σ =NC
R
FL FD
α
FL FD α
FL/FD
CL/CD
V∞ W−→ω
−→R
−→W =
−→V∞ −−→ω ∧
−→R
−→ω ∧−→R
W
V∞
−→ω ∧−→R θ
W
θ
W = V∞
√1 + 2λ sin θ + λ2
α = arctan
(
sin θ
cos θ + λ
)
αmax = arctan
(
1√
λ2 − 1
)
α
θλ
λλ
−→V∞
−→R −→ω
−→W
!"#$%&' () &"##'*+ +,& *-!./&0*$'11' *$2&' 3 "4' 5'&%$ "* /' %.#' /"&&$',+ !
"#$ %&"#$ '%()*+ ,-&%%").&+)(* &/ .#*+0# ,# %(/$$1#2 3/) #$+ &/ 3/&0+ ,# .(0,# #* +41(0)#
")*1&0)$1# #+ 5/),# %&06&)+7 $(*+ %01$#*+1#$ $/0 "& 8)9/0# :;<; =""#$ $(*+ ,1+#0>)*1#$ #*
%0(?#+&*+ FL #+ FD ,&*$ "# 0#%@0# +&*9#*+)#" &/ .#0."# ,# 0(+&+)(*; A# .(/%"# ,)$%(*)B"#
$/0 "& +/0B)*# C&00)#/$ #$+ ,(**1 &"(0$ %&0 "& 6(0.# >(+0).# FT '%0(?#.+)(* ,# "& %(0+&*.#2
,)0)91# D#0$ "-&D&*+ ,#$ %&"#$2 ,)>)*/1# ,# "& %0(?#.+)(* ,# "& +0&E*1#2 ,)0)91# D#0$ "-&00)@0#
,#$ %&"#$7; F&*+ 3/# "& .(>%($&*+# +&*9#*+)#""# ,# "& %(0+&*.# 0#$+# $/%10)#/0# G .#""# ,#
"& +0&E*1#2 "# .(/%"# >(+#/0 $#0& +(/?(/0$ %($)+)6; C&*$ "# .&$ .(*+0&)0#2 "& 0(+&+)(* $#0&
60#)*1#; A# +0&.1 ,#$ .(>%($&*+#$ ,/ .4&>% ,# D)+#$$# 01$/"+&*+ %&0+)+)(**# "# ,(>&)*#
#* 3/&+0# H(*#$ I ,#/J H(*#$ >(+0).#$2 $)+/1#$ ,&*$ "#$ ,#>)K.#0."#$ &>(*+ #+ &D&" L #+
,#/J H(*#$ ,# 60#)*&9# &/+(/0 ,# θ = 0 #+ θ = 180 2 "G (M "#$ D#.+#/0$
−→V∞ #+
−→ωR $(*+
.(")*1&)0#$ '8)9/0# :;<7; A& 6(0.# .(>%"1>#*+&)0# G FT #$+ "& 6(0.# *(0>&"# FN 2 *(0>&"#
&/ .#*+0# ,# 0(+&+)(*; A& .(00#$%(*,&*.# #*+0# "#$ 6(0.#$ ,# %(0+&*.#K+0&E*1# #+ "#$ 6(0.#$
*(0>&"#K+&*9#*+)#""# #$+ ,(**1# %&0 "#$ 0#"&+)(*$ ':;NO7;
FN = FL cos α − FD sin α
FT = FL sin α − FD cos α
':;NO7
A#$ 6(0.#$ FL2 FD2 FN #+ FT %#/D#*+ P+0# #J%0)>1#$ $(/$ 6(0># &,)>#*$)(**#""# '#*
6(*.+)(* ,-/*# "(*9/#/0 #+ ,-/*# D)+#$$# ,# 01610#*.#7 %&0 "#$ .(#Q.)#*+$ $/)D&*+$ I
CL =FL
1
2ρ (HR) V 2
∞
CD =FD
1
2ρ (HR) V 2
∞
CN =FN
1
2ρ (HR) V 2
∞
CT =FT
1
2ρ (HR) V 2
∞
':;NN7
!"#$%&! ' A-4S,0(")#**# &S&*+ /* .(>%(0+#>#*+ .S.")3/#2 )" #$+ %($$)B"# ,-1K
D&"/#0 $()+ ,#$ .(#Q.)#*+$ )*$+&*+&*1$2 $()+ ,#$ .(#Q.)#*+$ >(S#*$ .&"./"1$ $/0
/* +(/0 /*# 6()$ "& %10)(,).)+1 ,# "-1.(/"#>#*+ 1+&B")#; T&/6 )*,).&+)(* .(*+0&)0#2
"#$ 01$/"+&+$ ,(**1$ ,&*$ "# %01$#*+ ,(./>#*+ .(*.#0*#*+ %&0 ,16&/+ "#$ D&"#/0$
>(S#**#$2 .# 3/) %#0>#+ ,-&D()0 /*# #$+)>&+)(* 9"(B&"# ,/ .(>%(0+#>#*+ ,#
"& >&.4)*#;
U* & D/ %01.1,#>>#*+ 3/# "& %#06(0>&*.# ,# "-4S,0(")#**# #$+ 6&)B"# %(/0 "#$ %#+)+#$
D&"#/0$ ,# λ G .&/$# ,/ ,1.0(.4&9# ,S*&>)3/#; V" #* #$+ ,# >P># %(/0 "#$ 6(0+#$ D&"#/0$
,# λ2 >&)$ %(/0 ,#$ 0&)$(*$ ,)W10#*+#$; T) λ &/9>#*+#2 α ,)>)*/#; U02 "& .(>%($&*+#
+&*9#*+)#""# ,# "& 6(0.# ,# %(0+&*.# ,)>)*/# &D#. α; C# %"/$2 "# 0&%%(0+ %(0+&*.#K+0&E*1#
+#*,$ D#0$ H10( 3/&*, α +#*, D#0$ H10(; =* .(*."/$)(* I
! A#$ %#06(0>&*.#$ ,# "-4S,0(")#**# $(*+ 6&)B"#$ "(0$3/# "# %&0&>@+0# ,-&D&*.# λ %0#*,
,#$ D&"#/0$ 6&)B"#$ (/ 1"#D1#$;
! !"! #$%&'%#( )( *+&',%+&&(-(&, (, #.$.-/,$(0 .00+'%10
"#$ %#&''#()#$ *#)+,)%-./#$ $,.0 ,10#.(#$ *,() 2#$ 3-'#()$ &.0#)%42&-&)#$ 2# λ5λ ≈ 1.8 − 2.567 8-.$ /# /-$9 α #$0 $(:$-%%#.0 ;)-.2 *,() *),/()#) (. )-**,)0
2# *,)0-./#<0)-=.4# -$$#> 4'#34 #0 $(:$-%%#.0 *#0&0 *,() ?(# '#$ 24/,''#%#.0$
/,.2(&$-.0 -( 24/),/@-;# .A-&#.0 *-$ '&#(7
8#(B *-)-%C0)#$ 2# +,./0&,..#%#.0 *#)%#00#.0 2# /-)-/04)&$#) '#$ *#)+,)%-./#$ 2#
'A@D2),'&#..# E
"# !"# $"%& '" !()*" CM 5,( /,#:/&#.0 2# %,%#.06 #$0 24F.& #. .,)%-'&$-.0
'# %,%#.0 2# 0,)$&,.
−→M =
−→R ∧
−→F -( %,D#. 2( 2&-%C0)# 2# 'A@D2),'&#..# #0 2# '-
3&0#$$# 2( G(&2#7
CM =MZ
1
2ρ (HH) RV 2
∞
-3#/ MZ '# /,(*'# $(&3-.0 'A-B# 2# ),0-0&,. 5I7JI6
"# !"# $"%& '" )($++,% " CP 9 ?(& 24F.&0 '# )-**,)0 #.0)# '- *(&$$-./# P )#<
/(#&''&# *-) 'A-)1)# 2# ),0-0&,. #0 '- *(&$$-./# /&.40&?(# 0@4,)&?(# 2# 'A4/,('#%#.0
-%,.0 -( 0)-3#)$ 2# '- $()+-/# 2( ),0,)7
CP =P
1
2ρ (HH) V 3
∞
P = MZω
5I7J 6
P 40-.0 2&)#/0#%#.0 '&4 K MZ #0 ω9 '# /,#:/&#.0 2# *(&$$-./# #$0 2&)#/0#%#.0 )#'&4
-( /,#:/&#.0 2# /,(*'# *-) '# *-)-%C0)# 2A-3-./# 54?(-0&,. 5I7JL667
CP = λCM 5I7JL6
"- M&;()# I7N *)4$#.0# (.# 43,'(0&,. 0D*&?(# 2( /,#:/&#.0 2# *(&$$-./# #. +,./0&,. 2(
*-)-%C0)# 2A-3-./# λ7 O(&3-.0 'A&%*,)0-./# 2#$ #P#0$ *)&%-&)#$ 52D.-%&?(#$6 *-) )-**,)0
-(B #P#0$ $#/,.2-&)#$ 53&$?(#(B69 ,. 2&$0&.;(# $() /#00# /,()1# 0),&$ )4;&,.$ QJIR7 S,() 2#$
+-&1'#$ λ9 '#$ *-'#$ $(1&$$#.0 (. 24/),/@-;# 2D.-%&?(# &.0#.$# /,.2(&$-.0 K 2#$ /@-);#$
&%*,)0-.0#$ $() /#''#$</&7 "- *#)0# 2# )#.2#%#.0 #$0 2(# -(B '-);#$ $0)(/0()#$ 3,)0&/&0-&)#$
?(& $# 240-/@#.0 2# '- $()+-/# 2#$ *-'#$ #0 ?(& $,.0 /,.3#/04#$ #. -3-'7 S,() '#$ 3-'#()$
4'#34#$ 2# λ9 '- 0)-=.4# 2# 'A-)1)#9 '- %-$$# -T,(04# #0 '- 3&$/,$&04 *)42,%&.#.09 K /-($#
2#$ +-&1'#$ -.;'#$ 2A&./&2#./#7 U.0)# '#$ 2#(B )4;&,.$ &' D - (.# >,.# 2&0# 2# 0)-.$&0&,.9
/,))#$*,.2-.0 K (. 4?(&'&1)# #.0)# '#$ #P#0$ *)&%-&)#$ #0 '#$ #P#0$ $#/,.2-&)#$7 VA#$0 2-.$
/#00# >,.# ?(# $# $&0(# '# CP ,*0&%-'7 S,() 2#$ 4,'&#..#$ 8-))&#($9 '- *'-;# 2# λ .# 24*-$$#
*-$ JW X 2-.$ '# /-$ 2#$ @D2),'&#..#$9 λ < 57
λ = 2.14
N = 3
y+ = 1
∆t ≡ 1κω − SST
ρ = 1000kg/m3 V∞ = 2.3m/sλ = 1 5
λopt ≈ 2 CPopt ≈ 0.41λ
CP ≈ 0 λ ≈ 3.9
633 − 018C 32.8mm 56.3mm
175mm 300mmH/
σV∞ 2.3m/s 1.0m/sReC ∝ 105 ∝ 105
10 10CPopt 0.41 λ ≈ 2 0.235 λ ≈ 1.82
CP
CPopt = 0.235 λopt ≈ 1.82CPopt = 0.41 λopt ≈ 2.0
CM
CM
CM
θ ≈ 110λ = 1
λ = 1
λ = 2CM > 0
λ CM
λCM < 0
θ190◦ < θ < 355◦
CM = f (θ)80%
x y = 0 Cp
U∗
x = Vx/V∞
Cp (x, y = 0) U∗
x (x, y = 0)Cp
x = −R80%
x = +R
x = −R x = +R
λloc
λloc =ωR
Vloc
Vamont > Vaval λloc−amont < λloc−aval
Cptot U∗
x
CM < 0
FX FY
[N/m]
FX FY
FX
FY
FX
FY
FX FY
FX λ λ
λλ
θ = 360◦
λ
λ
λ = 1.0 λ = 1.0 θ = 360
λ = 2.0 λ = 2.0 θ = 360
λ = 3.0 λ = 4.0 λ = 5.0 θ = 360
θ = 270
λ
CQ =Vxrotor
V∞
Vxrotor
Q
Q = V∞S∞ = VxrotorSrotor = VxSx
S∞
Srotor
Sx xVx x
Vxrotor Srotor V∞
x
λλ
λλ
λ
FX FY
λ λ
λ
λ ≈ 2 CPopt
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Chapitre 3
C = 4.0Rα = 10
0.25 × 0.7m2
1 : 5
Re 1.75 × 105 5.00 × 105
rotor 0.175mCpale 0.032mV∞ 1.8m/s 2.3m/s 2.8m/s
V∞
V∞ = 2.3m/s
V∞ = 2.3m/s
0.25 < λ < 3.0λ
≈ 0.4m/s 0.5m
≈ 0.8m/s 1m
2 − 4m/s
rotor 0.175mCpale 0.032m
C 0.35mα
V∞ 1.8m/s 2.3m/s
1.8m/s
2.3m/s α = 10◦
V∞ = 2.3m/s
2.8m/s
CP
CP
CD λ
Ntotal ≈ R9/4
t Rt
Ntotal ≈ R2t
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S&7.#&- Q8T8
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&/&.14+# #/ -03.34&/0 .#4 :&.#-24 4-3:&/0#4 W
y+ = 1 #0 y+ = 2 (#2'#00#/0 )# 2#(2+4#/0#2 .& 4,-4>$,-$%# :346-#-4#8
y+ = 5 #0 y+ = 20 4# 430-#/0 )&/4 .& X,/# 0&'(,/< #0 /# (#2'#00#/0 (&4 )# (2#/)2#
#/ $,'(0# 0,-4 .#4 )+0&3.4 )# .*+$,-.#'#/08
≈ 392 000 ≈ 250 000
V∞ = 1.5m/s λ = 4.0
y+ y+ = 2
y+ ∆t ≡ 1◦
x y
∆t ≡ 1◦
px py
wssx wssy
∆t ≡ 1◦
∆t ≡ 0.5◦ ∆t ≡ 1◦ ∆t ≡ 0.5◦
∆t ≡ 1◦
∆t ≡ 1◦
∆t ≡ 0.5◦ ∆t ≡ 1◦ 2π/3π
∆t
∆t y+ = 1
∆t ≡ 1◦
∆t y+ = 1
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',0# *%'*3#%")!)(5# /0 $-.'-*)%.%") *3%, /0 #F#)R.%+ ,! )0*1("% $!*3"3% %#) !"!,F#3%
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!" #$%&'()*++* ,-&.+.* *+ /)()*0 10++*(
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*3#0,)!)# %E'3*(.%")!0E #-") (##0# /% ,! $!.'!4"% %E'3*(.%")!,% *3!,(#3% %" 53D*(%* >?Q?
'!* X0.%,!# NYOB Z0% $% #-() %" $-"740*!)(-" ,(1*% -0 $!*3"3%+ ,%# *3#0,)!)# "0.3*(G0%#
#0*%#)(.%") ,%# D!,%0*# %E'3*(.%")!,%# 6A %E$%')(-" /% ,2&F/*-,(%""% ,(1*% '-0* /%# )*R#
5!(1,%# D!,%0*# /% λ@B :2%**%0* *%,!)(D% %")*% ,%# D!,%0*# "0.3*(G0%# %) %E'3*(.%")!,%#/3'%"/ /0 '!*!.R)*% /2!D!"$% 6C!1,%!0 =BQ@B ;%.!*G0-"# G0% $%))% %**%0* %#) /% ,2-*<
/*% /% Y?[ /!"# ,! \-"% /%# %K%)# '*(.!(*%# 61.0 < λ < 2.0 '-0* ,2&F/*-,(%""% ,(1*% %)1.0 < λ < 3.0 '-0* ,2&F/*-,(%""% $!*3"3%@B 9!* $-")*%+ %,,% %#) !##%\ $-"#3G0%")% /!"# ,%#/%0E !0)*%# \-"%#+ $%,,% /% )*!"#()(-" %) $%,,% /%# %K%)# #%$-"/!(*%#B ]-)-"# !0##( G0% ,!
/(K3*%"$% /%# $-%T$(%")# /% '0(##!"$% %#) )*R# 4*!"/% %")*% ,! $-"740*!)(-" ,(1*% %) $%,,%
$!*3"3%B J%# /(K3*%"$%# #-") !.',(73%# "0.3*(G0%.%")B 9,0#(%0*# %E',($!)(-"# '%0D%")
U)*% !D!"$3%# A $% #0^%) _
`Q
HH
HH
HH
λCP
1.001.251.501.752.002.252.502.753.004.005.00
100
(
CPnum − CPexp
CPexp
)
105
κω − SSTλ
λ
! !"! #$%&'()*++* ,-&.+.* *+ /)()*0 10++*(
"# CP $
%&#'#( "# )*+,-.# "# *&/,+0*#1#2( 32"03( 4-5 *- 45+6313(/ "#7 4-5+37 "# *- 8#32#
"M-3 ,+2"03( /.-*#1#2( 9 02# 705#7(31-(3+2 "0 ,+#:,3#2( "# 40377-2,# ;,# 4+32(
#7( -2-*<7/ "-27 *- 703(# "0 ,=-43(5#>$
?/-21+327@ *- (#2"-2,# "#7 ,+05)#7 45/"3(#7 4-5 *# 1+"A*# 201/53B0# #7( ,+55#,(#@ #2 4-5C
(3,0*3#5 4+05 *# ,-7 "# *&=<"5+*3#22# *3)5#$ %- D3.05# !$E@ B03 (5-,# *#7 5+7-,#7 201/53B0#
#( #64/531#2(-*# "0 ,+#:,3#2( "# ,+04*# "# *&=<"5+*3#22#@ ,+55+)+5# -0773 ,#((# (#2"-2,#$
%#7 "+22/#7 201/53B0#7 #( #64/531#2(-*#7 7+2( )3#2 #2 4=-7#$ F2 +)7#58# -0773 B0# *#
"/4=-7-.# "#7 5+7-,#7 4+05 *#7 4+32(7 "# G+2,(3+22#1#2( "-27 *- H+2# "#7 #'#(7 4531-35#7
#7( )3#2 "/,53( -0(-2( 201/53B0#1#2( B0@/531#2(-*#1#2($ %#7 5+7-,#7 "#7 #'+5(7 *+2C
.3(0"32-06 #( (5-278#57-06 7+2( 45/7#2(/#7 "-27 *&I22#6# %$ %#7 /,-5(7 #2(5# *#7 "+22/#7
201/53B0#7 #( #64/531#2(-*7# 7+2( 4*07 4#(3(7 B0# 4+05 *# ,+#:,3#2( "# ,+04*#@ #2 4-5(3C
,0*3#5 "-27 *# ,-7 "# *&=<"5+*3#22# *3)5#$
J2# -2-*<7# 4*07 ,+14*A(# "#7 5/70*(-(7 +)(#207 #7( G+0523 "-27 *- 7#,(3+2 !$E "#7(32/#
-06 ,+14-5-37+27 #2(5# *#7 B0-(5# ,+1)32-37+27 4+773)*#7 K "+1-32# (022#* #( 13*3#0 32L23@
=<"5+*3#22# *3)5# #( ,-5/2/#$
!"!" #$%&%'('%) ') '*') +' ,-$./0' +' -12.$3-'('%) 4 (2)5$+'
+' .$66'.)7$%
%&#'#( "# )*+,-.# "# *&/,+0*#1#2( - /(/ ,3(/ ,+11# *&02# "#7 ,-07#7 4+05 #64*3B0#5
*#7 "3'/5#2,#7 #2(5# *&-445+,=# 201/53B0# #( ,#**# #64/531#2(-*# "-27 *# "+1-32# (022#*@
-3273 B0(5# *#7 ,+2L.05-(3+27 *3)5# #( ,-5/2/#$ M2 #'#(@ *# ,+2L2#1#2( 4#0( -0.1#2(#5
-5(3L,3#**#1#2( *- 4#5G+51-2,# "# *- 1-,=32#$ N# 1-23A5# ./2/5-*#@ *# O03"# +4(# (+0P+057
4+05 *# ,=#132 B03 *03 +44+7# *# 1+327 "# 5/737(-2,#$ N-27 *# ,-7 "&02 G+5( ,+2L2#1#2(@
*# "/)3( (5-8#57-2( *# 7<7(A1# 4#0( 7# 8+35 G+5(#1#2( -0.1#2(#5 4-5 *&#'#( "# )*+,-.# "0
O03"# 9 45+6313(/ "# *- 4-5+3$ M2 #'#( "-27 ,#((# ,+2L.05-(3+2 *&/,+0*#1#2( 8- .*+)-*#C
1#2( -8+35 (#2"-2,# 9 (5-8#57#5 *- 1-,=32# 4*0(Q( B0&9 *- ,+2(+052#5$ R-5 ,+27/B0#2(@ *#
,+2L2#1#2( P+0# 705 *- 40377-2,# 5/,04/5/# #2 -0.1#2(-2( *#7 4#5G+51-2,#7$ M2 13*3#0
32L23@ *# O06 32,3"#2( #2 -1+2( "# *- 1-,=32# 7# "/(+052# 45+.5#7738#1#2( "# ,#**#C,3 #2
*- ,+2(+052-2($ M2 13*3#0 ,+2L2/@ *# O03"# 2&/(-2( 4-7 *3)5# "# 7# "/(+052#5 -0(-2( B0&3* *#
8+0"5-3(@ 02# 4*07 .5-2"# 4-5(3# "# *&/,+0*#1#2( 4-77# -0 (5-8#57 *- 1-,=32# ;D3.05# !$S>$
?+07 "/L2377+27 *# 4-5-1A(5# "# ,+2L2#1#2( ./+1/(53B0# εg ,+11# *# 5-44+5( #2(5#
*,+1)5#1#2( "# *- 1-,=32# ;"3-1A(5# "0 5+(+5 "-27 *# ,-7 "# *&=<"5+*3#22# *3)5# #(
"37(-2,# 8#5(3,-*# #2(5# *#7 )+5"7 "# G03(# "#7 45+L*7 4+05 *&=<"5+*3#22# ,-5/2/#> #( *-
7#,(3+2 "# 4-77-.# "0 O03"# 32,3"#2( ;*-5.#05 "0 "+1-32#> K
εg =#2,+1)5#1#2( 7<7(A1#
*-5.#05 "+1-32#
;!$T>
%- "31320(3+2 "0 4-5-1A(5# εg 7# (5-"03( 4-5 02# "31320(3+2 "0 ,+2L2#1#2( #( ,+5C
5#74+2" "+2, 9 02# ,+2L.05-(3+2 4*07 45+,=# "&02 13*3#0 32L23$
%# U-)*#-0 !$E 32"3B0# *#7 ,+2L2#1#2(7 5#*-(3G7 #2 "+1-32# (022#*$ %- ,+*+22# V CPopt
,-*,0*/ W "+22# 7<7(/1-(3B0#1#2( *- 8-*#05 1-631-*# ,-*,0*/# "0 ,+#:,3#2( "# 40377-2,#@
3$#$ *# 1-63101 "#7 ;(<43B0#1#2(> X 8-*#057 ,-*,0*/#7 Y *- ,+*+22# V CPopt #7(31/ W ,+55#7C
4+2" 9 *- 8-*#05 1-631-*# "# *- ,+05)# "&32(#54+*-(3+2$ N-27 ,# (-)*#-0@ 4+05 -2-*<7#5
εg
α εg CPopt CPopt
C = 4.0RC = 2.5R
CPopt
CP
CP εg
p0 V0 Vc Vc
εp1 − p2 ε
ε (1 + CS)CT α
u0 uc
V0 Vc Va
CP
u0 uc
CT = u2c − u2
0
(uc − 1) (2u0 + uc − 1) = ε (u2c − u2
0)
ε = (1 + CS)Ap
Ac
CP =(p1 − p2) Vp
1
2ρV 3
a
ε CP = f (CT , α)
CPmax u0 = 0.33uc αuc α CT
p1−p2
CP VP p1 − p2
ε
CP = f (CT , α)
CPmax
CPmax
=1
(1 − ε)2
CPmax ε ε = 0.25
ε ε CP
!"#$%&' () *$+,-"%$./ 0' -1!20&.-$'//' "&3/3' 4 '55'%* 0' 6-. "7' !
"#$#%&
'()*#)%
%+&,-#()
$#./%
',/%)%%
'012314325
A0V
0=
ApV
pA
cV
a=
A0V
0+
(Ac−
A0)V
c
.67104883
p a+
1 2ρV
2 a=
p 1+
1 2ρV
2 pp a
+1 2ρV
2 a=
p 0+
1 2ρV
2 cp 2
+1 2ρV
2 p=
p 0+
1 2ρV
2 0
9326::6:;<3=>
u0
=V
0/V
au
c=
Vc/V
a
'06?>27;@15670207
CT≡
FT
1 2ρA
pV
2 a
=(p
1−
p 2)A
p
1 2ρA
pV
2 a
=u
2 c−
u2 0
*07A6;B3;86A;751;C6
FS
=0
FS
=C
S(p
1−
p 2)A
p=
CS
1 2C
TρV
2 aA
p=
CS
1 2ρV
2 aA
p(u
2 c−
u2 0)
'06?><6D80A;C6
CS
=0
ε≡
Ap/A
cε≡
(Ap/A
c)(1
+C
S)
+E"
p aA
c+
(pp2−
p p1)A
p−
p 0A
c=
p aA
c+
(pp2−
p p1)A
p−
p 0A
c+
FS
=
ρA
0V
2 0+
ρ(A
c−
A0)V
2 c−
ρA
cV
2 aρA
0V
2 0+
ρ(A
c−
A0)V
2 c−
ρA
cV
2 a
(uc−
1)(2
u0+
uc−
1)=
ε(u
2 c−
u2 0)
(uc−
1)(2
u0+
uc−
1)=
ε(u
2 c−
u2 0)
9326::6;4A08
up≡
Vp/V
a=
uo(u
0+
uc)/
(2u
0+
uc−
1)
'06?>F43::;1A6
CP
=(p
1−
p 2)V
p
1 2ρV
3 a
=u
0(u
c+
u0)2
(uc−
u0)
2u0+
uc−
1
'0776A2301D80A;C6
CP
max
u0(C
Pm
ax)
=1 3
uc(C
Pm
ax)
=1
+4 3
ε
(1−
ε)
CP
max89:;<;=
=C
Pm
ax8>;<;>
(1−
ε)2
!"#G>HI !"#$%&'($)*&+%!",(-&"+$+#%$,+.,()/0+&.%,'',(,'1&'23"+#$%&'1&'2'*,4
ε CPmax CPmax (ε = 0)× ε CPmax CPmax (ε = 0)××1.78
×1.12 ×2.05×1.24 ×2.37×1.39 ×2.79×1.58 ×3.32
CPmax
VP CP p1−p2
α p1 − p2 Vp
CS
ε
CD uc
CS
CD
Vc Vc = Va uc uc
CD
CS
FT FT
CD =FD
0.5ρ (Vauc)2 C H
CD =FD
0.5ρV 2a C H
FD =FD
u2c
CS =0.5ρ (Va)
2 C H CD
FT
=FD
FT
εg < 0.05
! !"! #$%&'()*++* ,-&.+.* *+ /)()*0 )+1)+)
"#$%&'(#% $)*)+, -./0/1 $)*)+, )/0/)
HH
HH
HH
λCP
2)34+ 5641/1+ 2)34+ 5641/1+
1.00 7879: 787;< 787=> ?
1.50 789@; ? ? 7899>
2.00 78<== @87 9 78=@= 78:!9
2.50 78<9! ? ? 78<7=
3.00 78=== !8<;! 78!>< 78!;<
4.00 787 > !8::: A7879 ?
5.00 A789!; !8@>! A78<= ?
!"# =8<B !"#$%"&'( )" *+%((,&$" &+-./%0+"( "& -%1%"+ $!&2&" "' %&2&% *!+/ *1+(%"+/(
*,/,-3'/"( )4,5,&$"6 78)/!1%"&&"( 1%9/" "' $,/.&."6
FAP =CPopt -641/1+
CPopt *)34+C=8 D
EFG
)/0/) /,H14)I,+ 0.623/0.414 = 1.50-./0/1 /,H14)I,+ 2.666/0.544 = 4.90
-./0/1 +JK14)H+/L6* 1.499/0.400 = 3.75
!"# =8:B :,$'"+/( )4,+;-"&','%!& )" *+%((,&$" *!+/ )%5"/("( $!&2;+/,'%!&(6
2+ M63*+6, =8: 41N,H+ *+N O6*+,4N P, EFG K.,4 *+N -./0Q,46L)./N L+NL1+N8 %/ H)*)+,
)/0/)R *S,L)*)N6L)./ P, -641/6Q+ 6H1*).4+ PS,/ T6-L+,4 @8< *+ 4+/P+H+/L P+ *6 H6-U)/+8 2+
-./0/+H+/L Q1/V4+ P+N O6*+,4N P+ EFG K*,N )HK.4L6/L+NR P+ *S.4P4+ P+ = W < X +**+N N./L
K*,N 1*+O1+N +/ /,H14)I,+ I,S+/ +JK14)H+/L6* C-T8 N+-L)./ =8@D8
#/+ 6,L4+ 4+H64I,+ -./-+4/+ *+N O6*+,4N P, K646HVL4+ PS6O6/-+ K.,4 *+N K.)/LN P+
T./-L).//+H+/L .KL)H6,J8 Y6/N ,/ H)*)+, )/0/)R *6 O6*+,4 P+ λopt O6,L +/O)4./ ! 6,NN)
3)+/ K.,4 *SUZP4.*)+//+ *)34+ I,+ -+**+ -641/1+8 %/ H)*)+, -./0/1R ./ 6 K+,LA[L4+ ,/ *1Q+4
P1-6*6Q+ P+ λopt O+4N !8!<
2
H6)NR N,4L.,LR ./ .3N+4O+ I,+ λopt O6,L P1N.4H6)N = K.,4
*SUZP4.*)+//+ -641/1+8 F)/N)R *+ -./0/+H+/L )/P,)L ,/ P1-6*6Q+ P, K.)/L P+ T./-L).//+H+/L
.KL)H6* O+4N P+N O6*+,4N P+ λ K*,N 1*+O1+N N) *SUZP4.*)+//+ +NL -641/1+8 G.,4 6KK,Z+4
-+LL+ 4+H64I,+R N.,*)Q/./N I,+ P+ H6/)V4+ Q1/146*+R PS6K4VN *+N P+,J H.PV*+N P+ \+4*+R
u0 = 0.33 CO)L+NN+ H.Z+//+ 6, N)**6Q+D )HK*)I,+ *S.KL)H,H P+ K,)NN6/-+ P+ *6 L,43)/+
)/NL6**1+ 6,NN) 3)+/ +/ H)*)+, )/0/) I,S+/ H)*)+, -./0/18 G64 6)**+,4NR +/ H)*)+, -./0/1R
/.,N 6O./N O, K41-1P+HH+/L I,+ CT 6,QH+/L+ +/ T./-L)./ P+ ε8 %/ PS6,L4+N L+4H+NR
P6/N -+ H)*)+,R *+ K.)/L P+ T./-L).//+H+/L .KL)H6* N+ N)L,+ K.,4 ,/ CT 3+6,-.,K K*,N
Q46/P I,+ -+*,) +/ H)*)+, )/0/)8 %L6/L P.//1 I,+ CT -4.]L 6O+- λR λopt PS,/+ -./0Q,46L)./
-641/1+ -./0/1 +NL /1-+NN6)4+H+/L N,K14)+,4 W -+*,) PS,/+ -./0Q,46L)./ -641/1+ ., *)34+
)/NL6**1+ +/ H)*)+, )/0/)8
!""#$%&' ()# $#' *!$#)+' ,%&&-#' ,!&' $# .!/$#!) 012 '%&3 $#' *!$#)+' #4#536*#7#&3 5!$5)$-#' 8 $#'
5%)+/#' ,96&3#+"%$!36%& ()6 "#+7#33#&3 ,# :!56$63#+ $! *6')!$6'!36%& ;$%/!$# ,#' +-')$3!3' &# '%&3 "!' )36<
$6'-#' "%)+ $9-*!$)!36%& ,# CPmax 5!+ %& '%)=!63# -*63#+ ,# +!6'%&&#+ ')+ ,#' ()!&363-' !)3+#' ()# 5#$$#'
+-#$$#7#&3 5!$5)$-#'1
CM
CM
CM
CM
FOCC =CMmax − CMmin
CMmoyen
CQ =Vxrotor
V∞
λopt
y = 0
V∞
Cptot U∗
x
! !"! #$%&'()*++* ,-&.+.* *+ /)()*0 )+1)+)
"# $ %&% '%()*&+% ,-. #. /$+%*$0. $-0(.*&. #$ 1.+2)+($*/. '. #345'+)#6.**. .& #.7
(%/$*67(.7 '. /.&&. $-0(.*&$&6)* )*& %&% $*$#57%78 9.1.*'$*&: #3$;)-& '.7 1+)<#7 '3$6#.
$ -*. 6*=-.*/. 7-+ #3.*7.(>#. '.7 .?)+&7 ,-6 73$11#6,-.*& $- 757&@(.8 "# 7.(>#. ')*/ 1.+A
&6*.*& '3%B$#-.+ /.-CA/68 D. E$>#.$- F8G +%7-(.: 1)-+ λopt: #.7 .?)+&7 ()5.*7 *-(%+6,-.7
,-$*&6<%7 7-+ #345'+)#6.**. .& 7-+ #. 757&@(. /$+%*$0.8 D3$*$#57. '. /.7 B$#.-+7 ()*&+.
,-. #3$;)-& '- /$+%*$0. *36*&+)'-6& 1$7 -* /4$+0.(.*& 7-11#%(.*&$6+. 6(1)+&$*& 1)-+
#345'+)#6.**. .* (6#6.- 6*<*68 D. /$+%*$0. $-0(.*&. '3.*B6+)* HIJ #$ 2)+/. #)*06&-'6*$#.
7-+ #345'+)#6.**. &$*'67 ,-. #$ 2)+/. &+$*7B.+7$#. *. B$+6. 1+.7,-. 1$7 '3-*. /)*<0-+$&6)*
K #3$-&+.8 L- +.0$+' '- MLN: ,-6 .7& '. #3)+'+. '. O8P: /.&&. 4$-77. '3.?)+&7 .7& $//.1&A
$>#.8 Q* (6#6.- /)*<*%: 1$+ /)*&+.: #3$;)-& '- /$+%*$0. $-0(.*&. '. 2$R)* 6(1)+&$*&. #.7
.?)+&7 7-+ #345'+)#6.**.: '3-* 2$/&.-+ P .*B6+)*8 J)&)*7: /)((. %B),-% 1+%/%'.((.*&:
,-. #.7 1+)<#7 '3$6#. /)*7&6&-$*& #. /$+%*$0. 7)*& $-776 >.$-/)-1 1#-7 7)##6/6&%7 .* (6#6.-
/)*<*%8 D.7 '67&+6>-&6)*7 /)++.71)*'$*&.7 K FX .& FY 1)-+ '6?%+.*&7 1$+$(@&+.7 '3$B$*/.
7)*& +./-.6##67 .* L**.C. D 7./&6)* D8S8
JTUQV"WTQ
9)*<*. #6>+. 9)*<*% /$+%*%. "*<*6 #6>+. "*<*6 /$+%*%.
λ = 2.0 λ = 4.0 λ = 2.0 λ = 2.0
FX 45'+)#6.**. [N ] ≈ 107 ≈ 341 ≈ 87 ≈ 116
FY 45'+)#6.**. [N ] ≈ 17 ≈ 93 ≈ 10 ≈ 12
FX 757&8 /$+%*$0.X [N ] Y ≈ 304 Y ≈ 76
FY 757&8 /$+%*$0.X [N ] Y ≈ 82 Y ≈ 16
X .?)+&7 *-(%+6,-.7 7->67 1$+ #3.*7.(>#. '.7 '.-C 1+)<#7
!"# F8GZ !"#$% &'()#*+',% -"'# ., -/#/(0$#, 12/3/&4, "-$*(/.5 6"(-/#/*%"& 1,% 4/%
1, #)7)#,&4,5
T* +.0$+' 7-+ #.7 /$+&.7 '.7 7&+-/&-+.7 B)+&6/6&$6+.7 '. #$ /)*<0-+$&6)* /$+%*%. .*
(6#6.- 6*<*6 [M60-+. F8OP\ ()*&+. ,-. #3%/)-#.(.*& .7& -* 1.- 1#-7 /)(1#.C. ,-. '$*7
#$ /)*<0-+$&6)* #6>+.8 D36*&.+$/&6)* '.7 7&+-/&-+.7 677-.7 '.7 1$#.7 .& '- /$+%*$0. 1.-&
]&+. K #3)+606*. '. /. 14%*)(@*.8 Q* /)(1$+$67)* K #$ M60-+. H8HO '- /4$16&+. H ,-6
/)++.71)*' K #345'+)#6.**. #6>+.: 6# 7.(>#. ,-. #$ &$6##. .& #$ 1)1-#$&6)* '.7 7&+-/&-+.7 7)6&
$*$#)0-. $- (](. 1)6*& '. 2)*/&6)**.(.*&8 D$ '6?%+.*/. ($;.-+.: 4)+(67 #36*&.+$/&6)*
'.7 7&+-/&-+.7 +)&)+A/$+%*$0. .7&: '$*7 #. /$7 /$+%*%: #$ 1+%7.*/. .* $B$# '- /.+/#. '.
+)&$&6)* '3-* 1#-7 0+$*' *)(>+. '. &)-+>6##)*78 D3.?.& '- 1$+$(@&+. '3$B$*/. .7& $*$#)0-.
K /.#-6 '. #345'+)#6.**. #6>+. Z #.7 7&+-/&-+.7 B)+&6/6&$6+.7 7. &+$*72)+(.*& .* 76##$0. $B./
#3$-0(.*&$&6)* '. λ8 T*. '.7/+61&6)* 1#-7 '%&$6##%. '.7 /$+&.7 '. B)+&6/6&% .7& '671)*6>#.
.* L**.C. "8
N)-+ /#)+. /. /4$16&+.: *)-7 1+)1)7)*7 '3-&6#67.+ #3$11+)/4. '. .̂+#. _F` 1+%7.*&%.
$-1$+$B$*& '$*7 #$ 7./&6)* F8H8 L 1$+&6+ '.7 B$#.-+7 .* (6#6.- /)*<*%: 6# .7& 1)776>#.
'3.7&6(.+ '3-*. 1$+& #. CP '. #$ &-+>6*. #6>+. 1+67. .* (6#6.- 6*<*6 .& '3$-&+. 1$+& #3$11)+&
'- /$+%*$0.8 J)-7 -&6#67)*7 #.7 +%7-#&$&7 677-7 '.7 /$#/-#7 *-(%+6,-.7 [.& %B.*&-.##.(.*&
'.7 .77$67\8 9.7 '.+*6.+7 1+.**.*& .* /)(1&. #$ B67/)76&%: #$ &-+>-#.*/. .& #$ 1+%7.*/. '-
+)&)+8
D. E$>#.$- F8! 6*'6,-.: 1)-+ #345'+)#6.**. #6>+.: '.7 B$#.-+7 /)++60%.7 K 1$+&6+ '. ε8D.7 /$#/-#7 '. #345'+)#6.**. #6>+. .* (6#6.- 6*<*6 1+%7.*&.*& .*/)+. -* 1.&6& /)*<*.(.*&
λ = 1.5 λ = 1.5 θ = 360
λ = 2.0 λ = 2.0 θ = 360
λ = 2.5 λ = 3.0 θ = 360
!! !"! #$%&'()*++* ,-&.+.* *+ /)()*0 )+1)+)
"#$%&'$( )*+,( -%.%($ %)/)% 01)& .1 &%-$.1+%*) )$-2,%'$( (&+ () 31%+ $) -%.%($ -*%)&
4*)/)25 (+ &*)+ 0*)4 261.(-()+ 4*,,%62(&7 8) 4*(94%()+ 0( #$%&&1)4( &%-%.1%,( : 4(.$% %&&$
0( .1 4*,,(4+%*) 0$ -%.%($ 4*)/)2 (&+ +,*$;2 4( '$% %)0%'$( .1 4*<2,()4( 0(& 1##,*4<(&
&$%;%(&7 =1 4*,,(4+%*) 0(& ;1.($,& %&&$(& 0(& (>#2,%()4(& 0*))( $) CP 0( .?*,0,( 0( !7@@A
4( '$% (&+ () 144*,0 1;(4 .(& (>#2,%()4(& 0( B<%*)* C@D () -%.%($ %)/)%A '$% 1))*)4()+ $)
CP 0( !7@EF7 =(& &*$,4(& 0( .?241,+ "1++()0$5 ()+,( .(& ;1.($,& )$-2,%'$(& @G "4*-#,%&(&
0*)4 ()+,( !7E!H (+ !7EEF5 (+ 4(++( ;1.($, (>#2,%-()+1.( 4*,,%62( 0( !7@EF *)+ 2+2 1)1.I&2(&
01)& .1 #,(-%J,( &(4+%*) 0( 4( 4<1#%+,(7
K*,,(4+%*)& #*$, .?<I0,*.%())( .%L,( MN !
OP8MQRSTB
CPmax−234546
= CPmax−2377896
(1 − ε)2
ε = Ap/Ac
R)3*7 CPmax−234546
ε CPmax−2377896
T8UA λ = 2.0A +$))(. !7FVV !7@F !7E!H
T8UA λ = 2.0A %)/)% !7V V !7 ! !7EEF
OWXA λ = 2.25A +$))(. !7V!! !7@F !7@@F
!"# V7YZ !""#$%&!' (# CP )!*" +,-.("!+&#''# +&/"#0
X*$, 2;1.$(, .?1##*,+ 0$ 41,2)16( () -%.%($ %)/)%A )*$& 31%&*)& 1##(. : .1 -2+<*0(
;%&1)+ : 4*,,%6(, .1 3*,4( 0( +,1[)2( 0( 4( 0(,)%(, () -%.%($ 4*)/)27 X*$, .1 +$,L%)( 41,2)2(
0( ,232,()4( 01)& .(& 0($> -%.%($>A .( Q1L.(1$ V7\ .%&+( .(& ;1.($,& 0(& '$1)+%+2& ,2.1+%;(&
1$ 41,2)16(7 X,24%&*)& '$( 4(..(& 0$ 41& %)/)% &*)+ %&&$(& 0$ 41.4$. 4*-#.(+7 =( CD 0$ 41&
4*)/)2 1 2+2 41.4$.2 () $+%.%&1)+ .1 ,(.1+%*) "V7F57 G1)& 4(++( 2'$1+%*)A uc (+ .1 3*,4( 0(
+,1[)2( 0$ 41,2)16( FD:234546
&*)+ .(& 0*))2(& 0?()+,2(7 T*$& 4*)&+1+*)& '$( .(& ;1.($,&
0( CD &*)+ &%-%.1%,(& 0?$)( 4*)/6$,1+%*) : .?1$+,( ] .?(,,($, (&+ 0?();%,*) ^7 R. &(-L.(
,1%&*))1L.( 0?10-(++,( '$( .1 -2+<*0( #,*#*&2( #*$, .?(&+%-1+%*) 0( CD : #1,+%, 0(&
0*))2(& () -%.%($ 4*)/)2 (&+ 144(#+1L.(7
K*)/)2 R)/)%
FX 41,2)16( "@ #,*/.&5 CTD E!V _H
FD #,*/. CTD F@ EY
uc C`D 7Y\_
V∞ C-a&D @7E @7E
C C-D !7EF !7EF
H C-D !7 _F !7 _F
CD C`D !7@H!_ ('7 "V7F5 !7@EVF ('7 "V7H5
FD #,*/. 4*,,%62 CTD ('7 "V7_5 V@ EY
CS C`D ('7 "V7Y5 !7VY !7HF
FAP C`D 7VY 7HF
FAP 41.4$. 4*-#.(+ C`D ` 7F
!"# V7\Z !""#$%&!' (# CP !"# $%&'(#!$)*++* ,-#.+.*/
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" #$%&'% () CD *+&',-. '/ *+& #0++'1/* (2-3$/)*% /$ 3$/*)% (* /2*40%& (* &%$56-* 7)' +2*89
*%:*%$'& +)% /* :$%-6$;* *6 ,'/'*) '6<6'= >*/06 /* ,0(?/* #%0#0+- #0)% 60&%* :$+ (2-&)(*.
:*&&* @0%:* *+& (2*63'%06 ABC= D//* *+& :06@0%,* $3*: /$ &%$56-* () :$%-6$;* '++)* () :$/:)/
*6 ,'/'*) '6<6' 7)' *+& (* EFC= D6<6. 60)+ :$/:)/06+ CS *& /* FAP G #$%&'% (* /2*40%&
(* &%$56-* *+&',- *& (* /$ @0%:* (* &%$56-* (* /$ &)%1'6* /'1%* #%'+* *6 ,'/'*) '6<6'. 7)'
*+& (* FH C= I* J"K *+&',- G #$%&'% (*+ (066-*+ :06<6-+ *+& (* =AF L '/ *+& :06@0%,* G
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'/ *+& 6-:*++$'%* (2-/$%;'% /* (0,$'6*. 7)2'/ +0'& 6),-%'7)* 0) *8#-%',*6&$/= "'6+'. /*+ :$/9
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."','("3:
ILH
x/C = 0 x/C = 1
e = 11.8mm
CP /CPmax
x/C = 0CPmax
λopt = 2.0
CP = 0.407 CP = 0.414CM
CM
α C
CL
CD
CD CL α
αα λ
C = 4.0RC = 4.0R
Cα λ
C = 2.5R, 3R, 4R, 5R, 6R
α
CP α
λ = 2.0α λ = 2.0
λ = 2.0αopt
αopt = 30◦
CPmax (α = 30◦) = 0.728 CPmax (α = 18◦) = 0.662
CP λopt = 2.0
α = 18◦ α = 30◦
θ = 120◦ θ = 160◦
CM
CM
∆p∗
∆p∗max
∆∗
p
λopt = 2.0 α = 18◦ α = 30◦
Cp
x/C ≈ 0.85
CD CL αλopt = 2.0 α ≈ 22◦ CD CL
CP < CP
α ≈ 22◦ CD
CL
CD
CL α = 18◦
α ≈ 30◦
CP CD CL
CD CL
CD
CL
CL
CL CD
CD
CL
CP (λ)
CP (C) λ
λCP
λ = 2.0 λ = 3.0 C = 4.0RCD CL
FX
CD = f (FX/C) CD
4.0R CL
CP
C = 4.0R
CP CD CL
C = 3.0RC = 4.0R C = 5.0R
C = 6.0R
C = 3.0R C = 4.0R
C = 5.0R C = 6.0R
t = f (α)
t = f (C)
α C
!"#$%&' () *%&"%+,$'* -."/+0$1&"%$12 -3 *4*%5/' -' "&+2",' '% /1-50'* *$/#0$6$+* !
CP−th =P
1
2ρApV 3
a
=(pp1 − pp2) ApVp
1
2ρApV 3
a
=
[
1 −(
V0
Va
)2]
1
2
(
1 +V0
Va
)
(1 + CS) "#$ %
&'()
Ap * +(),-./ 01 2.,.2
Vp * '-,(++( 0( 345).13(6(/, &1 ).3
V0 * '-,(++( 0( 345).13(6(/, 0&/+ 3( +-33&7( &'&3
Va * '-,(++( 0( 345).13(6(/, &6./,
(, CS ≡FX )&25/&7(
FX 2.,.2
"#$8%
.9 FX )&25/&7( 05:(/0 0( 34&/73( α (, 0( 3& ).20( C$
;1-+<1( 3( 2(/0(6(/, 01 +=+,>6( 05:(/0 0(+ :&2&6>,2(+ 75.65,2-<1(+ 01 )&25/&7(?
&@/ 0( +-6:3-@(2 3( 6.0>3(? /.1+ &33./+ (++&=(2 0( 05,(26-/(2 A1+<14B <1(3 :.-/, 3( ).6C
:.2,(6(/, 01 +=+,>6( D 2.,.2 E )&25/&7( F :(1, G,2( (+,-65 (/ 1,-3-+&/, 0(+ -/H.26&,-./+
.I,(/1(+ D 1/( H.-+ :.12 ,.1,(+ F :.12 3( 2.,.2 3-I2( <1- +(2&-(/, &3.2+ -/A(),5(+ 0&/+
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;&*1.=(D*=*1)# V)''),E;( @&**).=()*1 =;;W, 6 0*) .),,&0.;) A)=0;&0/ /'0, (</&.1=*1)#
!" #$%&'()*' +', -.)/', 0 1' 23/13435'/(
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B=D0), ,&0, 2&.<), @) 1.=(*, @5&*@), (*1).=D(,,)*1 )1 ,) ;&<A(*)*1 /&0. 2&.<). @), B=D0),
/'0, '&*D0), )1 /'0, .%D0'(W.),# X= -&0') /)01 ,) /.&/=D). ,0. @), <(''()., @) S('&<W1.),
=B); 1.W, /)0 @) /).1) /=. @(,,(/=1(&*# X5%*).D() <%;=*(40) B%-(;0'%) /=. '), B=D0), ,)
@(B(,) 6 /=.1 %D=') )*1.) 0*) %*).D() /&1)*1()'') )1 0*) %*).D() ;(*%1(40)#
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Annexe C
Annexe D
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7,$(O($*/ 21*$42#/ 21('%+,$ 2#/ 562&-0(#..#/>
Annexe E !"#$"%#&' (# )!*+,$"' (-#,*
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7892:;6
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! "#$%&'($)*% +,-&. /.#0$).'1 2$-!3 453 6('(-(
7! "8(9.' ":0(;<(&=%& 3 >(&#%; ?'=)'%%&)'=3 @A
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C! "$#('$)11$&.83 D%&8(',
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O! F&(9)#PQ&.$.$,Q% R;&.11 S.NT3 6%'$%& .G U%1%(&;< )' ?#%;$&);(# ?'%&=, P 6?M?23
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TC% RB88,2- 0)*88 ^*V /B)G,-. RB88,2- 0)*88 ^*V /B)G,-.
W"% RB//.- :*9;2-'3 ].E6,B9
W % 70*/).-.V2GE.8 3 70*/).-.V2GE.8 F,(2E FB)G,-. _7RFF`3 >4
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WT% 7/),-6)2' 3 FL. 5-6,-..),-6 ]B8,-.88 _5]`3 >4
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Annexe G
V∞
S∞
VSVw
Sw
V∞ Vw
Vw < V∞
!"
S∞ V∞ = S V = Sw Vw #$% &
'( )*+,- .- /+(012- FD 3-4/ 5/+- 26(742- 8 3(+/9+ .4 :97(1 .- ;4(1/9/2 .- <*46-<-1/ =
ρ V S (V∞ − Vw) = FD #$%>&
'( 349??(1,- 3-+.4- 3(+ 7- @49.- -?/ =
P = D V = ρ V 2S (V∞ − Vw) #$%A&
'( 349??(1,- +2,432+2- 3(+ 7( <(,B91- 1- 3-4/ 5/+- ;4C41- )+(,/9*1 #(4 <9-4D 2E(7-
8 & .- 7( 349??(1,- 3-+.4-% F*<<- ,-//- 349??(1,- -?/ 2E(7- 8 7( 6(+9(/9*1 .C21-+E9-
,912/9;4- .- 7( <(??- .C2,*47-<-1/ /+(6-+?(1/ 7- +*/*+ 3(+ ?-,*1.-G *1 3-4/ 2,+9+- =
P = ρ V S
(
V 2∞− V 2
w
2
)
#$%"&
H1 2E(7(1/ 7-? 349??(1,-? .*112-? 3(+ 7-? 2;4(/9*1? #$%A& -/ #$%"&G 7( 69/-??- (4 ?-91
.4 +*/*+ (33(+(0/ ,*<<- 2/(1/ 7( <*I-11- (+9/B<2/9;4- .-? 69/-??-? 8 7C(<*1/ -/ 8 7C(6(7 =
V =V∞ + Vw
2#$%J&
K1 -1 .2.49/ =
Vw = 2V − V ∞ #$%!&
H1 +(33*+/(1/ 7( 6(7-4+ .- 7( 69/-??- Vw .(1? 7C2;4(/9*1 #$%>& *1 *:/9-1/ 7C-D3+-??9*1
.- 7( /+(012- =
FD = 2ρ SVD (V∞ − Vw) #$%L&
K1 3-4/ .2M19+ > ,*-N,9-1/? .- /+(012-G 7C41 )*1.2 ?4+ 7( 69/-??- .(1? 7( <(,B91-
#2;4(/9*1 #$%O&& -/ 7C(4/+- )*1.2 ?4+ 7( 69/-??- .- 7C2,*47-<-1/ (<*1/ #2;4(/9*1 #$%P&& =
!!"#" $% &'()*+" ,)!)-+*".&+)!!"//" "& 0& &+)!! +*" -" 1"&2 !"
CDD =FD
1
2ρ SV 2
#$%&'
CD =FD
1
2ρ SV 2
∞
#$%('
)* +,-.-/0*, .0 12.0,-3* #$%4'5 .2/ 62+7 83298-2*,/ 62 ,10:*;2 <2+=2*, />27<1-?21 @+>2*
A3*8,-3* 62/ =-,2//2/ B
CDD = 4
(
V∞
V− 1
)
#$% C'
CD = CDDV 2
V 2∞
#$% '
D3??2 <3+1 .2/ 83298-2*,/ 62 ,10:*;25 3* <2+, 6;E*-1 62+7 <010?F,12/ 6>0=0*825 .>+*
-*,21*2 #;@+0,-3* #$% G'' 2, .>0+,12 <HI/-@+2 #;@+0,-3* #$% J'' B
λ =ω R
V#$% G'
λ0 =ω R
V∞
#$% J'
K. 2/, <3//-L.2 ?0-*,2*0*, 6>-*,136+-12 .0 *3,-3* 6>-*,21A;12*82 a @+- ,106+-, .0 6-?-*+M
,-3* 62 .0 =-,2//2 60*/ .0 ?08H-*2 <01 10<<31, N .0 =-,2//2 62 .>;83+.2?2*, 0?3*, B
V
V∞
=λ0
λ= (1 − a) #$% O'
)* 12;81-=0*, .2 83298-2*, 62 ,10:*;2 2* A3*8,-3* 62 a 3* 3L,-2*, B
CD = 4
(
λ0
λ
) (
1 −λ0
λ
)
#$% "'
3+
CD = 4a (1 − a) #$% !'
!!
"#$%&'( )*( &*)$+,-.( /0123 *+ /01435 )$ %6,(($.7* &876%8&8* 9*:,*.+ ;
P = 2ρ SV 2 (V∞ − V ) /01 <3
=*( 7-*>7,*.+( 9* %6,(($.7* ?$(8( (6& )$ :,+*((* 9$.( )* &-+-& /8@6$+,-. /01 A33 *+
B-.98 (6& )$ :,+*((* $C-.+ /8@6$+,-. /01 D33 (#87&,:*.+ ;
CPcol =P
1
2ρ SV 3
/01 A3
CP =P
1
2ρ SV 3
∞
/01 D3
E. B-.7+,-. 96 B$7+*6& 9#,.+*&B8&*.7* )* 7-*>7,*.+ 9* %6,(($.7* %*&96* %$& )* F6,9*
9*:,*.+ ;
CP = 4a (1 − a)2/01GH3
=*( 7-6&?*( %-&+8*( (6& )*( I,J6&*( 01G *+ 01K %*&C*++*.+ 9* 7-C%&*.9&* )#,.F6*.7*
96 B$7+*6& 9#,.+*&B8&*.7* (6& )*( 7-*>7,*.+( 9* %6,(($.7* *+ 9* +&$L.8*1 M.* :$)*6& 9*
a >1
2.#*(+ %$( %NO(,@6* %6,(@6#*))* *.+&$L.* 6.* :,+*((* *. $:$) Vw .8J$+,:*1 CD *(+
C$P,C6C *+ :$6+ )-&(@6* Vw = 01Q) (* +&-6:* @6* )* (O(+'C* 9#8@6$+,-.( .#*(+ %$( B*&C8 7$& ,) .#*P,(+* %$( 9#8@6$+,-.
%-6& )#,.7-..6* 7*.+&$)* 96 %&-?)'C* @6* 7-.(+,+6* a1 =#$(+67* 9* )$ +N8-&,* 9* R*+S *(+
9* 7N-,(,& 6.* :$)*6& 9* a @6, C$P,C,(* CP T +-6+* $6+&* :$)*6& 9* a .* %*6+ @6* 7-.96,&*
U 6.* %6,(($.7* %*&96*1 "* 7*++* B$V-. )* B-.7+,-..*C*.+ 9#6.* C$7N,.* ,98$)*5 71WUW91
7*))* @6, %-6&&$,+ &876%8&*& )#,.+8J&$),+8 9* )#8.*&J,*5 &*(+* 98X.,1 Q) *P,(+* 9*6P %-,.+( %-6&
)*(@6*)(
dCP
da= 05 U ($:-,& a = 1
3*+ a = 11 Y* 9*&.,*& %-,.+ .#$ %$( 9* (*.( %NO(,@6* 7$&
,) ,C%),@6* V = 0 *+ Vw = −V∞1 =$ :$)*6& C$P,C$)* 9* )$ %6,(($.7* *(+ 9-.7 -?+*.6*
%-6& 6. B$7+*6& 9#,.+*&B8&*.7* 9*
1
35 7* @6, 7-.96,+ U 6. &$%%-&+ 9* :,+*((*( V = 2V∞
39#-Z
Vw = V∞31 =$ %6,(($.7* C$P,C$)* *(+ ;
Pmax =16
27
(
1
2ρ S V 3
∞
)
/01G 3
Y*++* &*)$+,-. 7-.(+,+6* )$ ),C,+* 9* R*+S1 E))* C-.+&* @6* )#8.*&J,* C$P,C$)* @6* )#-.
%*6+ &876%8&*& .* %*6+ 98%$((*&
16
279* )#8.*&J,* @6, +&$:*&(*&$,+ )* &-+-& *. (-. $?(*.7*1
Annexe H
λ = 1.0
λ = 1.0
!" !"! λ = 1.0
# $%&'() *+,-.%) (. /&%0&' 1))2) (. *3&%4%.5
! θ ! "# 6
# 7.,8')9%2,918' (2 )1**&:. &2 ;<=
"
> ?8%0@ A&% B )9%2,92%.) &**8':@.) ,8'9%&%89&C
91D.)5
# E.) )9%2,92%.) )3&01',1).'9 .9 )8'9 A%8F.9@.) )2% *. ,.%,*. (. %89&918'5
"# ! θ ! $$ 6
# E. )1**&:. ,800.',. G )3@A&1))1%5
# E3@A&1))1)).0.'9 .)9 A*2) D1)14*. (2 ,89@ 1'9%&(8)5
# E. )1**&:. )3@9&*. D.%) *. ;<H
#
5
θ ≈ $$ 6
# E. )1**&:. G *31'9%&(8) &99.1'9 *. ;<H5
# E. )1**&:. ,800.',. G ). 9%&')?8%0.% A82% (.D.'1% 2' D8%9.I J K L21 982%'. (&')
*. ).') &'91-8%&1%.5
$$ ! θ ! %"" 6
# E. !"#$%& :%&'(19 (. A*2) .' A*2) .9 ,800.',. G ). (@9&,-.% A.2 G A.2 (2 ;<H
Jθ ≈ 72!K5# E. (@9&,-.0.'9 :@'M%. 2' &29%. !"#$%& ' ,8'9%&%89&91? J).') -8%&1%.K &2 ;<H
,89@ 1'9%&(8)5 N* )3@9&*. D.%) *. ;<= .9 :%8))195
θ ≈ %"" 6
# E. !"#$%& ). (@9&,-. ,80A*M9.0.'9 (2 ;<H (. *& A&*.5
# E. !"#$%& ' )3@9&*. )2% 9829 *31'9%&(8)5
θ ≈ %"$ 6
# O'. '82D.**. )9%2,92%. J!"#$%& ( K 982%'&'9 (&') *. ).') &'91-8%&1%. ). :@'M%. &2
;<H ,89@ .I9%&(8)5
# O'. &29%. )9%2,92%. J!"#$%& )K 982%'&'9 (&') *. ).') -8%&1%. .)9 :@'@%@. &2 ;<=
,89@ 1'9%&(8)5
%"$ ! θ ! %$ 6
# E. !"#$%& ( )3&:%&'(19 .9 &)A1%. *. !"#$%& ' D.%) *. ;<H5
# E. !"#$%& ). (@A*&,. D.%) *3&D&* (2 ,89@ 1'9@%1.2% &2 %898%5
θ ≈ %$ 6
# E. !"#$%& ( ,800.',. G ). (@9&,-.% (2 ;<H .9 .)9 4&*&P@ D.%) *3&D&* A&% *3@C
,82*.0.'95
%$ ! θ ! %& 6
# E. !"#$%& ( ,8'91'2. G :%&'(1% .9 G ). (@9&,-.%5
# E. !"#$%& ) @D8*2. &2 ;<=5
θ ≈ %& 6
# E. !"#$%& ' ). (@9&,-. (2 !"#$%& ( A%M) (. *& 01C,8%(. (. *& A&*. ,89@ 1'9%&(8)
.9 ). ,80A&,9. D.%) *. ;<=5
!" # $%& !' "()*'
!
!+ # $%& !,+**-.('
!!"#" $% & '(") *+'(,&,( ,'") -" ./$0-'+.,"!!" .,1'" 234 "! 5,.,"6 ,!7,!, !
" #$ !"#$% & !"# $% &'"&(' $' "!#)#*!+ ,!%" ' -!*+$"' )% !"#$% &.
!" # θ # $% /
0 1' !"#$% ' '+2'(!,,' ,)"#*'(('3'+# (' !"#$% (4 $52'(!,,' !+ '6#"53*#5 )% 789
2'" (:'6#")$! '# &!33'+&' ; ' $5#)&<'".
θ ≈ $% /
0 1' !"#$% ( ' $5#)&<' $% 789.
0 1' !"#$% ' =%* ' # )% 789 $% &!#5 '6#")$! &!+#*+%' ; >#"' )##)&<5.
$% # θ # & % /
0 1' !"#$% ' &!+#*+%' ; >#"' )##)&<5 )% 789.
0 1' !"#$% ( %*# (' $5,()&'3'+# $% !"#$% ' 2'" (:)2)( ? *( !+# )$-)&'+# .
0 1' !"#$% ) '# !"#$% & &!+#*+%'+# &!((5 . 1' !"#$% & +' :' # ,) '+&!"' &!3@
,(A#'3'+# $5#)&<5.
& % # θ # &'" /
0 1' !"#$% & ' # B)()C5 $% 78D 2'" (' 789 E )+ :)##)&<'" ; () %"F)&' '6#")$! G
,!%" C "'#"!%2'" (' !"#$% ' =%* &!+#*+%' ; >#"' )##)&<5 )% 789.
0 1' !"#$% & ) %+' F!"3' $' H)33' '# '+#"I*+' (' !"#$% ( $)+ !+ '6#"53*#5.
θ ≈ &'" /
0 1) ,)(' "'+&!+#"' (' (I&<'" $' J)"3)+ E '+ )+#*<!")*"'G.
&'" # θ # &(% /
0 1' !"#$% ' ' #")+ F!"3' '+ *(()K' '# %+ (I&<'" ' # '+2!C5 $' ) #"%&#%"'
,"*+&*,)(' ; θ ≈ 266!.0 1' !"#$% & :)##)&<' ; () %"F)&' *+#")$! .
0 L'" θ ≈ 266!4 %+' +!%2'((' #"%&#%"' 2!"#*&*#)*"' E !"#$% * G )+#*<!")*"' ' K5+A"'
,"A $' () 3*@&!"$' '6#")$! 4 =%* ' # '+#!%"5' ,)" (' !"#$% &. M' !"#$% * 52!(%'
'# :5#)(' 2'" (' 789.
θ ≈ &(% /
0 1) #"%&#%"' * &!%,' (' !"#$% & '+ $'%6 ? %+' ,)"#*' "' #' )##)&<5' )% 78D '#
(:)%#"' ' # #")+ ,!"#5' '+ )2)( ,)" (:5&!%('3'+#.
&(% # θ # )&% /
0 1' !"#$% * )##'*+# (' 7894 ' F% *!++' )2'& () #"%&#%"' ' '# ' #")+ F!"3' '+
*(()K' ; ,)"#*" $' θ ≈ 306!.0 1' !"#$% & :5#)(' 2'" (' 789 &!((5 ; () %"F)&' '6#")$! .
)&% # θ # )%" /
0 1I&<'" %&&' *F $' #"%&#%"' '+* '# &.
λ = 1.0
θ = 0 θ = 42 θ = 66
θ = 72 θ = 144 θ = 146
θ = 160 θ = 180 θ = 196
θ = 216 θ = 250 θ = 266
θ = 276 θ = 306 θ = 360
λ = 1.0
λ = 2.0
λ = 2.0
λ = 1.0
θ = 0
θ = 180θ θ
θ
θ
λ = 2.0
θ ≈
θ
θ ≈
θ
θ = 0 θ = 58 θ = 130
θ = 188 θ = 200 θ = 224
θ = 298 θ = 320 θ = 360
λ = 2.0
Annexe I
α = 10
λ = 1.5
λ = 1.5
!" !"! λ = 1.5
!"!#$% #
$ %&'()&('*+ ,-'&.).&/.'*+ /++*0 1'/23*+4 3* 5/ &/.55* 3* 5/ 6/5* 7-( 65(+ *2 /,/5 3(
)*')5* 3* '-&/&.-284 9(. -2& :&: 5;)<:*+ /( )-('+ 3* 5/ '-&/&.-2=
$ >'/23+ 5;)<*'+ 3* ?/'@/2 .++(+ 3* 5A/'B'*=
$ C2&*'/)&.-2 *2&'* 5*+ +&'()&('*+ .++(*+ 3*+ 6/5*+ *& 3*+ )/':2/1*+=
$ D*+ +&'()&('*+ 3( )/':2/1* +(6:'.*(' &-('2*2& 3/2+ 5* +*2+ .2,*'+* E )*55*+ 3(
)/':2/1* .2F:'.*('=
0◦ < θ < 54
◦#
$ G*)-2+&'()&.-2 3( +.55/1* E 6/'&.' 3( HIJ
"
=
&' ( θ ( )' #$ D* +.55/1* +A:6/.++.& 3( )-&: .2&'/3-+ 3* 5/ 6/5*=
$ D;)<*'+ :,*2&(*5+ 3* 6*&.&*+ +&'()&('*+ 3( B-(& 3( +.55/1*=
θ ≈ )' #
$ K2 ,-'&*L 7 8 )-@@*2)* E +* 1:2:'*' /( HIM
#
4 &-('2/2& 3/2+ 5* +*2+ /2&.<-'/.'*=
N/' '/66-'& E 5A<O3'-5.*22* 5.B'* *2 @.5.*( .2P2. 7λ = 1.084 )* !"#$%& /66/'/Q&
/,*) (2 '*&/'3 3A*2,.'-2 RS!=
)' ( θ ( *++ #
$ D* !"#$%& +* 3:,*5-66* &'T+ '/6.3*@*2& *& )-@@*2)* E +* 3:&/)<*' 6*( E 6*(
3( HIM 7θ ≈ 100!8=$ D* 3:&/)<*@*2& 1:2T'* (2 /(&'* !"#$%& ' )-2&'/'-&/&.F 7+*2+ <-'/.'*8 /( HIM
)-&: .2&'/3-+= C5 +A:&/5* ,*'+ 5* HIJ *& 1'-++.&=
θ ≈ *++ #
$ D* !"#$%& +* 3:&/)<* )-@65T&*@*2& 3( HIM *& '*+&* /&&'/6: *2&'* 5*+ 3*(L
B-(&+ 3( !"#$%& ' +-'&/2& 3( HIM *& 3( HIJ4 9(. 6'*23 &-(& 5A.2&'/3-+ 3* 5/
6/5*=
*++ ( θ ( *), #
$ D* !"#$%& +(.& 5/ &'/U*)&-.'* 3* 5/ 6/5* 3( )-&: .2&:'.*(' /( )*')5* 3* '-&/&.-2=
$ D* B-(& 3( !"#$%& ' +-'&/2& 3( HIM +* 3.++.6* 6*( E 6*( U(+9(AE +/ 3.+6/'.&.-2=
θ ≈ *), #
$ K2* 2-(,*55* +&'()&('* 7!"#$%& ( 8 +* 1:2T'* /( HIM )-&: *L&'/3-+=
*), ( θ ( *-' #
$ D* !"#$%& ( )-@@*2)* E +* 3:&/)<*' 3( HIM *& 1'/23.&=
θ ≈ *-' #
$ K2 2-(,*/( ,-'&*L 7)8 +* F-'@* /( HIJ )-&: *L&'/3-+=
*-' ( θ ( .*. #
$ D* !"#$%& ' +* 3.++.6* U(+9(AE 3.+6/'/Q&'*=
$ D*+ !"#$%& ( *& !"#$%& ) 1'/23.++*2& *& )* 3*'2.*' )-@@*2)* E +* 3:&/)<*' 3(
HIJ=
!" # $%& !' "()*'
!
!+ # $%& !,+**-.('
!!"#" $% & '(") *+'($&$( $'") ," -./0,'+-$"!!" & '1!1" "! 2$-$"3 $!4$!$ !!
" #$ !"#$% & $%%&'$ ($ %)*+'* (, -$*-.$ ($ *)+&+')/ 0&'% .$ 1&%%&2$ ($ .& 1&.$
.3$014-5$6
θ ≈ ! 7
" #$ !"#$% & $+ .$ !"#$% ' %$ 8,%')//$/+ &, 9:;6
! " θ " #$ 7
" #$ !"#$% &(' $+ ) %)/+ &%1'*<% .& %+*,-+,*$ '%%,$ (, -&*</&2$ '/8<*'$,*6
θ ≈ #$ 7
" #$ !"#$% &(' $+ ) %$ (<+&-5$/+ -)01.=+$0$/+ ($ .& 1&.$6
" #& 1&.$ 1$*-,+$ .3,/ ($% .>-5$*% ($ ?&*0&/ '%%, ($ .3&*@*$6
#$ " θ " %& 7
" #$ %'..&2$ -)00$/-$ A %$ 8)*0$* A 1&*+'* (, 9:;6
" #$ .>-5$* ($ ?&*0&/ %$ 0<.&/2$ &, %'..&2$ (, -)+< $B+*&()% ($ .& 1&.$66
θ ≈ %& 7
" C/ D)*+$B E* F %$ 8)*0$ A 0'G$/D$*2,*$ (, -)+< $B+*&()% ($ .& 1&.$6
%& " θ " '#$ 7
" #$ !"#$% * 1*)D)H,$ ($% .>-5$*% %,--$%%'8% ($ .,' 040$ $+ (, %'..&2$6
" I,B 1*)B'0'+<% (, -&*</&2$ %,1<*'$,*J .$% .>-5$*% '%%,% (, !"#$% * '/+$*&2'%%$/+
&D$- .$% %+*,-+,*$% (, -&*</&2$6
!" !"! λ = 1.5
!" θ = 0 #" θ = 54
$" θ = 84 %" θ = 166
&" θ = 180 '" θ = 194
(" θ = 212 )" θ = 260
*" θ = 294 +" θ = 360
!"# #$%& !"#$% &$ '("#)*)#+ ),%#!,#!,,+$%- λ = 1.5-
λ = 2.0
λ = 2.0
λ = 1.5
θ
θ ≈
θ
θ ≈
!" !"! λ = 2.0
!" # θ # $% #
$ %&' !"#$% & &( &' )*+,-&.(/
θ ≈ $% #
$ 0. 1+23&4- 5- *+2(&6 7&' 8 '& 5)(439& 3+1:,;(&1&.( 5- <=>/
$% # θ # " ! #
$ %4 '(2-3(-2& 5)(439)& & '& 1),4.?& @ ,4 '(2-3(-2& &' (4.5A' B-& ,& 2&'(& 5-
!"#$% & B-A 2&'(& 3+,,) 4- <=> '& 5):,43& ,& ,+.? 5& ,C&6(245+' D-'B-C4- <=E/
θ ≈ " ! #
$ %4 '(2-3(-2& 7&'(&8 '& 5)(439& 5- <=E &( &'( 45*&3()& &. 4*4, :42 ,C)3+-,&1&.(/
" ! # θ # "%" #
$ %4 '(2-3(-2& 7&'(&8 &'( 4':A2)& :42 ,4 '(2-3(-2& 5- 342).4?& A.F)2A&-2/
θ ≈ "%" #
$ %& 7 !"#$% &8 '& (24.'F+21& &. 'A,,4?& 4- <=E/
"%" # θ # "!& #
$ %& 'A,,4?& '& 5)*&,+::& 4*&3 ,4 2+(4(A+. 5& ,4 :4,&/ =&' '(2-3(-2&' 3+.(242+(4(A*&'
'+.( )*&.(-&,,&1&.( ,439)&'/
$ %4 :4,& :&23-(& ,&' ,439G2' 5& H4214./
!!"#" $% & '(") *+'($&$( $'") ," -./0,'+-$"!!" & '1!1" "! 2$-$"3 $!4$!$ !
!" θ = 0 #" θ = 90
$" θ = 154 %" θ = 162
&" θ = 174 '" θ = 216
(" θ = 242 )" θ = 360
!"# "#$% !"#$% &$ '("#)*)#+ ),%#!,#!,,+$%- λ = 2.0-
λ = 3.0
λ = 3.0
λ = 3.0 θ =
λ = 3.0
Annexe J
!" !"! #$%&'( )##()$*+,-
!!" #$%&'
x #$%&'()* y #$%&'()* x +,%&'()* y +,%&'()*
-...... .-...... .-..../. .-.. 0!.
.-1123 . .-.. 2". .-...//. 4.-.. .5.
.-1!21 . .-..250. .-.. .0. 4.-../12.
.-15055. .-. "! . .-.. 23. 4.-..35 .
.-13"!1. .-.0"35. .-..0"0. 4.-..5/ .
.-1/015. .-./"!0. .-..0!5. 4.-..!./.
.-1.2"/. .-."3/5. .-..//3. 4.-..!5 .
.-!53/1. .-.3232. .-../!1. 4.-..1/ .
.-!"./!. .-.2!".. .-.."30. 4.-..1!".
.-!. 1". .-.!.5!. .-..30". 4.-. ./".
.-52.22. .-.1/"1. .-..2.". 4.-. .!0.
.-5 5 0. .- .2/.. .-..5!2. 4.-. 21.
.-25 1/. .- !1". .-..11". 4.-. 0"1.
.-20321. .- / .!. .-. /3". 4.-. /3".
.-35!12. .- "0/!. .-.03 2. 4.-. 3"3.
.-3/001. .- 303 . .-.""11. 4.-. 312.
.-"!2 1. .- 2 .. .-.5.0". 4.-. " .
.-"" /. .- 2551. .- ..13. 4.-..1!0.
.-/153". .- 50 /. .- /503. 4.-../ 5.
.-/33"1. .- 5/2.. .- 51/!. .-..323.
.-/ "51. .- 50 0. .-00521. .-. 31/.
.-053"1. .- 2510. .-0! 11. .-.02/5.
.-0/550. .- 2 05. .-/" "!. .-./321.
.-0. 5 . .- 303/. .-"."2!. .-."05 .
.- 25!". .- "0. . .-"213!. .-."5.0.
.- /2"0. .- 011!. .-3/"22. .-."1. .
.- .552. .- 221. .-31!1 . .-."!1 .
.-.!0 0. .- .0".. .-22 !. .-."25/.
.-.315 . .-.!5/1. .-50.00. .-."0!0.
.-.".5 . .-.5 12. .-553 . .-./550.
.-.0305. .-.32"/. .-!03 5. .-./ 1..
.-. /"3. .-." 2. .-!21!0. .-.035..
.-..3/ . .-.023". .-1.!3.. .-. 13..
.-...!2. .-. /.3. .-1".!0. .-. /2".
.-..../. .-.. 0!. .-122"0. .-..!"0.
.-..../. .-.. 0!. .-1!"11. .-.." .
.-...//. 4.-.. .5. .-1120". .-.. 0.
.-.. .0. 4.-../12. -...... .-......
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?+*@!+0 A#B%&)+';)+!# C;);D;&%EF
!" !"! #$%&'( )*""+,$-(
!""#$%&'
x !"#$%&' y !"#$%&' x ()"#$%&' y ()"#$%&'
*+,,,,,, ,+,,,,,, -,+,,,,,. ,+,,,/,0
,+1102.3 ,+,,2*45 ,+,,,*5, -,+,,0,*2
,+1.5304 ,+,*343* ,+,,*,5, -,+,,/203
,+140523 ,+,*1121 ,+,,0,10 -,+,,1215
,+1/,/3* ,+,3/4*. ,+,,/34* -,+,**410
,+152*40 ,+,0050* ,+,*,*14 -,+,*03,1
,+13//50 ,+,5,53/ ,+,*54,/ -,+,*5*0,
,+1,5.., ,+,54401 ,+,*1.0/ -,+,*540.
,+..,*/4 ,+,22351 ,+,3243. -,+,*2*3.
,+.20*2* ,+,/34.* ,+,03/,0 -,+,*205,
,+.352.1 ,+,4,*4, ,+,5,4.5 -,+,*20/0
,+412*0, ,+,443.2 ,+,2,452 -,+,*2*24
,+4/2301 ,+,.5,0, ,+,/0*/, -,+,*5/5.
,+4023,5 ,+,1,031 ,+,4..13 -,+,*043/
,+4,2*5. ,+,1/*0* ,+,1./.* -,+,*33/0
,+/42,4, ,+*,*5*, ,+*3353* -,+,*,*31
,+/55... ,+*,/*/. ,+*51,0/ -,+,,43,2
,+/*55/4 ,+**,531 ,+*44/*3 -,+,,0533
,+2.0/2/ ,+**535* ,+3,41** ,+,,**53
,+2230,5 ,+**4/43 ,+3014,4 ,+,,/32*
,+23,0*5 ,+*3,.** ,+3432,, ,+,**//2
,+5.4/14 ,+*3042, ,+0,2423 ,+,*43,2
,+525/,2 ,+*3/2/2 ,+00.10, ,+,334,5
,+53*004 ,+*3131, ,+04*43/ ,+,3.,0,
,+0..032 ,+*0*.11 ,+5,5,0. ,+,00,.1
,+02/,21 ,+*0531/ ,+502.54 ,+,0441.
,+032,/* ,+*0/001 ,+5/4*/0 ,+,53,.1
,+312./1 ,+*04.5. ,+51.,0* ,+,521,*
,+3/..3* ,+*0./,4 ,+23.515 ,+,51*.0
,+350.*/ ,+*0.5*. ,+22./*/ ,+,2*.1.
,+33,510 ,+*04*/. ,+2..550 ,+,25,**
,+*1.2*4 ,+*05.*. ,+/*.,34 ,+,22511
,+*44//* ,+*0*020 ,+/545,3 ,+,2/001
,+*24443 ,+*3/443 ,+/4//*/ ,+,2/2*/
,+*0.445 ,+*3*,15 ,+4,2/.4 ,+,2/,*.
,+*3,/2. ,+**50/5 ,+405/0/ ,+,25.01
,+*,054. ,+*,//25 ,+4/050. ,+,2314/
,+,.4054 ,+,1.*,/ ,+413,51 ,+,2,503
,+,435/, ,+,..1// ,+.3,05, ,+,543*.
,+,21,/2 ,+,4121/ ,+.5.,42 ,+,5004,
,+,5404/ ,+,4,01. ,+.45.02 ,+,0.15/
,+,045/1 ,+,/*4,3 ,+1,,,** ,+,05,/,
,+,31321 ,+,20//* ,+1331*. ,+,3..2/
,+,3325. ,+,5/3.* ,+150,/5 ,+,305/0
,+,*4*,0 ,+,012,/ ,+1/,023 ,+,*4125
,+,*34,* ,+,00344 ,+142,,* ,+,*3045
,+,,1*20 ,+,3420. ,+1.405* ,+,,/.,3
,+,,/0*5 ,+,33302 *+,,,,,, ,+,,,,,,
,+,,5,4/ ,+,*40*2 6 6
,+,,30/2 ,+,*3434 6 6
,+,,**3. ,+,,.503 6 6
,+,,,00. ,+,,5014 6 6
-,+,,,,,. ,+,,,/,0 6 6
!"# #$%& !!"!##$%& '$!($)*+,-%& "- .*!/0 123345678 9 :!*"% -#+);+*% <1!-*:% =
>+*?!+0 @#A%&)+';)+!# B;);C;&%DE
Annexe K
Annexe L !"#!$%&"&'% (& )*+,($!)-&''& ./$0'0&1
2034)%/%3 .!"#)0"&'%/-$&3
!""! #$$!%! &'()'*"! +!, *-,.+"#", &/"-, &/01!,,'., 2
'()#*#/,'$ 1!, !3'*", $.(-*/4.! !" !%)-*/(!$"#+ 1#$, +! 1'(#/$! ".$$!+5
'()#*#/,'$ 1!, 1/,"*/6."/'$, &#*"-,/!$$!, $.(-*/4.!, 1. &'!7&/!$" 1! &'.)+! )'.*
.$! )#+! !" )'.* +89:1*'+/!$$! !$ (/+/!. &'$;$- !" /$;$/5
'()#*#/,'$ 1!, 1/,"*/6."/'$, &#*"-,/!$$!, $.(-*/4.!, 1!, !3'*", ,.* +89:1*'+/!$$!
!" ,.* +!, )*';+, 1. &#*-$#<! !$ (/+/!. &'$;$- !" /$;$/5
=>?
FX
FX
FY
FY
!!"#" $% &'()'*+"("!+ ," $-./,*'$0"!!" & *1!1"% *123$+ +2 &'()$1("!+ 0*"2 !
"#$%&'()**) '(+%) ,('()- .&*/*0 1 2340%(,)*56'
λ = 1.0 λ = 2.0 λ = 3.0
FX [N ] FY [N ] FX [N ] FY [N ] FX [N ] FY [N ]
,63 78 9 :;8< :!8<= !;899 9<8= 7!8!9
,(* ;:8: >9 8<: ?;8!< >;:8; ?;8 : >:!89?
,&#)**) @;8 ! >!8@: !;8?? ;98;9 7 8? :78?9
"#$%&'()**) '(+%) ,('()- .&*/*0 1 A-,0%(B-)
λ = 1.0 λ = 2.0 λ = 3.0
FX [N ] FY [N ] FX [N ] FY [N ] FX [N ] FY [N ]
,63 ;789: =8;7 978=! @78@9 <?899 778=;
,(* : 8! ><@8== 9<899 >;@8<7 9<89 >:?89<
,&#)**) @!87< >;?8 9 7=8 7 =8<= ;?897 ;!8@<
"#$%&'()**) .6%0*0) ,('()- .&*/*0 1 2340%(,)*56'
λ = 1.0 λ = 2.0 λ = 3.0
FX [N ] FY [N ] FX [N ] FY [N ] FX [N ] FY [N ]
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,(* =<89< >!;8 : 7;8@= >?=8? ?=8=< >:?8@7
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λ = 1.0 λ = 2.0 λ = 3.0 λ = 4.0
FX [N ] FY [N ] FX [N ] FY [N ] FX [N ] FY [N ] FX [N ] FY [N ]
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FY D#D58 .6%0*6F) [N ] 98 = 98<? >?=87? ><;8==
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