modélisation macroscopique des inondations fluviales et urbaines
TRANSCRIPT
HAL Id: tel-00490694https://tel.archives-ouvertes.fr/tel-00490694
Submitted on 9 Jun 2010
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Modélisation macroscopique des inondations fluviales eturbaines - Prise en compte des écoulements
directionnels et des échanges lit majeur - lit mineurPascal Finaud-Guyot
To cite this version:Pascal Finaud-Guyot. Modélisation macroscopique des inondations fluviales et urbaines - Prise encompte des écoulements directionnels et des échanges lit majeur - lit mineur. Dynamique des Fluides[physics.flu-dyn]. Université Montpellier II - Sciences et Techniques du Languedoc, 2009. Français.<tel-00490694>
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I>678<C D697 BAC E4C C<8D:AC A;\69 A= 54<C4=; 4DDA: ] BAC >FD6;>^CAC C<8D:<Q
_E4;7<EACON <: AC; =?EACC4<7A BS4DD76E>A7 EAC C6:9;<6=C D47 :A H<4<C BA 8?;>6BAC
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C<A97C BA EAC 69;<:C C6=; D7?CA=;?C B4=C EA E>4D<;7A+
%:9C<A97C E7<;^7AC DA78A;;A=; BA E>6<C<7 A=;7A :AC B<[?7A=;C 69;<:C BA 86B?:<C4Q
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76H9C;ACCA BAC 8?;>6BAC =98?7<@9AC A8D:6F?AC+++ 0S4=4:FCA BAC ;FDAC BA E6BAC
BA E4:E9: 9;<:<C?C A= 56=E;<6= BAC 6HXAE;<5C G<C?C 4 ?V4:A8A=; ?;? D7?CA=;?A D47
J`!M+ 0AC E6BAC BA E4:E9: ] 9C4VA E688A7E<4: D7<G<:?V<A=; V?=?74:A8A=; :4 74D<Q
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6HXAE;<5C BA EA;;A ;>^CA ?;4=; BA D76D6CA7 9= :6V<E<A: BA 86B?:<C4;<6= BAC ?E69:AQ
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4GAE :AC :6V<E<A:C BA E4:E9:+
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∂ (ρhb)
∂t+∂ (ρhbV )
∂x= 0 M ! >N
∂ (ρhbV )
∂t+∂(
β′
ρhbV 2)
∂x= −
∂
(
1
2ρgbh2
)
∂x− ρgbh∂zb
∂xM ! BN
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"+, =A0>62.1, ,.16 =6>B42+, ,0< 4> B>,+ /+, 89:.68;,+, ,02?>16+, R
V K9:! *!* R 4> 3>,,+ ?.4032A0+ /+ 47+>0 +,6 -.1,6>16+ M47+>0 +,6 -.1,2/=<=+
-.33+ 01 W02/+ 8.3.@;1+ +6 21-.3:<+,,2B4+ND
V K9:! *! R 4> :+16+ /0 5.1/ +,6 5>2B4+D
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∂S
∂t+∂Q
∂x= ql M !X>N
∂Q
∂t+
∂
(
β′Q2
S
)
∂x= −gS ∂z
∂x− g Q |Q|
C2SRhM !XBN
.S S <+:<=,+16+ 4> ,+-62.1 +1 6<>?+<, /+ 47=-.04+3+16D Q 4+ /=B26D ql 4+ /=B26
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S
b
h
x
V
zb
∂Q
∂x≈Qn+1
i+1 +Qni+1
2−Qn+1
i−1 +Qni−1
2∆2xi
∂S
∂t= b
∂h
∂t≈ Ao,i +Ao,i+1
∆2xj
hn+1i − hn
i
∆t
Qni hn
i
i n ∆2xi
i− 1 i+ 1 i
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i
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h n
iQ
n
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n
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t
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tn+1
tn
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S
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n
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∂ Q
∂ t∂
∂ x ( )β'Q2
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i + 2MN ∆t >@ A:D ;@ E@9AD ;@ ;=D6C8E=D:E=BF @E A0,i >: D<CO:6@ @F A>:F @FEC@ >@D
AB=FED ;@ 6:>6<> i − 1 @E i+ 0: 98E7B;@ ;@ 6:>6<> @9A>BP8@ AB<C 6:>6<>@C 6@EE@
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∂Q
∂t≈Qn+1
i+1 −Qni+1
∆tT*+Z:M
∂
(
β′Q2
S
)
∂x≈
[
β′Q2
S
]n+1/2
i+2
−[
β′Q2
S
]n+1/2
i
∆2xiT*+ZVM
∂z
∂x≈zn+1i+2 + zn
i+2
2− zn+1
i + zni
2∆2xi
T*+Z6M
.:FD 6@EE@ ;=D6C8E=D:E=BFN >@D AB=FED ;@ 6:>6<> i @E i + 2 DBFE ;@D AB=FED ;@
6:>6<> ;@ >: 7:<E@<C ;G@:< @E >@ ;8V=E FGP @DE A:C 6BFD8H<@FE A:D @QA>=6=E@9@FE
6BFF<+ 0@ >B?=6=@> 6:>6<>@ >@ ;8V=E ;@ >: 9:F=WC@ D<=J:FE@ Y
Q2 ≈ ΘQn+1i+1 Q
ni+1 − (1−Θ)Qn
i+1Qni+1 T*+[M
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∂Q
∂x^
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∂Q
∂xA
∂S
∂t! (- 6;8,-B-3.;/ 54.
6020B3<0. 59;/54. 3..?4. 54 2- H;B8?2-<3;/ 0C?3D-24/<4 E6;BB4.,;/5-/< -?I 0C?-=
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,;?B 06B3B4 24 .L.<O84 0C?3D-24/< A 642?3 53.6B0<3.0 P
∂S
∂t+∂Q
∂x= 0
∂Q
∂t+Q2
∂
(
β′
S
)
∂x= −gS ∂z
∂x
E !QG
"4<<4 H;B8?2-<3;/ ,4B84< 54 6-26?24B 2- 8-<B364 R-6;:34//4 A 4< 2- 6020B3<0 54.
;/54. P
A =
[
0 1
c2 − β′u2 0
]
E !SG
λ− = −√
c2 − β′u2λ+ = +
√
c2 − β′u2 E !TG
;M λ− 4< λ+ B4,B0.4/<4/< 24. 6020B3<0. 59;/54.@ u 2- D3<4..4 8;L4//4 54 2906;?=
2484/< 4< c 2- 6020B3<0 54. ;/54. 54 ,B4..3;/ Ec =√ghG! (4. 0C?-<3;/. E !TG
53UOB4/< 54. 6020B3<0. 59;/54. <70;B3C?4. P
λ− = β′
u−√
c2 − u2 (β′ − β′2)λ+ = β
′
u+√
c2 − u2 (β′ − β′2) E !*VG
(4 <B-60 54. 6020B3<0. 59;/54. E !TG 4< E !*VG 53D3.04. ,-B 2- 6020B3<0 54. ;/54.
54 ,B4..3;/ 4.< ,B0.4/<0 A 2- W1?B4 !+ E,;?B β′
= 1G! (97L,;<7O.4 XL,! *!Y
59?/ 06;?2484/< ?/3C?484/< Z?D3-2 ,4B84< 54 1-B-/<3B C?4 24. 6020B3<0. 59;/54.
B4.<4/< 50W/34. ,?3.C?4 2- 6020B3<0 54. ;/54. 54 ,B4..3;/ B4.<4 .?,0B34?B4 A 2-
D3<4..4 54 2906;?2484/<! $-/. 24 6-. 59?/ B01384 <;BB4/<342@ 2- D3<4..4 54 ,B;,-=
1-<3;/ 54. ;/54. /94.< ,2?. 6-26?2-:24 A ,-B<3B 54 2- H;B8?2-<3;/ E !TG ,?3.C?4
c2 − β′u2 < 0! %/ B01384 Z?D3-2@ 24. ;/54. 6-26?204. ,-B 24 2;136342 .4 ,B;,-14/<
5-/. ?/4 53B46<3;/ ;,,;.04 8-3. A 2- 8N84 D3<4..4! %/ B0-23<0@ 64 ,70/;8O/4
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06;?2484/<. Z?D3-?I ;M 2- D3<4..4 5906;?2484/< 4.< .4/.3:2484/< ,B;674 54 2-
6020B3<0 54. ;/54. 54 ,B4..3;/@ 29-,,B;I38-<3;/ B0-23.04 4/<B-[/4 ?/4 4.<38-<3;/
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-2
-1
0
1
2
-1.5 -1 -0.5 0 0.5 1 1.5
Nombre de Froude
λ / c
!" #$%&'(!)*$+ ,$'% (- %./*&- 0'1*!(
-1.5 -1 -0.5 0 0.5 1 1.5
-2
-1
0
1
2
Nombre de Froude
λ /
c
2" #$%&'(!)*$+ ,!% 3.4!') ,$'% (- %./*&- )$%5
%-+)*-(
-1.5 -1 -0.5 0 0.5 1 1.5
-2
-1
0
1
2
Nombre de Froude
λ /
c
!" #$%&'()*+$, )(*-%,)*+.- /$'% (- %01+&- *$%2
%-,*+-(
λ- analytique
λ+ analytique
λ- Mike 11
λ+ Mike 11
!"#$% !" # $%&'()*&+ ,- '. /0'01*)0 ,-2 &+,-2 -+ 3&+/)*&+ ,( +&451- ,-
61&(,- 7 /&48.1.*2&+ ,( /&48&1)-4-+) )90&1*:(- -) ,- /-'(* ,- ;*<- ==
! ! "#$%& $% "'(")( *$ *+
! !"!# $%&'()*)+( ,)- .&- )+ %/0'*) (1%%)+(')2
$, -./0 123 45,.0/673 8673/149423: ;2 ;6</8/2; 72 =2,0 09./029 ;23 8.3 27 94</>2
89/0/5,2 20 0699270/2; 12 ;. >?>2 >.7/@92 5,2 82,A 27 94</>2 B,C/.;! &/ ;D486,E
;2>270 >614;/34 =2,0 =.3329 27 94</>2 0699270/2;: ;D,0/;/3.02,9 16/0 .80/C29 ,72
6=0/67 =6,9 =29>20092 ., 8.;8,; 12 32 -./92! F,.71 ;D6=0/67 230 .80/C42: ;D45,.E
0/67 12 867329C.0/67 12 ;. 5,.70/04 12 >6,C2>270 230 >61/G42 =6,9 =6,C6/9
09./029 ;D2732>H;2 123 8.3 9278670943 I
∂Q
∂t+ fQ2
∂
(
β′
S
)
∂x= −gS ∂z
∂x− g Q |Q|
C2SRhJ !**K
6L ;2 029>2 f 92=9432702 ,72 -6780/67 14=271.70 1, 76>H92 12 M96,12 Fr! (.84;49/04 123 67123 3D489/0 .;693 J=6,9 β
′
= 1K I
λ = ±√
c2 − fu2 J !* K
(. -6780/67 f =2,0 ?092 14G7/2 12 12,A >.7/@923 1/N4927023 32;67 ;2 8O6/A 12
;D,0/;/3.02,9 JC6/9 P.H;2., !*K I
31%*42&('1+ 5&% ,/6&4( 31%*42&('1+ &2()%+&('7)
f =
1− Fr2 pourFr ≤ 10 pourFr > 1
f =
1 pourFr ≤ a1
(Fr + 1− a)bpourFr > a
=.9 14-.,0 I a = 1 20 b = 2
!"#$!% !* Q $4G7/0/67 12 ;. -6780/67 f (Fr) =6,9 R/S2 **
F,2;;2 5,2 36/0 ;. -69>,;.0/67 8O6/3/2: ;. -6780/67 f 230 8670/7,2 3,9 ;D27E
32>H;2 1, 16>./72 12 14G7/0/67 1, 76>H92 12 M96,12 =6,9 2>=?8O29 06,02
C.9/.0/67 H9,0.;2 1, 86>=6902>270 1, 38O4>. 7,>49/5,2!
31%*42&('1+ 5&% ,/6&4(! "2002 -69>,;.0/67 230 T ;D69/</72 1D,7 86>=6902E
>270 OU19.,;/5,2 299674! %7 94</>2 0699270/2; JFr > 1K: ;23 84;49/043 1D6712
8.;8,;423 =.9 R/S2 ** 12C/277270 λ = ±c! (23 67123 8.;8,;423 32 =96=.<270 1678T ;. >?>2 C/02332 1.73 ;23 12,A 1/9280/673 6==63423 86709./92>270 ., 86>=6902E
>270 OU19.,;/5,2 0O469/5,2 6L ;D2732>H;2 123 67123 32 =96=.<270 C293 ;D.C.;! %7
94</>2 B,C/.;: 67 >67092 5,2 ;23 84;49/043 8.;8,;423 =.9 ;2 ;6</8/2; 3670 >.A/>.;23
=6,9 Fr = 0 20 =6,9 Fr =√2/2 V 82;. />=;/5,2 5,2 ;23 67123 32 =96=.<270 27092 1
20
√
3/4 -6/3 ;. C/02332 12 =96=.<.0/67 123 67123 12 =9233/673 c! (2 86>=6902>270
1D,7 =6/70 89/0/5,2 JFr = 1K: =6,9 ;25,2; ,72 123 12,A 67123 92302 />>6H/;2:
230 />=633/H;2 T 3/>,;29 1.73 ;. >23,92 6L ;. 84;49/04 123 67123 8.;8,;423 =.9 ;2
;6</8/2; 72 =2,0 W.>./3 ?092 7,;;2!
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!"#$%&'(!) &%'*")&'(+*, -45 6789:;<=>74 <;=584<=>?5 @5895= A ;B:=>;>C<=5:8
D5 9>5:E F74=8G;58 ;< 674F=>74 D5 =8<4C>=>74 f + 05 F<;<H5 D5C @<8<9I=85C a 5= b 5C=;<>CCJ A ;B<@@8JF><=>74 D5 ;B:=>;>C<=5:8 K =7:=567>CL ;< C=<M>;>=J D: CFNJ9< 4:9J8>O:5
>9@7C5 a < 1 5= b > 2+ )4 5P5=L @7:8 @7:?7>8 :=>;>C58 ;5 CFNJ9< DB$MM7==Q&745CF:
>; F74?>54= D5 F74C58?58 ;5 F<8<F=I85 NR@58M7;>O:5 D: CRC=I95 SF5 O:> 85?>54= A
<?7>8
λ2/c2 ≥ 0L FB5C= A D>85 1 − fFr2 ≥ 0T+ 3:5; O:5 C7>= ;5 U5: D5 @<8<9I=85C
:=>;>CJL ;5C FJ;J8>=JC DB74D5C 85C=54= 7@@7CJ5C K ;5C 74D5C C5 @87@<H54= D74F A ;<
9V95 ?>=5CC5 5= D<4C D5:E D>85F=>74C 7@@7CJ5C+ 05 F79@78=5954= NRD8<:;>O:5
C>9:;J 5C= D74F 58874JL =<4= 54 8JH>95 W:?><; O:B54 8JH>95 =78854=>5;+ 0B7M=54=>74
DB:4 @7>4= F8>=>O:5 5C= JH<;5954= >9@7CC>M;5 <?5F F5==5 6789:;<=>74+
! ! "#$%&'(
05 ;7H>F>5; #)"Q($X 5C= :4 F7D5 D5 F<;F:; :4>D>954C>7445; DJ?5;7@@J @<8
;B-X $89R "78@C 76 )4H>4558C+ )4 F79@;J954= D5C F<;F:;C D5 ;>H45 DB5<: 54
8JH>95 @589<454= 7: =8<4C>=7>85L F5 ;7H>F>5; @5895= D5 97DJ;>C58 ;5C @NJ479I45C
D5 =8<4C@78= D5 CJD>954=C 7: D5 @7;;:<4=C+ .5:E 9J=N7D5C D5 F<;F:; D>PJ854=5C
C74= :=>;>CJ5C @7:8 ;5 8JH>95 @589<454= 5= ;5 8JH>95 =8<4C>=7>85+ 05C JO:<=>74C
9>C5C 54 U5: 5= ;< 9J=N7D7;7H>5 D5 8JC7;:=>74 C74= @8JC54=J5C D<4C YZ[\+
0< 97DJ;>C<=>74 D5C JF7:;5954=C 54 8JH>95 =8<4C>=7>85 C5 M<C5 C:8 ;5C JO:<Q
=>74C D5 F74C58?<=>74 D5 ;< 9<CC5 5= D5 ;< O:<4=>=J D5 97:?5954=+ 0BJF7:;5954=
5C= CJ@<8J 54=85 ;>= 9>45:8 S;5C ?<8><M;5C C74= 47=J5C <?5F ;B>4D>F5 MC @7:8 ]^<>4
"N<445;]T 5= ;>= 9<U5:8 S;5C ?<8><M;5C C74= 47=J5C <?5F ;B>4D>F5 FP @7:8 ]_;77D
%;<>4]T K C:8 FN<O:5 C7:CQ@<8=>5 D5 ;BJF7:;5954=L ;BJO:<=>74 D5 F74C58?<=>74 D5
;< 9<CC5 5C= J=<M;>5 `
∂SMC
∂t+∂QMC
∂xMC= ql,FP S*+ a<T
∂SFP
∂t+∂QFP
∂xFP+∂St
∂t= ql,MC + ql S*+ aMT
7b ql,(FP, MC, ∅) 85@8JC54=54= 85C@5F=>?5954= ;5 DJM>= :4>=<>85 DBJFN<4H5 D5@:>C
;5 ;>= 9<U5:8 ?58C ;5 ;>= 9>45:8L D5@:>C ;5 ;>= 9>45:8 ?58C ;5 ;>= 9<U5:8 5= 54=8<4=
D<4C ;< C5F=>74+ St 5C= ;5 ?7;:95 :4>=<>85 DB5<: F74=54: D<4C ;< C5F=>74 9<>C 45
@<8=>F>@<4= @<C <:E JF7:;5954=C+ 05C JO:<=>74C D5 F74C58?<=>74 D5 ;< O:<4=>=J
D5 97:?5954= C:8 FN<O:5 C7:C @<8=>5 CBJF8>?54= `
∂QMC
∂t+∂ (VMCQMC)
∂xMC=Ml,FP −gSMC
(
∂z
∂xMC+ Sf,MC + Sh,MC
)
S*+ c<T
∂QFP
∂t+∂ (VFPQFP )
∂xFP=Ml,MC − gSFP
(
∂z
∂xFP+ Sf,FP + Sh,FP
)
S*+ cMT
7b Ml,(FP, MC) 85@8JC54=54= 85C@5F=>?5954= ;5 W:E :4>=<>85 D5 O:<4=>=J D5 97:Q
?5954= DBJFN<4H5 ;<=J8<; D5@:>C ;5 ;>= 9<U5:8 ?58C ;5 ;>= 9>45:8 5= D5@:>C ;5 ;>=
9>45:8 ?58C ;5 ;>= 9<U5:8 5= Sh,(FP, MC) ;< @54=5 D5C @58=5C D5 FN<8H5 C>4H:;>I85C
;>J5C <:E 7MC=<F;5C 85C@5F=>?5954= @7:8 ;5 ;>= 9<U5:8 5= ;5 ;>= 9>45:8+ 05C JO:<Q
=>74C S*+ cT C74= J=<M;>5C 54 6<>C<4= ;BNR@7=NIC5 O:5 ;< F7=5 D5 ;< C:86<F5 ;>M85 z
! ! "#$%& $% "'(")( *$ *+
,-. /0 121, -34 .53., /0 -,6.758 9:;653/,1,8.! (,- ;<30.758- -34 /, /7. 178,34 ,.
-34 /, /7. 10=,34 -58. ,8-37., -511;,- >534 ;.0?/74 /0 658-,[email protected] 9, /0 10--,
,. 9, /0 <308.7.; 9, 153@,1,8. -34 /:,8-,1?/, 9, /0 -,6.758 ,8 .40@,4- A
∂SEQ
∂t+∂Q
∂x+∂St
∂t
∂xFP
∂x= ql
∂xFP
∂x∂QEQ
∂t+∂ (β′uQ)
∂x= ξ′
∂ (ulql)
∂x− gS
(
∂z
∂x+ Sf,EQ + Sh,EQ
)
B !*CD
5E
SEQ = SMC∂xMC
∂x+ SFP
∂xFP
∂xB !*FD
Q = QMC +QFP B !*GD
QEQ = QMC∂xMC
∂x+QFP
∂xFP
∂xB !*HD
β′
uQ = VMCQMC + VFPQFP B !*+D
S = SMC + SFP B ! ID
Sf,EQ = Sf,MC∂xMC
∂x+ Sf,FP
∂xFP
∂xB ! *D
Sh,EQ = Sh,MC∂xMC
∂x+ Sh,FP
∂xFP
∂xB ! D
5E x 4,>4;-,8., /:0?-67--, 634@7/7J8, 651138, K /:;653/,1,8. ,8 /7. 178,34 ,.
,8 /7. 10=,34 ,. ,-. 60/63/; K >04.74 9, xMC ,. xFP ,. 5E /,- .,41,- ql,FP L
ql,MC L Ml,FP ,. Ml,MC 58. ;.; -3>>471;- 908- /0 1,-34, 5E 7/- -:0883/,8.
9,3M K 9,3M Bql,MC∂xFP
∂x= −ql,FP
∂xMC
∂x,. Ml,MC
∂xFP
∂x= −Ml,FP
∂xMC
∂xD!
)8 .,41, 9:0>>54. /0.;40/ 9, <308.7.; 9, 153@,1,8. ξ′∂ (ulql)
∂x0 ;J0/,1,8.
;.; 0=53.; >534 >4,894, ,8 651>., /,- ;6N08J,- 9, <308.7.; 9, 153@,1,8. 03
87@,03 9,- 658O3,86,- ,. 9,- 9;O3,86,-!
(, -6N;10 9, P4,7--1088 BQC*RL QCIR ,. QS+RD ,-. 3.7/7-; >534 4;-5394, 6,-
;<30.758-! ", -6N;10 71>/767., 0 ;.; 658T3 9, 1087U4, K 658-,4@,4 /, 60406.U4,
658-,[email protected] 9,- ;<30.758-! (0 4,>4;-,8.0.758 9, /:,-.710.758 9,- 9;47@;,- >04W
.7,//,- 908- /:,->06, 9,- >N0-,- ,-. >4;-,8.;, K /0 XJ34, !F! (:,M>4,--758 >534
6,- 9;47@;,- ,-. A
∂U
∂t≈ (1− ψ) U
n+1i − Un
i
∆t+ ψ
Un+1i+1 − Un
i+1
∆tB ! Y0D
∂U
∂x≈ (1− θ) U
ni+1 − Un
i
∆x+ θ
Un+1i+1 − Un+1
i
∆xB ! Y?D
(, -Z-.U1, 9:;<30.758- B !*CD ;.08. 858 /78;074,L -0 4;-5/3.758 8;6,--7.,407. 9,-
.,1>- 9, 60/63/ 4,/0.7@,1,8. 71>54.08.- ,. 9,- >45?/U1,- 9, 658@,4J,86, >534W
407,8. 0>>040[.4, ,8 60- 9, -5/3.758- 97-658.783,- B6N56L 4,--03. NZ9403/7<3,L
! "#$%&'() * +',-) .&./&01($%#&2,)
t
xi i+1
tn+1
tnU
n
iU
n
i+1
Un+1
iU
n+1
i+1
∆x
ψ∆x (1-ψ)∆x
∂U∂t i+1
∂U∂t
∂U∂t i
!" #$%&'%() *) +! *%,'$-.%,!.%/& .)0(/$)++)
t
xi i+1
tn+1
tnU
n
iU
n
i+1
Un+1
iU
n+1
i+1
∆t
θ∆t
(1-θ)∆t
∂U∂x
n+1
∂U∂x
∂U∂x
n
1" #$%&'%() *) +! *%,'$-.%,!.%/& ,(!.%!+)
!"#$% *3 4 %567869: ;< =8>?@A ;: %5:6==@A77 <B6C6=? ;A7= #)"D($E
:B8*F* /A C67?A56B? :=B GHB:7<: :7 A99C6I<A7B CA B:8>76I<: ;?J:CG99?: 9A5 %5:6==D
@A77 K5A99G5B?: 9A5 /6LL:BB :B "<7L: MNOPF :B ">:7 MQP* /: 8AC8<C ;: CA C6L7:
;R:A< =<5 <7 H6:S 5:J6:7B ACG5= T 5?=G<;5: <7 =U=BV@: ;: CA SG5@: Ax = B*
/:= 8G7;6B6G7= A<W C6@6B:= 9:5@:BB:7B ;RGHB:765 C: @X@: 7G@H5: ;R?I<AB6G7=
I<: ;: JA56AHC:=* /: =8>?@A ;: %5:6==@A77Y <B6C6=? 9G<5 CA 5?=GC<B6G7 7<@?D
56I<:= ;:= ?I<AB6G7=Y :=B 8G77< 9G<5 SG<5765 <7: =GC<B6G7 678G55:8B: ;A7= C: 8A=
;R<7 ?8G<C:@:7B B5A7=856B6I<: MNZP* ,7: J:5=6G7 9G<5 C:= 5?L6@:= ;R?8G<C:@:7B
B5A7=856B6I<:= A ?B? 95G9G=?: MO[P @A6= 7R:=B A 956G56 9A= 6@9C?@:7B?: 9<6=I<:
C: @A7<:C ;: 5?S?5:78: ;: #)"D($E 7R:7 SA6B 9A= ?BAB* /: 9A==AL: :7 5?L6@:
BG55:7B6:C 9:<B XB5: T CRG56L67: ;RG=86CCAB6G7= 7<@?56I<:= I<6 B:7;:7B T =RA@9C6D
\:5 =A7= 9G<5 A<BA7B 8A<=:5 <7 A55XB ;< 95GL5A@@:* /: @A7<:C <B6C6=AB:<5 ;<
CGL686:C #)"D($E M3QP 67;6I<: I<: =6 CR?8G<C:@:7B 9:<B ;:J:765 BG55:7B6:CY 6C
8G7J6:7B ;R<B6C6=:5 <7: @?B>G;GCGL6: ACB:57AB6J:* /A B:8>76I<: :@9CGU?: ACG5=
:=B CA ]/G8AC %A5B6AC &7:5B6A] ;?J:CG99?: 9A5 5:A; :B AC* M !P* ":BB: @?B>G;GCGD
L6: 8G7=6=B: T 5?;<65: CR67_<:78: ;:= B:5@:= 67:5B6:C= ;A7= CR?I<AB6G7 ;: I<A7B6B?
;: @G<J:@:7B `
f
∂Q
∂t+
∂
(
β′Q2
S
)
∂x
= −gS(
∂z
∂x+ Sf,x
)
K * NF
/A SG78B6G7 ;: B5A7=6B6G7 f ?BA7B ;?\76: 9A5 `
f =
FT − Frm 9G<5Fr ≤ FT
0 9G<5Fr > FT
K * ZF
Ga m :=B <7 :7B6:5 ;G7B CR<B6C6=AB:<5 9:<B @G;6\:5 CA JAC:<5 :7B5: b :B b Q KCA
JAC:<5 9A5 ;?SA<B :=B ;: b!F :B FT :=B CA JAC:<5 =:<6C 9G<5 CAI<:CC: CA B5A7=6B6G7
:7B5: C:= SG5@<CAB6G7= =R:c:8B<:* /A @G;6\8AB6G7 ;: CR?I<AB6G7 ;: 8G7=:5JAB6G7 ;:
CA I<A7B6B? ;: @G<J:@:7B =: 5?9:58<B: =<5 C: 8AC8<C ;: CA @AB568: dA8GH6:77: A
;< =U=BV@: >U9:5HGC6I<: K *bZF :B =<5 CR:=B6@AB6G7 ;: CA J6B:==: ;: 95G9ALAB6G7
! ! "#$%& $% "'(")( *$ *
-3
-2
-1
0
1
2
3
-2 -1 0 1 2
Nombre de Froude
λ/c
λ+ analytique
λ- analytique
λ+ HEC-Ras
λ- HEC-Ras
!"#$% !+ , -./0123/4 56 07 8909:329 56; /456; 64 </4823/4 51 4/=>:6 56
?:/156 @ 8/=A7:73;/4 51 8/=A/:26=642 2B9/:3C16 62 56 86013 56 D%"EF'&
56; /456; 574; 0G98/106=642! (G6HA:6;;3/4 56 A 62 56; 8909:329; 5G/456; A/1:
β′
= 1 6;2 5/4496 A7: I
A =
0 1c2
f− u2 2u
J ! KL
λ− = u− c√f
λ+ = u+c√f
J ! +L
M0 8/4.3642 56 4/26: C16 06 870810 56 07 =72:386 N78/>36446 4G7 A01; 56 ;64; 0/:;C16
f = 0 87: 06 ;O;2P=6 5G9C1723/4; 4G6;2 A01; BOA6:>/03C16 C1745 0G9C1723/4 56
8/4;6:.723/4 56 07 C1742329 56 =/1.6=642 6;2 :6=A07896 A7: 0G9C1723/4 56 0G/456
53Q1;3.6! (G9./0123/4 51 :723/ 642:6 07 8909:329 56; /456; 62 86006 56; /456; 56
A:6;;3/4 64 </4823/4 51 4/=>:6 56 ?:/156 7 929 2:7896 ;1: 07 RS1:6 !+!
$1 <732 56 07 59R4323/4 56 07 </4823/4 56 2:74;323/4 f T 07 </:=10723/4 75/AE
296 A7: D%"EF'& 6;2 356423C16 U 07 </:=10723/4 807;;3C16 A/1: 14 98/106=642
3==/>306 JFr = 0L! V01; 06 :9S3=6 5G98/106=642 ;6 :7AA:/8B6 51 :9S3=6 8:323C16
JFr = 1L 62 A01; 06 8/=A/:26=642 598:32 A7: 07 </:=10723/4 D%"EF'& ;G987:26
51 8/=A/:26=642 2B9/:3C16! %4 6Q62T 07 2:74;323/4 51 :9S3=6 W1.370 .6:; 06 :9E
S3=6 2/::642360 ;6 2:75132 64 2B9/:36 A7: 07 53=34123/4 5G146 56; 8909:329; 5G/456;
J64 .7061: 7>;/016L N1;C1GU 56.643: 41006 A/1: 14 98/106=642 8:323C16! "6 8/=E
A/:26=642 6;2 0G34.6:;6 56 86013 598:32 A7: 06 0/S38360 A13;C16 06; 561H 8909:329;
5G/456; 87081096; 2645642 .6:; 0G34R43 J64 .7061: 7>;/016L C1745 06 4/=>:6 56
!"#$%&'( ) *&+,( -%-.%/0'#$"%1+(
234567 8796 :73; <)
(9 3=>?@7 8433798?7A Bf = 0CD AE=F5G8?49 67 F5G98?8= 67 @45:7@798 67:?798
H7AA7 67 AE4967 6?I5;?:7) !7887 GJJ34K?@G8?49 7;8 JLM;?F57@798 949 N5;8?O=7) (9
7I78D AE=F5G8?49 67 AE4967 6?I5;?:7 37:?798 P 9=>A?>73 A7; 873@7; 6E?9738?7 GA43;
F5E?A; ;498 J3=64@?9G98; 79 3=>?@7 8433798?7A) $G3 G?AA753;D AE=F5G8?49 67 AE4967
6?I5;?:7 8796 P G@438?3 AE4967 67 H357 A43; 67 ;G J34JG>G8?49) !78 G@438?;;7@798
67 AG J4?987 67 H357 7;8 A7 JA5; ;45:798 379H4983= 6G9; AG JG38?7 67; H453; 6E7G5
4Q A7 HLG@J 6E7KJG9;?49 67 H357 N457 59 3RA7 J3=64@?9G98 78 4Q AE=H45A7@798 M
7;8 >=9=3GA7@798 S5:?GA) (9 37:G9HL7 P AEG@498 67; H453; 6E7G5D AE=H45A7@798 7;8
JA5; T3=F57@@798 79 3=>?@7 8433798?7A 78 A7; U497; 6E7KJG9;?49 67 H357 ;498 JA5;
TG?VA7;D 97 J73@788G98 G?9;? F5E59 TG?VA7 GVG887@798 65 J?H 67 H357) .7; =F5G8?49;
87AA7; F5EG64J8=7; 6G9; A7 A4>?H?7A "(!W'#X :498 P AE79H49837 65 H4@J4387@798
JLM;?F57 >=9=3GA7@798 7K?;8G98 79 H3=G98 59 GVG887@798 65 J?H 67 H357 A43;F57
AE=H45A7@798 7;8 8433798?7A)
! !" #$%$ &'()
.7 A4>?H?7A %;?; 2A4Y 7;8 59 A4>?H?7A 67 @46=A?;G8?49 59?6?@79;?4997AA7 67;
=H45A7@798; P ;53TGH7 A?V37 79 3=>?@7 J73@G9798 45 83G9;?84?37 6=:7A4JJ= JG3
ZGAA?9>T436 X4T8YG37) .G @=8L464A4>?7 67 HGAH5A 7;8 J3=;798=7 6G9; A7 @G957A
67 3=T=379H7 65 A4>?H?7A [\]^) .7; LG58753; 78 A7; 6=V?8; ;53 AG U497 @46=A?;=7 ;498
HGAH5A=; 79 3=;4A:G98 A7; =F5G8?49; 67 XG?98W_79G98 `
∂S
∂t+∂Q
∂x= ql B ) aGC
∂Q
∂t+
∂
∂x
(
Q2
S
)
= −gS(
∂z
∂x− Sf
)
B ) aVC
!4@@7 6G9; A7 HG; 67 b?c7 <<D A7 873@7 6EGJJ438 AG8=3GA ql 9E7;8 JG; J3?; 79H4@J87 6G9; AE=F5G8?49 67 F5G98?8= 67 @45:7@798 H7 F5? J758 d837 P AE43?>?97
6E59 H4@J4387@798 ?9H4L=3798 B:4?3 X7H8?49 ) )<C)
.G @46=A?;G8?49 67; =H45A7@798; 79 3=>?@7 83G9;?84?37 58?A?;7 A7 ;HL=@G 67
$37?;;@G99 B:4?3 X7H8?49 ) ) C J453 3=;45637 A7; =F5G8?49; 67 XG?98W_79G98)
.G @=8L464A4>?7 7@JA4M=7 9E7;8 F57 83e; V3?e:7@798 6=H3?87 6G9; A7 @G957A 65
A4>?H?7A [\]^) .7; LMJ48Le;7; G;;4H?=7; P H7 8MJ7 67 @46=A?;G8?49 97 ;498 JG;
7KJA?H?8=7;)
.7 HGAH5A 67 A?>97 6E7G5 79 3=>?@7 J73@G9798 78 79 3=>?@7 83G9;?84?37 J758
J4;73 J34VAe@7 6G9; A7 HG; 6E59 =H45A7@798 8433798?7A f 948G@@798 P HG5;7 67;
?9;8GV?A?8=; F57 J758 >=9=373 A7 ;HL=@G 67 $37?;;@G99 G5 9?:7G5 65 JG;;G>7
S5:?GAg8433798?7A 45 8433798?7AgS5:?GA) $453 37@=6?73 P H7 J34VAe@7D AG ;7H8?49
6E=H45A7@798 7;8 H49;?6=3=7 H49;8G987 6G9; A7 873@7 H49:7H8?T 67 AE=F5G8?49 67
H49;73:G8?49 67 AG F5G98?8= 67 @45:7@798 B ) aVC `
∂
∂x
(
Q2
S
)
≈ 1
S
∂Q2
∂xB ) hC
! ! "#$%& $% "'(")( *$ +
-4
-3
-2
-1
0
1
2
3
4
-2 -1 0 1 2
Nombre de Froude
λ/c
λ+ analytique
λ- analytique
λ+ Isis Flow
λ- Isis Flow
-Fr m
ax
-Fr m
in
Fr m
ax
Fr m
in
!"#$% !" # $%&'()*&+ ,- '. /0'01*)0 ,-2 &+,-2 -+ 3&+/)*&+ ,( +&451- ,-
61&(,- 7 /&48.1.*2&+ ,( /&48&1)-4-+) )90&1*:(- -) ,- /-'(* ,- ;2*2 6'&<!
=-))- .881&>*4.)*&+ 2- 108-1/()- .( +*%-.( ,( /.'/(' ,- '. 4.)1*/- ?./&5*-++-
-) ,- '@->81-22*&+ ,- '. %*)-22- ,- 81&8.A.)*&+ ,-2 &+,-2!
A =
[
0 1c2 2u
]
B !CDE
λ− = u−√u2 − c2
λ+ = u+√u2 − c2 B !CFE
;' -2) *+,*:(0 :(@(+- )1.+2*)*&+ 81&A1-22*%- -2) 10.'*20- -+)1- '-2 0:(.)*&+2 ,-
G.*+)HI-+.+) -) /-''-2 *+/'(.+) '@.881&>*4.)*&+ B ! JE! K@->81-22*&+ ,- /-))-
3&+/)*&+ ,- )1.+2*)*&+ +@-2) 8.2 8102-+)0- ,.+2 '- 4.+(-' ,- 10301-+/- ,( '&H
A*/*-' LCMN! O0.+4&*+2P '-2 %.'-(12 2-(*' 8.1 ,03.() -+)1- '-2:(-''-2 '. 3&+/)*&+ ,-
)1.+2*)*&+ -+)1- -+ ?-( 2&+) 280/*Q0-2 R '. )1.+2*)*&+ 2@-S-/)(- 8&(1 (+ +&451-
,- 61&(,- /&481*2 -+)1- Frmin = 0, 75 -) Frmax = 0, 9! K-2 0:(.)*&+2 /&48'T)-2
,- G.*+)HI-+.+) 2&+) ,&+/ ()*'*20-2 8&(1 (+ +&451- ,- 61&(,- *+301*-(1 U Frmin-) '@.881&>*4.)*&+ -2) /&48'T)-4-+) .88'*:(0- .(H,-'U ,- Frmax! K@0%&'()*&+ ,(
1.)*&
λ
c-+ 3&+/)*&+ ,( +&451- ,- 61&(,- . 0)0 )1./0- 2(1 '. QA(1- !"! K@->81-2H
2*&+ ,- '. 3&+/)*&+ ,- )1.+2*)*&+ +@0).+) 8.2 810/*20- ,.+2 '- 4.+(-' ,( '&A*/*-'P *'
-2) *48&22*5'- ,- ,0)-14*+-1 '. 3&14('.)*&+ ,-2 /0'01*)02 ,@&+,-2 2(1 '@-+2-45'-
,( ,&4.*+- ,- %.1*.)*&+ ,( +&451- ,- 61&(,- 7 '- 1.)*& +@. 8.1 /&+20:(-+) 8.2
0)0 )1./0 ,.+2 '- ,&4.*+- ,@.88'*/.)*&+ ,- '. 3&+/)*&+ ,- )1.+2*)*&+ 8&(1 (+
+&451- ,- 61&(,- /&481*2 -+)1- Frmin = 0, 75 -) Frmax = 0, 9!V&(1 (+ 0/&('-4-+) W(%*.' '- )-14- c2 810,&4*+- ,-%.+) u2 7 '-2 /0'01*)02
,@&+,-2 B !CFE +- 2&+) ,&+/ 8.2 ,0Q+*-2 8&(1 (+ +&451- ,- 61&(,- /&481*2 -+)1-
! "#$%&'() * +',-) .&./&01($%#&2,)
0, 9 34 1* /5 657389 :59 ;<=584 ;3 Frmax >3 :384 ;?>@ :5A B493 @?>A396<3 <C573 D
0, 9 34 ;?E4 ><@3AA5E93F3>4 6<9EG39 Frmax ≥ 1* -H58493A @?>495E>43A A89 Frmin 34
Frmax :3863>4 B493 EF:?A<3A :59 7H3I:93AAE?> ;3 75 =?>@4E?> ;3 495>AE4E?>*
%?89 8> >?FJ93 ;3 K9?8;3 E>=<9E389 D FrminL 73A @<7<9E4<A ;H?>;3A @57@87<3A :5973 7?CE@E37 A?>4 E;3>4EM83A 58I @<7<9E4<A 4N<?9EM83A* %?89 8> <@?873F3>4 @9E4EM83
OFr = 1PL 73A ;38I @<7<9E4<A ;H?>;3 A?>4 <C573A D cL @3 M8E >3 @?993A:?>; :5A
58 @?F:?943F3>4 NQ;9587EM83 4N<?9EM83 ?R 8>3 ;3A @<7<9E4<A 3A4 >8773* 0> :384
<C573F3>4 F?>4939 M8H3> 9<CEF3 4?993>4E37L 73A @<7<9E4<A @57@87<3A :59 73 7?CE@E37 >3
A?>4 <C573A 58I @<7<9E4<A 4N<?9EM83A M83 :?89 Fr = ±√2* /3 495@< ;3A @<7<9E4<A
;H?>;3A 4N<?9EM83A 34 @57@87<3A :59 73 7?CE@E37 ;5>A 7H3A:5@3 ;3A :N5A3A O6?E9
KEC893 *SP :39F34 ;HE;3>4EG39 ;38I @?F:?943F3>4A ;ET<93>4A* %?89 1 < Fr2 < 2L75 @<7<9E4< ;H?>;3 ><C54E63 O93A:3@4E63F3>4 :?AE4E63P @57@87<3 :59 73 7?CE@E37 3A4
A893A4EF<3 O93A:3@4E63F3>4 A?8AU3A4EF<3P :59 95::?94 58I @<7<9E4<A 4N<?9EM83A V
73 @?F:?943F3>4 E>639A3 3A4 ?JA3965J73 :?89 Fr2 > 2* %59 5E77389AL 7?9AM83 73
>?FJ93 ;3 K9?8;3 58CF3>43L 7H8>3 ;3A @<7<9E4<A ;H?>;3A @57@87<3 :59 73 7?CE@E37
43>; 639A 0* "3 @?F:?943F3>4 3A4 E>@?N<93>4 563@ 73 @?F:?943F3>4 4N<?9EM83
:8EAM83 75 6E43AA3 ;3 :9?:5C54E?> ;3A ?>;3A 58CF3>43 563@ 73 >?FJ93 ;3 K9?8;3*
! !" #$%&'()*$% +$(, '-) &$.-) /0
/3A @?;3A ;3 @57@87 8>E;EF3>AE?>>37A A?>4 58W?89;HN8E 3>@?93 759C3F3>4 84EU
7EA<AL >?45FF3>4 ;5>A 73A ;?F5E>3A ;3 7HE>C<>E39E3 34 ;3 7H<657854E?> ;8 9EAM83
;HE>?>;54E?>* /3A =5EJ73A J3A?E>A 3> ;?>><3A 4?:?C95:NEM83A O@?F:5954E63F3>4
58I F?;X73A ;3 :78A N5843 ;EF3>AE?>P 34 75 93754E63 95:E;E4< ;3 @57@87 @?FU
:3>A3>4 73A NQ:?4NXA3A AEF:7EG@549E@3A A89 73AM83773A @3A 7?CE@E37A A?>4 J5A<A*
-H58493A 5::9?@N3A A?>4 3F:7?Q<3A ;5>A @3945E>A 7?CE@E37A* %59 3I3F:73L 73 7?U
CE@E37 "59:5 ;<637?::< :59 7H,>E639AE4< %?7Q43@N>EM83 ;3 "5457?C>3 O)A:5C>3P
YZZ[ 3A4 8> 3I3F:73 ;3 @?;3 84E7EA5>4 8>3 5::9?@N3 58I 6?78F3A G>EA M8E :39F34
;3 495E439 ;3 75 FBF3 F5>EX93 73A <@?873F3>4A \86E58I 34 4?993>4E37A*
-5>A 73 @5A ;H<@?873F3>4A :78A @?F:73I3A ?8 :?89 73AM837A 73A NQ:?4NXA3A
;3A @?;3A 8>E;EF3>AE?>>37A A?>4 49?: AEF:7EG@549E@3AL 7H8A5C3 ;3 @?;3 ;3 @57@87
JE;EF3>AE?>>37 3A4 :9E6E7<CE<*
!" #$%&' %& ()*(+* ,
-XA 7?9A M83 7H<@?873F3>4 >3 :384 :78A B493 @?>AE;<9< @?FF3 8>E=?9F3 A89
8>3 A3@4E?> 3> 495639AL E7 @?>6E3>4 ;3 =5E93 5::37 D 75 F?;<7EA54E?> JE;EF3>AE?>U
>3773 :784]4 M8HD 75 F?;<7EA54E?> 8>E;EF3>AE?>>3773* /3A ;?>><3A ><@3AA5E93A D 75
@?>A498@4E?> ;8 F?;X73 O>?45FF3>4 3> 439F3A ;3 4?:?C95:NE3P A?>4 57?9A :78A
>?FJ938A3A* /3A F?;X73A @?>A498E4A A?>4 C<><9573F3>4 :78A C95>;A M83 73A F?U
;X73A ^- 34 ><@3AAE43>4L :59 @?>A<M83>4L ;3A ;89<3A ;3 AEF8754E?> :78A 7?>C83A*
&7 3A4 58W?89;HN8E :?AAEJ73 ;3 49?8639 8> >?FJ93 EF:?945>4 ;3 @?;3A ;3 @57U
@87 -* .E3> M83 75 M85AE 4?457E4< ;3A 7?CE@E37A 9<A?763>4 73A <M854E?>A ;3 .599<
!"! #$%&' %& #()#*) % +
λ- Is
is =
λ+ Is
is
t
xλ+ théo
λ+ Isis
λ− théo
λ− Isis
!" Fr = 1
t
xλ+ théo
λ+ Isis
λ− théo
λ− Isis
#" 1 < Fr2 < 2
λ+ Isis = λ
+ théo
λ- Isi
s =
λ- t
hé
o
t
xλ+ théo
λ+ Isis
λ− théo
λ− Isis
$" Fr2
= 2
t
xλ+ théo
λ+ Isis
λ− théo
λ− Isis
%" Fr2 > 2
!"#$% !, - ./0123/4565784 9/3 :2;217523 9<849/3 5=2817>?/3 /5 :6;:?;2/3 061
@373 A;8B 9643 ;</306:/ 9/3 0=63/3!
! "#$%&'() * +',-) .&./&01($%#&2,)
y
x
yj+1
yj
xi
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(i,j) (i+1,j)
(i+1,j+1)
!" #!$%%!&' ()*+,)+*- ,!*)-.
($'/
y
x
yj+1
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y
x
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Pi
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," #!$%%!&' /2/ ()*+,)+*-
!"#$% *34 5 6789:; <= >?@A7@>= <=B <@C9?=ADB DE>=B <= :;@FF;G=
<= 6;@ADHI=A;ADJ F; :9D8K<= <= :;@FF;G= <= F; LKA= :K<9F@B9= =D F; :9D8K<=
<= ?9BKFMD@KA <=B 9NM;D@KAB >KB9=B >=MO=AD O;?@=? <= :;A@P?= @:>K?D;AD= <QMA
7K<= R FQ;MD?=* -=MS G?;A<B DE>=B <= :;@FF;G=B BKAD R <@BD@AGM=? T F=B :;@FF;G=B
BD?M7DM?9B =D F=B :;@FF;G=B AKA BD?M7DM?9B UOK@? V@GM?= *34W* %KM? F=B :;@FF;G=B
BD?M7DM?9BJ F=B >K@ADB <= 7;F7MF BKAD <@B>KB9B <= :;A@P?= K?G;A@B9= X DKMB F=B >K@ADB
>;?D;G=;AD MA :Y:= @A<@7= 9D;AD <@B>KB9B BM? MA= :Y:= 7KM?Z=* )A ?=O;A78=J
>KM? F=B :;@FF;G=B AKA BD?M7DM?9BJ ;M7MA= K?G;A@B;D@KA >;?D@7MF@P?= <=B >K@ADB
<= 7;F7MF AQ=BD A97=BB;@?=* -;AB F= 7;B <=B :;@FF;G=B BD?M7DM?9BJ F; ?9BKFMD@KA <=
BEBDP:=B <Q9NM;D@KAB :MFD@<@:=AB@KAA=FB =BD G9A9?;F=:=AD [;@D= >;? F; ?9BKFMH
D@KA <= >FMB@=M?B >?KZFP:=B MA@<@:=AB@KAA=FB* "= DE>= <= :;@FF;G= >=MD DKMD=H
[K@B YD?= >?KZF9:;D@NM= >KM? ?=>?9B=AD=? <=B DK>KG?;>8@=B ?9=FF=B* %;? ;@FF=M?BJ
F=B D=78A@NM=B <= ?9BKFMD@KAB <= BEBDP:=B <Q9NM;D@KAB :MFD@<@:=AB@KAA=FB BKMB
[K?:= <QMA= BM77=BB@KA <= >?KZFP:=B MA@<@:=AB@KAA=FB >=MO=AD >?K<M@?= <=B
BKFMD@KAB [K?D=:=AD ;A@BKD?K>=B =D >;?D@7MF@P?=:=AD B=AB@ZF=B R FQK?@=AD;D@KA <M
:;@FF;G= \ !]J \ ^]*
-;AB F; :=BM?= K_ F= 7K<= <= 7;F7MF <9O=FK>>9 <;AB F= 7;<?= <= 7=DD= D8PB=
MD@F@B=?; MA :;@FF;G= <= 7;F7MF AKA BD?M7DM?9J F=B 7K<=B <= 7;F7MF Z@<@:=AB@KAA=FB
D?;O;@FF;AD BM? <=B :;@FF;G=B BD?M7DM?9B U`@a= 3 <M -#&J ""#) - <M b""#)
<= FQ,A@O=?B@D9 <M `@BB@BB@>@J cKF[ - <M #$"# <= FQ,A@O=?B@D9 <= /@PG=J =D7W
A= B=?KAD >;B >?9B=AD9B* /=B 7K<=B <= 7;F7MF >?9B=AD9B 7@H;>?PB BKAD T `@a= 3
V`J '=F=:;7J 6c -* /= 7K<= <= 7;F7MF 6c - B=?O@?; <= Z;B= ;M 7K<= <= 7;F7MF
3-H - <9O=FK>>9 <;AB F= 7;<?= <= 7=DD= D8PB= T 6c3 - U>KM? 68;FFKd c;D=?
3-H -W*
!"!# $%&' # ()*+ $*,') ($
/; BK7@9D9 -#& ; <9O=FK>>9 >FMB@=M?B FKG@7@=FB <= :K<9F@B;D@KA Z@<@:=AB@KAH
A=FF= <=B 97KMF=:=ADB X 78;7MA BQ;>>F@NM;AD BM? MA DE>= <= :;@FF;G= <@C9?=AD =D
MD@F@B;AD <=B :9D8K<KFKG@=B <= 7;F7MF D?PB <@C9?=AD=B <QMA FKG@7@=F R FQ;MD?=* /;
:9D8K<KFKG@= <= 7;F7MF <M FKG@7@=F `@a= 3 VFKd `K<=F V`J D=FF= NM= <97?@D=
<;AB F= :;AM=F <= ?9[9?=A7= \3e]J =BD >?9B=AD9= @7@* )A =C=DJ 7= FKG@7@=F U>;?:@
!"! #$%&' %& #()#*) % +
,-./ 01202343 051 6- %789 -3: 6- 3-.6 ; 3- <53-1 3.1 .= >5?665@- =2= 3:1.,:.14
-: 0-.: 4@56->-=: A:1- ,2.064 ; B?C- DD 051 6- <?5?3 E. 62@?,?-6 B?C- F622E!
#- 62@?,?-6 5 4:4 E4G-62004 5H= E- 02.G2?1 >2E46?3-1 6-3 4,2.6->-=:3 -= >?6?-.
>51?=I ,J:?-1I -3:.51?-= 2. 6213 EK?=2=E5:?2=!
)-3 4L.5:?2=3 14326.-3 051 6- 62@?,?-6 32=: ,-66-3 E- '5?=:MN-=5=: -= E-./
E?>-=3?2=3 O
∂h
∂t+∂q
∂x+∂r
∂y= Ql P !" 59
∂q
∂t+
∂
∂x
(
q2
h
)
+∂
∂y
(qr
h
)
=
− gh∂z∂x− g
√
q2 + r2q
C2h2+ Cx +Dx +Wx + Tx + Sx P !" <9
∂r
∂t+
∂
∂x
(qr
h
)
+∂
∂y
(
r2
h
)
=
− gh∂z∂y− g
√
q2 + r2r
C2h2+ Cy +Dy +Wy + Ty + Sy P !" ,9
2Q CxI DxI WxI TxI Sx P1-30-,:?G->-=: CyI DyI WyI TyI Sy9 1-0143-=:-=: 65
R21,- E- #21?26?3I 65 G51?5:?2= E- 65 E-=3?:4I 6K-S-: E. G-=:I 6K-S-: E- 65 :.1<.M
6-=,- -: 6- T./ E- L.5=:?:4 E- >2.G->-=: -=:15=: 051 .=?:4 E- 3.1R5,- E5=3 65
E?1-,:?2= Ox P1-30-,:?G->-=: Oy9 U Ql -3: 6- G26.>- -=:15=: 051 .=?:4 E- 3.1R5,-
-: 051 .=?:4 E- :->03! #-3 4L.5:?2=3 32=: 14326.-3 051 .=- >4:V2E- 5./ G26.>-3
H=?3 3.1 .= >5?665@- =2= 3:1.,:.14 5G-, E-3 464>-=:3 E- R21>- L.-6,2=L.-! )5
051:?- ,2=G-,:?G- E. G-,:-.1 T./ -3: ,56,.64- 051 6K?=:-1>4E?5?1- EK.= 326G-.1
E- W2- -: 6K?=:4@15:?2= 305:?56- -3: 1456?34- 051 .= 3,V4>5 E. 01->?-1 21E1-! )5
>4:V2E262@?- E. 326G-.1 E- W2- 3-15 E4:5?664- ; 65 3-,:?2= !X! !D!
Y2.1 65 E?3,14:?35:?2= :->021-66-I .= 3,V4>5 -/06?,?:- EK&.6-1 -3: 5E20:4!
)- >5=.-6 E- 6K.:?6?35:-.1 ZD[\ -/06?,?:- 65 ,2=E?:?2= E- 3:5<?6?:4 L.? E2?: A:1-
G41?H4- O
Cr =√
gh∆t
∆x≤ 0.5 P !""9
2Q Cr 1-0143-=:- 6- =2><1- E- #2.15=:I ∆t 6- 053 E- :->03 -: ∆x -3: 50012,V4
051 65 [email protected] E- 65 06.3 0-:?:- ?=:-1R5,- E- 6K464>-=: ,2=3?E414! )5 ,2=E?:?2=
E- 3:5<?6?:4 P !""9 0-.: =- 053 A:1- 3.]35=:- 02.1 @515=:?1 65 3:5<?6?:4 E. >2E^6-
E5=3 65 >-3.1- 2Q 65 G?:-33- EK4,2.6->-=: -3: =4@6?@4- -: 2Q 6K50012/?>5:?2= E-
∆x 0-.: A:1- :120 @15=E- E5=3 ,-1:5?=-3 ,[email protected]:?2=3 E. >5?665@-! Y51 5?66-.13I
6- 053 E- :->03 E- ,56,.6 E2?: A:1- H/4 051 6K.:?6?35:-.1 U ,- ,V2?/ 0-.: 3- 14G46-1
E?],?6- E5=3 6- ,53 E- >2E46?35:?2= ,2>06-/- 2. 305:?56->-=: :1^3 4:-=E.-!
)5 01?3- -= ,2>0:- EK2.G15@-3 V_E15.6?L.-3 P,2.13 EK-5. ,5=56?343I 02=:3I
-:,!!!9 =K5 053 4:4 014G.- E5=3 6- 62@?,?-6! )5 >2E46?35:?2= E-3 4,2.6->-=:3 E5=3
E-3 `2=-3 2Q ,-1:5?=3 2.G15@-3 0-.G-=: 3- >-::1- -= ,V51@- 0-.: 3- 14G46-1
E?],?6- G2?1- ,2=E.?1- ; E-3 143.6:5:3 =2= 1-0143-=:5:?R3!
! "#$%&'() * +',-) .&./&01($%#&2,)
%345 6347385 9:5;5 <355;<=;>;?= @;A >B8@@;A C; <B@<4@ B7;< 4?; DB8E@; FB4=;45
CG;B4H 4?; B6653<F; 6B5=8<4@8I5; B :=: BC36=:;* '538A 7B@;45A A;48@ A3?= ;>6@3J:;A
A45 <FBK4; <;@@4@; 6345 C:L?85 A8 4?; 8?=;5DB<; ;A= ;? ;B4 34 ?3? ;= A8 4?;
>B8@@; C; <B@<4@ ;A= 8?3?C:;H >348@@:; 34 AI<F;* %345 4?; <;@@4@; 8?3?C:;H @;A
:K4B=83?A A3?= <3?A8C:5:;A CB?A @;45 8?=:95B@8=: M 6345 @;A <;@@4@;A >348@@:;AH A;4@A
@;A =5B?AD;5=A C; >BAA; A3?= 658A ;? <3>6=; M B4<4? <B@<4@ ?G;A= DB8= 6345 @;A
<;@@4@;A AI<F;A* -4 DB8= C;A <58=I5;A 658A ;? <3>6=;H B4<4? :<FB?9; ?G;A= 5:B@8A:
;?=5; 4?; <;@@4@; AI<F; ;= 4?; <;@@4@; >348@@:;* %B5 B8@@;45A @35AK4G4?; <;@@4@;
;A= AI<F;H @B DB8E@; K4B?=8=: CG;B4 K4G;@@; <3?=8;?= ;A= 5;=85:; C4 >3CI@; <; K48
DB4AA; B8?A8 @; E8@B? C; >BAA;* /G4=8@8AB=83? C4 @398<8;@ N8O; P 6345 >3C:@8A;5 C;A
:<34@;>;?=A A45 C;A A45DB<;A C; DB8E@; 6;?=; Q6345 C;A :=4C;A C; 548AA;@@;>;?=
6B5 ;R;>6@;S 6;4= T=5; 5;?C4; C8U<8@; 6B5 <;==; B6653<F; C; 9;A=83? C;A D53?=A
C; >348@@B9;A*
!"! #$%$&'( )
/; @398<8;@ ';@;>B< -H C:7;@366: 6B5 )-VH ;A= 4? <3C; C; >3C:@8AB=83?
E8C8>;?A83??;@@; C;A :<34@;>;?=A* /;A 8?D35>B=83?A 5;@B=87;A B4R >:=F3C;A ?4W
>:58K4;A ;>6@3J:;A 6B5 <; <3C; C; <B@<4@ A3?= C8U<8@;>;?= B<<;AA8E@;A* )? ;X;=H
@GB@9358=F>; C; 5:A3@4=83? C;A :K4B=83?A ;A= 653=:9: CB?A @B >;A45; 34 ';@;>B<
- ;A= 4? @398<8;@ <3>>;5<8B@*
Y45 4? >B8@@B9; ?3? A=54<=45: =58B?94@B85;H @; @398<8;@ 5:A34= @;A :K4B=83?A
C; YB8?=WZ;?B?= A34A D35>; ?3? <3?A;57B=87; 6B5 4?; >:=F3C; B4R :@:>;?=A
L?8A [\\]H [\P] ^
∂h
∂t+−→V
−−→grad (h) + h div
(−→V)
= 0 Q *\_BS
∂u
∂t+−→V
−−→grad (u) = −g ∂z
∂x+ Fx +
1
hdiv(
hνt−−→grad (u)
)
Q *\_ES
∂v
∂t+−→V
−−→grad (v) = −g ∂z
∂y+ Fy +
1
hdiv(
hνt−−→grad (v)
)
Q *\_<S
3` Fx ;= Fy 5;65:A;?=;?= @;A =;5>;A A345<;A 65;?B?= ?3=B>>;?= ;? <3>6=; @;A
D53==;>;?=A ;= 3` νt 5;65:A;?=; @B 78A<3A8=: =45E4@;?=;* /;A :K4B=83?A Q *\_S A3?=
5:A3@4;A ;? 4=8@8AB?= @B >:=F3C; C; %;=537W1B@;5O8? ;= @B !"!#$%&'!( )&"&*+*
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n+½
21
Mike 11
Mike 21
Débit d’échange calculé explicitement à partir des variables du modèle 1D
Hauteur d’eau du modèle 2D servant de condition limite du 1D
i Etape de calcul n°i
!"#$% !++ . '/0123 45 /36/76 45 89:5 ;6<<4 =<7> 65? @<A/B9<A? 45 BC=5
D?B3A43>4 69A:D
!"!#!$ %&'()*&' +, )-., /0)1'+12+ 3*'4/
#5? @<A/B9<A? E<AB 65 695A 5AB>5 7A5 5FB>129B1 4G7A 2<4H65 +% 5B 7A5 <7 =67,
?957>? 239665? 45 /36/76 %! )5 E<A/B9<AA525AB 45 /5 695A 5AB>5 65? 457F 2<4H65?
5?B 65 2I25 J75 =<7> 65 /<45 K<6E LM<9> '5/B9<A !"!+N! %3A? 7A =>5295> B52=?O
89:5 + >1?<7B 65? 1J73B9<A? 5AB>5 65? B52=? n 5B n + 1! )5 41P9B 45 B>3A?E5>B
45=79? 65 2<4H65 +% M5>? 65 % 5?B 5FB>3=<61 Q =3>B9> 45? M3>93P65? 47 2<4H65
+% 37 =3? 45 B52=? n R
∂Qn+1/2
∂t= −
(
gS∂hn
∂x+ g
Qn |Qn|C2SRh
)
L !-SN
%3A? /5BB5 E<>2763B9<AO 65 B5>25 4G3//161>3B9<A /<AM5/B9M5 3 1B1 A1T69T15!
)G5FB>3=<63B9<A 47 41P9B B>3A?9B3AB 4G7A 2<4H65 Q 6G37B>5 5?B >1369?15 T>U/5 Q
7A ?/0123 5F=69/9B5 5B A1/5??9B5 4<A/ 45 >5?=5/B5> 63 /<A49B9<A 45 #<7>3AB =<7>
T3>3AB9> 63 ?B3P969B1 47 2<4H65! )3 M3657> 23F92365 3429??9P65 =<7> 65 A<2P>5 45
#<7>3AB 5?B 9A49J715 1T365 Q ! ( =3>B9> 47 41P9B 45 B>3A?E5>BO 63 23??5 1/03AT15
5AB>5 /03/7A 45? 2<4H65? 39A?9 J75 63 J73AB9B1 45 2<7M525AB 5AB>3AB5 43A? 65
2<4H65 % =57M5AB IB>5 5?B9215?! %3A? 7A ?5/<A4 B52=?O 89:5 ++ >1?<7B 65?
1J73B9<A? 5AB>5 65? B52=? n 5B n + 1! V<7> /5 E39>5O 63 /<B5 45 63 ?7>E3/5 69P>5
43A? 65 2<4H65 % 5?B 7B969?15 /<225 /<A49B9<A 37F 6929B5? 47 2<4H65 +%! )5
?/0123 45 /36/76 /<7=61 45? 457F 2<4H65? 5?B =>1?5AB1 ?7> 63 WT7>5 !++!
! "#$%&'() *+ ,'-.) /&/0&12($%#&3-)
!"!#! $%&'()%& *+ (,-+ ./0(+10/ /)&2.
"45 6789:;785 <4=>4::48: ?4 =4<=@548:4= A45 ?@B7=?4>48:5 ?C A;: >;84C= D4=5
A45 9EF><5 ?G4H<F85;78 ?4 9=C4 4: D;94ID4=5F+ 045 9F=F9:@=;5:;JC45 ?4 AG@9EF8K4
578: ;8:@K=@45 FCH @JCF:;785 ?4 L;M4 NN O=45<49:;D4>48: L;M4 *NP D;F C8 :4=>4
57C=94+ 0F ?@Q8;:;78 ?4 :4AA45 5:=C9:C=45 <4C: 54 RF;=4 5C= AG4854>BA4 ?GC8 B;4R
?C >7?SA4 N. O4: ?789 5C= <AC5;4C=5 549:;785 ?4 9FA9CAP 4: 5C= <AC5;4C=5 94AACA45
?C >7?SA4 *.+ 04 9E7;H ?4 AF <7=:;78 ?4 >7?SA4 N. 4: ?45 94AACA45 F5579;@45 T
C84 5:=C9:C=4 ?G@9EF8K4 45: RF;: ?4 >F8;S=4 FC:7>F:;JC4 <F= A4 97?4+ $ <F=:;= ?4
AF K@7>@:=;4 ?45 ?4CH >7?SA45U AF AF=K4C= 4: AF 97:4 ?C ?@D4=57;= 578: 9FA9CA@45+
.4CH R7=>CAF:;785 578: <=7<75@45 <7C= 9FA9CA4= A4 ?@B;: :=F85;:F8: ?GC8 >7?SA4
T AGFC:=4 V
Qe
DL= ham
√
ham
[
1−(
havham
)1,5]0,385
O*+ WFP
Qe
DL=
ham√ham !"#
havham
≤ 23
32
√3hav
√ham − hav !"#
havham
> 23
$%&'()*
!+ Qe ,-. /, 01)2. 0, .#34-5,#.6 D ,-. /, 7!,872,4. 0, 019,#-!2# $D = 1, 838*6 L /3
/3#:,"# 0;17<34:, ,. ham $#,- ,7.29,=,4. hav* /3 <3".,"# 3" 0,--"- 0" 019,#-!2#
> /;3=!4. $#,- ,7.29,=,4. > /;393/*& ?3 5!#="/3.2!4 $%&'(3* !-, #!)/@=, 0@-
/!#- A", /, 019,#-!2# 4;,-. 3- 4!B1& C4 ,D,.6 -2 /3 7!., 0, /3 -"#537, /2)#, >
/;393/ ,-. 2451#2,"#, > /3 7!., 0" 019,#-!2#6 /3 5!#="/, $%&'(3* 4;,-. 3- 01E42, F
/, =34",/ 0, #151#,47, 0, G2H, I/!!0 JK%L 4, #172-, 3- 7!==,4. 7, 73- ,-.
.#32.1&
C4.#, /,- 0,"M =!0@/,-6 -,"/- /,- 17<34:,- 0, =3--, -!4. #2- ,4 7!= ., F /3
A"34.2.1 0, =!"9,=,4. -!#.34., 4;1.34. 3- 24.#!0"2., 034- /, =!0@/, 393/& N3#
32//,"#-6 2/ 7!492,4. 0, 4!.,# A", /3 A"34.2.1 0, =!"9,=,4. 3--!721, > /3 =3--,
-!#.34., 0;"4 =!0@/, ,-. 3 #2!#2 7!4-,#91, 034- /, =!0@/,& O,/3 -, .#30"2. 3#
"4, 4!4 7!4-,#93.2!4 0, /3 A"34.2.1 0, =!"9,=,4.6 /, <14!=@4, 1.34. 0;3".34.
/"- 377,4."1 A", /3 <3".,"# 0;,3" 034- /, =!0@/, 0#3241 ,-. 532)/, !" A", /,
4!=)#, 0, I#!"0, ,-. 1/,91&
O!==, !"# /, 7!" /3:, 0,- =!0@/,- OOPCKQ ,. OOPC%Q6 /;".2/2-3.2!4 0,
/!2- 0, 019,#-!2# !"# =!01/2-,# /,- 17<34:,- ,4.#, /,- 0,"M =!0@/,- ,= R7<, /3
#, #1-,4.3.2!4 0,- <14!=@4,- .,/- A", /, 7!"#.S72#7"2. 0, =1340#,&
!"!#!# $%&'(%)*'%+, -* -.)%' /,'(/ 0%1/ 22 /' 0%1/ 2
N!"# /,- T!47.2!4- 0, .B , U-.3403#0 /24HU6 /, 01)2. 0, .#34-5,#. ,4.#, /,- =!S
0@/,- ,-. -!2. 02-.#2)"1 "425!#=1=,4. > .!".,- /,- =32//,- %Q6 -!2. #1 3#.2 ,4.#,
/,- 7,//"/,- ,4 5!47.2!4 0, /3 <3".,"# 0;,3" 3" 01)". 0" 3- 0, .,= -& V2 /;".2S
/2-3.,"# 7<!2-2. "4, #1 3#.2.2!4 ,4 5!47.2!4 0, /3 <3".,"# 0;,3"6 /3 5!#="/3.2!4
!"! #$%&' %& #()#*) +%, % -.
/0123456 6/5 6789:;<6 =
Qi = Qh
3/2
in∑
i=1
h3/2
i
> !-?@
#6556 A:B709351:4 6/5 <53C916 64 /088:/345 D06 96 E<C15 /6 B<83B515 64 B6/86F5345
9G<D0351:4 E6 #H<I; =
Q = SCi1/2R
1/2
h > !"J@
&4 A31/345 9GH;8:5HK/6 D06 96 F3439 6/5 93BL6 >b≫ h@M :4 8605 388B:FH6B 96 B3;:4
H;EB3091D06 Rh 83B 93 H30560B EG630 h =
Rh =S
χ≈ bh
b+ 2h≈ h > !"+@
&4 145<LB345 > !"+@ N > !"J@M :4 :C51645 =
Q = bCi1/2h
3/2> !" @
)6 734069 E6 B<A<B64F6 E6 O1P6 Q9::E 14E1D06 D06 96 E<C15 Q 8605 R5B6 F:4/1E<B<
F:776 8B:8:B51:4469 N h3/2S F6 D01 6/5 96 F3/ 8:0B 04 B<L176 FB151D06! #6864E345
F693 4G6/5 3 8B1:B1 83/ 2B31 /1 93 86456 E0 A:4EM 93 93BL60B 65 96 F:6TF1645 E6 #H<I;
46 /:45 83/ F:4/5345/ /0B 93 I:46 E6 E1/5B1C051:4 E0 E<C15 E6 5B34/A6B5!
U069 D06 /:15 96 5;86 E6 V:4F51:4 6421/3L<M 96 E<C15 EG<FH34L6 6/5 F39F09<
14E<864E377645 E0 83/ E6 5678/ E6 FH3D06 7:EK96! )3 F:4/6B2351:4 E6 93
73//6 8605 46 83/ R5B6 2<B1W<6 D034E 96 2:9076 <FH34L< ><L39 30 8B:E015 E0
E<C15 83B 96 83/ E6 5678/@ 6/5 /08<B160B 30 2:9076 E1/8:41C96 E34/ 96/ 731996/
F:4F6B4<6/ E0 7:EK96 EB314<! &4 6X65M E34/ 04 569 F3/M 96 2:9076 645B345 E34/
96 7:EK96 3239 6/5 /08<B160B N F6901 /:B5345 E0 7:EK96 37:45 S 96 C1934 E6 73//6
L9:C39 /0B 9G64/67C96 E6/ E60Y 7:EK96/ 4G6/5 39:B/ 890/ 2<B1W<! %34/ 93 76/0B6
:Z B164 4G6/5 76451:44< E34/ F6 /64/ E34/ 96 734069 E6 B<A<B64F6 [+ \M 19 6/5
8://1C96 D06 O1P6 Q9::E 46 2<B1W6 83/ 93 F:4/6B2351:4 E6 93 73//6 30 5B326B/
EG046 9131/:4!
!"!" #$%&'()* *
)6 9:L1F169 '$]&^,+% % [+.\M E<269:88< 83B %69A5 _;EB3091F/M 6/5 04 F:E6
E6 F39F09 F:089< +%, %! )3 7<5H:E:9:L16 E6 F:0893L6 6789:;<6 83B ':C6P,+% %
8:0B <53C91B 96/ <D0351:4/ B</:906/ F:4/1EKB6 890/160B/ 2:9076/ E6 F:45B`96 >2:1B
Q1L0B6 !+ @! )3 I:46 N 7:E<91/6B 6/5 /<83B<6 645B6 96 B</630 041E1764/1:4469 65
96 731993L6 F3B5</164 C1E1764/1:4469! )6/ <D0351:4/ E6 D034515< E6 7:0267645
/:45 <53C916/ 65 B</:906/ 14E<864E377645 8:0B 96/ E60Y 7:EK96/! &4 B6234FH6
046 041D06 <D0351:4 E6 73//6 6/5 05191/<6 8:0B FH3D06 731996 E0 7:EK96 C1E1,
764/1:4469 >14F90345 <2645069967645 046 731996 E0 7:EK96 041E1764/1:4469@ =
dVi,j (z)
dt+Qk−1/2 −Qk+1/2
+∆y(
qi−1/2 − qi+1/2
)
+∆x(
rj−1/2 − rj+1/2
)
= 0 > !"-@
r j+1/2
qi+1/2
r j-1/2
qi-1/2
Q k+1/2
Q k-1/2
r jr +1/2
qi-1/2
Modèle 1D
Modèle 2D
Vi,j i j
h1D Q1D
h2D q2D r2D
Qk−1/2 Qk+1/2
!"! #$%$ &' ()%*$ &'+ %,,*-./'+ %01 2-(03'+ 4565+ ,-0*
('+ #70%$5-6+ /8,'*9-(570'+ :;
!"!# $%&'()*+%& ,%)- (.* '%/.* '%),(0* 123 2
(< =>?@A< B< ?CDE>B>F>GH<I B< J>KLFMG< N&O & <ID PKMIH?<=D MKIIH H?L>AO
DM=D PK< F< =>?@A< B< J>B<I N&O & <QHIDM=D! .<ADMH=<I ?CDE>B<I L<A?<DD<=D
=CM=?>H=I B< ACBKHA< F< =>?@A< B)ERL>DESI<I ACMFHIC<I <D I>=D <= J< I<=I LFKI
TM@F<I! (< J>KLFMG< <ID UMJHFHDC LMA F)<?LF>H B<I ?V?<I ?CDE>B<I =K?CAHPK<I
B< ACI>FKDH>= B<I CPKMDH>=I L>KA F<I B<KQ J>B<I B< JMFJKF! '= <W<DX BM=I F< JMI
BK J>KLFMG< LMA F<I <QDAC?HDCI BK ?>BSF< N&X F)MLLA>JE< J>=IHIDM=D Y JMFJKF<A
F< ZKQ B< DAM=IU<AD <= J>=IHBCAM=D JEMPK< ?>BSF< J>??< F)K= B<I CDMD BK LA>O
@FS?< B< *H<?M== Y ACI>KBA< [J>??< J)<ID F< JMI L>KA F< F>GHJH<F \>FU] L<A?<D
B< I)MWAM=JEHA BK JMFJKF <QLFHJHD< B< ^MAHM@F< =CJ<IIMHA< BM=I F< JMI B< 3H_<
4F>>B!
(< J>KLFMG< LA>L>IC BM=I J<DD< DESI< KDHFHI<AM B>=J FM ?V?< ?CDE>B>F>GH<
B< JMFJKF L>KA F)<=I<?@F< B<I CJ>KF<?<=DI! (< U>A?MFHI?< I<AM LMA MHFF<KAI CDM@FH
IKA K= ^>FK?< B< J>=DA`F< LA<=M=D <= J>?LD< F)CJ>KF<?<=D K=H <D F)CJ>KF<?<=D
@HBH?<=IH>==<F [J>??< J)<ID F< JMI L>KA F< F>GHJH<F +>@<_ N& &]! 0=< MLLA>JE<
MKQ ^>FK?<I T=HI I<AM LAH^HFCGHC<X F<I ZKQ MKQ H=D<AUMJ<I <=DA< F<I ?MHFF<I B<
JMFJKF I<A>=D JMFJKFCI LMA FM ACI>FKDH>= B)K= LA>@FS?< B< *H<?M== a F)CDMD B<
F)MAD B< J<DD< MLLA>JE< <ID BCDMHFFC Y FM I<JDH>= !"!N!
!" #$%$ &' ()%*$ &'+ %,,*-./'+ %01 2-(03'+ 456+
,-0* ('+ 780%$6-5+ /9,'*:-(680'+
()KDHFHIMDH>= B)K=< ?CDE>B< MKQ ^>FK?<I T=HI L>KA ACI>KBA< B<I IRIDS?<I
ERL<A@>FHPK<I B< F>HI B< J>=I<A^MDH>= M CDC LA>L>IC< <= Nb"b LMA c>BK=>^ d e!
(< BCDMHF B< FM ?CDE>B< <D B<I CPKMDH>=I MII>JHC<I I<AM LACI<=DC MK JEMLHDA< f!
6CM=?>H=IX F<I LAH=JHLMF<I CDML<I B< JMFJKF I>=D LACI<=DC<I HJH! '= CJAH^M=D F<I
CPKMDH>=I B< J>=I<A^MDH>= [B< FM ?MII< <D B< FM PKM=DHDC B< ?>K^<?<=D BM=I F<
JMI B<I CPKMDH>=I B< +MH=DO2<=M=D] L>KA K= ^>FK?< B< J>=DA`F<X >= L<KD ?>=DA<A
PK< F< IRIDS?< ERL<A@>FHPK< Y ACI>KBA< <ID B< FM U>A?< d" e g
ˆ
Ω
∂U
∂tdΩ +
ˆ
Γ
(−→F .−→n
)
dΓ = S [ !ff]
>h UX
−→F <D S A<LACI<=D<=D A<IL<JDH^<?<=D F< ^<JD<KA B<I ^MAHM@F<I J>=I<A^C<IX F<
^<JD<KA B<I ZKQ Y DAM^<AI F)H=D<AUMJ< <D F< ^<JD<KA B<I D<A?<I I>KAJ<X
−→n A<LACI<=D<
F< ^<JD<KA =>A?MF [I>ADM=D] Y F)H=D<AUMJ<X Ω FM IKAUMJ< <= LFM= BK ^>FK?< B<
J>=DA`F< <D Γ FM F>=GK<KA BK J>=D>KA BK ^>FK?< B< J>=DA`F<! ()<ILMJ< ?>BCFHIC <ID
BHIJACDHIC <= ^>FK?<I T=HI [CGMF<?<=D MLL<FCI J<FFKF<I >K ?MHFF<I B< JMFJKF] F>AI
B< FM J>=IDAKJDH>= BK ?>BSF<! .<DD< BHIJACDHIMDH>= A<ID< GC=CAMF<?<=D J>=IDM=D<
D>KD MK F>=G BK JMFJKF! ()CPKMDH>= [ !ff] <ID H=DCGAC< IKA JEMPK< ?MHFF< B< JMFJKF
<D F<I ZKQ I>=D ACCJAHDI BM=I F< ACUCA<=DH<F MII>JHC Y F)H=D<AUMJ< [^>HA 4HGKA< !N:]!
()CPKMDH>= BHIJACDHIC< I)CJAHD MF>AI g
! "#$%&'() *+ ,'-.) /&/0&12($%#&3-)
F
Fi,j
Li,j
G
n
α
x
y
Cellule j
Cellule i
!"#$% *+4 5 6789:; <= >?@A7@>= <B 7;C7BC <=D EBF G CH@AI=?J;7= =AI?= <=BF
:;@CC=D <= 7;C7BC+
Ai
(
Un+1i −U
ni
)
∆t+
∑
j
[
Pi,jFn+1/2
i,j Li,j +(
Sn+1/2
i,j
)
i
]
= 0 K*+LMN
OP Ai ?=>?9D=AI= C; DB?J;7= =A >C;A <= C; :;@CC= <= 7;C7BC iQ Uni =DI C; :OR=AA=
<= U DB? C; :;@CC= G CH@ADI;AI nQ ∆t =DI C= >;D <= I=:>DQ Pi,j =DI C; :;I?@7=
<= 78;AS=:=AI <= ?9J9?=AI@=CQ Fn+1/2
i,j =DI C; :OR=AA= <B EBF <;AD C; <@?=7I@OA
AO?:;C= G CH@AI=?J;7= =AI?= C=D >;D <= I=:>D n =I n + 1Q Li,j =DI C; COASB=B? <=
CH@AI=?J;7= =AI?= C=D :;@CC=D i =I jQ =I(
Sn+1/2
i,j
)
i?=>?9D=AI= C; 7OAI?@TBI@OA G C;
:;@CC= i <B I=?:= DOB?7= =AI?= C=D :;@CC=D i =I j+ 0= 7;C7BC <B I=?:= Fn+1/2
i,j =DI
?9;C@D9 >;? C; ?9DOCBI@OA <HBA >?OTCU:= <= (@=:;AA G CH@AI=?J;7= =AI?= C=D :;@CC=D
<= 7;C7BCQ <;AD C; <@?=7I@OA AO?:;C= G CH@AI=?J;7=+ "=II= ;>>?O78= >?9D=AI= AOV
I;::=AI C=D ;W;AI;S=D <= I?;@I=? <= C; :X:= :;A@U?= C=D 7;D OP C= >?OYC <= C;
DOCBI@OA =DI 7OAI@AB =I 7=BF OP @C =DI <@D7OAI@AB KA= A97=DD@I;AI >;? 7OAD9ZB=AI
>;D <H;>>?O78=D >;?I@7BC@U?=D >OB? C=D 78O7D OB C=D <@D7OAI@AB@I9D <= 7OAI;7IN =I
<= S;?;AI@? C= 7;?;7IU?= 7OAD=?W;I@J <=D 9ZB;I@OAD ;BF <9?@W9=D >;?I@=CC=D ?9DOV
CB=D [*\]+
!"! #$%$ &' ()%*$ &'+ %,,*-./'+ %01 2-(03'+ 4565+ ,-0*
('+ #70%$5-6+ /8,'*9-(570'+ :;
!"!# $% &'()*+,% -% ./%,011
0< =>?@ABCD ED *FDCG<< HD EIJ<FK =G> AG L?C@F<GFH?< E)M<D INMGKF?< GMO
EI>FPIDH =G>KFDAADH DK E)M<D L?<EFKF?< F<FKFGAD L?<HKG<KD =G> C?>LDGMO Q:RS T
∂U
∂t+∂F
∂x= S U !VWGX
U (x; 0) =
UL =?M> x < x0UR =?M> x > x0
U !VW@X
?Y UL DK UR H?<K ADH IKGKH ZGML[D DK E>?FK EM =>?@ABCD ED *FDCG<<! %MLM<D
L?<EFKF?< GMO AFCFKDH <)F<KD>PFD<K EG<H AG EIJ<FKF?< EM =>?@ABCD \ ADH P?AMCDH
IKG<K FC=AFLFKDCD<K HM==?HIH F<J<FH! (G >IH?AMKF?< E)M< KDA =>?@ABCD =GHHD =G>
=AMHFDM>H IKG=DH HMLLDHHFPDH!
]! (D =>?JA ED AG PG>FG@AD HM> L[GNMD LDAAMAD DHK >DL?<HKFKMI ^ =G>KF> EDH PGADM>H
L?<KD<MDH EG<H ADH LDAAMADH P?FHF<DH \ AG PGADM> C?_D<<D HM> AG LDAAMAD IKG<K
K?MKD`?FH L?<HD>PID! (G >DL?<HK>MLKF?< EM =>?JA DHK >IGAFHID ^ =G>KF> E)M<D
`?<LKF?< EIKD>CF<ID =AMH ?M C?F<H L?C=ADOD UMKFAFHG<K IPD<KMDAADCD<K AG
PGADM> ED AG PG>FG@AD EG<H ADH LDAAMADH P?FHF<DHX! %=>BH >DL?<HK>MLKF?< ED
AG PG>FG@AD HM> L[GNMD LDAAMADa AD =>?JA ED LDKKD ED><FB>D <)DHK =AMH <ILDHb
HGF>DCD<K L?<HKG<K =G> C?>LDGMO! (D =>?@ABCD ED *FDCG<< DHK GA?>H EFK
cdI<I>GAFHIc U,*dX!
! (DH =>?@ABCDH ED *FDCG<< dI<I>GAFHIH <)GECDKKD<K ZI<I>GADCD<K =GH ED
H?AMKF?< G<GA_KFNMD! (G >IH?AMKF?< EM ,*d =DMK K?MKD`?FH eK>D `GFKD =G>
M<D CIK[?ED ED K_=D =>IEFLKDM>bL?>>DLKDM> <ILDHHFKG<K =AMHFDM>H IKG=DH \
AG CIK[?ED ED 30+.(b/G<L?Lf QWVS D< DHK M< DODC=AD! 5A DHK IZGADCD<K
=?HHF@AD ED K>G<H`?>CD> AD =>?@ABCD ED *FDCG<< ZI<I>GAFHI D< M< =>?@ABCD
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! !- #$%&' () ./&'0$01 () *2/3)*)'0 (&'+ %& ($4)50$2' Ox ! ! ! ! ,
! ! #$%&' () ./&'0$01 () *2/3)*)'0 (&'+ %& ($4)50$2' Oy ! ! ! ! ,6
! !6 #$%&' ()+ 7245)+ &88%$./1)+ &/ 9/$() ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,6
! !6!" :1+/%0&'0) () %& 7245) );)451) 8&4 %) 82$(+ ! ! ! ! ! ,6
! !6!- :1+/%0&'0) () %& 41&50$2' (/ 72'( ! ! ! ! ! ! ! ! ! ! ,,
! !6! :1+/%0&'0) () %& 7245) () 74200)*)'0 +/4 %) 72'( ! ! ! ,,
! !6!6 :1+/%0&'0) ()+ 7245)+ () 84)++$2' ! ! ! ! ! ! ! ! ! ! ! ,<
! !6!, :1+/%0&'0) ()+ 7245)+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,=
! !, >./&0$2'+ +2/+ 724*) $'01?4&%) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,=
! !< @18&4&0$2' ()+ 1./&0$2'+ "A )0 -A ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,B
! != >./&0$2'+ +2/+ 724*) ($C14)'0$)%%) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,D
! !=!" >./&0$2'+ "A ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,D
! !=!- >./&0$2'+ -A ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! <-
!3 ?%%/$+).$' -'& &7&)9:'& -@%=.*)+,(& -*(& 1' $%A%$'()+'1 -'
1@+()'$A*/' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! B>
!
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
!"#$% &%'$(#)*#+!
1D 78 9:;;<9=:>= ? @ABC<D@8E8>= D>FGFE8>7F<>>8@
2D 78 9:;;<9=:>= ? @ABC<D@8E8>= HFGFE8>7F<>>8@
i 78 9:;;<9=:>= ? @AB@BE8>= G8 C:@CD@ IE:F@@8 :D ><8DGJ > i
i,j 78 9:;;<9=:>= ? @AF>=89K:C8 8>=98 @87 B@BE8>=7 G8 C:@CD@ > i 8= > j
0F7=8 G87 F>GFC87
,-./+0% &%'$(#)*#+! 1!#*2
Ai 1D9K:C8 8> ;@:> G8 @: E:F@@8 > i m2
b 0:9L8D9 :D EF9<F9 G8 @ABC<D@8E8>= mCr 2<EH98 G8 "<D9:>= ∅
F M8C=8D9 NDO G:>7 @: GF98C=F<> Ox
F′
M8C=8D9 NDO G:>7 @: GF98C=F<> Ox′
F′′
M8C=8D9 NDO G:>7 @: GF98C=F<> ><9E:@8 ? @AF>=89K:C8
FQ P@DO G8 QD:>=F=B G8 E<DR8E8>= ? =9:R897 @AF>=89K:C8 kgm s−1
Ff,xS Ff,y "<E;<7:>=87 G8 @: K<9C8 G8 K9<==8E8>= 7D9 @8 K<>G kgm s−2
FM P@DO G8 E:=FT98 ? =9:R897 @AF>=89K:C8 GD R<@DE8 G8 C<>=9U@8 kgFp P<9C8 G8 ;9877F<> :;;@FQDB8 :D R<@DE8 G8 C<>=9U@8 kgm s−2
Fp,xS Fp,y "<E;<7:>=87 G8 @: K<9C8 G8 ;9877F<> kgm s−2
FxS Fy "<E;<7:>=87 GD R8C=8D9 K<9C8 :;;@FQDB :D NDFG8 kgm s−2
G M8C=8D9 NDO G:>7 @: GF98C=F<> Oy
g $CCB@B9:=F<> L9:RF=:=F<>>8@@8 ms−2
h #:D=8D9 GA8:D mKxS Ky "<8VCF8>= G8 1=9FCW@89 m1/3 s−1
Li,j 0<>LD8D9 G8 @AF>=89K:C8 8>=98 @87 E:F@@87 G8 C:@CD@ > i 8= j mMx XD:>=F=B G8 E<DR8E8>= G:>7 @: GF98C=F<> Ox kgm s−1
m ,:778 GA8:D kg−→n M8C=8D9 7<9=:>= ><9E:@ ? @AF>=89K:C8 ∅
nxS ny "<E;<7:>=87 GD R8C=8D9
−→n ∅−→nxS
−→ny M8C=8D97 D>F=:F987 G:>7 @87 GF98C=F<>7 Ox 8= Oy ∅
Pr ,:=9FC8 G87 =89E87 G8 ;9877F<>7 :;;@FQDB7 @8 @<>L G87
F>=89K:C87 F>=B9F8D987
Pi,j ,:=9FC8 G8 ;:77:L8 GD 9BKB98>=F8@ L@<H:@ :D 9BKB98>=F8@
:==:CYB ? @AF>=89K:C8 8>=98 @87 C8@@D@87 > i 8= j
P %<FG7 GD R<@DE8 G8 C<>=9U@8 kgm s−2
q .BHF= D>F=:F98 G:>7 @: GF98C=F<> Ox m2 s−1
q′′
.BHF= D>F=:F98 7DFR:>= @: ;98EFT98 GF98C=F<> GD 9BKB98>=F8@
:77<CFB ? @AF>=89K:C8
m2 s−1
Rf,xS Rf,y "<E;<7:>=87 G8 @: 9B:C=F<> GD K<>G kgm s−2
Rh,xS Rh,y (:Z<>7 YZG9:D@FQD87 G8 @ABC<D@8E8>= mr .BHF= D>F=:F98 G:>7 @: GF98C=F<> Oy m2 s−1
r′′
.BHF= D>F=:F98 7DFR:>= @: 78C<>G8 GF98C=F<> GD 9BKB98>=F8@
:77<CFB ? @AF>=89K:C8
m2 s−1
S M8C=8D9 G87 =89E87 7<D9C8 G:>7 @8 9BKB98>=F8@ L@<H:@
!
!"#$%& '&()*+,-+$. /.+-0
S′
"#$%#&' (#) %#'*#) )+&'$# (,-) .# '/0/'#-%1#. ,))+$1/ ,&
$,-,. 23
S′′
"#$%#&' (#) %#'*#) )+&'$# (,-) .# '/0/'#-%1#. ,))+$1/ 4
.51-%#'0,$#
Sp,x6 Sp,y "#$%#&') (#) %#'*#) )+&'$# .1/) ,&7 0+'$#) (# 8'#))1+-)
S0,x6 S0,x′′ 6 S0,y 9+*8+),-%#) (# ., 8#-%# (& 0+-( ∅
Sf,x6 Sf,x′′ 6 Sf,y 9+*8+),-%#) (# ., 8#-%# (# 0'+%%#*#-% ∅
T "#$%#&' (#) %#'*#) .1/) 4 ., )/8,',%1+- (#) /$+&.#*#-%)
Te6 T′′
e "#$%#&') (#) %#'*#) (5/$:,-;# #-%'# .#) /$+&.#*#-%) 23 #%
<3
T′
u "#$%#&' (#) %#'*#) .1/) 4 ., )/8,',%1+- (#) /$+&.#*#-%)
8,' &-1%/ (# .+-;&#&'
t 9++'(+--/# (# %#*8) sU "#$%#&' (#) =,'1,>.#) (# .5/$+&.#*#-% (,-) .# '/0/'#-%1#.
;.+>,.
U′
"#$%#&' (#) =,'1,>.#) (# .5/$+&.#*#-% (,-) .# '/0/'#-%1#.
,))+$1/ 4 .51-%#'0,$#
u 9+*8+),-%# (& =#$%#&' =1%#))# (,-) ., (1'#$%1+- Ox ms−1
u′′
9+*8+),-%# (& =#$%#&' =1%#))# )&1=,-% ., 8'#*1?'#
(1'#$%1+- (& '/0/'#-%1#. ,))+$1/ 4 .51-%#'0,$#
ms−1
−→V "#$%#&' =1%#))# (# .5/$+&.#*#-% ms−1
v 9+*8+),-%# (& =#$%#&' =1%#))# (,-) ., (1'#$%1+- Oy ms−1
v′′
9+*8+),-%# (& =#$%#&' =1%#))# )&1=,-% ., )#$+-(#
(1'#$%1+- (& '/0/'#-%1#. ,))+$1/ 4 .51-%#'0,$#
ms−1
x 2?'# $++'(+--/# (5#)8,$# (,-) .# '/0/'#-%1#. ;.+>,. m
x′
2?'# $++'(+--/# (5#)8,$# (,-) .# '/0/'#-%1#. ,))+$1/ ,&
$,-,. 23
m
x′′
2?'# $++'(+--/# (5#)8,$# (,-) .# '/0/'#-%1#. ,))+$1/ 4
.51-%#'0,$#
m
y <?*# $++'(+--/# (5#)8,$# (,-) .# '/0/'#-%1#. ;.+>,. mz 9+%# (# ., )&'0,$# .1>'# mzb 9+%# (& 0+-( mα @-;.# (# '+%,%1+- (& '/0/'#-%1#. ;.+>,. ,& '/0/'#-%1#.
,))+$1/ 4 .51-%#'0,$#
rad
αx6 αy @-;.# #-%'# .5:+'1A+-%,.# #% .# 0+-( rad∆t B,) (# %#*8) (# $,.$&. s∆x B,) (5#)8,$# (# $,.$&. mφ B+'+)1%/ ∅
λ 9/./'1%/ (# 8'+8,;,%1+- (#) +-(#) ms−1
Γ C-%#'0,$# (& =+.&*# (# $+-%'D.# mΩ E&'0,$# #- 8.,- (& =+.&*# (# $+-%'D.# m2
ρ F,))# =+.&*1G&# (& H&1(# kgm−3
τx6 τy 9+-%',1-%#) (# $1),1..#*#-% )&' .# 0+-( kgm−1 s−2
I1)%# (#) =,'1,>.#)
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
0789 :;< =>?;@ABC< :; @;AA; ADE<; ;<A F; :GH;F=II;J;9A :789 =8ABF :; J=:GFB5
<KAB=9 @=8IFG 4.56. =IGLKAB=99;FM I;LJ;AAK9A 9=AKJJ;9A FK J=:GFB<KAB=9 :;<
G@=8F;J;9A< ;9 FBA JB9;8L <K9< KH=BL N :GO9BL O9;J;9A FK A=I=PLKIDB; :; @;
:;L9B;LM ;A @; KH;@ =8 <K9< :G>=L:;J;9A+ %KL KBFF;8L<M @;A =8ABF :=BA I;LJ;AAL;
:; J=:GFB<;L F;< G@=8F;J;9A< KH;@ 89; ILG@B<B=9 IL=@D; :; @;FF; =>A;98; KH;@ 89
J=:EF; >B:BJ;9<B=99;F @FK<<BQ8;M JKB< KH;@ :;< :8LG;< :; <BJ8FKAB=9< J=B9:L;<+
.K9< @; @K:L;M F; F=PB@B;F 1R6. SH=BL 1;@AB=9 6+*+*T ;<A F; @=:; >B:BJ;9<B=99;F
:; @=JIKLKB<=9+ )9 ;U;AM F; @D=BV :; @; F=PB@B;F @=JJ; LGCGL;9@; ;<A F; IF8<
F=PBQ8; :K9< FK J;<8L; =WM :789; IKLA BF K GAG :GH;F=IIG K8 <;B9 :8 FK>=LKA=BL;
:7K@@8;BF :; @;AA; ADE<; S;A IKL @=9<GQ8;9A F7K@@E< K8V @=:;< <=8L@;< ;<A CK@BFBAGT X
;A :7K8AL; IKLA F; @=:; :; @KF@8F 1R46. IL=I=<G B@B I=8LLK L;JIFK@;L 1R6.
K8ILE< :; FK <=@BGAG YB9P;L )9HBL=99;J;9A Z &9CLK<AL8@A8L;< SQ8B O9K9@; @;AA;
ADE<; :K9< F; @K:L; :; FK >=8L<; "&[() 9 6\\]5!*!TM K8 J=B9< I=8L FK J=:G5
FB<KAB=9 :; @;LAKB9;< @=9OP8LKAB=9< :7G@=8F;J;9A ILG<;9AG;< N FK <;@AB=9 *+4+ 0K
JGAD=:=F=PB; K8V H=F8J;< O9B< ;JIF=^G; IKL 1R6. <;LK @=9<;LHG;+
!" #$%&'()*)+,( -. /,(/'0) -' /,.01*2'
0;< A=I=PLKIDB;< I=8L F;<Q8;FF;< F; @=:; :; @KF@8F 1R6. <;LK L;JIFK@G IKL
F; 9=8H;K8 @=:; :; @KF@8F @=LL;<I=9:;9A N :;< G@=8F;J;9A< =LB;9AG< <=BA IKL
:;< @D;9K8V SFBA JB9;8LM @K9KFM ;A@+T <=BA IKL :;< KV;< L=8AB;L< ;9 _=9; 8L>KB9;
SH=BL [BP8L; *+4T+ .K9< F; 9=8H;K8 F=PB@B;FM F;< KV;< :7G@=8F;J;9A< <;L=9A K ILB=LB
J=:GFB<G< <=8< C=LJ; 89B:BJ;9<B=99;FF;+ 0;< @=9`8;9@;< ;A F;< :G`8;9@;< ILG5
<;9AK9A :;< @=9OP8LKAB=9< :7G@=8F;J;9A Q8B <=9A PG9GLKFF;J;9A B9@=JIKAB>F;
KH;@ 89; J=:GFB<KAB=9 89B:BJ;9<B=99;FF;M F;8L J=:GFB<KAB=9 <=8< C=LJ; @=8IFG;
4.56. 97;<A IKL @=9<GQ8;9A IK< ;9HB<KPG; :K9< F; @K:L; :; @;AA; ADE<;+
.K9< F; @K< PG9GLKFM FK @GFGLBAG :;< =9:;< ;<A L;FKABH;J;9A B<=AL=I; ;A F78<KP;
:789 JKBFFKP; GPKF;J;9A B<=AL=I; ;<A IKL @=9<GQ8;9A @=9<;BFFG I=8L =>A;9BL 89;
>=99; ILG@B<B=9 98JGLBQ8;+ .K9< :; 9=J>L;8V @K<M FK ILG<;9@; :; @K9K8V :;
CKB>F; FKLP;8L SI=8HK9A KFF;L ?8<Q87K8 JEAL;T BJI=<;LKB :78ABFB<;L :; 9=J>L;8<;<
JKBFF;< I=8L J=:GFB<;L @=LL;@A;J;9A 89; _=9; :7GA8:;+ ";AA; LEPF; :; JKBFFKP;
;<A :=9@ PG9GLKF;J;9A 9GPFBPG; ;A F;< JKBFF;< :; @KF@8F< <=9A GABLG;< :K9< FK
:BL;@AB=9 :; @;< @K9K8V+ .K9< F; @K< :789 <@DGJK 98JGLBQ8; ;VIFB@BA;M BF @=9HB;9A
IKL KBFF;8L< :; HGLBO;L FK @=9:BAB=9 :; <AK>BFBAG <8BHK9A; a
|Cr| =|λ|
∑
j Li,j∆t
Ai≤ 1 pour tout i S*+4T
=W Cr L;ILG<;9A; F; 9=J>L; :; "=8LK9AM λ FK IF8< PLK9:; @GFGLBAG :7=9:; :K9<
FK :BL;@AB=9 9=LJKF; N F7B9A;LCK@;M ∆t F; IK< :; A;JI< :; @KF@8F ;A
∑
j Li,j FK
F=9P8;8L @8J8FG; :;< B9A;LCK@;< :; FK JKBFF; i ;A Ai FK <8LCK@; ;9 IFK9 :; FK
JKBFF; i+ ";AA; @=9:BAB=9 :; <AK>BFBAG I;LJ;A :; :G:8BL; F; IK< :; A;JI< :;
@KF@8F ;9 C=9@AB=9 :;< K8AL;< IKLKJEAL;< a
∆t ≤ Ai
|λ|∑
j Li,jS*+6T
Axe d’écoulement orienté par le bâti
Axe d’écoulement orienté par le lit mineur
Zone bâtie empêchant les écoulements
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
Ecoulement 1DEcoulement 2DE
cou
lem
en
t
de
su
rfa
ceE
cou
lem
en
t p
rofo
nd
Ecoulement 2D
A
B
A BStructure tourbillonnaire Zone morte C Court-circuit
C
!"#$% *+6 7 189:;8:9< =:>>?=@< A< BC@;?:B<D<E8 AFE= BF AG9<;8G?E 89FE=H<9=FB<
7 #I>+ *+6 J 0C<F: ><:8 K89< F==GDGB@< L :E M:GA< GE;?D>9<==GNB< OBF DF==<
H?B:DGP:< ρ <=8 ;?E=8FE8<Q+
7 #I>+ *+* J 0F 9@>F98G8G?E A<= HG8<==<= <=8 :EGR?9D< =:9 BF H<98G;FB< AFE=
:E @B@D<E8 6.+ "<88< SI>?8ST=< <=8 ;?:9FDD<E8 <D>B?I@< AFE= B<= ;?A<=
A< ;FB;:B NGAGD<E=G?EE<B= <8 <E >F98G;:BG<9 AFE= 1U6.+ )BB< F A?E; @8@
9<>9G=< >?:9 B< A@H<B?>><D<E8 A< 1U46.+
7 #I>+ *+V J 0F =<;8G?E A: ;FEFB 4. <=8 =:>>?=@< A< R?9D< 9<;8FEW:BFG9< X
BF ><E8< A<= @B@D<E8= A< ;FB;:B 4. <=8 >F9 ;?E=@P:<E8 FBGWE@< FH<; BCFY<
A: ;FEFB 4.+ .FE= :E ;FEFB A< R?9D< 9<;8FEW:BFG9<Z B<= @P:F8G?E= >?:9
;<98FGE= >S@E?DTE<= =< =GD>BG[<E8 OE?8FDD<E8 >?:9 B<= 8<9D<= =?:9;<=QZ
DFG= BF D@8S?A< WB?NFB< 9<=8< BF DKD<+ "<88< SI>?8ST=< =GD>BG[;F89G;< E<
D?AG[FE8 >F= BF D@8S?A?B?WG< WB?NFB<Z <BB< F A?E; @8@ :8GBG=@<+ 0C<Y8<E5
=G?E A: ;?A< 1U46. >?:9 A<= ;?E[W:9F8G?E= E?E 9<;8FEW:BFG9< ><:8 K89<
<EHG=FW@< :B8@9G<:9<D<E8+
7 #I>+ *+ J 0C@;?:B<D<E8 :EGAGD<E=G?EE<B ><:8 K89< =@>F9@ <E A<:Y \?E<=
=G8:@<= 9<=><;8GH<D<E8 F: A<==:= <8 <E A<==?:= A< BF >BFGE< ACGE?EAF8G?E
OH?G9 ]GW:9< *+6Q+ )E FN=<E;< AC@;?:B<D<E8 AFE= BF >BFGE< ACGE?EAF8G?EZ
BC@;?:B<D<E8 89FE=H<9=FB AFE= B< ;FEFB 4. <=8 E@;<==FG9<D<E8 E:B+ 0?9=
AC:E A@N?9A<D<E8Z :E< =89:;8:9< 8?:9NGBB?EEFG9< =< R?9D< AFE= BF \?E<
GER@9G<:9< DFG= A?E8 BF HG8<==< 89FE=H<9=FB< D?I<EE< <=8 E:BB<+
%F9 FGBB<:9=Z ?E A@[EG8 B<= 8<9D<= =:GHFE8= OH?G9 ]GW:9< *+*Q J
7 !"#$%&'# !"($ #)$# J BGDG8< <E89< BC@B@D<E8 :EGAGD<E=G?EE<B <8 BC@B@D<E8
NGAGD<E=G?EE<B
7 !"#$%&'# #*"($ #)$# J BGDG8< A: H?B:D< A< ;?E89^B< X B< ;F= @;S@FE8Z <BB<
><:8 K89< A@E?DD@< 4. ?: 6. =:GHFE8 B< 8I>< AC@;?:B<D<E8 AFE= BC@B@D<E8
P:C<BB< A@BGDG8<
7 &*# +) '&!&, -. J AG9<;8G?E A< BC@;?:B<D<E8 :EGAGD<E=G?EE<B+ "<8 FY< <=8
?9G<E8@ F9NG89FG9<D<E8 X R′
(
O;x′
; y′
; z)
Z B< 9@R@9<E8G<B F==?;G@ F: ;FEFB 4.
X
Y
Z
Ω
Γ
interface extérieure 1D
interface
extérieure 2D
interface
intérieure
axe du canal 1D
Ox′
R (O;x; y; z)
Γ Ω
δt = t2 − t1
δt
x
y
x’
y’
x
y
y’
x’Element 1D
V
V
u
v
v’
u’
Element 2D
R (O;x; y; z)
R′
(
O;x′
; y′
; z)
m (t2)−m (t1) = FM
m (t) tFM m (t)
m (t) =
ˆ
Ω
(ρh) (x, y, t) dΩ
ρ hh = z−zb z zb
f(x, y, t)g(x, y, t) =(fg)(x, y, t) f g FM
Γ δt
FM =
ˆ t2
t1
ˆ
Γ
(
ρh(−→V
−→n))
(x, y, t) dΓdt
−→n Γ−→VOx Oy
! ! "#$%&'()* +, -.(-%/%&'() 0
1234 56789:92 5;35 <= >;9?2 53:@=A42 B
FM =
ˆ t2
t1
ˆ
Γ
(ρh (unx)) (x, y, t) dΓdt C !DE
+
ˆ t2
t1
ˆ
Γ
(ρh (vny)) (x, y, t) dΓdt
F2 G:<=A H2 ?=552 539 <2 @;<3?2 H2 8;A49I<2 56789:4 B
ˆ
Ω
[(ρh) (x, y, t2)− (ρh) (x, y, t1)] dΩ =
ˆ t2
t1
ˆ
Γ
(ρh (unx)) (x, y, t) dΓdt+
ˆ t2
t1
ˆ
Γ
(ρh (vny)) (x, y, t) dΓdt C !JE
! !" #$%&' () *+&',$,- () ./+0).)', (&'1 %& ($2)3,$/'
Ox
F= @=9:=4:;A H2 K3=A4:47 H2 ?;3@2?2A4 539 <2 @;<3?2 H2 8;A49I<2 254 7L=<2
M <= 5;??2 H25 N3O 2A49=A4 24 5;94=A4 1=9 <25 :A429>=825 24 H25 >;9825 562O29P=A4
539 <2 N3:H2! F2 G:<=A H2 K3=A4:47 H2 ?;3@2?2A4 1234 A;4=??2A4 Q492 74=G<: 2A
8;A5:H79=A4 3A:K32?2A4 <= K3=A4:47 H2 ?;3@2?2A4 H=A5 <= H:9284:;A Ox 24 <=
9753<4=A42 H25 >;9825 53:@=A4 82442 ?Q?2 H:9284:;A B
Mx (t2)−Mx (t1) = FQ +∑
Fx C !RE
;SMx 9219752A42 <= K3=A4:47 H2 ?;3@2?2A4 H=A5 <= H:9284:;A Ox 8;A42A32 H=A5
<2 @;<3?2 H2 8;A49I<2 M 3A :A54=A4 tT FQ <2 N3O H2 K3=A4:47 H2 ?;3@2?2A4
49=@295=A4 <25 :A429>=825 H3 @;<3?2 H2 8;A49I<2 24
∑
Fx <= 9753<4=A42 H25 >;9825
562O298=A4 539 <2 N3:H2 H=A5 <= H:9284:;A Ox! Mx (t) 56789:4 B
Mx (t) =
ˆ
Ω
(ρhu) (x, y, t) dΩ C !UE
F2 N3O H2 K3=A4:47 H2 ?;3@2?2A4 M 49=@295 <6:A429>=82 Γ 12AH=A4 <= H3972 δt56789:4 B
FQ =
ˆ t2
t1
ˆ
Γ
(
ρhu(−→V
−→n))
(x, y, t) dΓdt C !VWE
,A 789:@=A4 <25 N3O 5;35 <= >;9?2 H25 8;?1;5=A425 53:@=A4 <25 H:9284:;A5 Ox 24
OyT <67K3=4:;A 56789:4 B
FQ =
ˆ t2
t1
ˆ
Γ
(ρhu (unx)) (x, y, t) dΓdt C !VVE
+
ˆ t2
t1
ˆ
Γ
(ρhu (vny)) (x, y, t) dΓdt
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
07 89:;< =7 >?;<@9@A =7 BC?D7B7<@ =;<E :; =9F7G@9C< Ox E7 @F;=?9@ H;F :IA>?;@9C<
E?9D;<@7 J
ˆ
Ω
[(ρhu) (x, y, t2)− (ρhu) (x, y, t1)] dΩ =
ˆ t2
t1
ˆ
Γ
(ρhu (unx)) (x, y, t) dΓdt+
ˆ t2
t1
ˆ
Γ
(ρhu (vny)) (x, y, t) dΓdt+∑
Fx K*+46L
CM
∑
Fx F7HFAE7<@7 :I9<@ANF;:7 =7E OCFG7E 7P7FGA7E E?F :7 Q?9=7 =;<E :; =9F7G@9C<
Ox H7<=;<@ δt+
! ! "#$%& '( )*%&+#+, '( -.*/(-(&+ '%&0 $% '#1(2+#.&
Oy
.7 :; BRB7 B;<9SF7T :7 Q?P =7 >?;<@9@A =7 BC?D7B7<@ =;<E :; =9F7G@9C< OyEIAGF9@ J
ˆ
Ω
[(ρhv) (x, y, t2)− (ρhv) (x, y, t1)] dΩ =
ˆ t2
t1
ˆ
Γ
(ρhv (unx)) (x, y, t) dΓdt+
ˆ t2
t1
ˆ
Γ
(ρhv (vny)) (x, y, t) dΓdt+∑
Fy K*+4*L
CM
∑
Fy F7HFAE7<@7 :I9<@ANF;:7 =7E OCFG7E 7P7FGA7E E?F :7 Q?9=7 =;<E :; =9F7G@9C<
Oy H7<=;<@ δt+
! !3 "#$%& '(0 4.12(0 %55$#)*,(0 %* 6*#'(
0; FA;:9E;@9C< =I?< 89:;< =7E OCFG7E 7P7FGA7E E?F :7 Q?9=7 H7FB7@ =I7PH:9G9@7F
:7E @7FB7E
∑
Fx 7@
∑
Fy+ 07E OCFG7E GC<E9=AFA7E EC<@ J :7 HC9=ET :; FA;G@9C< =?
OC<=T :7E OCFG7E =7 OFC@@7B7<@E E?F :7 OC<= 7@ :7E OCFG7E =7 HF7EE9C<+ 0; FA;G@9C<
=? OC<= 7E@ CF97<@A7 H7FH7<=9G?:;9F7B7<@ U G7 =7F<97F @;<=9E >?7 :7E GC<@F;9<@7E
=7 G9E;9::7B7<@ U :ICF9N9<7 =7E OCFG7E =7 OFC@@7B7<@ EC<@ CF97<@A7E H;F;::S:7B7<@
;? OC<=+ %C?F G7F@;9<7E ;HH:9G;@9C<ET =I;?@F7E OCFG7E H7?D7<@ AD7<@?7::7B7<@ R@F7
HF9E7E 7< GCBH@7 K9<Q?7<G7 =? D7<@T 7@G+L+
! !"!# $%&'()*+), -, (* ./01, ,2,01%, 3*0 (, 3/4-&
07 HC9=E =? Q?9=7 GC<@7<? =;<E :7 DC:?B7 =7 GC<@FV:7 EI7P7FG7 H;F =AW<9@9C<
D7F@9G;:7B7<@ 7@ D7FE :7 8;ET 9: 7E@ =C<<A H;F J
P =
ˆ
Ω
g (ρh) (x, y, t) dΩ K*+4!L
P
Rf
α
gOx Oy
Rf,x Rf,y Ox Oy
Rf,x =
ˆ
Ω
g (ρh) (x, y, t) tanαxdΩ
=
ˆ
Ω
g (ρh) (x, y, t)S0,xdΩ
Rf,y =
ˆ
Ω
g (ρh) (x, y, t) tanαydΩ
=
ˆ
Ω
g (ρh) (x, y, t)S0,ydΩ
αx S0,xOx αy S0,y Oy
Ox Oy
Ff,x =´
Ω−τxdΩ
Ff,y =´
Ω−τydΩ
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
78 τx 9:;9<=:>?: @A B7>?9AC>?: D: BC=AC@@:E:>? DA>= @A DC9:B?C7> Ox F9:=;:B?CG:5
E:>? τy DA>= @A DC9:B?C7> OyH+ -> E7>?9: IJ: @:= B7>?9AC>?:= D: BC=AC@@:E:>?
;:JG:>? K?9: :=?CE<:= ;A9 L 6M N
τx = ρgRh,xSf,x
τy = ρgRh,ySf,y
F*+4OH
78 Rh,x :? Sf,x 9:;9<=:>?:>? @: 9AP7> QPD9AJ@CIJ: D: @R<B7J@:E:>? :? @A ;:>?:
D: @A @CS>: D: BQA9S: DA>= @A DC9:B?C7> Ox F9:=;:B?CG:E:>? Rh,y :? Sf,y DA>= @A
DC9:B?C7> OyH+ %A9 AC@@:J9=T 7> :=?CE: @A ;:>?: D: @A @CS>: D: BQA9S: DA>= BQABJ>:
D:= DC9:B?C7>= DR<B7J@:E:>? ;A9 :U?:>=C7> D: @A V79EJ@: D: ,A>>C>S51?9CBW@:9 :>
D:JU DCE:>=C7>=+
Sf,x =
√u2 + v2
K2xR
4/3h,x
u
Sf,y =
√u2 + v2
K2yR
4/3h,y
v
F*+4XH
78 Kx 9:;9<=:>?: @: B7:YBC:>? D: 1?9CBW@:9 DA>= @A DC9:B?C7> Ox F9:=;:B?CG:E:>?
Ky DA>= @A DC9:B?C7> OyH+ %7J9 D:= <B7J@:E:>?= >A?J9:@=T C@ :=? 9:@A?CG:E:>?
DCYBC@: D: DC=?C>SJ:9 @:= B7:YBC:>?= D: V97??:E:>? DA>= @:= DCZ<9:>?:= DC9:B?C7>=
D: @R:=;AB:+ .A>= @A =JC?:T @:= ?:9E:= Kx :? Ky =7>? =J;;7=<= CD:>?CIJ:= N Kx =Ky = K+ -> 7[?C:>? D7>B IJ: @:= V79B:= D: V97??:E:>? DA>= @:= DC9:B?C7>= Ox :?
Oy ;:JG:>? =R<B9C9: N
Ff,x = −ˆ
Ω
ρg
√u2 + v2
K2R1/3h,x
udΩ F*+4\AH
Ff,y = −ˆ
Ω
ρg
√u2 + v2
K2R1/3h,y
vdΩ F*+4\[H
! !"!" #$%&'()*(+ ,+% -./0+% ,+ 1/+%%2.*
0:= V79B:= D: ;9:==C7> =RA;;@CIJ:>? ;:9;:>DCBJ@AC9:E:>? AJU C>?:9VAB:= DJ
G7@JE: D: B7>?9]@:+ 0: ;97^@ D:= ;9:==C7>= <?A>? =J;;7=< QPD97=?A?CIJ: F#P;+
*+4HT @A ;9:==C7> _ J>: ;97V7>D:J9 z D7>><: =R<B9C? p(z) = ρgz+ 0RC>?:>=C?< D: @A
V79B: D: ;9:==C7> =R:U:9`A>? =J9 @: aJCD: =J9 @RC>?:9VAB: Γ =R<B9C? A@79= N
Fp =
ˆ
Γ
(
ˆ h(x,y,t)
0
ρgzdz
)
dΓ
=
ˆ
Γ
(
[
1
2ρgz2
]h(x,y,t)
0
)
dΓ
=
ˆ
Γ
1
2
(
ρgh2)
(x, y, t) dΓ F*+6bH
! ! "#$%&'()* +, -.(-%/%&'() 01
-234 56789:4 9;< =2>?2<7@6;< A7@< 9;< A:4;=6:2@< Ox ;6 Oy A; 97 45<3967@6; A;<
B24=;< A; ?4;<<:2@C :9 =2@D:;@6 A; 4;D;@:4 E 9F5=4:634; D;=624:;99; G
−→Fp = − Fp
−→n H !IJK
'9 ;<6 7924< ?2<<:89; A; B7:4; 7??747L64; 9;< =2>?2<7@6;< A;< B24=;< A; ?4;<<:2@
A7@< =M7N3; A:4;=6:2@ AF;<?7=; G
Fp,x =−→Fp
−→nx = −ˆ
Γ
1
2
(
ρgh2)
(x, y, t)−→n
−→nxdΓ H !II7K
Fp,y =−→Fp
−→ny = −ˆ
Γ
1
2
(
ρgh2)
(x, y, t)−→n
−→nydΓ H !II8K
! !"!# $%&'()*+), -,& ./01,&
,@ =2@<:A547@6 97 >7<<; D293>:N3; ρ =2>>; =2@<67@6; HOP?! !IKC 9;< 45Q
<3967@6;< A;< B24=;< <34 9; D293>; A; =2@64R9; A7@< 9;< A:4;=6:2@< Ox ;6 Oy<F5=4:D;@6 G
∑
Fx = ρ (Rf,x + Ff,x + Fp,x) H !I K
∑
Fy = ρ (Rf,y + Ff,y + Fp,y) H !ISK
! !" #$%&'()*+ +)%+ ,)-./ (*'01-&2/
&236;< 9;< B24=;< 7P7@6 565 :A;@6:T5;<C :9 ;<6 ?2<<:89; A; 455=4:4; 9;< 8:97@< A;
>7<<; ;6 A; N37@6:65 A; >23D;>;@6! U;< 5N376:2@< H !1KC H !JIK ;6 H !J K <2@6
V5@5479;>;@6 5=4:6;< <23< B24>; D;=624:;99; G
ˆ
Ω
ρ [U (t2)−U (t1)] dΩ +
ˆ t2
t1
ˆ
Γ
ρ [Fnx +Gny] dΓdt =
ˆ t2
t1
ρSdt+
ˆ t2
t1
ρSp,xdt+
ˆ t2
t1
ρSp,ydt H !I07K
2W
U =
hhuhv
H !I08K
F =
huhu2
huv
H !I0=K
G =
hvhuvhv2
H !I0AK
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
S =
0Rf,x + Ff,x
Rf,y + Ff,y
7*+6 89
Sp,x =
0Fp,x
0
7*+6 :9
Sp,y =
00Fp,y
7*+6 ;9
08< =8>?8< <@A>B8 CDDEFGAH< <A> EIFJ=8>:CB8 Γ 7Fp,x 8= Fp,y9 <@J= ;HJH>CE8?8J=
>8;>@ADH< CK8B E8< =8>?8< L8 MAN F 8= G+ %C> CFEE8A><O @J D8A= <F?DEFP8> EIHB>F5
=A>8 7*+6 9 8J LFKF<CJ= DC> EC ?C<<8 K@EA?FGA8 <ADD@<H8 B@J<=CJ=8 7#QD+ *+69 R
ˆ
Ω
[U (t2)−U (t1)] dΩ +
ˆ t2
t1
ˆ
Γ
[Fnx +Gny] dΓdt =
ˆ t2
t1
ˆ
Ω
SdΩdt 7*+6S9
TF8J GA8 E@A>L8< U HB>F>8O B8< HGAC=F@J< <@J= KCECVE8< ?W?8 E@><GA8 E8< KC>FCVE8<
L8 EIHB@AE8?8J= <@J= LF<B@J=FJA8<+ "8==8 HB>F=A>8 8<= CDD8EH8 EC :@>?8 X:CFVE8X
L8< HGAC=F@J< L8 D>@DC;C=F@J+
! !" #$%&'&()*+ ,-. $/0&()*+. 12 -( 32
$PJ L8 B@J<FLH>8> E8< L8AN =QD8< LIHB@AE8?8J= 7AJF5 8= VFLF?8J<F@JJ8E9O @J
BY8>BY8 C >HHB>F>8 E8 <Q<=Z?8 LIHGAC=F@J< 7*+6S9 D@A> BYCGA8 HB@AE8?8J=+ )J
HB>FKCJ= R
ˆ
Ω
UdΩ =
ˆ
Ω1D
U1DdΩ1D +
ˆ
Ω2D
U2DdΩ2D 7*+6[9
8= 8J FL8J=FPCJ= E8< MAN U =>CK8>< E8< FJ=8>:CB8< 8N=H>F8A>8< L8 BYCGA8 <@A<5K@EA?8
L8 B@J=>\E8 7Γ1D 8= Γ2D9O @J >HHB>F= EIHGAC=F@J 7*+6S9 <@A< EC :@>?8 R
ˆ
Ω1D
[U1D (t2)−U1D (t1)] dΩ1D+
ˆ t2
t1
ˆ
Γ1D
[Fnx +Gny] dΓdt =
ˆ t2
t1
ˆ
Ω1D
S1DdΩ1Ddt+
ˆ t2
t1
Tdt 7*+6!C9
ˆ
Ω2D
[U2D (t2)−U2D (t1)] dΩ2D+
ˆ t2
t1
ˆ
Γ2D
[Fnx +Gny] dΓdt =
ˆ t2
t1
ˆ
Ω2D
S2DdΩ2Ddt−ˆ t2
t1
Tdt 7*+6!V9
! ! "#$%&'()* +, -.(-%/%&'() 01
23 T4 S1D =[
S1,1 S1,2 S1,3
]T56 S2D =
[
S2,1 S2,2 S2,3
]T75879:
;5<65<6 75;85=6>?5@5<6 A5; 657@5; BC9=DE<F5 5<675 A5; B5GH 9=2GA5@5<6;4 A5 657@5
;2G7=5 ;G7 A5 ;2G;:?2AG@5 B5 =2<67IA5 J+ 56 A5 657@5 ;2G7=5 ;G7 A5 ;2G;:?2AG@5
B5 =2<67IA5 K+ L
S1D =
0Rf,x 1D + Ff,x 1D
Rf,y 1D + Ff,y 1D
=
0
gh1DS0,x 1D − g√
u21D + v21D
K21DR
1/3h,x 1D
u1D
gh1DS0,y 1D − g√
u21D + v21D
K21DR
1/3h,y 1D
v1D
M !K1N
S2D =
0Rf,x 2D + Ff,x 2D
Rf,y 2D + Ff,y 2D
=
0
gh2DS0,x 2D − g√
u22D + v22D
K22DR
1/3h,x 2D
u2D
gh2DS0,y 2D − g√
u22D + v22D
K22DR
1/3h,y 2D
v2D
M ! ON
S1D,2 56 S1D,3 75879;5<65<6 A5; =2@82;E<65; B5 AE 79E=6>2< BG P2<B 56 B5; P27=5;
B5 P72665@5<6 75;85=6>?5@5<6 ;G>?E<6 ACEH5 Ox 56 ACEH5 Oy! -E7 DQ826DR;54 AE
85<65 BG P2<B 5;6 EA>F<95 E?5= ACEH5 BG =E<EA J+ MSQ8! !TN U AE =2@82;E<65 B5
AE 79E=6>2< BG P2<B BE<; AE B>75=6>2< 67E<;?57;EA5 V ACEH5 BG =E<EA J+ 5;6 8E7
=2<;9WG5<6 <GAA5! +E<; A5 79P975<6>5A R′
(
O;x′
; y′
; z)
B9X<> V AE XFG75 !K4 A5
?5=65G7 B5; 657@5; ;2G7=5 82G7 AC9=2GA5@5<6 G<>B>@5<;>2<<5A 85G6 B2<= ;C9=7>75
;2G; AE P27@5 L
S′
1D =
0
ghS0,x′ − g
√
(u′)2+ (v′)
2
K2R1/3
h,x′
u′
−g
√
(u′)2+ (v′)
2
K2R1/3
h,y′
v′
M ! JN
! !" #$%&'()*+ +)%+ ,)-./ 0(12-/*'(/33/
(< =D57=D5 V 96EYA>7 AE P27@5 B>Z975<6>5AA5 B5; 9WGE6>2<; M !K[N 82G7 =DEWG5
;2G;:?2AG@5 B5 =2<67IA5!
! !"!# $%&'()*+, #-
\C9=7>6G75 B5; 9WGE6>2<; B5 8728EFE6>2< G<>B>@5<;>2<<5AA5; PE>6 E885A EGH
DQ826DR;5; ;G>?E<65; M?2>7 ]>FG75 !TN L
^ SQ8! !_ L AE AE7F5G7 B5 AC9A9@5<6 5;6 PE>YA5 B5?E<6 ;E A2<FG5G7 L dΩ1D ≈bdx
′
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
7 #89+ *+: ; <=>?@A<BCBDE BFE ADGHGCBDFG@DDB< ; Fnx +Gny = F′
nx′
7 #89+ *+I ; <BF JAK F@DE ?@DFELDEF <B <@DM HB ?NLOAB FB?EG@D BD EPLQBPF HB
<=>?@A<BCBDE ;
´
Γ1D(x′)F′
(
x′
)
dΓ ≈(
bF′
)(
x′
)
"@C9EB EBDA HBF N89@ENRFBF ?G5HBFFAFS <=>OALEG@D T*+6ILU F=>?PGE ;
ˆ x′
2
x′
1
b [U1D (t2)−U1D (t1)] dx′
+
ˆ t2
t1
[(
bF′
)(
x′
2
)
−(
bF′
)(
x′
1
)]
dt =
ˆ t2
t1
ˆ x′
2
x′
1
(
bS′
1D
)
dx′
dt+
ˆ t2
t1
Tdt T*+*6U
)D FA99@FLDE OAB <BF QLPGLV<BF H=>?@A<BCBDE F@DE ?@DEGDABF BE H>PGQLV<BF FAP
<=><>CBDE ADGHGCBDFG@DDB<S G< BFE 9@FFGV<B HB P>L<GFBP AD H>QB<@99BCBDE BD F>PGB
HB 'L8<@P LA 9PBCGBP @PHPB ;
limt1→t2
[U1D (t2)−U1D (t1)] ≈ˆ t2
t1
∂U1D
∂tdt T*+**U
limx′
1→x′
2
[(
bF′
)(
x′
2
)
−(
bF′
)(
x′
1
)]
≈ˆ x
′
2
x′
1
∂bF′
∂x′dx
′
T*+*WU
0B 9LF H=BF9L?B >ELDE ?@DFGH>P> ?@CCB GDXDGCBDE 9BEGES <=>OALEG@D T*+*6U 9BAE
L<@PF FB P>>?PGPB F@AF <L Y@PCB FAGQLDEB ;
ˆ x′
2
x′
1
ˆ t2
t1
∂bU1D
∂tdtdx
′
+
ˆ t2
t1
ˆ x′
2
x′
1
∂bF′
∂x′dx
′
dt =
ˆ t2
t1
ˆ x′
2
x′
1
(
bS′
1D
)
dx′
dt+
ˆ t2
t1
ˆ x′
2
x′
1
T′
udx′
dt T*+*ZU
@[ Tu PB9P>FBDEB <L CLEPG?B HBF EBPCBF H=>?NLDMB BDEPB <BF >?@A<BCBDEF ADG5 BE
VGHGCBDFG@DDB< 9LP ADGE> HB <@DMABAP HA Q@<ACB 4.+ 0BF HAP>BF BE <BF <@DMABAPF
>ELDE GDXDGCBDE 9BEGEBFS G< BFE 9@FFGV<B HB FA99PGCBP <BF GDE>MPL<BF 9@AP >ELV<GP <L
Y@PCB HG\>PBDEGB<<B HBF >OALEG@DF HB 9P@9LMLEG@D FAP <B F@AF5Q@<ACB HB ?@DEP]<B
4. ;
∂bU1D
∂t+∂bF
′
∂x′= bS1D +T
′
u T*+* U
0BF QB?EBAPF U1DS F′
BE S1D F@DE M>D>PL<BCBDE PBY@PCA<>F BD YLGFLDE GD5
EBPQBDGP <B H>VGE ADGELGPB 9<AE]E OAB <L QGEBFFB HB <=>?@A<BCBDE+ "NLOAB QB?EBAP
HB T*+* U F=>?PGE L<@PF ;
U′
1D =
h1Dq1Dr1D
T*+*:LU
! ! "#$%&'()* +, -.(-%/%&'() 01
F′
=
q1Dq21Dh1D
+1
2g.h21D
q1Dr1Dh1D
2 ! 345
S′
1D =
0
gbh1D
S0,x′ −√
q21D + r21D
h21DK2R
1/3
h,x′
q1D
gbh1D
−√
q21D + r21D
h21DK2R
1/3
h,y′
r1D
2 ! 365
T′
u = Pr +Te 2 ! 375
=
Pr,1
Pr,2
Pr,3
+
Te,1
Te,2
Te,3
89 Pr :;< =: >:6<:?@ 7:; A8@6:; 7: B@:;;C8D EBB=CF?G:; ;?@ =: H?C7: E? DC>:E?
7:; CD<:@AE6:; CD<G@C:?@:; :< Te =:; <:@I:; 7JG6KEDL: :D<@: =:; G68?=:I:D<; ?DCM
:< 4C7CI:D;C8DD:=! NE B@:ICO@: 68IB8;ED<: 7: Pr 68@@:;B8D7ED< P =JGF?E<C8D
7: 68D;:@>E<C8D 7: =E IE;;: ;?@ =: >8=?I: 7: 68D<@Q=: ?DC7CI:D;C8DD:=R =: 4C=ED
7:; A8@6:; EBB=CF?G:; ;?@ =: H?C7: DJCD<:@>C:D< BE; :< 6:<<: 68IB8;ED<: :;< 78D6
D?==: S
Pr,1 = 0 2 ! T5
Pr,2 :< Pr,3 B:?>:D< U<@: 7G<:@ICDG; :D 68D;C7G@ED< ?D G68?=:I:D< CII84C=:!
+ED; 6: 6E;R =:; <:@I:; 7: H?V :D<@: =:; 7:?V G68?=:I:D<; ;8D< D?=;R 7: IUI:
F?: =E >C<:;;: 7: =JG68?=:I:D< ?DC7CI:D;C8DD:= S
∂bF′
∂x′= S
′
1D +T′
u
∂
∂x′
01
2gbh21D
0
=
0gbh1DS0,x′
0
+
0Pr,2
Pr,3
+
000
2 ! W5
N:; <:@I:; Pr,2 :< Pr,3 B@:DD:D< 78D6 =:; :VB@:;;C8D; ;?C>ED<:; S
Pr,2 =∂
∂x′
(
1
2gbh21D
)
− gbh1DS0,x′ 2 !XY5
Pr,3 = 0 2 !X15
,D :VB=C6C<ED< =:; <:@I:; 7: Pr,2R 8D 84<C:D< S
Pr,2 = gbh1D∂h1D∂x′
+1
2gh21D
∂b
∂x′+ gbh1D
∂zb
∂x′2 !XZ5
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.5!.
)6 789:86; 8<<8=8>;=? @ABC8@9;B h = z − zbD 9@ E9?6; F
Pr,2 = gbh1D∂ (z1D − zb1D
)
∂x′+1
2gh21D
∂b
∂x′+ gbh1D
∂zb1D
∂x′G*+H*I
= gbh1D∂z1D∂x′
− gbh1D∂zb1D
∂x′+1
2gh21D
∂b
∂x′+ gbh1D
∂zb1D
∂x′
0ABJKL@?M?6; B;86; 9MMKN9@?D @8 :L=78J? @9N=? ?:; OK=9PK6;8@? ?; QK6J
∂z1D∂x′
= 0+
Pr,2 <=?6Q QK6J @A?R<=?::9K6 :L9E86;? F
Pr,2 =1
2gh21D
∂b
∂x′G*+HHI
0? J8@JL@ Q?: JKM<K:86;?: Q? @8 M8;=9J? Te :?=8 <=B:?6;B 8L JO8<9;=? H+
! !"!# $%&'()*+, #-
-6 <=B:?6;? 9J9 @?: BSL8;9K6: :KL: 7K=M? Q9TB=?6;9?@@? Q86: @? J8: QAL6 EK@LM?
Q? JK6;=U@? ?6 JKK=QK66B?: J8=;B:9?66?:+
.86: @? J8: KV 9@ 6AW 8 <8: QABJKL@?M?6; 4.D @ABSL8;9K6 G*+!XNI <?L; :?
=BBJ=9=? F
ˆ x2
x1
ˆ y2
y1
[U2D (t2)−U2D (t1)] dydx+
ˆ t2
t1
ˆ y2
y1
[F2D (x2)− F2D (x1)] dydt+
ˆ t2
t1
ˆ x2
x1
[G2D (y2)−G2D (y1)] dxdt =
ˆ t2
t1
S2Ddt G*+HYI
0?: E8=98N@?: Q? @ABJKL@?M?6; B;86; :L<<K:B?: JK6;96L?: ?; QB=9E8N@?: :L= JO8SL?
:KL:5EK@LM? Q? JK6;=U@?D 9@ ?:; <K::9N@? Q? =B8@9:?= L6 QBE?@K<<?M?6; ?6 :B=9? Q?
'8W@K= F
ˆ x2
x1
ˆ y2
y1
ˆ t2
t1
∂φU2D
∂tdtdydx+
ˆ t2
t1
ˆ y2
y1
ˆ x2
x1
∂F2D
∂xdxdydt+
ˆ t2
t1
ˆ x2
x1
ˆ y2
y1
∂G2D
∂ydydxdt =
ˆ t2
t1
S2Ddt G*+H I
)6 JK6:9QB=86; SL? @?: Q9M?6:9K6: QL EK@LM? Q? JK6;=U@? ?; SL? @8 QL=B? QA965
;BC=8;9K6 :K6; 96Z69;B:9M8@?:D @ABSL8;9K6 G*+H I :ABJ=9; F
∂U2D
∂t+∂F2D
∂x+∂G2D
∂y= S2D G*+H[8I
! ! "#$%&'()* +, -.(-%/%&'() 0
12
U2D =
h2Dq2Dr2D
3 !4567
F2D =
q2Dq22Dh2D
+1
2gh22D
q2Dr2Dh2D
3 !4587
G2D =
r2Dq2Dr2Dh2D
r22Dh2D
+1
2gh22D
3 !4597
S2D =
0
gh2D
(
S0,x −√
q22D + r22D
h22DK2R
1/3h,x
q2D
)
gh2D
(
S0,y −√
q22D + r22D
h22DK2R
1/3h,y
r2D
)
3 !45:7
;<=>?@AB1C 3 !457 :DA 8:EE: >?B :DA F=D1E?: G1?F E:D H@BEE:D 9: 8@E8?E 6B9BH:CI
DB1CC:EE:D C1C 81?GE=:D J 9:D H@BEE:D ?CB9BH:CDB1CC:EE:D! K:D =>?@AB1CD 81FF:DI
G1C9:CA @?L =>?@AB1CD 6B9BH:CDB1CC:EE:D 8E@DDB>?:D 9: *@BCAIM:C@CA!
+@CD E: 8@D 12 ?C =81?E:H:CA ?CB9BH:CDB1CC:E AF@N:FD: E: N1E?H: 9: 81CAFOE:P
E<=81?E:H:CA 6B9BH:CDB1CC:E C<188?G: G@D E<:CD:H6E: 9: 8: N1E?H:! (C F=@EBD:
E<@C@E1QB: @N:8 E@ G1F1DBA= 9=8FBA: 9@CD R4ST :A 1C BCAF19?BA ?C A:FH: 9: G1F1DBA=
F:GF=D:CA@CA E: U@BA >?: E<:CD:H6E: 9? N1E?H: 9: 81CAFOE: C<:DA G@D 9BDG1CB6E:
G1?F E<=81?E:H:CA 6B9BH:CDB1CC:E! ;@ G1F1DBA= :DA 9=VCB: G@F E: F@AB1 9: E@ D?FI
U@8: :C GE@C 9BDG1CB6E: G1?F E<=81?E:H:CA 36B9BH:CDB1CC:E7 D?F E@ D?FU@8: :C GE@C
A1A@E: 9? N1E?H: 9: 81CAFOE:! ,C BCAF19?BD@CA 9@CD 3 !45@7 E: A:FH: 9: G1F1DBA=P
E:D A:FH:D 9<=8W@CQ: :CAF: E:D =81?E:H:CAD 3>?B D1CA C=8:DD@BF:H:CA 1GG1D=D
@?L A:FH:D Te @GG@F@BDD@CA 9@CD E<=>?@AB1C G1?F E<=81?E:H:CA ?CB9BH:CDB1CI
C:E 3 ! 077 :A E:D U1F8:D 9: GF:DDB1C D<:L:FX@CA E: E1CQ 9:D BCA:FU@8:D BCA=FB:?F:DP
BE @GG@F@YA Z
∂U2D
∂t+∂F2D
∂x+∂G2D
∂y= S2D −Te 3 !4[@7
12
U2D =
φh2Dφq2Dφr2D
3 !4[67
F2D =
φq2D
φq22Dh2D
+1
2gφ.h22D
φq2Dr2Dh2D
3 !4[87
! "#$%&'() *+ ,-./0&1$'&-2 "-3%0/) 4.56.
G2D =
φr2D
φq2Dr2Dh2D
φr22Dh2D
+1
2gφ.h22D
7*+!89:
S2D =
0
gφh2D
(
S0,x −√
q22D + r22D
h22DK2R
1/3h,x
q2D
)
+1
2gh22D
∂φ
∂x
gφh2D
(
S0,y −√
q2 + r2
h2K2R1/3h,y
r2D
)
+1
2gh22D
∂φ
∂y
7*+!8;:
0< 9=>?@A@B? 9;C D<E@<FG;C 9<?C 7*+!H: ?I;CA J<C KB?AE<9@KAB@E; <D;K 7*+!8: JL@C5
MLI;GG; KBEE;CJB?9 <L K<C J<EA@KLG@;E φ = 1+ &G <JJ<E<NA 9B?K ML; G;C =ML<A@B?C
JBLE GI=KBLG;O;?A F@9@O;?C@B??;G B?A G< OPO; QBEO;R MLI@G S <@A BL ?B? L? =KBL5
G;O;?A L?@9@O;?C@B??;G CLE G; DBGLO; 9; KB?AETG;+ )? ;U;AR @?AEB9L@E; φ = 1 ;A
T′
e = 0 9<?C G;C =ML<A@B?C 7*+!8: Q<@A <JJ<E<NAE; G;C =ML<A@B?C 7*+!H:+ )? @?A=5
VE<?A G;C A;EO;C φ ;A b 9<?C G< 9=>?@A@B? 9;C O<AE@K;CR B? BFA@;?A JBLE GI=KBLG;5
O;?A L?@9@O;?C@B??;G W
∂U1D
∂t+∂F1D
∂x= S1D +Te 7*+!X<:
U1D =
bh1Dbq1Dbr1D
7*+!XF:
F1D =
bq1D
bq21Dh1D
+1
2gb.h21D
bq1Dr1Dh1D
7*+!XK:
S1D =
0
gbh1D
S0 −√
q21D + r21D
h21DK2R
1/3
h,x′
q1D
+1
2gh21D
∂b
∂x
−gb√
q21D + r21D
h1DK2R1/3
h,y′
r1D
7*+!X9:
0I=KE@ALE; 9; 7*+!8: ;A 7*+!X: KBEE;CJB?9 <L A;EO; T′
e JEYC <LZ =ML<A@B?C [
JBEBC@A= 9=OB?AE=;C 9<?C \!4]+ -? J;LA OB?AE;E ML; K;C CSCAYO;C 9I=ML<A@B?C
CB?A 9;C CSCAYO;C ^SJ;EFBG@ML;C 9; GB@C 9; KB?C;ED<A@B? \6H]+
!"! #$$%#&'(#) *)+ +,+'-.)+ */$0(1'&23+ *13+ 4)
#$5$#)3'&)4 *) 4/&3')#51%) 67
!" #$$%&'()&* +*, ,-,(./*, +0$1)2('34, +24, 5*
&$6$&*4('*5 +* 50'4(*&62%*
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(O;x; y) >D: EA?>DD@;F>! 4@ IA:G8=8<8R;> => I@;<<@R> => <@ Q8E> =/A:9=> D>F@
CFAD>E:A> S <@ D>?:;8E "!T! 4>D AU9@:;8ED => CF8C@R@:;8E D8E: =;D?FA:;DA>D D9F <>D
I@;<<>D => ?@<?9< @;ED; 8V:>E9>D! WXXY @ I8E:FA U9/;< A:@;: C8DD;V<> =/>D:;I>F <>
Z9H S :F@B>FD ?G@U9> ;E:>F[@?> C@F <@ FAD8<9:;8E =/9E CF8V<\I> => #;>I@EE =@ED
<@ =;F>?:;8E E8FI@<> S </;E:>F[@?>!
2E ?G>F?G> =8E? S A:@V<;F <> D]D:\I> =/AU9@:;8ED AU9;B@<>E: S L !"ON L89
S L !"MNN =@ED <> FA[AF>E:;>< @DD8?;A S </;E:>F[@?>! +; Pi,j >D: <@ I@:F;?> => C@DD@R>
=9 FA[AF>E:;>< R<8V@< @9 FA[AF>E:;>< @DD8?;A S </;E:>F[@?> @<8FD ^
U′′
= P−1i,j U
φh
φhu′′
φhv′′
= P−1i,j
φhφhuφhv
L !7_N
8` U′′
>D: <> B>?:>9F =>D B@F;@V<>D ?8ED>FBA>D A?F;: =@ED <> FA[AF>E:;>< @DD8?;A
S </;E:>F[@?>! *@ED ?> FA[AF>E:;><P <>D ?8IC8D@E:>D =9 B>?:>9F B;:>DD> D/A?F;B>E:
LB8;F 5;R9F> !6N ^
u′′
= u cosα+ v sinα
v′′
= −u sinα+ v cosαL !7aN
b89F </A?89<>I>E: 9E;=;I>ED;8EE>< ?8II> C89F </A?89<>I>E: V;=;I>ED;8EE><P
<@ I@:F;?> => C@DD@R> P−1
D/A?F;: =8E? ^
P−1 =
1 0 00 cosα sinα0 − sinα cosα
L !7XN
8E >E =A=9;: </A?F;:9F> => P ^
P =
1 0 00 cosα − sinα0 sinα cosα
L !7 N
1CF\D F8:@:;8E =@ED <> FA[AF>E:;>< @DD8?;A S </;E:>F[@?>P <>D D]D:\I>D =/AU9@K
:;8ED L !"MN >: L !"ON D> FAD9I>E: S =>D D]D:\I>D 9E;=;I>ED;8EE><D => <@ [8FI> ^
∂U′′
∂t+∂F
′′
∂x′′= S
′′ ±T′′
e L !7"N
8` ^
U′′
=
φh
φhu′′
φhv′′
=
φh
φq′′
φr′′
L !77@N
!"#$%&'( )* +,-./%0#&%,1 !,2$/.( 3-45-
u’
u’’
v’’
Li,j
v’
n
α’
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V
u
v
α’’
!"
u
u’’
v’’Li,j
v
n
α’’
V
#"
!"#$% )* 6 '7879:;<=:> >=7 ? >@=;<:98AB:
F′′
=
φq′′
φq′′2
h+1
2gφh2
φq′′
r′′
h
C)*DDEF
S′′
=
0
gφh(
S0,x′′ − Sf,x′′
)
+1
2gh2
∂φ
∂x′′
0
C)*DDBF
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# QA9<=9 J:G E=>A;G J: MAGG: :< J: NIA;<=<7 J: MHIO:M:;<S => :G< QHGG=E>:
J@7<AE>=9 >:G 7NIA<=H;G J: Q9HQATA<=H; J:G 7BHI>:M:;<G I;=4 :< E=J=M:;G=H;;:>*
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8A=9: QA9 I;: M7<XHJH>HT=: I;=NI: COH=9 !XAQ=<9: YF*
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!"!4!" @<A&200,%- 720 .%'A%0*-+20 7( :2.+2(& ;(< F∗
! ! ! 3B
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@0+,'*+,%- 7( ;(< 72 '*002 F ∗
1 ! ! ! ! ! ! ! ! ! ! ! ! ! 3D
@0+,'*+,%- 7( +2&'2 0%(&.2 (∆S0)LR! ! ! ! ! ! ! ! ! ! B"
@0+,'*+,%- 7( ;(< 72 1(*-+,+/ 72 '%(:2'2-+ F ∗
2 ! ! ! B"
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! !# NIA%+JO020 72 +&*:*,) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! D#
! !" P&*-0?2&+ 72 '*002 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! D"
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! !"!" S1(,),C&*62 720 '*,))20 #H 2+ "H Q:%,& $,6(&2 !BQCR
2+ Q.RR ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! D4
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! ! P&*-0?2&+ 72 1(*-+,+/ 72 '%(:2'2-+ )%-6,+(7,-*)2 ! ! ! ! ! ! DT
!D E,/)(&1 -,/ ,A)'.+()./ -0 8616$ "F ! ! ! ! ! ! ! ! ! ! ! ! ! ! C>
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!
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2D 89 :;<<=:>;?> @ ABCD=EA9F9?> MGHGF9?8G=??9A
2D→1D 89 :;<<=:>;?> ;E >:;?8I9:> H9<EG8 A; F;GAA9 L2 K9:8 A; F;GAA9 J2
b 89 :;<<=:>;?> @ A; AGFG>9 HE F;GAA;N9
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i,j 89 :;<<=:>;?> @ ABG?>9:I;D9 9?>:9 A98 CACF9?>8 H9 D;ADEA ? i 9> ? j
L 89 :;<<=:>;?> @ ABC>;> N;EDO9 HE <:=MAPF9 H9 (G9F;??
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t 89 :;<<=:>;?> @ ABCD=EA9F9?> >:;?8K9:8;A
/G8>9 H98 G?HGD98
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H98 CQE;>G=?8
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A=?NG>EHG?;A9
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n89 :;<<=:>;?> ;E nieme
<;8 H9 >9F<8 H9 D;ADEA
n+1/289 :;<<=:>;?> @ A; F=R9??9 9?>:9 A98 <;8 H9 >9F<8 n 9> n+ 1
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F=EK9F9?> >:;?8K9:8;A9
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cs -.).",/. *& ("1($2$/,1' *! %31% ms−1
E∆t
4(."$/&!" *& %$)%!) *&0 5!6 7 /"$8&"0 )&0 ,'/&"#$%&0
,'/.",&!"&0 (&'*$'/ )$ *!".& ∆t
F 9&%/&!" *& 5!6 7 /"$8&"0 ):,'/&"#$%&
Fa; Fb; Fc <1"%&0 *& ("&00,1' &6&"%.&0 7 ):,'/&"#$%& kgm s−2
f <1'%/,1' *1'/ )$ *.=',/,1' &0/ (".%,0.& )1"0 *& 01'
!/,),0$/,1'
g >%%.)."$/,1' 2"$8,/$/,1''&))& ms−2
h ?$!/&!" *:&$! m
!
!"#$%& '&()*+,-+$. /.+-0
Id "#$%&'( &)(*$&$+
K "#$%&'( )(, -('$(.%, /%0/%(, )( A
K∆t
1/+%#$(.% )2&*$+3%#$&0* )(, $(%4(, )( 5%0$$(4(*$ /(*)#*$
6# ).%+( ∆t
K 70(8'&(*$ )( 9$%&':6(% m1/3 s−1
L∆t
1/+%#$(.% )( %+,06.$&0* )( 6# /#%$&( '0*,(%-#$&-( )(,
+;.#$&0*, /(*)#*$ 6# ).%+( ∆t
L <0*3.(.% )( 62&*$(%5#'( &*$+%&(.%( mLi,j <0*3.(.% )( 62&*$(%5#'( (*$%( 6(, 4#&66(, )( '#6'.6 * i ($ j mN (i) =04>%( )( '(66.6(, #?#*$ .*( &*$(%5#'( (* '044.* #-(' 6#
'(66.6( * i
Pi,j "#$%&'( )( /#,,#3( ). %+5+%(*$&(6 360>#6 #. %+5+%(*$&(6
#$$#'@+ A 62&*$(%5#'( (*$%( 6(, '(66.6(, * i ($ j
Q B+>&$ )( 62+'0.6(4(*$ m3 s−1
q B+>&$ .*&$#&%( )#*, 6# )&%('$&0* Ox m2 s−1
Rh,xC Rh,y D#?0*, @?)%#.6&;.( )( 62+'0.6(4(*$ mr B+>&$ .*&$#&%( )#*, 6# )&%('$&0* Oy m2 s−1
S E('$(.% )(, $(%4(, ,0.%'( )#*, 6( %+5+%(*$&(6 #,,0'&+ A
62&*$(%5#'(
S′
"#$%&'( )(, $(%4(, ,0.%'( )+F*&( /#% S′
= K−1
S0
S0 E('$(.% )(, $(%4(, ,0.%'( 6&+, A 6# 3+04+$%&( )#*, 6(
%+5+%(*$&(6 #,,0'&+ A 62&*$(%5#'(
S 9('$&0* )( 62+'0.6(4(*$ m2
S′
k "0?(**( )( 6# kieme'04/0,#*$( ). -('$(.% S
′
6( 60*3 ).
'@(4&* )2&*$+3%#$&0*
t 700%)0**+( )( $(4/, sU E('$(.% )(, -#%&#>6(, )( 62+'0.6(4(*$ )#*, 6( %+5+%(*$&(6
360>#6
u 704/0,#*$( ). -('$(.% -&$(,,( )#*, 6# )&%('$&0* Ox ms−1
V E06.4( )2(#. m3
v 704/0,#*$( ). -('$(.% -&$(,,( )#*, 6# )&%('$&0* Oy ms−1
W "#$%&'( )(, &*-#%&#*$, )( D&(4#**
x GH%( '00%)0**+( )2(,/#'( )#*, 6( %+5+%(*$&(6 #$$#'@+ A
62&*$(%5#'(
m
z 70$( )( 6# ,.%5#'( 6&>%( mzb 70$( ). 50*) mβ I*36( (*$%( 62#J( ). '#*#6 GB ($ 62&*$(%5#'( &*$+%&(.%( rad
∆S0 K*$+3%#6( ). $(%4( ,0.%'( A 62&*$(%5#'( m3 s−2
∆t L#, )( $(4/, )( '#6'.6 s∆x L#, )2(,/#'( )( '#6'.6 mΛ "#$%&'( )(, -#6(.%, /%0/%(, )( A
λ 7+6+%&$+ )( /%0/#3#$&0* )(, 0*)(, ms−1
φ L0%0,&$+ ∅
ρ "#,,( -06.4&;.( kgm−3
Ψ 70(8'&(*$ )( $.%>.6(*'( ∅
χ L+%&4H$%( 40.&66+ )( 62+'0.6(4(*$ m
<&,$( )(, -#%&#>6(,
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
/8 9:;<=>?@<8 A8 B:CD>=E8 A8@ FGC=H?:I@ A8 D;:D=E=H?:I AFJ8>:DDF =C BK=L
D?H;8 D;FBFA8IH D8;<8H AM8IJ?@=E8; >= <:AF>?@=H?:I CI?A?<8I@?:II8>>8 A8@ FB:CL
>8<8IH@ B=I=>?@F@N B:CD>F8 O >= <:AF>?@=H?:I P?A?<8I@?:II8>>8 A8@ FB:C>8<8IH@
A=I@ >= Q:I8 AM8RD=I@?:I A8 B;C8+ /M:PS8BH?9 A8 B8 BK=D?H;8 8@H A8 D;F@8IH8; >8@
<FHK:A8@ IC<F;?GC8@ 8<D>:TF8@ D:C; ;F@:CA;8 >8@ A?UF;8IH@ @T@HV<8@ AMFGC=L
H?:I@ B:IB8;IF@+ /M=>E:;?HK<8 E>:P=> 8@H H:CH AM=P:;A D;F@8IHFN DC?@ >8@ D;?IB?L
D=>8@ FH=D8@ A8 B=>BC> @:IH AFH=?>>F8@+
/= <FHK:A:>:E?8 A8 B=>BC> 8<D>:TF8 A=I@ B8HH8 D=;H?8 = A:IIF >?8C O CI8
DCP>?B=H?:I WXYZ+
!" #$%&'()*+, -, './&$0)(&1
/MFGC=H?:I [\+]*^ 8@H A?@B;FH?@F8 DC?@ ;F@:>C8 O BK=GC8 D=@ A8 H8<D@ A8 B=>BC>
8I CH?>?@=IH >= <FHK:A8 A8@ D=@ 9;=BH?:II=?;8@ [J:?; -8BH?:I _+]+\+X^+ "8HH8 =DL
D;:BK8 B:I@?@H8 O ;F@:CA;8 >M?IHFE;=>?HF AC D;:P>V<8 @:C@ 9:;<8 AMCI8 @CBB8@@?:I
A8 D;:P>V<8@ ` >= @:>CH?:I ?I?H?=>8 AMCI @:C@LD;:P>V<8 FH=IH >= @:>CH?:I aI=>8 AC
@:C@LD;:P>V<8 D;FBFA8IH W_bZ+
c B:I@H;CBH?:I AC <=?>>=E8 @C; >8GC8> >8@ FGC=H?:I@ A?@B;FH?@F8@ @8;:IH ;F@:L
>C8@ [B8HH8 FH=D8 8@H ;F=>?@F8 CI8 @8C>8 9:?@ =C AFPCH AC B=>BC>^N
c ;F@:>CH?:I A8 >= D=;H?8 B:I@8;J=H?J8 A8@ FGC=H?:I@ 8I D;8I=IH 8I B:<DH8
>8@ H8;<8@ @:C;B8 >?F@ O >= EF:<FH;?8N
c ?IHFE;=H?:I A8@ H8;<8@ A8 9;:HH8<8IHN
c B=>BC> A8@ dCR O H;=J8;@ >8@ ?IH8;9=B8@ ?IHF;?8C;8@ [D;?@8 8I B:<DH8 A8@
H8;<8@ AMFBK=IE8^+
$ >= aI A8 BK=GC8 FH=D8 A8 >M=>E:;?HK<8N CI8 J=>8C; ?IH8;<FA?=?;8 A8 >= J=;?=P>8
8@H B=>BC>F8 e Un,c
N Un,f
N +++ [J:?; 7?EC;8 *+X^+
5=HKF<=H?GC8<8IHN >M=>E:;?HK<8 A8 ;F@:>CH?:I D8CH fH;8 FB;?H @:C@ >= 9:;<8
@C?J=IH8 e
Un+1 =
(
E∆t
K∆t
L∆t)
Un
[*+X^
:g E∆t
;8D;F@8IH8 >M:DF;=H8C; A8 B=>BC> A8@ dCR O H;=J8;@ >8@ ?IH8;9=B8@ ?IHF;?8C;8@
D8IA=IH >= AC;F8 ∆tN K∆t
>M:DF;=H8C; AM?IHFE;=H?:I A8@ 9;:HH8<8IH@ @C; ∆t 8HL∆t
>M:DF;=H8C; A8 ;F@:>CH?:I A8 >= D=;H?8 B:I@8;J=H?J8 A8@ FGC=H?:I@+ /M:;A;8 A8
B=>BC> A8@ A?UF;8IH8@ FH=D8@ IM=N HKF:;?GC8<8IHN D=@ AM?<D:;H=IB8 AV@ >:;@ GC8
>8 D=@ A8 H8<D@ A8 B=>BC> H8IA J8;@ 0 [∆t → 0^+ "K=BCI8 A8@ FH=D8@ AFB;?H8@
B?L=D;V@ 8@H ?IHFE;F8 @C; >M8I@8<P>8 AC D=@ A8 H8<D@+ $aI AM=<F>?:;8; >= D;FB?L
@?:I AC <:AV>8N ?> 8@H D:@@?P>8 AMCH?>?@8; CI @BKF<= AM:;A;8 _ D:C; >M?IHFE;=H?:I
H8<D:;8>>8 A8@ 9;:HH8<8IH@ e
Un+1 =
(
E∆t
K∆t/2
L∆t
K∆t/2
)
Un
[*+_^
/M=CE<8IH=H?:I A8 >M:;A;8 8@H D:@@?P>8 ` B8>= @M=BB:<D=EI8 EFIF;=>8<8IH AMCI8
=CE<8IH=H?:I I:H=P>8 AC I:<P;8 AMFH=D8@ A8 B=>BC> 8H >?<?H8 AM=CH=IH >= ;=D?A?HF
AM8RFBCH?:I AC B:A8 A8 B=>BC>+ 0I8 <FHK:A:>:E?8 D:C; >?<?H8; B8@ D;:P>V<8@ 8@H
D;F@8IHF8 A=I@ Wb*Z+
!"! #$%&'()*+, -, './&$0)(&1 2"
Un
Un,l=U
n+1
Un,c
Un,m
Un,t
équilib
rage de la m
asse
transfert de q
uantité d
e
m
ouveme
nt lon
gitudinal
e
Un,f
résolution de la partie conservativ
e
équilibrage de la quantité de
des équations
mouvement transversale
!"#$% !" 3 /45678 9:; 6<8=:; 9: 48>4?>
@A?B 4:B<8CD; 48; =8B<C4?>C:B; E4AD9C<CAD; 8?F >C7C<:;G :F<B67C<6; 9? 48D8>G
:<4!HG >8 76<5A9A>AIC: I6D6B8>: 9: B6;A>?<CAD D: =:?< J<B: 8==>CK?6:! $:; 76L
<5A9:; D?76BCK?:; ?<C>C;6:; =A?B 4:; 48; ;AD< 6I8>:7:D< =B6;:D<6:;!
$MCD<6IB8<CAD <:7=AB:>>: 9:; 6K?8<CAD; :;< B68>C;6: N =8B<CB 9M?D ;45678 :FL
=>C4C<:! $: 4BC<OB: 9: PA?B8D< =B6;:D<6 N >8 ;:4<CAD Q!" 9AC< =8B 4AD;6K?:D< J<B:
R6BCS6 =A?B I8B8D<CB >8 ;<8TC>C<6 9? ;45678! $8 UAB7?>8<CAD EQ!VH :;< :7=>AW6:
=A?B 96<:B7CD:B >: =8; 9: <:7=; 9: 48>4?> :D UAD4<CAD 9? 78C>>8I: :< 9:; R8L
BC8T>:; 9: >M64A?>:7:D<! @A?B 458K?: 78C>>: 9: 48>4?>G >: =8; 9: <:7=; 78FC78>
897C;;CT>: :;< 48>4?>6 X >: =8; 9: <:7=; ?<C>C;6 =A?B >MC<6B8<CAD :;< >: =>?; =:<C< 9:
<A?< >:; =8; 9: <:7=; 897C;;CT>:;! -8D; ?D =B:7C:B <:7=;G >8 =8B<C: 4AD;:BR8<CR:
9:; 6K?8<CAD; :;< B6;A>?:! $:; 6K?8<CAD; ?DC9C7:D;CADD:>>:; EQ! YH :< TC9C7:DL
;CADD:>>:; EQ! ZH B664BC<:; 98D; >: B6U6B:D<C:> 9: >MCD<:BU84: ;AD< 9C;4B6<C;6:; 9: >8
78DCOB: ;?CR8D<: [
Un+1i = U
ni −
∆t
Ai
∑
j∈N(i)
[
P−1i,j F
n+1/2
i,j Li,j +(
Sn+1/2
i,j
)
i
]
E !QH
A\ Pi,j B:=B6;:D<: >8 78<BC4: 9: =8;;8I: 9? B6U6B:D<C:> =AB<8D< >: ;W;<O7: 9M6K?8L
<CAD; ER (O;x; y; z) =A?B >:; 64A?>:7:D<; TC9C7:D;CADD:>; :< R′
(
O;x′
; y′
; z)
=A?B >:; 64A?>:7:D<; ?DC9C7:D;CADD:>;H 8? B6U6B:D<C:> 8;;A4C6 N >MCD<:BU84:G ∆t:;< >: =8; 9: <:7=; 9: 48>4?>G Ai >8 ;?BU84: :D =>8D 9: >8 78C>>: 9: 48>4?> iG Li,j
>8 >ADI?:?B 9: >MCD<:BU84: :D<B: >:; 78C>>:; 9: 48>4?> i :< j :< N (i) >: DA7TB:
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
89 :9;;<;9= >?>@A <@9 B@A9CD>:9 9@ :EFF<@ >G9: ;> :9;;<;9 iH Uni 9=A ;> FE?9@@9
89= G>CB>I;9= 8JK:E<;9F9@A =<C ;> :9;;<;9 i >< L>= 89 A9FL= nH Fn+1/2
i,j 9=A ;9 M<N
8>@= ;> 8BC9:ABE@ @ECF>;9 O ;JB@A9CD>:9 9@AC9 ;9= F>B;;9= 89 :>;:<; i 9A j L9@8>@A
;> 8<CK9 ∆t 9A(
Sn+1/2
i,j
)
iC9LCK=9@A9 ;JB@M<9@:9 =<C ;> F>B;;9 89 :>;:<; i L9@8>@A
;> 8<CK9 ∆t 8< A9CF9 =E<C:9 :>;:<;K O ;JB@A9CD>:9 9@AC9 ;9= F>B;;9= i 9A j+ "EFF9
FE@ACK O ;> =9:ABE@ P+*H ;9= =?=AQF9= 8JKR<>ABE@= SP+*TU 9A SP+*VU L9<G9@A WAC9
AC>@=DECFK= 9@ <@ =?=AQF9 KR<BG>;9@A <@B8BF9@=BE@@9; 8>@= ;> 8BC9:ABE@ @ECX
F>;9 O ;JB@A9CD>:9 SP+Y*U+ /> CK=E;<ABE@ 89 :9 =?=AQF9 8JKR<>ABE@= LE<C :Z>R<9
B@A9CD>:9 L9CF9A 89 :>;:<;9C ;9= A9CF9= Fn+1/2
i,j 9A
(
Sn+1/2
i,j
)
i+
!" #$%&'()*&+ ,- '. /.0)*- 1&+%-02.)*2- ,-% $3(.4
)*&+%
2>@= ;9 CKDKC9@AB9; >==E:BK O ;JB@A9CD>:9 9@AC9 89<N F>B;;9= 89 :>;:<;H ;9= =?=X
AQF9= 8JKR<>ABE@= <@BX 9A IB8BF9@=BE@@9; =E@A AE<= 89<N 89= =?=AQF9= 8JKR<>X
ABE@= <@B8BF9@=BE@@9;= SGEBC -9:ABE@ P+*U+ /9= =?=AQF9= 8JKR<>ABE@ 89 LCEL>[>X
ABE@ KA>@A AE<= 89<N <@B8BF9@=BE@@9;=H <@9 FKAZE8E;E[B9 <@BR<9 L9CF9A ;9<C
CK=E;<ABE@+ .@ L9<A FE@AC9C R<9 :9= =?=AQF9= 8JKR<>ABE@= =E@A Z?L9CIE;BR<9=+
.@ =JB@AKC9==9 8>@= :9AA9 L>CAB9 <@BR<9F9@A >< LC9FB9C L>= DC>:ABE@@>BC9
S;JELKC>A9<C L∆t
89 S*+\UU R<B :E@:9C@9 ;> CK=E;<ABE@ 89 ;> L>CAB9 :E@=9CG>ABG9
89= KR<>ABE@= 9A 89= A9CF9= =E<C:9 ;BK= O ;> [KEFKACB9 ]
∂U
∂t+∂F
∂x= S0 S*+*>U
>G9:
U =
φhφqφr
S*+*IU
F =
φq
φq2
h+1
2gφh2
φqr
h
S*+*:U
S0 =
0
−gφh∂zb
∂x+1
2gh2
∂φ
∂x0
S*+*8U
.@ =9 LCELE=9 89 :>;:<;9C ;9 M<N O ;JB@A9CD>:9 9@AC9 89<N F>B;;9= 89 :>;:<; L>C ;>
CK=E;<ABE@ 8J<@ LCEI;QF9 89 (B9F>@@+ 29<N >LLCE:Z9= =9@=BI;9F9@A 8B^KC9@A9=
9NB=A9@A ] :>;:<;9C ;> G>;9<C 89= G>CB>I;9= :E@=9CGK9= U L<B= 9@ 8K8<BC9 ;> G>;9<C
8< M<N F E< :>;:<;9C 8BC9:A9F9@A ;9 M<N O ;JB@A9CD>:9+ /> LC9FBQC9 >LLCE:Z9
@K:9==BA9 AE<A9DEB= 8J9=ABF9C ;> G>;9<C 89 :Z>R<9 L>C>FQAC9 S:EA9 8< DE@8 9A
LECE=BAKU >< @BG9>< 89 ;JB@A9CD>:9+ /J9=ABF>ABE@ 89 :9AA9 G>;9<C 8EBA WAC9 D>BA9
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%34
UL
Etat Gauche
U*
UR
Zone intermédiaire
d’état constant
Etat DroitOnde de
raréfaction
Onde de
raréfaction
x
x
t
U
∆t
λ1,L
λ*
1λ*
3λ
3,R
!"#$% !" 5 %6789687: ;8<<=;>: ?: @A ;=@86B=C <=87 @: ?>D:@=<<:E:C6 ?8 ;=@F
D:87 ?G>6A6 A<<7=9H>! IA86 J %6789687: ?: @A ;=@86B=C K LA; J %6789687: ?: @A
9=CMN87A6B=C ?G=C?: ?AC; @G:;<A9: ?:; <HA;:;
<A7 @: OBAB; ?G8C: P=C96B=C ?><:C?AC6 ?:; DA@:87; ?AC; @:; EAB@@:; ?: 9A@98@
;><A7>:; <A7 @GBC6:7PA9:! 'A ?>6:7EBCA6B=C ?: 9:66: P=C96B=C :;6 7:@A6BD:E:C6
?>@B9A6: ?AC; @A E:;87: =Q @A DA@:87 ?: 9HAR8: <A7AES67: T @GBC6:7PA9: CGA <A; ?:
;BNCBM9A6B=C <HU;BR8:! 'A ;:9=C?: E>6H=?: ;:7A ?=C9 <7BDB@>NB>:! ,G8C <=BC6 ?:
D8: A@N=7B6HEBR8:V @A 7>;=@86B=C ?:; W8X T 67AD:7; 8C: BC6:7PA9: :;6 7>A@B;>: <A7
@: OBAB; ?G8C ;=@D:87 ?: #B:EACC ?G>6A6 A<<7=9H> ?=C6 @A E>6H=?=@=NB: ?: 9A@98@
:;6 <7>;:C6>: T @A ;:96B=C !"!4! 0: ;=@D:87 A >6> C=EE> Y ;=@D:87 /=7.% Z <=87
Y !"!#$%& '(("!)$*+%, -%+%, #!./," Z! 'A ;6789687: ?: @A ;=@86B=C :;6 OA;>: ;87
@GHU<=6HS;: ?: ?:8X =C?:; ?: 7A7>PA96B=C ;><A7AC6 8C: [=C: ?G>6A6 9=C;6AC6 ?:;
>6A6; NA89H: :6 ?7=B6 ?8 <7=O@SE: ?: #B:EACC \D=B7 ]BN87: !"^!
!"!# $%&'()*+,%- .*&*.+/&,0+,1(2
'G>R8A6B=C ?: 9=C;:7DA6B=C \ ! A^ :;6 E8@6B<@B>: T NA89H: <A7 @A EA67B9:
_A9=OB:CC: A J
A∂U
∂t+A
∂F
∂x= AS0 \ !`^
!
"#$%&'() !* (+,-./'&-0 1), +2/$'&-0, 1) %(-%$3$'&-0 %$(
.$ 4+'#-1) 1), 5-./4), 6&0&,
78 A 9:; <= >=;?@A9 B=A7C@9DD9 EFGD@9 H=? I
A =∂F
∂U=
0 1 0c2 − u2 2u 0−uv v u
J!*KL
78 c 9:; <= M@;9::9 E9 H?7H=N=;@7D E9: 7DE9: E9 H?9::@7D E=D: <O9=P =P ?9H7: I
c =
[
d
dh
(
gh2
2
)]1/2
= (gh)1/2 J!* L
)D ?9>H<=Q=D; A H=? :7D 9RH?9::@7D J!*KLS <OFTP=;@7D J!*UL H9P; V;?9 FA?@;9 :7P:
<= W7?>9 I
∂F
∂t+A
∂F
∂x= AS0 J!*XL
"9;;9 FA?@;P?9 A7??9:H7DE Y <= W7?>9 A=?=A;F?@:;@TP9 E9 J!*!=L FA?@;9 =M9A <9
M9A;9P? ZPR*
.= >=;?@A9 Λ E9: M=<9P?: H?7H?9: E9 S <= >=;?@A9 ! E9: M9A;9P?: H?7H?9:
E9 9; <= >=;?@A9 @DM9?:9 E9 ! :7D; EFGD@9: H=? I
Λ =
u− c 0 00 u 00 0 u+ c
K =
1 0 1u− c 0 u+ cv 1 v
K−1 =
1
2c
u+ c −1 0−2cv 0 2c−(u− c) 1 0
J!*[L
.9: AF<F?@;F: EO7DE9 A7??9:H7DE9D; =PR M=<9P?: H?7H?9: E9 <= >=;?@A9 B=A7C@9DD9
9; :7D; E7DA E7DDF9: H=? I
λ1 = u− cλ2 = uλ3 = u+ c
J!*\]L
.OFTP=;@7D J!*XL 9:; >P<;@H<@F9 Y N=PA^9 H=? K−1
I
K−1 ∂F
∂t+K
−1A∂F
∂x= K
−1AS0 J!*\\L
)D @D;?7EP@:=D; <= >=;?@A9 @E9D;@;F JId = KK−1
L E=D: J!*\\LS @< 9:; H7::@C<9 E9
W=@?9 =HH=?=_;?9 Λ = K−1
AK I
K−1 ∂F
∂t+ΛK
−1 ∂F
∂x= ΛK
−1S0 J!*\`L
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%34
'5 67858976 dW = K−1dF :;<=;8 >; <??@<9<; A !B"C D7ED F5 G7<=; H
∂W
∂t+Λ
∂W
∂x= ΛS
′
A !BIC
7J F; 8;<=; D7E<@; S′
;D8 >?K69 :5< H
S′
= K−1
S0 A !B C
';D L;@8;E<D dW ;8 S′
D768 :5< @76D?ME;68 >766?D :5< F;D ;N:<;DD976D H
dW =1
2c
λ3dF1 − dF2−2cvdF1 + 2cdF3−λ1dF1 + dF2
, S′
=1
2c
−S00S0
A !B4C
7J Fk=(1,2,3) @7<<;D:76> O F5 keme@7=:7D568; >E L;@8;E< F ;8 7J S0 @7<<;D:76>
O F5 >;EN9P=; @7=:7D568; >E L;@8;E< S0! 'Q?ME58976 A !BIC ;D8 ?ME9L5F;68; 5EN
<;F58976D >9R?<;689;FF;D DE9L568;D H
dWk
dt= λkS
′
k pourdx
dt= λk, k = 1, ..., 3 A !BSC
7J FQ96>9@; k @7<<;D:76> O F5 keme @7=:7D568; >;D L;@8;E<D >; FQ?ME58976 A !BIC
;8 7J H
dWk =3∑
l=1
K−1l,k dFl A !B3C
7J K−1l,k ;D8 F5 L5F;E< O F5 leme @7F766; ;8 O F5 keme F9T6; >; F5 =58<9@; K−1
;8
Fl ;D8 F5 leme
@7=:7D568; >E L;@8;E< UEN F! %9 F; L;@8;E< dW :;E8 V8<; 968?T<?
5F7<D F;D Wk D768 F;D 96L5<9568D >E :<7WFP=; >; #9;=566!
!"!" #$%&'()%*+$ ,- ,+$' .-/ 0)()0%&(*/%*12-/
*F 5 ?8? =768<? ME; FQ968?T<58976 >; dW 6; :;E8 @76>E9<; O >;D 96L5<9568D
>; #9;=566 96>?:;6>568D D5EG >56D @;<8596D @5D :5<89@EF9;<D A:5< ;N;=:F; :7E<
F;D DXD8P=;D 2× 2 >; F79D >; @76D;<L58976DC YISZ! '5 =?8[7>; :<?D;68?; @9\5:<PD
;D8 :5< @76D?ME;68 W5D?; DE< F5 G7<=EF58976 T?6?<5F; A !BSC!
/7E< ;D89=;< F; UEN Fn+1/2
i,j ] FQ?ME58976 A !BSC >798 V8<; 968?T<?; ;68<; F; :7968
M(
xi+1/2, tn+1/2)
;8 F; :9;> Ak >; F5 keme @5<5@8?<9D89ME; :5DD568 :5< M H
dx/dt = λk AL79< ^9TE<; !IC! -6 G59D568 FQ[X:78[PD; ME; F5 @?F?<98? >Q76>; λk
;D8 @76D8568; DE< F; :5D >; 8;=:D >; @5F@EF] FQ968?T<58976 >; FQ?ME58976 A !BSC
@76>E98 O H
ˆ M
Ak
dWk =
ˆ tn+1/2
tn
λkS′
kdt ≈∆t
2λkS
′
k, k = 1, 2, 3 A !B_C
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
!"#$% *+8 9 -:;<=> ?@ ?<ABCDCEB ?F :;@=CB ?GCBD<HI>DCEB J@ JEBH ?@K :>I>:D<L
ICKDCMF@K+
EN S′
k @KD J> =EO@BB@ ?@ S′
k J@ JEBH ?F :;@=CB ?GCBD<HI>DCEB+ )B KFPKDCDF>BD Q*+R S
?>BK Q*+RTSU EB EPDC@BD V
ˆ M
Ak
3∑
l=1
K−1l,k dFl =∆t
2λkS
′
k, k = 1, 2, 3 Q*+RWS
/@K :E@X:C@BDK ?@ J> =>DIC:@ K−1
KEBD @FY >FKKC KFZZEK<K :EBKD>BDK J@ JEBH ?@K
:>I>:D<ICKDCMF@K [*\]+ /@K :E@X:C@BDK ?@ J> keme JCHB@ ?@ J> =>DIC:@ KEBD ?EB:
:>J:FJ<K ^ Z>IDCI ?@K _>J@FIK ?>BK J> :@JJFJ@ ?>BK J>MF@JJ@ @KD KCDF< J@ ZC@? Ak ?@
J> keme :>I>:D<ICKDCMF@ V
K−1l,k =
K−1l,k (UL) si λk > 0
K−1l,k (UR) si λk < 0Q*+\`S
2>BK J> =@KFI@ EN J@K :E@X:C@BDK ?@ J> =>DIC:@
−1@D J@K λk KEBD KFZZEK<K aDI@
:EBKD>BDK J@ JEBH ?F :;@=CB ?GCBD<HI>DCEBU JG<MF>DCEB Q*+RWS Z@FD aDI@ <:ICD@ KEFK
J> bEI=@ V
ˆ M
Ak
3∑
l=1
K−1l,k dFl ≈3∑
l=1
K−1l,k
ˆ M
Ak
dFl =∆t
2λkS
′
k, k = 1, 2, 3 Q*+\RS
@D Z>I :EBK<MF@BD V
3∑
l=1
K−1l,k (Fl,M − Fl,Ak) =
∆t
2λkS
′
k, k = 1, 2, 3 Q*+\\S
EN Fl,M @D Fl,AkI@ZI<K@BD@BD J> leme :E=ZEK>BD@ ?F _@:D@FI F I@KZ@:DC_@=@BD
@B M @D @B Ak+ Ak @KD KCDF< ?F :cD< H>F:;@ ?@ JGCBD@Ib>:@ MF>B? λk @KD ZEKCDC_@
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%33
45 67 895: 6;<=5 >7?@6 4AA4 4B5 @:C?5=D4! /?; 8<@B:>74@5 E
Fl,Ak=
Fl,L si λk > 0Fl,R si λk < 0
F !"GH
'? ;:B<A75=<@ 67 BIB5JK4 6L:>7?5=<@B 3×3 F !""H M4;K45 64 8?A87A4; A? D?A47;
64 F 4@ 5<75 M<=@5 M!
!"!# $%&'(& )( *+',+(- .(/ F 0 &123,+-4%'+
&@ 8N4;8N4 O 8?A87A4; A? D?A47; 64 F O AL=@54;P?84 4@5;4 647Q K?=AA4B 64 8?A87A!
&@ P?=5 ALNIM<5NJB4 >74 A? B<A75=<@ 64 F !""H 4B5 :C?A4 O A? D?A47; K<I4@@4 Fn+1/2i+1/2
4@5;4 A4B 54KMB n 45 n + 1! '? ;:B<A75=<@ 64 F !""H @:84BB=54 64 6:54;K=@4; A?
K<I4@@4 67 54;K4 B<7;84 S′
k!
!"!#!$ %&'()*'(+, -. '/0)/ &+.01/
'? D?A47; K<I4@@4 67 54;K4 B<7;84 6?@B F !""H 4B5 4B5=K:4 4@ P<@85=<@ 64B
:5?5B C?78N4 45 6;<=5R 64B D?;=?SA4B O AL=@54;P?84 45 64 A? 8?;?85:;=B5=>74 8<@B=6:T
;:4 E
S′
k = S′
k
(
ULR,
(
∂zb
∂x
)
LR
,
(
∂φ
∂x
)
LR
)
F !" H
<U A4B D?;=?SA4B 8<;;4BM<@6?@5 O A? 5<M<C;?MN=4 O AL=@54;P?84 B<@5 4B5=K:4B M?; E
(
∂zb
∂x
)
LR
≈ zbR− zbL
(λ3 − λ1)∆t
2
F !"VH
(
∂φ
∂x
)
LR
≈ φR − φL
(λ3 − λ1)∆t
2
F !"WH
-@ =@5:C;?@5 F !"VH 45 F !"WH 6?@B A? 6:X@=5=<@ 64B S′
k F !YVHR <@ <S5=4@5 A4B
4QM;4BB=<@B B7=D?@54B E
S′ =
1
2c
S′
1
S′
2
S′
3
=
1
2c
1∆t
2
−S0S
F !"3H
<U E
S = −g (φh)LR
zbR− zbL
λ3 − λ1+1
2g (hLR)
2 φR − φL
λ3 − λ1F !"ZH
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
!"!#!" $%&'())*+, -() .+/&+)0,1() -2 3(.1(2' 42% F∗
/89 :;<8=>9 ?>@?>89 A8 5 BC;DC 9=??@9B89 E@D9C;DC89F <; 9=G9CHC=CH@D A8 I*+JK
8C I*+L K A;D9 I*+LLK E@DA=HC ;= 9M9CNO8 9=H:;DC P
λ3 (F∗1 − F1,R)− (F ∗2 − F2,R) = ∆t
2 λ1S′
1 = −λ1S1 pourdx
dt= λ1
−v (F ∗1 − F1,A) + (F ∗3 − F3,A) = 0 pourdx
dt= λ2
−λ1 (F ∗1 − F1,L) + (F ∗2 − F2,L) = ∆t2 λ3S
′
3 = λ3S3 pourdx
dt= λ3
I*+LJK
@Q
Fl,A =
Fl,L si λ2 > 0Fl,R si λ2 < 0
I*+RSK
8C @Q <8 C8>O8 F ∗l >8?>B98DC8 <; lemeE@O?@9;DC8 A= T=U F
∗V C>;:8>9 <WHDC8>X;E8+
2;D9 I*+LJKF <89 C8>O89 S′
1 8C S′
3 E@>>89?@DA8DC V <WHDCBY>;<8 A= C8>O8 9@=>E8 8DC>8
<WHDC8>X;E8 8C <8 ?H8A A8 <; E;>;ECB>H9CHZ=8 V <;Z=8<<8 H<9 9@DC ;99@EHB9+ /W89CHO;[
CH@D A89 :;>H;G<89 V <WHDC8>X;E8 DW; ?\M9HZ=8O8DC ?;9 A8 98D9 8C H< 89C DBE899;H>8
AW8D X;H>8 =D8 ;??>@UHO;CH@D 8D X@DECH@D A89 :;>H;G<89 A;D9 <89 E8<<=<89 A>@HC8
8C Y;=E\8+ %<=9H8=>9 ;??>@UHO;CH@D9 ?8=:8DC ]C>8 =CH<H9B89 ^*L_+ %@=> <; >B9@<=[
CH@D A8 I*+LJKF <W\M?@C\N98 Z=8 S1 = S3 89C >B;<H9B8+ /; >B9@<=CH@D A= 9M9CNO8
AWBZ=;CH@D I*+LJK ?8>O8C AWBC;G<H> <W8U?>899H@D A89 F ∗l P
F ∗1 =1
λ3 − λ1[F2,L − F2,R + λ3F1,R − λ1F1,L]
−g (φh)(1)LR
zbR− zbL
λ3 − λ1+ 1
2g(
h(1)LR
)2 φR − φL
λ3 − λ1F ∗2 =
1
λ3 − λ1[λ3F2,L − λ1F2,R + λ1λ3 (F1,R − F1,L)]
+ (λ3 + λ1)
[
−g (φh)(2)LR
zbR− zbL
λ3 − λ1+ 1
2g(
h(2)LR
)2 φR − φL
λ3 − λ1
]
F ∗3 = F3,A + v (F ∗1 − F1,A)
I*+R`K
%;> ;H<<8=>9F <; ABC8>OHD;CH@D A89 :;>H;G<89 9=> E\;Z=8 E8<<=<8 A8 E;<E=< ?;>
<8 GH;H9 A8 I*+RK X;HC HDC8>:8DH> <8 C8>O8 9@=>E8 V <WHDC8>X;E8 8DC>8 <89 A8=U E8<<=<89
A8 E;<E=< A@DC <; X@>O8 HDCBY>;<8 ?8=C 9WBE>H>8 P
(∆S0)LR = −g (φh)(S)LR (zbR− zbL
) +1
2g(
h(S)LR
)2
(φR − φL) I*+RLK
%@=> E\;Z=8 E8<<=<8F E8 C8>O8 89C ?@DAB>B ?;> <; EB<B>HCB AW@DA8 8DC>;DC ?;> <WHD[
C8>X;E8+ /; ABaDHCH@D A89 F ∗l I*+R`K 8C A= C8>O8 9@=>E8 V <WHDC8>X;E8 X;HC HDC8>:8DH>
<; :;<8=> A89 :;>H;G<89 V <WHDC8>X;E8 I(φh)(1)LRF
(
h(1)LR
)2
F (φh)(2)LRF
(
h(2)LR
)2
F (φh)(S)LR
8C
(
h(S)LR
)2
K+
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%34
!"!#!# $%&'()&'*+ ,-% .)/')01-% 2 13'+&-/4)5-
'56789:;89<= >67 ?;@9;AB67 C B59=86@D;E6 678 @F;B97F6 C G;@89@ >6 E;7 GHI79JK67
><=8 B; 7<BK89<= 678 D;E9B6:6=8 >F86@:9=;AB6!
$%&'()&'*+ ,6 768 ,- ()%%- F ∗1 &= E<=79>L@6 K= FE<KB6:6=8 G6@:;=6=8
6=8@6 >6KM :;9BB67 >6 E;BEKB G<K@ B67JK6BB67 zb 68 φ 7<=8 JK6BE<=JK67! '5FJK;89<=
>6 E<=89=K98F 75FE@98 ;B<@7 N
(φhu)L = (φhu)R = (φhu)∗= q O !PPQ
-= E<:A9=;=8 O !PPQ ;?6E B56MG@6779<= >K RKM F ∗1 >;=7 O !PSQT <= <A896=8 N
F ∗1 =1
λ3 − λ1
[
q (uL − uR) +1
2g[(
φh2)
L−(
φh2)
R
]
]
+
1
λ3 − λ1
[
(λ3 − λ1) q − g (φh)(1)LR (zbR− zbL
) +1
2g(
h(1)LR
)2
(φR − φL)
]
O !P Q
'5FE<KB6:6=8 F8;=8 7KGG<7F G6@:;=6=8T B6 RKM F ∗1 C 8@;?6@7 B59=86@D;E6 678
FU;B C q! &= <A896=8 ><=E B; @6B;89<= 7K9?;=86 N
q (uL − uR) +1
2g[(
φh2)
L−(
φh2)
R
]
− g (φh)(1)LR (zbR− zbL
) +1
2g(
h(1)LR
)2
(φR − φL) = 0 O !PVQ
/<K@ >F86@:9=6@ B56MG@6779<= >K 86@:6 qT B5FJK;89<= >6 E<=76@?;89<= >6 B;
JK;=898F >6 :<K?6:6=8 678 K89B97F6! '6 A9B;= 678 @F;B97F 7K@ >6KM :;9BB67 >6
E;BEKB 798KF67 >6 G;@8 68 >5;K8@6 >5K=6 >97E<=89=K98F >;=7 B; E<86 >K D<=> 68 B;
G<@<798F! -= @FU9:6 G6@:;=6=8T B; ?;@9;89<= >6 JK;=898F >6 :<K?6:6=8 6=8@6 B67
>6KM E6BBKB67 678 6M;E86:6=8 FU;B6 C B; @F7KB8;=86 >67 D<@E67 6M6@EF67 7K@ B6 RK9>6
7K9?;=8 B5;M6 H<@9W<=8;B! /<K@ B6 E;7 E<=79>F@FT B5FE<KB6:6=8 678 7KGG<7F ;BB6@ >6
B; U;KEH6 ?6@7 B; >@<986! '67 E<=XUK@;89<=7 K89B97F67 G<K@ >F86@:9=6@ B6 A9B;= >6
JK;=898F >6 :<K?6:6=8 7<=8 G@F76=8F67 C B; XUK@6 ! !
'6 E;BEKB >67 D<@E67 678 >F8;9BBF G<K@ B6 E;7 S E<@@67G<=>;=8 C zbL< zbR
68
φL > φR!
Y<@E6 6M6@EF6 G;@ B67 7K@D;E67 .ZY- 68 0,[\ N
Fa = −gˆ φL
φR
[
ˆ zL
zbR
(zL − z) dz]
dφ = −12g (φL − φR) (zL − zbR
)2
O !P]Q
Y<@E6 6M6@EF6 G;@ B; 7K@D;E6 Y\^_ N
Fb = −gφR
ˆ zbR
zbL
(zL − z) dz = −gφR (zbR− zbL
)
(
zL −zbR
+ zbL
2
)
O !P3Q
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
A
B
C
D
E
F
G
H
I
J
K
L
zb,L
zb,R
φR
φL
!" zbL< zbR
φL > φR
A
B
C
D
E
F
G
H
J
K
zb,L
zb,R
φR
φL
#" zbR< zbL
φL > φR
A
B
CD
E
FG
HJ
Kzb,L
zb,R
φR
φL
I
L
$" zbR< zbL
φR > φL
A
B
CD
EF
GH
J
zb,L
zb,R
φR
φL
I
!" zbL< zbR
φR > φL
!"#$% ! " #$%&'()*+,$%- (+,.,-/0- 1$() /+*2.,) .0 2,.*% 30- 4$)50-
6$)50 070)5/0 1*) .0- -()4*50- 869: 0+ ;<=> ?
Fc = −gˆ φL
φR
[
ˆ zbR
zbL
(zL − z) dz]
dφ
= −g (φL − φR) (zbR− zbL
)
(
zL −zbR
+ zbL
2
)
@ !ABC
6$)50 +$+*.0 070)5/0 -() .0 D(,30 ?
Fa + Fb + Fc = −1
2g (φL − φR)h
2L − gφR (zbR
− zbL)
(
zL −zbR
+ zbL
2
)
@ !AEC
8% )/',F0 10)F*%0%+G .0 2,.*% 30 H(*%+,+/ 30 F$(I0F0%+ -J/5),+ ?
(
φhu2 +1
2gφh2
)
R
−(
φhu2 +1
2gφh2
)
L
= Fa + Fb + Fc @ ! KC
8% ,%+)$3(,-*%+ @ !AEC 3*%- @ ! KCG $% /+*2.,+ .* )0.*+,$% -(,I*%+0 ?
q (uR − uL) =1
2g[(
φh2)
L−
(
φh2)
R
]
− 1
2g (φL − φR)h
2L − gφR (zbR
− zbL)
(
zL −zbR
+ zbL
2
)
@ ! LC
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%34
. 567897 :; <=<> 769?@AA><>A8 5@;7 B>? C6? " D E @A 5>;8 F86GB97 BH>I57>??9@A
JFAF76B> ?;9K6A8> L
q (uL − uR) = −1
2g[(
φh2)
L−
(
φh2)
R
]
+1
2g (φL − φR)
(
h(0)LR
)2
+ gφ(0)LR (zbR
− zbL)
(
z(0)LR −
zbR+ zbL
2
)
M ! "N
@O
φ(0)LR = min (φL;φR) M ! P6N
z(0)LR =
zL ?9 zbL< zbR
zR ?9 zbL> zbR
M ! PGN
h(0)LR =
hL ?9 φL > φR
hR ?9 φL < φRM ! PCN
'6 C@<G9A69?@A :> M !QRN >8 M ! "N <SA> D BH>I57>??9@A ?;9K6A8> L
g (zbR− zbL
)
[
φ(0)LR
(
z(0)LR −
zbR+ zbL
2
)
− (φh)(1)LR
]
+1
2g
[
(
h(1)LR
)2
−(
h(0)LR
)2]
(φR − φL) = 0 M ! N
'HFT;689@A M ! N >?8 K6B6GB> T;>BB>? T;> ?@9>A8 B>? K6B>;7? :> φRE φLE zbL>8 zbR
!
'> C6? 56789C;B9>7 φL = φR 5>7<>8 :HF86GB97 L
φ(0)LR
(
z(0)LR −
zbR+ zbL
2
)
= (φh)(1)LR M ! QN
6B@7? T;> B> C6? zbR= zbL
9<5B9T;> L
(
h(1)LR
)2
=(
h(0)LR
)2
M ! UN
-A 9A87@:;9?6A8 M ! QN >8 M ! UN :6A? M !P4NE BH>I57>??9@A :> F ∗1 :>K9>A8 C@<5BSV
8><>A8 :F8>7<9AF> L
F ∗1 =1
λ3 − λ1[F2,L − F2,R + λ3F1,R − λ1F1,L]
− gφ(0)LR
(
z(0)LR −
zbR+ zbL
2
)
zbR− zbL
λ3 − λ1
+1
2g(
h(0)LR
)2 φR − φL
λ3 − λ1M ! WN
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
!"#$%"#&' () "*+$* !&)+,* (∆S0)LR /8 98:;8 <=>:?8 @ ABCD98:EF?8 8<9
8<9C;G @ HF:9C: IB>D JCAFD I8 K>FD9C9G I8 ;=>L8;8D9 <>: I8>M ;FCAA8< I8 HF:9
89 IBF>9:8 IB>D8 IC<?=D9CD>C9G I8 AF ?=98 I> E=DI 89 I8 AF H=:=<C9G+ )D :GNC;8
H8:;FD8D9O A8 JCAFD I8 K>FD9C9G I8 ;=>L8;8D9 <BG?:C9 P
F2,L − F2,R + (∆S0)LR = 0 Q*+* R
(
φhu2 +1
2gφh2
)
L
−(
φhu2 +1
2gφh2
)
R
+ (∆S0)LR = 0
)D CD9:=I>C<FD9 ABGK>F9C=D I8 ?=D9CD>C9G Q*+SSR 89 AF IGTDC9C=D I> 98:;8 <=>:?8
@ ABCD98:EF?8 Q*+S!R IFD< Q*+* RO =D =J9C8D9 P
q (uL − uR) +1
2g[(
φh2)
L−
(
φh2)
R
]
− g (φh)(S)LR (zbR− zbL
) +1
2g(
h(S)LR
)2
(φR − φL) = 0 Q*+*UR
)D :8;HAFVFD9 q (uL − uR) HF: <=D 8MH:8<<C=D Q*+*!RO Q*+*UR I8LC8D9 P
g
[
φ(0)LR
(
z(0)LR −
zbR+ zbL
2
)
− (φh)(S)LR
]
(zbR− zbL
)
+1
2g
[
(
h(S)LR
)2
−(
h(0)LR
)2]
(φR − φL) = 0 Q*+WXR
)D >9CAC<FD9 A8< ?F< HF:9C?>AC8:< φR = φL 89 zbR= zbL
O =D H8>9 IGI>C:8 I8 Q*+WXR
K>8 P
(φh)(S)LR = φ
(0)LR
(
z(0)LR −
zbR+ zbL
2
)
Q*+WYR
(
h(S)LR
)2
=(
h(0)LR
)2
Q*+W!R
/B8MH:8<<C=D I> 98:;8 <=>:?8 @ ABCD98:EF?8 8<9 I=D? P
(∆S0)LR = −gφ(0)LR
(
z(0)LR −
zbR+ zbL
2
)
(zbR− zbL
)
+1
2g(
h(0)LR
)2
(φR − φL) Q*+WSR
!"#$%"#&' () -). (* /)%'"#"0 (* $&)1*$*'" F ∗2 0D JCAFD I8 K>FD9C9G
I8 ;=>L8;8D9 8D :GNC;8 H8:;FD8D9 8<9 :GFAC<G <>: 9:=C< ;FCAA8< I8 ?FA?>A Qi− 1Oi 89 i+1R FL8? >D8 IC<?=D9CD>C9G I8 H=:=<C9G 89 I8 ?=98 I> E=DI 8D9:8 A8< ;FCAA8<
i 89 i+ 1 QL=C: 7CN>:8 *+WR+
/8 JCAFD I8 K>FD9C9G I8 ;=>L8;8D9 <>: AF ?8AA>A8 i <BG?:C9 P
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%34
zb
zb
i-1
i+1
φi+1
φi-1
φi
zb
i
!"#$% !5 6 0789:;<=>?78 ;>?@?ABC D7;< E=?<C @C F?@=8 GC H;=8>?>B GC I7;JCIC8>
D7;< GB>C<I?8C< (φh)(2)LR C> h
(2)LR
(φhu)n+1i − (φhu)
ni =
∆t
∆xi
(
F ∗2 +λ3
λ3 − λ1∆S0
)n+1/2
i−1/2−
∆t
∆xi
(
F ∗2 +λ1
λ3 − λ1∆S0
)n+1/2
i+1/2
= 0 K !5 L
-8 M7IF?8=8> K !4NL C> K !5 LO 78 7F>?C8> D7;< @P?8>C<E=MC C8><C @CA MC@@;@CA i− 1C> i Q
(λ3 − λ1)(
F ∗2 +λ3
λ3 − λ1∆S0
)n+1/2
i−1/2=
λ3F2,i−1 − λ1F2,i + λ1λ3 (F1,i − F1,i−1)
+ (λ3 + λ1)
[
−g (φh)(2)LR
(
zbi− zbi−1
) 1
2g(
h(2)LR
)2
(φi − φi−1)
]
− λ3gφ(0)LR
(
z(0)LR −
zbi+ zbi−1
2
)
(
zbi − zbi−1
)
+ λ31
2g(
h(0)LR
)2
(φi − φi−1) K !55L
/;?AH;P?@ 8PR = D=A GC G?AM78>?8;?>B C8><C @CA MC@@;@CA i− 1 C> i Q(
φhu2 +1
2gφh2
)
i−1=
(
φhu2 +1
2gφh2
)
i
6=(
φhu2 +1
2gφh2
)
i+1
(φhu)i−1 = (φhu)i = (φhu)i+1 = qφi−1 = φi 6= φi+1
zbi−1= zbi 6= zbi+1
K !5SL
!
"#$%&'() !* (+,-./'&-0 1), +2/$'&-0, 1) %(-%$3$'&-0 %$(
.$ 4+'#-1) 1), 5-./4), 6&0&,
)7 879:;<=8>?79 @!*ABC <?7> @!*AACD ;7 ;E98F79 G
(
F ∗2 +λ3
λ3 − λ1∆S0
)n+1/2
i−1/2=
1
λ3 − λ1
[
λ3
(
φhu2 +1
2gφh2
)
i
− λ1(
φhu2 +1
2gφh2
)
i
]
=
(
φhu2 +1
2gφh2
)
i
@!*AHC
.F I=J <F K=?7989L <F M;=NFMF79 F79:F OF> PFOO=OF> i F9 i+ 1 >QLP:89 G
(λ3 − λ1)(
F ∗2 +λ1
λ3 − λ1∆S0
)n+1/2
i+1/2
=
λ3
(
φhu2 +1
2gφh2
)
i
− λ1(
φhu2 +1
2gφh2
)
i+1
+
λ1λ3[
(φhu)i+1 − (φhu)i]
+
λ1
[
−gφ(0)LR
(
z(0)LR −
zbi+1+ zbi
2
)
(
zbi+1− zbi
)
+1
2g(
h(0)LR
)2
(φi+1 − φi)
]
+
(λ3 + λ1)
[
−g (φh)(2)LR
(
zbi− zbi−1
)
+1
2g(
h(2)LR
)2
(φi − φi−1)
]
@!*A C
)7 879:;<=8>?79 OQLK=?98;7 <F P;7987=89L @!*RRC <?7> @!*A CD ;7 ;E98F79 G
(λ3 − λ1)(
F ∗2 +λ1
λ3 − λ1∆S0
)n+1/2
i+1/2
=
λ3
(
φhu2 +1
2gφh2
)
i
− λ1(
φhu2 +1
2gφh2
)
i+1
+
(λ3 + λ1)
[
−g (φh)(2)LR
(
zbi− zbi−1
)
+1
2g(
h(2)LR
)2
(φi − φi−1)
]
+
λ1
[
−gφ(0)LR
(
z(0)LR −
zbi+1+ zbi
2
)
(
zbi+1− zbi
)
+1
2g(
h(0)LR
)2
(φi+1 − φi)
]
@!*ASC
.F E8O?7 <F K=?7989L <F M;=NFMF79 F7 :LT8MF UF:M?7F79 @!*! C 8MUO8K=F G
(
F ∗2 +λ3
λ3 − λ1∆S0
)n+1/2
i−1/2=
(
F ∗2 +λ1
λ3 − λ1∆S0
)n+1/2
i+1/2
@!*BVC
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%34
-5 6789:5;5< = !>?@A = !4B@ C< = !4D@A 75 <E7FGC H
0 = λ1
[
q (ui − ui+1) +1
2g[
(
φh2)
i−
(
φh2)
i+1
]
]
+ (λ3 + λ1)
[
−g (φh)(2)LR
(
zbi− zbi−1
)
+1
2g(
h(2)LR
)2
(φi − φi−1)
]
+ λ1
[
−gφ(0)LR
(
z(0)LR −
zbi+1+ zbi
2
)
(
zbi+1− zbi
)
+1
2g(
h(0)LR
)2
(φi+1 − φi)
]
= !>I@
-5 EC8JK;L;5< q (ui − ui+1) M;5N = !>I@ J;E KOCPJECNN:75 = ! "@A 75 79<:C5< H
0 = −g (φh)(2)LR
(
zbi− zbi−1
)
+1
2g(
h(2)LR
)2
(φi − φi−1) = !>"@
&5 JCF< M756 C5 6756KFEC QFC H
(φh)(2)LR = 0 = !>R@
(
h(2)LR
)2
= 0 = !> @
'OC5NC89KC MCN G;E:;9KCN S KO:5<CET;6C ;U;5< V<V MV<CE8:5VA KC NUN<W8C = !RI@ JCF<
NOV6E:EC H
F ∗1 =1
λ3 − λ1[F2,L − F2,R − λ1F1,L + λ3F1,R]
+1
λ3 − λ1
[
1
2g (φR − φL)
(
h(0)LR
)2
− gφ(0)LR (zbR− zbL
)
(
z(0)LR −
zbR+ zbL
2
)]
= !>4;@
F ∗2 =1
λ3 − λ1[λ3F2,L − λ1F2,R − λ1λ3 (F1,L − F1,R)] = !>49@
F ∗3 = v (F ∗1 − F1,A) + F3,A = !>46@
7X φ(0)LRA z
(0)LR C< h
(0)LR N75< MVY5:N J;E = ! R@!
!"! #$%&'()')*+( ,- .+&/0-1 2+134
'C NUN<W8C MOVQF;<:75N = !"D@ F<:K:NV J7FE V<;K:E KCN ZFP = !>4@ CN< 675N<EF:<
NFE KO[UJ7<[WNC MOF5C 75MC JE7GC5;5< MC 6[;QFC 6CKKFKC MC J;E< C< MO;F<EC MC
KO:5<CET;6C 675N:MVEVC! *K 67EECNJ75M M756 ;F ZFP M;5N K; \75C :5<CE8VM:;:EC
MOV<;< 675N<;5<! '; 8V<[7M7K7]:C ;FP G7KF8CN Y5:N 5V6CNN:<C MC MV<CE8:5CE KCN
ZFP S KO:5<CET;6C C5<EC MCFP 8;:KKCN MC 6;K6FKA !"! S <E;GCEN K; M:N675<:5F:<V
:5:<:;KC MF JE79KW8C MC #:C8;55! 0C<<C :5<CET;6C 5OV<;5< J;N 5V6CNN;:EC8C5<
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
U
x
U
x
U
x
t
x
t
x
t
x
U
x
U
x
t
x
t
xCon!guration C4
Con!guration C1 Con!guration C2 Con!guration C3
Con!guration C5
λ1,L
λ1
λ3
λ3,R
∗
∗
!"#$% *+! 8 %9:;<;9= >?@A<;B? C? @A D9=? ;=<?>EFC;A;>? GA> >AGG9>< H @I;=<?>JAK?
?=<>? @?: EA;@@?: C? KA@KL@ M:LG?>G9:F? AB?K @IAN? C?: <?EG:O
!"! #$%&'()*&+ ,- '. /.#)*- 0&+%-#1.)*1- ,-% $2(.)*&+%34
5678 9:;;: <=7: 5>?;6; 9=78;67;@ A:8 ?BC6;D=78 5:8 ECF G !HIJ 7: K:CL:7; 5=79
K68 M;N: C;DAD8?:8 K=CN ;=C;:8 A:8 9=7OPCN6;D=78 5>=75:8! %CDL67; A6 9=7OPCN6;D=7
5>=75:8 :; A6 K=8D;D=7 N:A6;DL: 5: A6 <=7: 5>?;6; 9=78;67; K6N N6KK=N; Q A>D7;:NR69:@
9D7B 968 K:CL:7; M;N: D5:7;DO?8 GL=DN SDPCN: !HJ!
':8 9=7OPCN6;D=78 0T :; 0I 9=NN:8K=75:7; 6C N?PDU: ;=NN:7;D:A@ A:8 9=7OPCV
N6;D=78 0" :; 0 6C N?PDU: 9ND;DBC: :; A6 9=7OPCN6;D=7 0W 6C N?PDU: ECLD6A!
'6 5?;:NUD76;D=7 5:8 9?A?ND;?8 5>=75:8 λ∗1 :; λ∗3 5678 A6 <=7: 5>?;6; 9=78;67;
:8; 7?9:886DN: K=CN 5?;:NUD7:N A6 9=7OPCN6;D=7! ':8 D7L6ND67;8 5: #D:U677 8=7;
C;DAD8?8 A: A=7P 5: 9X6BC: 96N69;?ND8;DBC: K=CN 5?;:NUD7:N A6 LD;:88: 5: A>?9=CA:V
U:7; u∗ :; A6 9?A?ND;? 5:8 =75:8 5: KN:88D=7 c∗ 5678 A6 <=7: D7;:NU?5D6DN: 5>?;6;
9=78;67; Y
u∗ + 2c∗ = uL + 2cLu∗ − 2c∗ = uR − 2cR
G !HHJ
)X?=NDBC:U:7;@ DA 9=7LD:7; 5: KN:75N: :7 9=UK;: A:8 ;:NU:8 8=CN9: 5678 A: 8Z8V
;[U: 5>?BC6;D=78 G !HHJ U6D8 A>:FK?ND:79: U=7;N: BC: A>=7 K:C; A:8 7?PADP:N Q
9: 8;65:! '=N8 5: A6 O76AD86;D=7 5C KN?8:7; U?U=DN:@ DA 6 ?;? UD8 :7 ?LD5:79:
BC: A6 KND8: :7 9=UK;: 5:8 ;:NU:8 8=CN9: 5678 A6 R=NUCA6;D=7 5:8 D7L6ND67;8 5:
#D:U677 :8; 7?9:886DN: K=CN A6 U=5?AD86;D=7 5: 9:N;6D78 N?PDU:8 K:NU67:7;8 6C
7DL:6C 5: A6 ADUD;: 5: A6 <=7: D7=756\A:! '6 KND8: :7 9=UK;: 5:8 ;:NU:8 8=CN9:
D7EC: 7=;6UU:7; 8CN A: 96A9CA 5: A6 LD;:88: 5: A>?9=CA:U:7; u∗ 5678 A6 <=7:
D7;:NU?5D6DN: :; 7?9:88D;: 5: N?6AD8:N 9:N;6D7:8 N:9X:N9X:8 9=UKA?U:7;6DN:8 K=CN
:8;DU:N A6 L6A:CN 5:8 L6ND6\A:8 Q A>D7;:NR69:! '6 N?8=AC;D=7 5: G !HHJ 5=77: Y
u∗ =uL + uR
2+ cL − cR
c∗ =uL − uR
4+cL + cR
2
G !H4J
':8 9?A?ND;?8 5>=75:8 λ∗1 :; λ∗3 8=7; 5=79 ?P6A:8 Q Y
λ∗1 = u∗ − c∗λ∗3 = u∗ + c∗
G !H3J
'>XZK=;X[8: 5: 5:CF =75:8 5: N6N?R69;D=7 8?K6N67; C7: <=7: D7;:NU?5D6DN:
5>?;6; 9=78;67; 5:8 ?;6;8 P6C9X: :; 5N=D; 5C KN=\A[U: 5: #D:U677 5=D; M;N:
L?NDO?:! ,678 A: 968 =] DA Z 6 C7 9X=9 :7;N: A>?;6; P6C9X: :; A>?;6; D7;:NU?5D6DN:
Gλ1,L > λ∗1J =C :7;N: A>?;6; 5N=D; :; A>?;6; D7;:NU?5D6DN: Gλ3,R < λ∗3J@ A6 LD;:88: cs5: KN=K6P6;D=7 5C 9X=9 5=D; M;N: :8;DU?:! -7 :^:;@ 5678 C7: ;:AA: 9=7OPCN6;D=7
5>=75:@ A:8 9?A?ND;?8 8=7; :7 N?6AD;? ?P6A: Q A6 LD;:88: 5: KN=K6P;D=7 5C 9X=9! '6
9?A?ND;? 5C 9X=9 :8; 6A=N8 5?;:NUD7?: :7 C;DAD867; A:8 N:A6;D=78 5: 86C; 5: #67_D7:V
`CP=7D=;! /=CN A: 968 5>C7 9X=9 :7;N: A>?;6; P6C9X: :; A>?;6; D7;:NU?5D6DN:@ =7
=\;D:7; Y
cs =F ∗1 − F1,LU∗
1 − U1,L=h∗u∗ − hLuL
h∗ − hLG !Ha6J
cs =F2,L − F ∗2U2,L − U∗2
=hLu
2L +
12gh
2L − h∗u∗2 − 1
2gh∗2
hLuL − h∗u∗G !Ha\J
!"#$%&'( )* '+,-./&%-0 1(, +2/#&%-0, 1( $'-$#3#&%-0 $#'
.# 4+&"-1( 1(, 5-./4(, 6%0%,
.78 97:; <=:>?@AB8 C7:D7B? E?F7 :?@G@8<78 @B9@H<F7II7B?* &A:?7JA@8 8@ h∗ = hL
A: 8@ hLuL = h∗u∗ GK:B7 978 JAFI:G>?@AB8 78? @B9<?7FI@B<7 L G7 8AGD7:F :?@G@87
>GAF8 GK>:?F7 JAFI:G>?@AB*
.> MAIC>F>@8AB 97 λ∗1N λ∗3N λ1,L 7? λ3,R C7FI7? 97 9<?7FI@B7F G> MABOP:F>?@AB
9KAB97 7? C>F MAB8<=:7B? G> JAFI:G>?@AB Q 7ICGAR7F CA:F 78?@I7F G7 S:; Q ?F>D7F8
G> 9@8MAB?@B:@?< @B@?@>G7* 1>B8 G7 M>8 9K:B F<P@I7 MF@?@=:7N G> JAFI:G>?@AB MG>88@=:7
"..! 78? M7GG7 7ICGAR<7 T
λ1,L > 0
F1 = F1,LF2 = F2,L
!U
λ1,L < 0 Vλ∗1 > 0
F1 =1
λ3 − λ1[λ3F1,L − λ1F1,R − λ1λ3 (zL − zR)]
F2 =1
λ3 − λ1[λ3F2,L − λ1F2,R − λ1λ3 (F1,L − F1,R)]
!W
λ∗1 < 0 Vλ∗3 > 0
F1 = F ∗1F2 = F ∗2
!X
λ∗3 < 0 Vλ3,R > 0
F1 =1
λ3 − λ1[λ3F1,L − λ1F1,R − λ1λ3 (zL − zR)]
F2 =1
λ3 − λ1[λ3F2,L − λ1F2,R − λ1λ3 (F1,L − F1,R)]
!)
λ3,R < 0
F1 = F1,RF2 = F2,R
!Y
Z)*[\]
.78 S:; Q GK@B?7FJ>M7 Z)*[\] A^?7B:8 C>F F<8AG:?@AB 97 G> C>F?@7 MAB87FD>?@D7 978
<=:>?@AB8 7? GK<?>? >: 9<^:? 9: C>8 97 ?7IC8 C7FI7??7B? G7 M>GM:G 97 GK<?>?
@B?7FI<9@>@F7 Q C>F?@F 97 GK<=:>?@AB 9@8MF<?@8<7 Z)*X]* !7? <?>? @B?7FI<9@>@F7 78?
BA?< Un,c
ZDA@F 6@P:F7 )*U]*
!" #$%&'()%*+$ ,-. %-(/-. ,- 0(+%%-/-$%
.> I<?_A9AGAP@7 7ICGAR<7 CA:F CF7B9F7 7B MAIC?7 G78 ?7FI78 97 JFA??7I7B?
9>B8 G7 MA97 ,`W1 > <?< MAB87FD<7 9>B8 ,`UW1* (GG7 MAB8@8?7 Q F<8A:9F7 >B>a
GR?@=:7I7B? G78 <=:>?@AB8 9@H<F7B?@7GG78 AF9@B>@F78 8:@D>B?78 T
dφq
dt= −gφ
√
q2 + r2
K2h2R1/3
h,x
q Z)*[U>]
dφr
dt= −gφ
√
q2 + r2
K2h2R1/3
h,y
r Z)*[U^]
.> CAFA8@?< ZA: G> G>FP7:F 9: M>B>G U1] B7 D>F@>B? C>8 >D7M G7 ?7IC8N M78 <=:>a
?@AB8 8AB? <=:@D>G7B?78 Q T
dq
dt= −g
√
q2 + r2
K2h2R1/3
h,x
q Z)*[W>]
dr
dt= −g
√
q2 + r2
K2h2R1/3
h,y
r
Rh
x
y
χx=b
x
χy=b
y
Sx=b
xh
Sy=b
yh
x
y
χx=b
x
χy=b
y+2h
1D
Sy=b
yh1D
Sx=b
xh1D
x
y
Sx=b
xh
Sy=b
yh
χx=b
x
χy=b
y+z
b,2D-z
b,1D
Périmètre mouillé
Rh =S
χ
S = bh b
χ
h≪ b
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
89:; <=>? @AABCBDE F D@ D@>G?H> 8? DIE;9HD?C?:= J χ ≈ b K D? >@L9: ML8>@HDBNH? ?A=
@D9>A EG@D F D@ M@H=?H> 8I?@H J Rh ≃ h+2@:A D? ;@A 8?A C@BDD?A O2P D? C<C? >@BA9::?C?:= Q?H= <=>? R@B= Q9H> D@ 8E=?>S
CB:@=B9: 8H >@L9: ML8>@HDBNH? Q9H> D? ;@D;HD 8?A R>9==?C?:=A 8@:A D@ 8B>?;=B9:
=>@:AT?>A@D? @H ;@:@D O2+ ): >?T@:;M?P Q9H> D?A R>9==?C?:=A 8@:A D@ 8B>?;=B9:
D9:GB=H8B:@D?P DIMLQ9=MUA? h≪ b ?A= F Q>B9>B R@HAA?+ ): ?V?=P D?A C@BDD?A O2 A9:=
H=BDBAE?A Q9H> C98EDBA?> 8?A ;M?:@HW 8IE;9HD?C?:= 8? R@BXD? D@>G?H>P D? QE>BCU=>?
C9HBDDE :? Q?H= QDHA <=>? @AABCBDE F D@ ABCQD? D@>G?H> 8? DIE;9HD?C?:=+ &D ;9:TB?:=
@D9>A 8? =?:B> ;9CQ=? 8H ;9:=@;= ?:=>? DI?@H ?= D?A Q@>9BA 8H ;@:@D O2+ %9H> D@
Q>BA? ?: ;9CQ=? 8?A R>9==?C?:=A 8@:A D@ 8B>?;=B9: D9:GB=H8B:@D? AH> D?A C@BDD?A
O2P D? >@L9: ML8>@HDBNH? Q>?:8 89:; DI?WQ>?AAB9: AHBT@:=? J
Rh =bh
b+ 2max (h; zb2D− zb1D
)Y*+Z*[
/? >@L9: ML8>@HDBNH? Rh E=@:= 8E=?>CB:E Q9H> DI?:A?CXD? 8?A C@BDD?A 8? ;@D;HDP
BD ?A= @D9>A Q9AABXD? 8? >EA9H8>? D?A ENH@=B9:A 8BVE>?:=B?DD?A 9>8B:@B>?A >EGBAA@:=
D?A R>9==?C?:=A+
!"!# $%&'()*+', -.& %/)0*+',& -+1%2.,*+.((.& '2-+,0+2.&
2@:A =9HA D?A ;@AP D@ Q>BA? ?: ;9CQ=? 8?A =?>C?A 8? R>9==?C?:= >?TB?:= F
>EA9H8>? H:? ENH@=B9: 8BVE>?:=B?DD? 9>8B:@B>? 8? D@ R9>C? J
df
dt= −αf Y*+Z\[
9] f = q 9H f = r+ "?==? ENH@=B9: ?A= ENHBT@D?:=? F J
df
f= −αdt Y*+Z^[
"?==? >?D@=B9: ?A= B:=EG>E? AH> D? Q@A 8? =?CQA J
ˆ fn+1
fn
1
fdf =
ˆ tn+1
tn
−αdt Y*+ZZ[
/@ >EA9DH=B9: @:@DL=BNH? 8? ;?==? ENH@=B9: 89::? J
[ln f ]fn+1
fn = −α∆tln fn+1 − ln fn = −α∆t
ln
(
fn+1
fn
)
= −α∆t
fn+1
fn= exp (−α∆t)
fn+1 = fn exp (−α∆t) Y*+Z_[
! ! "#$"%$ &'( )$%* + ,-#.'-( $'( /0,'-)#"'( /0,1-/'%-'(23
4567 89: ;<=889: >&? 5@ < A
α2D = g
√
q2 + r2
K2h7/3 !"#$%
&'() *+, -./**+, 012 '3 4'3,+)5+ *6+78)+,,/'3 9( α2D 8'() *+ 4.*4(* 9+, :)';;+<
-+3;, 9.3, *. 9/)+4;/'3 ;).3,5+),.*+ .( 4.3.*" &'() *. 9/)+4;/'3 *'3=/;(9/3.*+2
'3 . >
α = g
√
q2 + r2
K2h7/3
(
b
b+2max(h;zb2D−zb1D )
)1/3 !"?@%
&'() 4A.B(+ -./**+ 9+ 4.*4(*2 /* +,; .*'), 8',,/C*+ 96/3;D=)+) *+, ;+)-+, 9+ :)';<
;+-+3; 8.) *6/3;+)-D9/./)+ 9+ !"#?%2 α D;.3; 9DE3/ 8.) !"#$% '( !"?@%" &'() *+
4., 96(3+ -./**+ F12 *+ 5+4;+() 9+, 5.)/.C*+, Un,c
+,; ;).3,:')-D 8.) *6+78)+,,/'3
,(/5.3;+ >
hn,f = hn,c !"?0%
qn,f = qn,c exp
−g
√
(qn,c)2+ (rn,c)
2
K2 (hn,c)7/3
∆t
!"?F%
rn,f = rn,c exp
−g
√
(qn,c)2+ (rn,c)
2
K2 (hn,c)7/3
∆t
!"?G%
! "#$%&$ '() *&+ , -.#/(.) $() 01-(.2#%() 01-34
.0(&.()
H+ ,'*5+() 9+ I/+-.33 9D5+*'88D 8)D4D9+--+3; 8+)-+; *+ 4.*4(* 9+, J(7
K ;).5+), *+, /3;+):.4+, +7;D)/+()+,2 !"! *+, /3;+):.4+, ,D8.).3; 9+(7 -./**+, 9+
4.*4(* F1 '( 9+(7 -./**+, 9+ 4.*4(* 01" H+, J(7 +3;)+ (3+ -./**+ 01 +; *. -./**+
F1 B(/ *. 4'3;/+3; )+,;+3; K 9D;+)-/3+)" L* +,; ).88+*D B(+ *6(3 9+, 'CM+4;/:, E7D,
8'() *. -D;A'9'*'=/+ 9( 4'9+ 4'(8*D 01<F1 +,; B(+ *+ 4.*4(* 9+, J(7 +3;)+ *+,
-./**+, 01 +; F1 3+ )D9(/,+3; 8., *+ 8., 9+ ;+-8, 9+ 4.*4(*"
&.) ,'(4/ 9+ ,/-8*/4/;D2 '3 ,+ 8*.4+ 9.3, *+ )D:D)+3;/+* .,,'4/D .( 4.3.* 01
R′
(
O;x′
; y′
; z)
5'/) N/=()+ G"!%"
! !" #$%&'()*+* ,+ '-./.01
H+, AO8';AP,+, (;/*/,D+, 8'() *+ 4.*4(* 9+, J(7 +3;)+ *+, -./**+, 01 +; F1
,'3; *+, ,(/5.3;+, >
Q RO8" !"0 > K *. E3 9( 8., 9+ ;+-8,2 *. 4';+ 9+ *. ,():.4+ */C)+ ,() (3+
-./**+ 01 +,; /9+3;/B(+ K 4+**+ ,() *. -./**+ F1 B(/ *. 4'3;/+3; ,/ +**+ +,;
3'3 3(**+
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
8 #9:+ *+! ; <= >?@:?A=BCD ED <= FGCDAAD CH=BAFDHA=<D =I >=B=< J2 K <= LB
EI :=A ED CD@:A DAC <= @M@D AIH IBD @=G<<D J2 DC AIH <= @=G<<D !2 NIG <=
>?BCGDBC
! !" #$%&'()$* +) ,%'')
/D CH=BAODHC ED @=AAD DBCHD <DA EDIP @=G<<DA ED >=<>I< DAC HQ=<GAQ K :=HCGH ED
<RS9:?CSTAD *+J+ "DCCD S9:?CSTAD :DH@DC ED >=<>I<DH <= >?CD ED <= AIHO=>D <GUHD
E=BA >S=NID >D<<I<D =:HTA <D CH=BAODHC ED @=AAD+ -IGF=BC <= >?CD HD<=CGFD ED <=
AIHO=>D <GUHD DC EI O?BE E=BA <DA >D<<I<DAV :<IAGDIHA >?BLWIH=CG?BA E?GFDBC MCHD
EGACGBWIQDA XF?GH 7GWIHD *+YZ+ /RD=I QC=BC =AAG@G<QD K IB [IGED GB>?@:HDAAGU<D
h1D
h2D
!" #$$%&'()(*+ ,( -! )!.--( /0
h1D
h2D
1" 23!*$4(3+ ,( -! )!.--( 50 6(3$ -! )!.--(
/0
h1D
h2D
&" 23!*$4(3+ ,( -! )!.--( /0 6(3$ -! )!.--(
50
!"#$% *+Y 8 "?BLWIH=CG?B EDA S=ICDIHA ERD=I E=BA <DA @=G<<DA ED >=<>I< K <=
LB ED <RQC=:D ED HQA?<ICG?B EDA QNI=CG?BA >?BADHF=CGFDA
X#9:+ \+!ZV G< DAC :?AAGU<D ED H=GA?BBDH DB CDH@DA ED F?<I@D :<IC]C NIRDB CDH@DA
ED @=AAD+
! !"!# $%%&'()*)+, -) ./ */0..) "1 23405 60785) !92/::
/D F?<I@D ERD=I >?BCDBI E=BA <= @=G<<D !2 =F=BC <D CH=BAODHC DAC QW=< K ;
V2D = A2D (h2D)n,f
X*+Y*Z
-G >D F?<I@D ERD=I DAC GBOQHGDIH =I F?<I@D EGA:?BGU<D E=BA <= @=G<<D J2 =F=BC
A?B EQU?HED@DBCV <= @=G<<D !2 AR=AAT>SD ;
! ! "#$"%$ &'( )$%* + ,-#.'-( $'( /0,'-)#"'( /0,1-/'%-'(23
V2D < V1D,dispo = A1D
(
zb2D− zb1D
− (h1D)n,f)
4 !567
$8 98:;<:= >?<8: >8@A B8 C8DBB< E& F B8 G@ >: ;=8@AH<=; >< C8AA< <A; 8BI=A
@:BB< J hn,m2D = 0! $8 98:;<:= >?<8: K;8@; @:BB< >8@A B8 C8DBB< E&L B8 M:8@;D;K
>< CI:N<C<@; O <A; KP8B<C<@; @KQ<AA8D=<C<@; @:BB< J qn,m2D = 0 <; rn,m
2D = 0! $8QI;< >< B8 A:=H8Q< BDR=< >8@A B8 C8DBB< S& 8T=UA B< ;=8@AH<=; >< C8AA< <A; 8BI=A B8
A:DN8@;< J
(z1D)n,m
= (z1D)n,f
+V2D
A1D4 !5V7
(DC:B;8@KC<@; 8: ;=8@AH<=; >< C8AA< ><T:DA B8 C8DBB< E& N<=A B8 C8DBB< S&L
B8 M:8@;D;K >< CI:N<C<@; >< B?KQI:B<C<@; E& <A; ;=8@AHK=K<! $8 M:8@;D;K ><
CI:N<C<@; >8@A B8 C8DBB< S& 8T=UA ;=8@AH<=; >< C8AA< <A; >I@Q KP8B< F J
(q1D)n,m
= (q1D)n,f
+(q2D)
n,fA2D
A1D4 !5W7
(r1D)n,m
= (r1D)n,f
+(r2D)
n,fA2D
A1D4 !557
$8 C8DBB< S& @?<A; ;I:;<HIDA T8A >KRI=>8@;<L B?KQI:B<C<@; CIO<@ >8@A B8 >D=<QX
;DI@ ;=8@AN<=A8B< <A; >I@Q @KQ<AA8D=<C<@; @:B! $8 M:8@;D;K >< CI:N<C<@; >8@A
B8 >D=<Q;DI@ ;=8@AN<=A8B< 8T=UA ;=8@AH<=; <A; >I@Q @KQ<AA8D=<C<@; @:BB< J
rn,m1D = 0 4 !527
! !"!" #$%&'&()*+, -,. /*&'',. 01 ,2 "1 345&) 6&+%), !73(8 ,2 3988
$?<8: K;8@; QI@AD>K=K< QICC< :@ Y:D>< D@QICT=<AADRB< 4ZOT! 3!E7L B< NIB:C<
PBIR8B QI@;<@: >8@A B8 C8DBB< S& <; B8 C8DBB< E& @< N8=D< T8A >:=8@; B< ;=8@AH<=;
>< C8AA<! $< ;=8@AH<=; >< C8AA< <@;=< B<A ><:[ C8DBB<A >ID; >I@Q T<=C<;;=< ><
NK=DG<= B?9OTI;9UA< A:= B?KP8BD;K ><A QI;<A F Q98M:< D@A;8@; 4ZOT! !S7! $< NIB:C<
>?<8: V QI@;<@: 8: ;<CTA D@;<=CK>D8D=< A:= B<A ><:[ C8DBB<A <A; KP8B F J
V = A1D (h1D)n,f
+A2D (h2D)n,f
= A1D
(
(z1D)n,f − zb1D
)
+A2D
(
(z2D)n,f − zb2D
)
4 !2\7
<; T<:; ];=< Q8BQ:BK F T8=;D= ><A N8=D8RB<A F B8 G@ >< B?K;8T< 9OT<=RIBDM:<! '@
D@;=I>:DA8@; B8 =<B8;DI@ (z1D)n,f
= (z2D)n,f
= z DCTIAK< T8= B?9OTI;9UA< !S
>8@A B?KM:8;DI@ 4 !2\7L I@ IR;D<@; J
V = A1D (z − zb1D) +A2D (z − zb2D
) 4 !2S7
"<;;< =<B8;DI@ <A; KM:DN8B<@;< F J
z =A1D (z1D)
n,f+A2D (z2D)
n,f
A1D +A2D4 !2E7
!
"#$%&'() !* (+,-./'&-0 1), +2/$'&-0, 1) %(-%$3$'&-0 %$(
.$ 4+'#-1) 1), 5-./4), 6&0&,
.7 89:;<7 =>?@ABC>C D7E;FA :? <?F::7 G1 87>A :? <?F::7 H1 7A= D9@I CJ?: K L
V1D→2D = A1D
(
(z1D)n,f − z
)
= −A2D(
(z2D)n,f − z
)
M!* NO
.? P;?@=F=C D7 <9;87<7@= I9@=7@;7 D?@A I7 89:;<7 7A= CJ?:7<7@= =>?@ABC>C7*
$E>QA CP;F:FR>?J7S :7A I9<E9A?@=7A D7 :? P;?@=F=C D7 <9;87<7@= A;> IT?P;7
<?F::7 D7 I?:I;: AUCI>F87@=S D?@A :7 I?A D; =>?@AB7>= D7E;FA :? <?F::7 H1 87>A :?
<?F::7 G1 L
(q1D)n,m
= (q1D)n,f
+ (q2D)n,f (z2D)
n,f − z(h2D)
n,f
A2DA1D
M!* !?O
(q2D)n,m
= (q2D)n,f − (q2D)
n,f (z2D)n,f − z
(h2D)n,f
M!* !RO
(r1D)n,m
= (r1D)n,f
+ (r2D)n,f (z2D)
n,f − z(h2D)
n,f
A2DA1D
M!* !IO
(r2D)n,m
= (r2D)n,f − (r2D)
n,f (z2D)n,f − z
(h2D)n,f
M!* !DO
7= E9;> :7 =>?@AB7>= D7 :? <?F::7 G1 87>A :? <?F::7 H1 L
(q1D)n,m
= (q1D)n,f − (q1D)
n,f (z1D)n,f − z
(h1D)n,f
M!* V?O
(q2D)n,m
= (q2D)n,f
+ (q1D)n,f (z1D)
n,f − z(h1D)
n,f
A1DA2D
M!* VRO
(r1D)n,m
= (r1D)n,f − (r1D)
n,f (z1D)n,f − z
(h1D)n,f
M!* VIO
(r2D)n,m
= (r2D)n,f
+ (r1D)n,f (z1D)
n,f − z(h1D)
n,f
A1DA2D
M!* VDO
! !" #$%&'&()*+, -, '* $%*./&/0 -, 12%3,1,./ /)*.43,)5
4*',
.7 =>?@AB7>= D7 P;?@=F=C D7 <9;87<7@= D?@A :? DF>7I=F9@ =>?@A87>A?:7 7A= R?AC
A;> :UTWE9=TQA7 !*H* )@ 7X7=S :? A=>;I=;>7 A;EE9AC7 D7 :UCI9;:7<7@= P;F A7 <7=
7@ E:?I7 7A= E>CA7@=C7 K :? YJ;>7 N*H* .7 RF:?@ D7 P;?@=F=C D7 <9;87<7@= D?@A
:? DF>7I=F9@ =>?@A87>A?:7 K :U?Z7 D; I?@?: G1 A;> :U7@A7<R:7 D7A <?F::7A AUCI>F= L
(QdMt)n,m
= A2D (r2D)n,m
+A1D (r1D)n,m
= A2D (h2D)n,m
(v2D)n,m
+A1D (h1D)n,m
(v1D)n,m
M!* [O
! ! "#$"%$ &'( )$%* + ,-#.'-( $'( /0,'-)#"'( /0,1-/'%-'(23
$4 56789:465 ;4 <=785>5? ;4 @A=B4@485 ;A>5 C46@45564 ;4 B?6>D46 EFGHCA5GI94 !J K
(v2D)n,t
= (v1D)n,t
= vn,t! #C6I9 6??<=>E>L67M4N E7 <=785>5? ;4 @A=B4@485 9=6
E49 ;4=O P4EE=E49 495 ?M7E4 Q K
(QdMt)n,t
= A2D (h2D)n,tvn,t +A1D (h1D)
n,tvn,t
R !2ST
#= PA=69 ;= 56789:465N E7 <=785>5? ;4 @A=B4@485 56789B4697E4 9=6 E49 ;4=O @7>EE49
;4 P7EP=E 64954 E7 @U@4 K
(QdMt)n,m
= (QdMt)n,t
R !2VT
'8 >856A;=>9785 E49 ;?D8>5>A89 ;4 (QdMt)n,m
45 (QdMt)n,t
45 EFGHCA5GI94 !J
;789 R !2VTN A8 AL5>485 <=4 E7 B>54994 ;4 EF?PA=E4@485 ;789 E7 ;>64P5>A8 56789B46W
97E4 495 ?M7E4 Q K
vn,t =A2D (h2D)
n,m(v2D)
n,m+A1D (h1D)
n,m(v1D)
n,m
A2D (h2D)n,t
+A1D (h1D)n,t R !22T
vn,tC4=5 U564 P7EP=E?4 48 =5>E>9785 R !22T 45 E4 :7>5 <=4 E49 56789:4659 ;4 <=78W
5>5?9 ;4 @A=B4@485 84 @A;>D485 C79 E49 G7=54=69 ;F47= 9=6 PG7<=4 P4EE=E4 K
(h1D)n,t
= (h1D)n,m
45 (h2D)n,t
= (h2D)n,m
! X8 48 ;?;=>5 ;A8P <=4 E7 <=785>5?
;4 @A=B4@485 ;789 E7 ;>64P5>A8 56789B4697E4 CA=6 E7 @7>EE4 Y& 495 ?M7E4 Q K
(r1D)n,t
= (h1D)n,tvn,t
R !YZZ7T
45 CA=6 E7 @7>EE4 J& K
(r2D)n,t
= (h2D)n,tvn,t
R !YZZLT
$F?PA=E4@485 56789B4697E 56789CA654 ;4 E7 <=785>5? ;4 @A=B4@485 EA8M>5=W
;>87E4! $4 56789:465 ;4 <=785>5? ;4 @A=B4@485 56789B4697E4 9F7PPA@C7M84 ;A8P
;F=8 56789:465 ;4 <=785>5? ;4 @A=B4@485 EA8M>5=;>87E4! $4 BAE=@4 ;F47= <=>
567B4694 PG7<=4 >8546:7P4 >85?6>4=64 C48;785 E4 C79 54@C9 495 ?M7E Q K
VL = LL (h2D)n,t ∣∣vn,t
∣
∣∆t R !YZY7T
VR = LR (h2D)n,t ∣∣vn,t
∣
∣∆t R !YZYLT
"4 56789:465 ;4 @7994 9F7PPA@C7M84 ;F=8 56789:465 ;4 <=785>5? ;4 @A=B4@485
EA8M>5=;>87E4! [A=6 =8 ?PA=E4@485 56789B4697E ;4 E7 6>B4 ;6A>54 B469 E7 6>B4
M7=PG4N E49 <=785>5?9 ;4 @A=B4@485 EA8M>5=;>87E49 7C6I9 56789:465 9F?P6>B485 K
(q2D)n,t
= (q2D)n,m
+1
A2D(VL (u1D)
n,m − VR (u2D)n,m
) R !YZJ7T
(q1D)n,t
= (q1D)n,m
+1
A1D(VR (u2D)
n,m − VL (u1D)n,m
) R !YZJLT
[A=6 =8 ?PA=E4@485 56789B4697E ;789 EF7=564 ;>64P5>A8N E49 <=785>5?9 ;4 @A=B4W
@485 EA8M>5=;>87E49 7C6I9 56789:465 9F?P6>B485 K
(q2D)n,t
= (q2D)n,m
+1
A2D(VR (u1D)
n,m − VL (u2D)n,m
) R !YZ\7T
(q1D)n,t
= (q1D)n,m
+1
A1D(VL (u2D)
n,m − VR (u1D)n,m
) R !YZ\LT
!
"#$%&'() *+ (,-./0'&.1 2)- ,30$'&.1- 2) %(.%$4$'&.1 %$(
/$ 5,'#.2) 2)- 6./05)- 7&1&-
Lit mineurLit majeur
Limite lit mineur / lit majeur
Vecteur vitesse 1D
Vecteur vitesse 2D
Echange de quantité de mouvement
entre les lits
!"#$% *+ 8 '9:;<=>9? @> AB:;?C?D @> EFBG>E>;? H IJC;?>9=:K> @>< DKFBI>E>;?<
L2 >? M2
! ! "#$%&'(#) *( +,$%)-). *( /0,1(/(%) 20%3-),*-%$2(
/>< ?9:;<=>9?< @> AB:;?C?D @> EFBG>E>;? IF;NC?B@C;:I> F;? @>BO F9CNC;>< P
I>< ?9:;<=>9?< :<<FKCD< :BO ?9:;<=>9?< @> E:<<> >;?9> I>< E:CII>< @> K:IKBI Q@DRH
K:IKBID< S9DKD@>EE>;?T >? I>< ?9:;<=>9?< ICD< :BO =9F??>E>;?< ICABC@>UICABC@> I>
IF;N @>< C;?>9=:K>< C;?D9C>B9><+ ">< ?9:;<=>9?< <F;? KFEE:;@D< S:9 I: =F9K> @>
KC<:CII>E>;? H IJC;?>9=:K>+ 2B =:C? @B N9:@C>;? @> GC?><<> >;?9> IJDKFBI>E>;? L2
>? IJDKFBI>E>;? M2 H IJF9CNC;> @> K>??> =F9K> @> KC<:CII>E>;? H IJC;?>9=:K>V B;>
WF;> ?B9XBI>;?> <> =F9E> :B ;CG>:B @> IJC;?>9=:K>+ $B ;CG>:B @> K>??> WF;>V @><
<?9BK?B9>< ?FB9XCIIF;;:C9>< <F;? H IJF9CNC;> @JDKY:;N>< >;?9> IJDKFBI>E>;? L2 >?
IJDKFBI>E>;? M2V XC>; AB> I>< ?9:;<=>9?< @> E:<<> <FC>;? ;BI< >; EFZ>;;> QGFC9
7CNB9> *+ T+ &I Z : >; >[>?V B; ?9:;<=>9? <CEBI?:;D @>SBC< I: E:CII> L2 G>9<
I: E:CII> M2 >? @> I: E:CII> M2 G>9< I: E:CII> L2+ 0; EF@\I> @> ?B9XBI>;K>
D?:XIC<<:;? I> @DXC? @> ?9:;<=>9? @JB; DKFBI>E>;? G>9< IJ:B?9> >; =F;K?CF; @B
N9:@C>;? @> GC?><<> : D?D KYFC<C+ />< EF@\I>< @> ?B9XBI>;K> <F;? 9>I:?CG>E>;?
;FEX9>BOV ?FB?>=FC<V I> KYFCO <J><? SF9?D <B9 B; EF@\I> 9>I:?CG>E>;? <CESI> >?
<B]<:EE>;? S9DKC< >; 9>N:9@ @>< CES9DKC<CF;< C;YD9>;?>< H B; EF@\I> @:;< @><
KF;^NB9:?CF;< 9D>II><+ /: =F9EBI:?CF; S9FSF<D> S:9 _!`V DABCG:I>;? H B; EF@\I>
@> IF;NB>B9 @> EDI:;N>V : D?D 9>?>;B>+ /> @DXC? @> ?9:;<=>9? ICD H I: ?B9XBI>;K>
><? <DS:9D @B @DXC? @> ?9:;<=>9? ICD :B KY:;N>E>;? @> =F9E> @B K:;:I L2+ />
@DXC? @> ?9:;<=>9? ICD H I: ?B9XBI>;K> >;?9> IJDKFBI>E>;? L2 >? IJDKFBI>E>;? M2
<BC? I: 9>I:?CF; P
qt = v (h2D)n,t
Q*+La*T
Fb v ><? I: GC?><<> @> IJDKFBI>E>;? @:;< I: @C9>K?CF; ?9:;<G>9<:I> H IJC;?>9=:K>
KF;<C@D9D>+ %:9 YZSF?Y\<> @B EF@\I>V K>??> GC?><<> ><? <BSSF<D> S9FSF9?CF;;>II>
H I: @C[D9>;K> @> GC?><<> >;?9> I>< @>BO DKFBI>E>;?<+ /: =F9EBI:?CF; S9FSFU
<D> <BSSF<> AB> I: GC?><<> @:;< KY:AB> DKFBI>E>;? ><? S:9:II\I> H IJC;?>9=:K>+
! ! "#$"%$ &'( )$%* + ,-#.'-( $'( /0,'-)#"'( /0,1-/'%-'(23
β
Interface gauche
Interface droite
u1
v1
u1,L
u1,R
u2
v2
u2,L
u2
v2
u2,R
β
!"#$% !45 6 789:;<=>9? @;A B;<=;C8A B>=;AA; @D?A E; 8;FG8;?=>;E DAA9<>G DCH
>?=;8FD<;A >?=G8>;C8;A
/E <9?B>;?= @9?< @; I8;?@8; ;? <9JI=;K @D?A ED F98JCED=>9?K ED <9JI9AD?=; @;
ED B>=;AA; I9C8 <LDMC; G<9CE;J;?= MC> ;A= ID8DEENE; O EP>?=;8FD<; >?=G8>;C8; QB9>8
)>RC8; !45S!
$; @GT>= @; =8D?AF;8= O =8DB;8A <LDMC; >?=;8FD<; APG<8>= @9?< U
qt,L = Ψ |((u1D,L)n,m − (u2D,L)
n,m) cosβ| (h2D)n,t
Q !45VDS
qt,R = Ψ |((u1D,R)n,m − (u2D,R)
n,m) cosβ| (h2D)n,t
Q !45VTS
9W Ψ ;A= C? <9;X<>;?= @; I89I98=>9??DE>=G ;= 9W β ;A= EPD?RE; F98JG ID8 EPDH;
@C <D?DE 4& ;= C?; >?=;8FD<; >?=G8>;C8; QB9>8 )>RC8; !45S! YZ[ D J9?=8G MC; ΨBD8>; I;C DB;< ED RG9JG=8>; @;A G<9CE;J;?=A ;= I89I9A; @P;JIE9\;8 Ψ = 0.016@D?A EP;?A;JTE; @;A <9?]RC8D=>9?A 8;?<9?=8G;A! "9??D>AAD?= E; @GT>= @; =8D?AF;8=
O =8DB;8A <LDMC; >?=;8FD<; >?=G8>;C8;K ED E9?RC;C8 @; <;A >?=;8FD<;A ;= E; IDA @;
=;JIAK >E ;A= I9AA>TE; @P;? @G@C>8; EPGB9EC=>9? @; MCD?=>=G @; J9CB;J;?= AC8
<LDMC; JD>EE; @; <DE<CE U
(q1D)n,l
= (q1D)n,t
+∆t (LLqt,L + LRqt,R)(u2D)
n,t − (u1D)n,t
A1DQ !45ZDS
(q2D)n,l
= (q2D)n,t −∆t (LLqt,L + LRqt,R)
(u2D)n,t − (u1D)
n,t
A2DQ !45ZTS
(r1D)n,l
= (r1D)n,t
+∆t (LLqt,L + LRqt,R)(v2D)
n,t − (v1D)n,t
A1DQ !45Z<S
(r2D)n,l
= (r2D)n,t −∆t (LLqt,L + LRqt,R)
(v2D)n,t − (v1D)
n,t
A2DQ !45Z@S
Maille 2Di
Maille 2Di+1
Maille 2Di-1
Maille 2Di
Maille 2Di+1
Maille 2Di-1
Maille 1Di+1
Maille 1DiMaille virtuelle 1D
i-1
i − 1i
ub − 2cb = uR − 2cR
b R
!"! #$%&'()$%*)$+, %-*)$*./ #/ .* 0+,/ #1()2#/ 33
45 46789: ;8 <:=>?@A5 45 '95AB88 C ?198D5:EBF5 4=8D ?B G=?;D9=8 <5:A5D 45
46D5:A985: ?5G H;I C D:BJ5:G ?B ?9A9D5 4; A=4@?5!
.5G F=849D9=8G B;I ?9A9D5G G=8D 467895G G;: ?5G ?9A9D5G 4; AB9??BK5 >949A58L
G9=885?! #B8G ?5 FBG =M 9? N B 6KB?5A58D ;85 AB9??5 O# 58 ?9A9D5 4; A=4@?5P
?1;D9?9GBD9=8 45 ?B F=849D9=8 B;I ?9A9D5G 58 46>9D 5GD <:=>?6ABD9Q;5! /8 5R5DP <B:
G=;F9G 45 G9A<?9F9D6P G5;? ?5 46>9D 58D:B8D G;: ?B AB9??5 O# 5D G;: ?B AB9??5 S#
5GD F=88;! .B :6<B:D9D9=8 58D:5 ?5G 45;I AB9??5G 5GD 46D5:A9865 <B: ?5 ?=K9F95?! .5
46>9D ;89DB9:5 q D:B8G9DB8D <B: ?B ?9A9D5 4; A=4@?5 5GD 7I6 <B: ?1;D9?9GBD5;: T ?5
46>9D Q D:B8G9DB8D <B: ?5G 45;I AB9??5G 5GD 4=8F 6KB? C U
Q = (L1D + L2D) q = Q1D +Q2D V !OWXY
+8 EB9D ?1ZN<=DZ@G5 Q;5 ?5 46>9D 58D:B8D G;: ;85 AB9??5 5GD <:=<=:D9=885? C ?B
G5FD9=8 A=;9??65 U
Q1D = q1DL1D =L1Dh1D
L1Dh1D + L2Dh2DQ V !OW3Y
Q2D = q2DL2D =L2Dh2D
L1Dh1D + L2Dh2DQ V !OOWY
+8 58 464;9D 4=8F ?5 46>9D ;89DB9:5 ;D9?9G6 C ?198D5:EBF5 45G F5??;?5G O# 5D S# U
q1D =(L1D + L2D)h1DL1Dh1D + L2Dh2D
q V !OOOY
q2D =(L1D + L2D)h2DL1Dh1D + L2Dh2D
q V !OOSY
!" #$%&'()$%*)$+, %-*)$*./ 0/ .* 1+,/ 02()30/
.B 49GF:6D9GBD9=8 45 ?B [=85 416D;45 5GD ;85 6DB<5 E=84BA58DB?5 45 ?B F=8GD:;FL
D9=8 41;8 A=4@?5 >949A58G9=885?! .B JB?5;: 45G JB:9B>?5G FB?F;?65G 85 G5:B F=88;5
Q;5 G;: ?5G AB9??5G 45 FB?F;?! .B 49GF:6D9GBD9=8 V5D 4=8F ?5 AB9??BK5Y 45J:B <B:
F=8G6Q;58D \D:5 B4B<D65 BJ5F ;85 458G9D6 BFF:;5 4B8G ?5G [=85G <=;: ?5GQ;5??5G
;85 <:6F9G9=8 <?;G 9A<=:DB8D5 5GD J9G65! -=;: F5DD5 6DB<5P F15GD K686:B?5A58D
?15I<6:958F5 4; A=46?9GBD5;: Q;9 <5:A5D 45 46D5:A985: ?B 458G9D6 45 AB9??BK5
86F5GGB9:5 <=;: ;85 <:6F9G9=8 J9G65!
.B [=85 416D;45 5GD 49GF:6D9G65 G=;G E=:A5 45 <=?NK=85G! ]958 Q;5 ?B A6DZ=L
4=?=K95 5D ?5 ?=K9F95? 45 :6G=?;D9=8 45G 6Q;BD9=8G 85 ?19A<=G58D <BGP ?5G <=?NK=85G
=8D ^ =; F=D6G! /8 5R5DP ?5G ?=K9F95?G 45 AB9??BK5G F=;:B8D 85 <5:A5DD58D K686:BL
?5A58D <BG 45 D:B9D5: 45G <=?NK=85G BJ5F <?;G 45 F=D6G! .B G5;?5 F=8D:B98D5 9A<=L
G65 <B: ?B A6DZ=4=?=K95 <=;: ?B F=8GD:;FD9=8 4; AB9??BK5 S# 5GD Q;5 ?5G AB9??5G
4=9J58D \D:5 F=8J5I5G! /8 5R5DP ?B A6DZ=4=?=K95 45 FB?F;? 9458D975 ?1=:958DBD9=8
41;85 98D5:EBF5 C <B:D9: 45 ?B <=G9D9=8 45G F58D:5G 45 K:BJ9D6 45G AB9??5G 45 <B:D
5D 41B;D:5 45 F5DD5 98D5:EBF5! #B8G ?5 FBG 45 AB9??5G F=8FBJ5GP ?5G 45;I F58D:5G 45
K:BJ9D6 <5;J58D \D:5 4; A\A5 F=D6 45 ?198D5:EBF5 5D B98G9 <:=J=Q;5: ;85 5::5;:
!!
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/$ 5,'#.2) 2)- 6./05)- 7&1&-
89:; <=>?@A:B9B@>: 8A <=@:BA?C9DA+ /A D>8A 8A D9<DE< @8A:B@FA 9EB>G9B@HEAGA:B <9
I?J;A:DA 8A G9@<<A; D>:D9KA; I>E? JK@BA? B>EBA A??AE? E<BJ?@AE?A+
/A G9@<<9LA I>E? <=JD>E<AGA:B E:@8@GA:;@>::A< MG9@<<9LA 2N A;B D>:;B?E@B
;E? <A G9@<<9LA I>E? <A; JD>E<AGA:B; O@8@GA:;@>::A<; MG9@<<9LA P2N+ /A D>EI<9LA
A:B?A <A; 8AEQ G>8R<A; ;EII>;A HE=E:A G9@<<A 2 A;B @:BJL?9<AGA:B @:D<E;A 89:;
E:A G9@<<A P2+ ): ASABT DA<9 IA?GAB 8A ;=9;;E?A? HEA <A; JDU9:LA; <9BJ?9EQ 8=E:A
G9@<<A 2 :A ;A C>:B HE=9KAD E:A E:@HEA G9@<<A P2+
-E? <9 O9;A 8E G9@<<9LA P2T <A <>L@D@A< I?>8E@B 9EB>G9B@HEAGA:B <A G9@<<9LA
2 V I9?B@? 8E B?9DJ 8E D9:9< 2 C>E?:@ ;>E; C>?GA 8A I><W<@L:A;+ %>E? DU9HEA
;>GGAB 8A <9 I><W<@L:AT <9 D>BA 8E C>:8 zb,1D A;B ?A:;A@L:JA 9@:;@ HEA <9 <9?LAE?
b 8E D9:9< 2 ;E@K9:B <9 O@;;ADB?@DA C>?GJA I9? <=9QA 8E D9:9< 9K9:B AB 9I?R;
<A I>@:B D>:DA?:J >E IA?IA:8@DE<9@?AGA:B 9E B?>:X>: I>E? <A; AQB?JG@BJ; 8A <9
I><W<@L:A MK>@? 7@LE?A *+ P9N+
/A; @:BA?C9DA; 8E G9@<<9LA 2 ;>:B D>:;B?E@BA; <A <>:L 8A; @:BA?C9DA; P2 @:Y
BA?;ADBJA; I9? <9 I><W<@L:A I>?B9:B <A D9:9< 2 MK>@? 7@LE?A *+ PON+ /9 I><WY
<@L:A 8A ;EII>?B 8E D9:9< 2 :A 8>@B 8>:D I9; I9;;A? I9? <=E: 8A; ;>GGAB; 8E
G9@<<9LA; P2+ /A; G9@<<A; 2 ;>:B D>:;B?E@BA; V I9?B@? 8A <AE?; @:BA?C9DA; MK>@?
7@LE?A *+ PDN+
/9 D>:;B?EDB@>: 8E G9@<<9LA P2 8>@B ZB?A ?J9<@;JA A: I?JK>W9:B <9 D>:;B?EDB@>:
E<BJ?@AE?A 8E G9@<<9LA 2+ ): ASAB 89:; DA?B9@:A; D>:FLE?9B@>:;T E:A G9@<<A 2
IAEB D>:BA:@? E:A @:BA?C9DA 8=E:A G9@<<A P2 MK>@? 7@LE?A *+ [9N >E <9 G9@<<A 2
IAEB ZB?A E: I><WL>:A D?>@;J MK>@? 7@LE?A *+ [ON+
"A; D>:FLE?9B@>:; ;>:B I?>O<JG9B@HEA;T ;>@B I>E? <9 LA;B@>: 8A; G9@<<A; 8A
D9<DE< I9? <A D>8A ;>@B D9? A<<A; @:K9<@8A:B DA?B9@:A; UWI>BUR;A; ?J9<@;JA; MK>@?
-ADB@>: *+*+ N+ />?; 8A <9 K9<@89B@>: 8E G>8E<A 8A D>:;B?EDB@>: 8E G9@<<9LA 2T
@< A;B 9II9?E HEA <9 D>:;B?EDB@>: 8=E: G9@<<9LA P2 98JHE9B IA?GAB 8A ;=9S?9:Y
DU@? 8A; I?>O<RGA; D@BJ;+ ): ASABT <=E;9LA 8A G9@<<A; P2 D>EI<JA; ;A:;@O<AGA:B
I<E; L?9:8A; HEA <A; G9@<<A; 2 HE=A<<A; D>:B@A::A:B 9@:;@ HEA <=EB@<@;9B@>: 8=E:
G9@<<9LA P2 9W9:B JBJ D>:XE I>E? ;EII>?BA? <A G9@<<9LA 2 A;B D>:;A@<<J MK>@?
7@LE?A *+ *N+
/A; JHE9B@>:; E:@8@GA:;@>::A<<A M[+*\N AB O@8@GA:;@>::A<<A M[+*]N ;>:B 8>:D
8@;D?JB@;JA; ;E? <A G9@<<9LA 9@:;@ >OBA:ET DU9HEA G9@<<A 2 JB9:B 9;;>D@JA V <9
G9@<<A P2 HE@ <9 D>:B@A:B+
/A; 8@SJ?A:BA; JB9IA; 8A ?J;><EB@>: 8A; JHE9B@>:; 8A I?>I9L9B@>: I9? <9 GJY
BU>8A 8A; K><EGA; F:@; >:B B>EBA; JBJ AQI<@D@BJ;+ /A D>8A 8A D9<DE< -^ P2
D>??A;I>:8 V <9 I?>L?9GG9B@>: A: C>?B?9: 8A DA; 8@SJ?A:BA; JB9IA; 8A D9<DE<+ /A
DU9I@B?A ;E@K9:B ;=@:BJ?A;;A V <9 K9<@89B@>: AB V <=JK9<E9B@>: 8A; IA?C>?G9:DA; 8A
DA :>EKA9E D>8A 8A D9<DE<+
interface des mailles 2Dpolyligne de support du canal 1Dlargeur du canal 1D
polyligne de support du canal 1Dlargeur du canal 1Drives du canal 1D
interface extérieure 1D
maille 1D
interface intérieure 1D
interface des mailles 2D
interface des mailles 1Dinterface intérieure
polyligne de support du canal 1Dlargeur du canal 1D
inte
rfa
ce 2
D in
clu
sed
an
s u
ne
ma
ille
1D
interface des mailles 2Dinterface des mailles 1D
interface intérieure
polyligne de support du canal 1D
largeur du canal 1Dmaille 1D croisée
polyligne de support du canal 1D
interface intérieure 1D (pas de résolution d’un problème de Riemann)
interface des mailles 2D (résolution d’un problème de Riemann)
interface extérieure 1D (résolution d’un problème de Riemann)
!"#$%&' (
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+,*&( AT4B ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "";
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!5 6789/&.*:., /9$%&,9., ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "53
!;!" F(.$0,4+,%* -& -,.4%.,+,G (E450,)(*+1' ! ! ! ! ! ! ! ! ! ! ! ! ! ";9
!;!; H50,( " C D1'1?( -& $%(I$,(*+ -( 0&?%.,+5 ! ! ! ! ! ! ! ! ! ! ! ";;
!;!;!" D%*J?&01+,%* K D1*1' / *& L AM&B ! ! ! ! ! ! ! ! ! ! ";;
!;!;!; D%*J?&01+,%* K N1041,*?. L AN0B ! ! ! ! ! ! ! ! ! ! ! ";=
!;!;!= D%*J?&01+,%* K O0,P&(. L AO0B ! ! ! ! ! ! ! ! ! ! ! ! ";
!;!;!: Q%-5',.1+,%* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ";R
!;!;! D%(I$,(*+ -( 0&?%.,+5 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "=;
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!;!=!; D%*J?&01+,%* K T,+ ),*(&0 17 cm L A"<$)B ! ! ! ! ! "=R
!;!=!= D%*J?&01+,%* ST,+ ),*(&0 28 cmU A;R$)B ! ! ! ! ! ! ":9
!;!=!: F&05( -( .,)&'1+,%* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ":;
!;!: H50,( = C D%&-(. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "::
!;!:!" D%*J?&01+,%* K D%&-(. / @9 01440%$V5. L A@9$B ! ! "::
!;!:!; D%*J?&01+,%* K D%&-(. / @9 5'%,?*5. L A@9'B ! ! ! ! " 9
!;!:!= F&05( -( .,)&'1+,%* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! " ;
!;! H50,( : C W+&-( -( '1 .+0&$+&0( -(. 5$%&'()(*+. +01*.2(0.1&E " =
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<;>@ ;B4 ?62=1@3 M >A B68<;>@ $00# =@3B;A43 71AB NDOP, E1AB >A B;?6A7 4;2=BJ
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A18; .>=3@:;>@; 7K%9@6A62:; 7; U6A4=;88:;@ F*T.%UI, %=@SB >A; 7;B?@:=4:6A
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0;B ?1B 4;B4B B6A4 73A6223B Ti,(sub, sup, trans, jump), 0;B @3B>8414B B6A4 ?62=1@3B
M 81 B68>4:6A 1A18[4:V>; F6> B;2:H1A18[4:V>;I ;4 1>R @3B>8414B ?18?>83B 1<;? 8;
B68<;>@ $00# =@3B;A43 71AB NDOP, E1AB ?51V>; B;?4:6AJ 81 936234@:; ;2=86[3;
;B4 73?@:4;, 01 936234@:; 1 343 73<;86==3; =6>@ 3<18>;@ 8;B =;@W6@21A?;B 7>
B68<;>@ &6@%. ;A @39:2; =;@21A;A4 FT1,(sub, sup, trans, jump)I, EK1>4@;B ?62=1@1:H
B6AB 6A4 3918;2;A4 343 @318:B3;B B>@ 81 Q1B; 7;B ?1B 4;B4B =@6=6B3B 71AB N"DP \
T2,(sub, trans) ;4 71AB NDOP \ T3J T4 ;4 T5, /A; 34>7; 7; ?6A<;@9;A?; 1 3918;2;A4
343 @318:B3; =6>@ 8;B 7;>R B68<;>@B &6@%. ;4 $00# \ T6,(1, 2, 3), 0K;L?1?:43 7> B68H
<;>@ &6@%. 1 343 4;B43; ;A ?62=1@1A4 8;B @3B>8414B 7> 267S8; 1<;? 7;B 76AA3;B
;R=3@:2;A418;B \ T7,
!"!" #$%&'()(*+ &*,-,)(*.,%**(' / +012(0. &*( 3%*( -(
4%0%.,+5 210,16'( 7 8(.+ T1
01 =@;2:S@; 936234@:; ?6AB:B4; ;A >A 3?6>8;2;A4 =;@21A;A4 71AB >A ?1A18
B1AB =;A4; ;4 1<;? >A; =6@6B:43 <1@:1Q8;, 0;B V>14@; ?1B 4;B4B 1BB6?:3B 76:<;A4
=;@2;44@; 7; 2;44@; ;A 3<:7;A?; 8;B =;@W6@21A?;B 7> B68<;>@ 73<;86==3 ;A =@3H
B;A?; 7; =6@6B:43 <1@:1Q8;, /A =@6X8 =1@1Q68:V>; 1 343 ?56:B: =6>@ 81 =6@6B:43 \
φ (x) =
ax2 + bx+ c B:x ∈ [Xm − L0;Xm + L0]φ0 B:A6A
F+, I
a =φ0 − φm
L20, b = −2aXm, c = φ0 + a
(
X2m − L20
)
6] Xm ;B4 8K1QB?:BB; 7> 2:8:;> 7> @34@3?:BB;2;A4J 2L0 B1 86A9>;>@J φ0 ;B4 81
=6@6B:43 M 8K;R43@:;>@ 7> @34@3?:BB;2;A4 ;4 φm ;B4 81 =6@6B:43 1> ?;A4@; F<6:@
!"! #$%&'$(&)* '+ ,)%#-+. /).$, "0
Xm
2L0
φ0
x
φm
!"
Xm
2L0
φ0
x
φm
φ' m
X’m
2L0
#"
!"#$% !" 1 ,23456 78 749:;<;=: 78> ?6@;6AB8> C=D@ B8 26> <8>< T1!
E;FD@8 !"6G! %8> C6@65H<@8> D<;B;>4> C=D@ B8> >;5DB6<;=:> >=:< @8F@=DC4> 76:>
B8 <6AB86D !"!
/=D@ 4<6AB;@ B6 >=BD<;=: 78 @4I4@8:28 C=D@ 28 26> <8><J =: D<;B;>8 BK4LD6<;=: 78
2=:>8@?6<;=: 78 B6 LD6:<;<4 78 5=D?858:< 8: @4F;58 C8@56:8:< M
d
dx
(
q2
φh+1
2gφh2
)
=1
2gh2
dφ
dxN !OG
-: 2=5A;:6:< BK4LD6<;=: 78 2=:<;:D;<4 6?82 BK4LD6<;=: N !OG =: 6A=D<;< P M
φdh
dx
(
1− E@
2)
= hE@2dφ
dxN !QG
/=D@ D: @4F;58 RD?;6BJ B6 7;>2@4<;>6<;=: 78 N !QG 8>< @46B;>48 78 BK65=:< ?8@>
BK6?6B 8: D<;B;>6:< B6 54<3=78 7K-DB8@ 8SCB;2;<8 N8< ?;28 ?8@>6 C=D@ D: 42=DB858:<
<=@@8:<;8BG M
hd = hu − huu2u
φ (ghu − u2u)(φu − φd) N !TG
%6 @4I4@8:28 7D 26> <8>< T1,sup 8>< 4<6AB;8 8: ;5C=>6:< D: 74A;< D:;<6;@8 q =1m2 s−1 8< D:8 36D<8D@ 7K86D P B6 B;5;<8 65=:< h = 0, 2m! /=D@ B8 26> <8><
T1,subJ B8 74A;< D:;<6;@8 ?6D< 4F6B858:< q = 1m2 s−1 8< B6 2=:7;<;=: P B6 B;5;<8
6?6B 8>< h = 1, 1m! /=D@ B8 26> <8>< <@6:>2@;<;LD8 T1,transJ =: C8D< 5=:<@8@ LD8
B8 C=;:< 2@;<;LD8 =U E@ = 1 2=@@8>C=:7 :428>>6;@858:< P D: F@67;8:< 78 C=@=>;<4
:DB B=26B;>4 8: Xm! %6 B;F:8 7K86D C8D< V<@8 74<8@5;:48 8: 7;>2@4<;>6:< N !QG
78CD;> Xm ?8@> B8> 8S<@45;<4> 7D 7=56;:8 78 26B2DB 8: 622=@7 6?82 B8 @4F;58
7K42=DB858:<! /=D@ B8 26> <8>< 6?82 D: @8>>6D< 3W7@6DB;LD8J B6 B;F:8 7K86D P
BK65=:< 7D F@67;8:< 78 C=@=>;<4 :DB 8>< 26B2DB48 78 B6 5V58 56:;H@8 LD8 C=D@
B8 26> <8>< <@6:>2@;<;LD8! $ B6 B;5;<8 6?6B 7D 7=56;:8J D:8 2=:7;<;=: RD?;6B8 8><
;5C=>48 M h = 0, 811m! %6 C=>;<;=: 7D @8>>6D< 3W7@6DB;LD8 8>< 74<8@5;:48 8:
26B2DB6:< B6 B;F:8 7K86D 78CD;> Xm ?8@> BK6?6B 8< 78 BK8S<@45;<4 6?6B ?8@> BK65=:< X
N !QG 8>< 7;>2@4<;>48 8: 674LD6<;=: 6?82 B8 @4F;58 7K42=DB858:<! %K;5CDB>;=: I
!" #$%&'()* +, )-./0(%(.
.123456 .7897:;<=749 ><56?@
g %;;A5A@<=749 8@<B7=<=74996556 9, 81m s−2
φ0 &4@4C7=A D 5E6F=A@76?@ G? @A=@A;7CC6269= 1H7972?2 G6 5< I4@4C7=A J
φm K I4?@ 56C ;<C =6C=C T1,sup 6= T1,jump 0, 6K I4?@ 56C ;<C =6C=C T1,sub 6= T1,trans 0, 5
L0 0498?6?@ G? @A=@A;7CC6269= 25mxm &4C7=749 G6 5< I4@4C7=A 27972<56 G<9C 56 G42<796 φm 50mL L7269C749 G? G42<796 100m∆x L7269C749 G6C 2<7556C G6 ;<5;?5 0, 1mq LA37= ?97=<7@6 72I4CA D 5< 5727=6 <249= 1m2s−1
$<?=6?@ GE6<? 72I4CA6 D 5< 5727=6 <B<5 J
hav K I4?@ 56 ;<C =6C= T1,sup 0, 2mK I4?@ 56 ;<C =6C= T1,sub 1, 1m
!"#$!% +, M &<@<2N=@6C I4?@ 5< I@627N@6 8A42A=@76 J #<C =6C=C 69 @A8726
I6@2<969= OA;4?56269= ?97G7269C749965 G<9C ?9 ;<9<5 <B6; @A=@A;7CC6269= K
T1,(sub, sup, trans, jump)P,
;4@@6CI49G<9= D ;6C G6?F I@4:5C 6C= A8<56269= ;<5;?5A6 J
I =Q2
S+1
2ghS !"!#
$% &%''()* +,-&()./0)% %'* .12(./'3 4 .5(6'2/''% -5/7*%&'%2*/17 -%' -%)8 9&1:.'
-5/;9).'/17" $( 91'/*/17 -) &%''()* +,-&()./0)% %'* *&<' '%7'/6.% 4 .( 21*% -% .(
')&=(2% ./6&% 4 .5(>(." ?7 %@%*A '/ .( 21*% %'* .3B<&%;%7* ')&C%'*/;3%A .% &%''()*
'%&( .12(./'3 *&19 %7 (;17* %* 71/%&( .% &3*&32/''%;%7* 21;;% 91)& .% 2('C*%'*
%7 &3B/;% D)>/(.# (.1&' 0)% '/ %..% %'* '1)' %'*/;3%A .% &%''()* '%&( .12(./'3 +1&'
-) -1;(/7% -% 2(.2)." E1)& .% 2(' *%'* 7);3&/0)%A )7 '%217- &3*&32/''%;%7* ( 3*3
/;9.(7*3 4 .5(>(. -) 9&%;/%& -(7' .% ;1-<.% -% ;(7/<&% 4 16*%7/& )7 321).%;%7*
%7 &3B/;% D)>/(. 4 .5(;17* -% 2% -%&7/%& %* -172 )7 &%'()* .12(./'3 %7*&% .%' -%)8
&3*&32/''%;%7*'" $% ;/./%) -) '%217- &3*&32/''%;%7* %'* .12(./'3 4 x = 150mA .(
91&1'/*3 () 2%7*&% >()* φ′
m = 0, 7 %* .( .17B)%)& *1*(.% -% .( >(&/(*/17 %'* -%
50m >1/& F/B)&% !"G6#" $( 21*% -% .( ')&=(2% ./6&% /;91'3% 4 .5(>(. 91)& .% 2(.2).
-% .( '1.)*/17 -% &3=3&%72% 21&&%'917- %8(2*%;%7* 4 .( 21*% -% .( ')&=(2% ./6&% 4
.5(;17* -% .( '%217-% >(&/(*/17 -% 91&1'/*3 -(7' .( '/;).(*/17 7);3&/0)%"
$%' '/;).(*/17' '17* &3(./'3%' 4 9(&*/& -5)7% 217-/*/17 /7/*/(.% () &%91' %* ')&
)7 ;(/..(B% )7/=1&;% -% 9(' -5%'9(2% ∆x = 0, 1mA H)'0)54 3*(6./''%;%7* -5)7
&3B/;% 9%&;(7%7* ')& .5%7'%;6.% -) -1;(/7%" $%' :B)&%' !"I 4 !"! 9&3'%7*%7* .%'
9&1:.' 16*%7)' (>%2 .%' '1.>%)&' J$$K %* E1&LM 91)& .%' 0)(*&% 217:B)&(*/17'
-) *%'* T1" E(& '1)2/ -% 2.(&*3A '%).% )7% ;(/..% -% 2(.2). ')& GN %'* &%9&3'%7*3%
')& .%' :B)&%' !"I 4 !"O" E1)& .%' 2(' *%'*' D)>/(.A *1&&%7*/%. %* (>%2 )7 &%''()*
+,-&()./0)%A .( +()*%)& -5%() %* .% -36/* )7/*(/&% '17* 21&&%2*%;%7* 2(.2).3' 9(&
.%' -%)8 '1.>%)&'" P3(7;1/7'A .( +()*%)& -5%() -(7' .% &3*&32/''%;%7* %'* ;/%)8
!"! #$%&'$(&)* '+ ,)%#-+. /).$, "01
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60 70 80 90 100
x (m)
h (
m)
Reference
HLLC
Poras
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 10 20 30 40 50 60 70 80 90 100
x (m)
q (
m²/
s)
Reference
HLLC
Poras
0.15
0.20
0.25
0.30
0.35
20 30 40 50x (m)
h (
m)
Reference
HLLC
Poras
4.6
4.7
4.8
4.9
5.0
5.1
20 30 40 50x (m)
u (
m/s
)
Reference
HLLC
Poras
!"#$% !2 3 456789:9;< <6==9;<>98 7;>?>:9;@>6;;98 ?A;@ 7; =B<=B5>@@9:9;<
CT1,supD! ,687<>6;@ ?9 =BEB=9;59 9< @>:78B9@ AF95 89@ @68F97=@ G%%H 9< /6=$, I
9; JA7< K LA75J9M JA7<97= ?N9A7 I 9; JA7< K ?=6><9M ?BO>< 7;><A>=9 I 9; OA@ K
LA75J9M ?B<A>8 ?9 8A 8>L;9 ?N9A7 ?A;@ 89 =B<=B5>@@9:9;< I 9; OA@ K ?=6><9M F><9@@9
?9 8NB56789:9;< ?A;@ 89 =B<=B5>@@9:9;<!
9@<>:B9 AF95 89 @68F97= /6=$, P67= 89 =BL>:9 <6==9;<>98 CF6>= Q>L7=9 !2M 9; OA@
K LA75J9D! %9 @68F97= G%%H @67@R9@<>:9 8A F><9@@9 ?9 8NB56789:9;< ?A;@ 59<<9
=BL>6;! /67= 89 5A@ <9@< <=A;@5=><>S79M 89@ P=6T8@ ?9 F><9@@9 9< ?9 JA7<97= ?N9A7
@6;< 56==95<9:9;< 9@<>:B@ PA= 89@ ?97U @68F97=@! %A TL7=9 !V P=B@9;<9 89 ?BO><
5A8578B AF95 89@ @68F97=@ G%%H 9< /6=$,! %9@ 5A@ <9@<@ T1,(sub, sup, trans, jump)
@6;< 9; =BL>:9 P9=:A;9;<M 89 ?BO>< ?6>< ?6;5 W<=9 89 :W:9 @7= 8N9;@9:O89 ?7
?6:A>;9! /67= 59@ @>:78A<>6;@M 89 @68F97= G%%H @7=9@<>:9 89 ?BO>< <A;?>@ S79 89
@68F97= /6=$, 5A85789 7; ?BO>< 56;@<A;<! H9 PJB;6:X;9 P97< 9UP8>S79= P67=S76>
8A F><9@@9 9@< @67@ 9@<>:B9 ?A;@ 89 =B<=B5>@@9:9;< P67= 89 5A@ <9@< <6==9;<>98!
%N>;9Y5A5><B ?7 @68F97= G%%H A B<B B<7?>B9 ?A;@ Z[2\M 9< 89@ A7<97=@ 6;< :6;<=B
S79 59=<A>;@ <9=:9@ ?7 O>8A; ?9 S7A;<><B ?9 :67F9:9;< ;9 @6;< PA@ 9@<>:B@
56==95<9:9;<!
!"!# $%&'()*)+, '+-.-*)+/-&++)( /'0 '+) 1&//) 2 3)/, T2
%9 5A@ <9@< P=B5B?9;< A :>@ 9; BF>?9;59 8A P9=E6=:A;59 ?7 @68F97= /6=$,
9; P=B@9;59 ?9 P6=6@><B FA=>AO89! %9@ 5A@ <9@<@ S7> @7>F9;< ?6>F9;< P9=:9<<=9
?NBFA879= 89@ P9=E6=:A;59@ ?7 @68F97= P=6P6@B AF95 7; <9=:9 @67=59 8>B K 8A
P9;<9 ?7 E6;?! %9@ @PB5>T5A<>6;@ P=6P6@B9@ ?A;@ Z[2\ 6;< B<B =B7<>8>@B9@ CF6>=
!" #$%&'()* +, )-./0(%(.
0.8
0.9
1.0
1.1
0 10 20 30 40 50 60 70 80 90 100
x (m)
h (
m)
Reference
HLLC
Poras
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 10 20 30 40 50 60 70 80 90 100
x (m)q
(m
²/s)
Reference
HLLC
Poras
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0 10 20 30 40 50 60 70 80 90 100
x (m)
u (
m/s
)
Reference
HLLC
Poras
!"#$% +,1 2 -34567879: ;5<=>6 59=?=879@=49976 ?>9@ 59 AB:AB3=@@7879:
CT1,subD, .465:=49@ ?7 ABEBA7937 7: @=856B7@ ><73 67@ @46<75A@ $00# 7: &4A%. F 79
G>5: H I>53G7J G>5:75A ?K7>5 F 79 G>5: H ?A4=:7J ?BL=: 59=:>=A7 F 79 L>@J <=:7@@7
?7 6KB34567879:,
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50 60 70 80 90 100
x (m)
h (
m)
Reference
HLLC
Poras
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 10 20 30 40 50 60 70 80 90 100
x (m)
u (
m/s
)
Reference
HLLC
Poras
!"#$% +,M 2 -34567879: :A>9@3A=:=N57 59=?=879@=49976 ?>9@ 59 AB:AB3=@@7879:
CT1,transD, .465:=49@ ?7 ABEBA7937 7: @=856B7@ ><73 67@ @46<75A@ $00# 7: &4A%. F
I>53G7J G>5:75A ?K7>5 F ?A4=:7J <=:7@@7 ?7 6KB34567879:,
!"! #$%&'$(&)* '+ ,)%#-+. /).$, "01
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50 60 70 80 90 100x (m)
h (
m)
Reference
HLLC
Poras
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 10 20 30 40 50 60 70 80 90 100x (m)
q (
m²/
s)
Reference
HLLC
Poras
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 10 20 30 40 50 60 70 80 90 100x (m)
u (
m/s
)
Reference
HLLC
Poras
!"#$% ! 2 .344567 89:;56<=>63 :5?4 6? ;@7;@A=443B3?7 CT1,jumpD! ,E<67=E?4:3 ;@F@;3?A3 37 4=B6<@34 5G3A <34 4E<G36;4 H%%I 37 /E;$, J 3? 8567 K L56A83M
856736; :N356 J 3? 8567 K :;E=73M :@O=7 6?=75=;3 J 3? O54M G=73443 :3 <N@AE6<3B3?7!
(5O<356P !Q 37 !RD!
%5 SL6;3 !T AEBU5;3 <5 856736; :N356 37 <3 :@O=7 6?=75=;3 A5<A6<@4 K U5;7=; :34
:36P 4E<G36;4 5G3A <5 4E<67=E? 43B=V5?5<97=>63 EO73?63 3? ;@4E<G5?7 <N@>657=E?
:3 <5 <=L?3 :N356 3? ;@L=B3 U3;B5?3?7 W
dh
dx
(
1− X;
2)
= −dzb
dxC !YD
%34 :36P 4E<G36;4 A5<A6<3?7 AE;;3A73B3?7 <5 AE73 :3 <5 46;F5A3 <=O;3 B5=4 <3 4E<G36;
H%%I =?7;E:6=7 6?3 G5;=57=E? =?5773?:63 :6 :@O=7 6?=75=;3! %5 BZB3 ;5=4E? >63
UE6; <3 A54 7347 U;@A@:3?7 U367 3PU<=>63; A3 U8@?EB[?3 CGE=; ,3A7=E? !"!"D! +?3
5?5<943 U<64 UE644@3 :3 A3A= U367 Z7;3 7;E6G@3 :5?4 \]Q^!
!"!# $%&'%() *) +,((,-)
/E6; @G5<63; <34 U3;FE;B5?A34 :6 4E<G36; /E;$, 3? ;@L=B3 ?E? U3;B5?3?7M
7;E=4 A54 73474 5G3A ;6U76;3 :3 O5;;5L3 E?7 @7@ ;@5<=4@4!
!"!#!" $%&'%() %*+,+-)*.+/**)00) 12)3 &/(/.+'4 21(+150) 6T37
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O5;;5L3 46; 6? FE?: U<57 5G3A 6?3 UE;E4=7@ G5;=5?7 <=?@5=;3B3?7 :3 0 K 1! %34U5;5B[7;34 :3 <5 4=B6<57=E? 4E?7 ;3L;E6U@4 56 75O<356 !]! IEBB3 3PU<=>6@
! "#$%&'() *+ (,-./'$'-
0.999
1.000
1.001
1.002
1.003
1.004
0 10 20 30 40 50 60 70 80 90 100
x (m)
Φ.q
(m
²/s)
Reference
HLLC
Poras
0.9995
1.0000
1.0005
1.0010
1.0015
1.0020
0 10 20 30 40 50 60 70 80 90 100x (m)
Φ.q
(m
²/s)
Reference
HLLC
Poras
0.995
1.000
1.005
0 10 20 30 40 50 60 70 80 90 100
x (m)
Φ.q
(m
2/s
)
Reference
HLLC
Poras
0.995
1.000
1.005
0 10 20 30 40 50 60 70 80 90 100x (m)
Φ.q
(m
²/s)
Reference
HLLC
Poras
!"#$% *+0 1 ,234567689 48:;:768<:38865 ;=8< 48 >?9>?2:<<67689
@T1,(sub, sup, trans, jump)A+ B?C:9< ;6 >?D?>6826 69 <:745?< =E62 56< <35E64><
#//" 69 %3>$- F 68 G=49 H I=42G6J 2=< 96<9 93>>689:65 F 68 G=49 H ;>3:96J 2=<
96<9 K4E:=5 F 68 C=< H I=42G6J 2=< 96<9 9>=8<2>:9:L46 F 68 C=< H ;>3:96J 2=< 96<9
=E62 48 >6<<=49 GM;>=45:L46+
-M7C356 -:I8:N2=9:38 O=564>
g $22?5?>=9:38 I>=E:9=9:3886556 9, 81m s−2
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E7279>6 K?4K4?=74996556269= <> K<B ED6BK<;6! %6 =62KB E6 ;<5;>5 K4>? B72>56?
1000 s 69 >=757B<9= 56B E6>X B45A6>?B 6B= K?@B69=@ E<9B 56 =<356<> !W! /4>? =4>=6B
56B B72>5<=749BP 56 B45A6>? /4?$, @;49427B6 4.5% E6 5< E>?@6 E6 ;<5;>5!
!"! #$% &'%& '()*+,-'.&$/ 0T71
%D79=@?]= 6= 5< A<57E7=@ E6 5D<KK?4;[6 <A6; K4?4B7=@ K4>? 5< 24E@57B<=749 C
8?<9E6 @;[6556 E6B 7949E<=749B 69 K?@B69;6 E6 F496B >?3<97B@6B 49= @=@ E@249J
=?@6B ^SW_! $ 5< B6;=749 K?@;@E69=6P 5< A<57E7=@ E> B45A6>? /4?$, 6B= E@249=?@6
B>? E6B ;<B =6B=B OB627JQ <9<51=7Z>6B! $:9 E6 A@?7:6? 5< A<57E7=@ E> B45A6>? /4J
?$, K4>? >96 ;49:8>?<=749 ?@6556P 56B ?@B>5=<=B 7BB>B E6B E6>X B45A6>?B 49= @=@
;49H?49=@B C E6B 26B>?6B 6XK@?7269=<56B! %6B 6XK@?769;6B 49= @=@ ?@<57B@6B 69
! "#$%&'() *+ (,-./'$'-
1.99
2.00
2.01
2.02
2.03
2.04
2.05
2.06
2.07
2.08
2.09
0 10 20 30 40x (m)
z (
m)
Reference
HLLC
Poras
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0 10 20 30 40x (m)
q (
m²/
s)
Reference
HLLC
Poras
0.999
1.000
1.001
1.002
1.003
1.004
1.005
1.006
1.007
0 10 20 30 40x (m)
Φ.q
(m
²/s)
Reference
HLLC
Poras
!"#$% !"# $ %&'()'& *+, -',- ./0& 12+3+14,' 5' */36'&7'3*' 8T6,19 : ∆x = 1m!
;/10-)/3, 5' &<=<&'3*' '- ,)(01<', +6'* 1', ,/16'0&, >??@ '- %/&A; B '3 C+0- D
7+0*C'E */-' 5' 1+ ,0&=+*' 1)F&' B '3 C+0- D 5&/)-'E 5<F)- 03)-+)&' B '3 F+,E 5<F)-
-/-+1!
1.99
2.00
2.01
2.02
2.03
2.04
2.05
2.06
2.07
2.08
2.09
0 10 20 30 40x (m)
z (
m)
Reference
HLLC
Poras
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0 10 20 30 40x (m)
q (
m²/
s)
Reference
HLLC
Poras
0.9999
1.0000
1.0001
1.0002
1.0003
1.0004
1.0005
1.0006
1.0007
0 10 20 30 40x (m)
Φ.q
(m
²/s)
Reference
HLLC
Poras
!"#$% !"G $ ;'*/35 *+, -',- ./0& 12+3+14,' 5' */36'&7'3*' 8T6,29 :∆x = 0, 1m!
;/10-)/3, 5' &<=<&'3*' '- ,)(01<', +6'* 1', ,/16'0&, >??@ '- %/&A; B '3 C+0- D
7+0*C'E */-' 5' 1+ ,0&=+*' 1)F&' B '3 C+0- D 5&/)-'E 5<F)- 03)-+)&' B '3 F+,E 5<F)-
-/-+1!
!"! #$%&'$(&)* '+ ,)%#-+. /).$, ""0
1.99
2.00
2.01
2.02
2.03
2.04
2.05
2.06
2.07
2.08
2.09
0 10 20 30 40x (m)
z (
m)
Reference
HLLC
Poras
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0 10 20 30 40x (m)
q (
m²/
s)
Reference
HLLC
Poras
0.99999
1.00000
1.00001
1.00002
1.00003
1.00004
1.00005
1.00006
1.00007
0 10 20 30 40x (m)
Φ.q
(m
²/s)
Reference
HLLC
Poras
!"#$% !"1 2 (34565789 :;6 <96< =4>3 ?@;A;?B69 C9 :4AD93E9A:9 FT6,3G H ∆x =0, 01m! ,4?><54A6 C9 3IJI39A:9 9< 658>?I96 ;D9: ?96 64?D9>36 K%%L 9< /43$, M 9A
N;>< O E;>:N9P :4<9 C9 ?; 6>3J;:9 ?5Q39 M 9A N;>< O C345<9P CIQ5< >A5<;539 M 9A Q;6P
CIQ5< <4<;?!
*4389 C9 ?@9339>3
∆x FmG N /43$, N K%%L R /43$, R K%%L
1 4, 86.10−4 3, 76.10−4 2, 43.10−10 3, 01.10−3
0, 1 4, 86.10−5 3, 94.10−5 2, 19.10−10 3, 10.10−4
0, 01 2, 18.10−6 6, 65.10−6 9, 73.10−6 4, 08.10−5
&'()%'# !0 2 -D4?><54A C9 ?; A4389 C9 ?@9339>3 6>3 ?; N;><9>3 C@9;> 9< ?9 CIQ5<
9A J4A:<54A C> =;6 C@96=;:9 FT6G!
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54 653 IF8KA3 A8>LD476D>27M,
74DAD32>4 7> ?86BA5 FN67D4 65 A2 IA2D>5 D42AD5>>5 67 (895 27 #*.' :'42AD5; I87F
A5 IF8O54 57F8IN5> 'P&%#( QR S, T53 @A893 92FFN3 8>4 N4N D?IA2>4N3 62>3 7>
92>2A 65 7m 65 A8>L 54 6=7>5 A2FL57F 65 3, 5 U 5m I87F F5IFN35>45F 7>5 V8>5
7F@2D>5, :<8DF WDL7F5 +, +;, 0=XY6F8LF2IX5 ?537FN 534 74DAD3N 98??5 98>6D4D8>
U A2 AD?D45 2?8>4, 02 9845 65 A2 37FJ295 AD@F5 92A97AN5 U t = 20 s 2<59 A53 657M
38A<57F3 534 98?I2FN5 2<59 A53 ?537F53 5MINFD?5>42A53 :<8DF WDL7F5 +, R;, 'A >=Y
2 I23 65 6DZNF5>95 3DL>DK924D<5 5>4F5 A53 IF8KA3 92A97AN3 2<59 A53 657M 38A<57F3,
&2F 2DAA57F3[ A53 X27457F3 6=527 92A97AN53 98FF53I8>65>4 27M ?537F53 5MINFD?5>E
42A53 H A53 5MIAD924D8>3 IFN35>4N53 62>3 Q!\S I87F A53 6DZNF5>953 8@35F<N53 F5345>4
<2AD653,
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FD?5>42A U J8>6 IA24 65 A=*].%P
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I5F?54 65 FNLA5F A5 6N@D4 4F2>3D42>4 62>3 A5 92>2A 54 7> 6N<5F38DF 5> @59 65
92>2F6 54 7> 342@DAD32457F I5F?5445>4 65 AD?D45F A53 47F@7A5>953 62>3 A5 92>2A
65 ?537F5 :<8DF WDL7F5 +, ^;, 05 ?867A5 U ?23C753 I5F?54 65 FNLA5F 7> 6N@D4
98?IFD3 5>4F5 5 l s−1 54 60 l s−1 4873 A53 5 l s−1 2<59 7>5 IFN9D3D8> 65 5%, % A=2<2A
67 92>2A 65 ?537F5[ 7> 6N<5F38DF I574 _4F5 D>342AAN 2K> 65 ?86DK5F A2 98>6D4D8>
$% &'()* +!,-(.!)* /0123-*03* 45673(.(8-* 4* 9(.,1*))-*3
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7.55
7.57
7.59
7.61
7.63
7.65
7.67
0 1 2 3 4 5s (m)
z (m)
HLLC
PorAS
Experimental
!"
7.55
7.57
7.59
7.61
7.63
7.65
7.67
0 1 2 3 4 5s (m)
z (m)
HLLC
PorAS
Experimental
#"
!"#$% !./ 0 *123 43 56 789:6;3 5<=93 > t = 20s 78<?6@2 53 A91B5 . C6D 32 53
A91B5 " C=D A189 53 ;67 2372 48 E1;3 CT7D!
!" $%& '% (!)!* #" +,'%*& - .!/0%&/
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!! "#$%&'() *+ (,-./'$'-
0102+ /34 5673846984 5: ;0802 53 734:<3 498= 67 cm 53 20<>3 ?9:< 43 cm 53 @0:=
4:< :83 298>:3:< 5A3816<98 12m+ $B8 53 ?9:196< 7956B3< 234 ;98B>:<0=6984
5AC;9:23738=D 534 E<6F:34 3= 534 ?0<?068>4 98= C=C 684=022C4 5084 23 ;0802 53
734:<3+
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<638;34 0 C=C <C0264C3 ?9:< ;023< 23 ;93M;638= 53 N<9==3738= 5084 23 ;0802 O 8:
3= 5084 23 ;0802 <3;9:13<= 53 ?0<?068>4 9: 53 E<6F:34+ /0 43;9853 4C<63 5A3KL
?C<638;3 <3?<C438=06= :8 C;9:23738= 5084 :8 ;0802 <3;=08>:206<3 53 5673846984
10<60E234 013; 5CE9<53738=+ )8 3I3=D 20 ?<376J<3 4C<63 53 734:<34 ?3:= G=<3
0446762C3 P5: ?968= 53 1:3 53 20 795C2640=698Q O :8 ;0802 <3;=08>:206<3 4084 5CL
E9<53738=+ /9<4 53 20 =<9646J73 4C<63 5A3K?C<638;34D 534 468:946=C4 53 N<CF:38;3
3= 5A07?26=:53 10<60E234 98= C=C :=6264C34+ )8B8D :83 53<86J<3 4C<63 5A3K?C<638;34
4A34= 68=C<344C3 0:K C;9:23738=4 =<08413<40:K 0B8 53 102653< F:026=0=613738= 234
@H?9=@J434 <C0264C34 4:< 20 4=<:;=:<3 53 2AC;9:23738= =<08413<402 P196< -3;=698
R+!Q+
,=08= 5988C34 234 S:;=:0=6984 53 20 ;9=3 53 20 4:<N0;3 26E<3D 234 1023:<4 @0:=3
3= E0443 98= C=C 734:<C34+ )8 ;@0F:3 ?968=D 20 79H3883 534 53:K 1023:<4 734:L
<C34 ;9<<34?985 O 20 1023:< <3=38:3 3= 2AC;0<= 38=<3 234 53:K 734:<34 N9:<86= 20
?<C;64698 ?9:< ;@0F:3 734:<3+
%0< 49:;6 53 ;20<=CD 234 3K?C<638;34 3= 234 795C2640=6984 98= C=C 8977C34 53
7086J<3 4H4=C70=6F:3+ /34 3K?C<638;34 498= ?0< 3K37?23 8977C34 T !UV;7UW*5 X
9Y T ! X ;9<<34?985 0: 8:7C<9 53 20 4C<63 5A3K?C<638;3D T V;7 X N06= <CNC<38;3
O 20 ;98B>:<0=698 5: ;0802 5AC;9:23738=D T W* X 6856F:3 :8 5CE6= 53 5 l s−1 3=
T 5 X 6856F:3 20 ?<C438;3 5A:8 5C13<496< O 2A0102 5: ;0802 53 734:<3+ /34 79L
5C2640=6984 <3?<38838= 23 897 53 2A3K?C<638;3 F:A32234 498= ;384C34 <3?<95:6<3
38 0Z9:=08= T U![ X 9: T U ![ X ?9:< 6856F:3< 23 ;953 53 ;02;:2 37?29HC
P-\![ 9: -\ ![Q 3= C138=:3223738= 38 0Z9:=08= :83 23==<3 N06408= <CNC<38;3
0: 706220>3 ?9:< 234 467:20=6984 ?9:< 234F:32234 ?2:463:<4 98= C=C =34=C4+
!"!" #$%&' ( ) *+,+-' ./ 01'20&'34 .' %/-15&4$
$B8 53 ;023< 23 ;93M;638= 53 <:>946=C ?9:< 234 56IC<38=4 70=C<60:K :=6264C4D :8
231C 53 26>83 5A30: 0 C=C <C0264C ?9:< 56IC<38=4 5CE6=4+ %9:< 234 734:<34 53 26>83
5A30: 4:< 234 ?0<?068>4 3= 534 E<6F:34D 234 70=C<60:K 98= C=C 564?94C4 53 7086J<3
O =0?6443< :8 =<98]98 5: ;0802 53 734:<3 4:< 23 N985 3= 234 ?0<964+ -:< 20 E043
534 ?<9B24 38 298>D 23 ;93M;638= 53 -=<6;^23< 34= ;02C+ %0< 20 4:6=3D 2AC;9:23738=
34= C>023738= 795C264C ?9:< C102:3< 20 ;0?0;6=C 5: 29>6;632 O <3?<95:6<3 20 26>83
5A30: 9E43<1C3 O ?0<=6< 5: ;93M;638= 53 <:>946=C 9E=38:+
!"!"!# $%&'()*+,-%& . $+&+/ 0 &) 1 23)4
%9:< 20 ;98B>:<0=698 T "0802 O 8: XD R 3K?C<638;34 98= C=C <C0264C34 ?9:< 534
5CE6=4 53 15 l s−1D 40 l s−1 3= 60 l s−1+ /34 734:<34 53 @0:=3:<4 5A30: O 56IC<38=34
0E4;64434 498= ?<C438=C34 5084 23 =0E230: *+ _ ` 234 734:<34 E<:=34 498= ?<C438=C34
O 5084 23 =0E230: $+ 38 0883K3 $+ %9:< ;@0F:3 5CE6=D 234 1023:<4 3K?C<6738=0234
!"! #$%&'(#)*#+ '&,-(+&#+ ."/
Q = 15 l s−1 Q = 40 l s−1 Q = 60 l s−1
,0123114 567 8 567 98 567 8 567 98 567 8 567 98 567
:;:: :;: < :;::/ :;.:: :;:: :;."/ :;::=
.;>/ :;: ? :;::" :;.:" :;::@ :;."/ :;::=
@; @ :;: @ :;::/ :;:?: :;:: :;..: :;::
=;=? :;:@= :;::/ :;:<" :;::@ :;:? :;::
!"#$!% !.: A B41CD41 94 8ECF4CD 9G4EC HICD JE 2IKLMCDEF3IK N *EKEJ O KC P!
94 8ECF4CD 9G4EC IKF QFQ 2IKRDIKFQ41 O JE J3MK4 9G4EC F8QID3SC4 5TI3D U3MCD4
!.<7! *4FF4 94DK3VD4 41F 2EJ2CJQ4 4K 2I603KEKF J41 QSCEF3IK1 94 2IK14DTEF3IK 94
JE 6E114 4F 94 JE SCEKF3FQ 94 6ICT464KF 4K DQM364 H4D6EK4KF W
dh
dx=S0 − Sf
1− Fr25 !.:7
IX S0 = 0 2ED J4 2EKEJ 41F O RIK9 HJEF 4F IX Sf 41F 9QLK3 HED W
Sf =V 2
K2R4/3
h
5 !..7
5 !.:7 41F 9312DQF31Q4 O JGE394 9GCK 128Q6E 9G#CJ4D 4YHJ323F4 4F DQ1IJC4 94 HDI284
4K HDI284 O HEDF3D 94 JE 8ECF4CD 9G4EC 641CDQ4 O JGETEJ! ZCDEKF 241 4YHQD34K241;
J4 DQM364 9GQ2ICJ464KF QFE3F 4K 4[4F \CT3EJ! (J EHHEDE]F SC4 HICD CK 2I4^234KF 94
+FD32_J4D K = 90m1/3 s−1 J41 J3MK41 9G4EC F8QID3SC41 HE114KF HED JE SCE13 FIFEJ3FQ
941 HI3KF1 94 641CD41 ECY 0I3F41 9G4DD4CD1 HDV1! %ICD J4 9Q03F Q = 60 l s−1; JE2ICD04 K4 2IDD41HIK9 HE1 HICD CK4 TEJ4CD 5HICD x = 4, 54m7! (J 41F DEHH4JQ SC4
J4 6I9CJ4 O 6E1SC4 SC3 DVMJ4 J4 9Q03F 9EK1 J4 2EKEJ 41F HDQ231 O 5% HDV1! -41
J3MK41 9G4EC F8QID3SC41 HICD J41 9Q03F1 63KIDQ1 4F 6E`IDQ1 1IKF HDQ14KFQ41 HED J41
LMCD41 ,!.E 4F ,!.0 4K EKK4Y4 ,! %ICD J4 2I4^234KF 94 +FD32_J4D K = 90m1/3 s−1;JE J3MK4 9G4EC F8QID3SC4 HICD J4 9Q03F 63KIDQ HE114 HED JG4K1460J4 941 HI3KF1
94 641CD41 ECY 0I3F41 9G4DD4CD1 HDV1! -E TEJ4CD 94 JE DCMI13FQ HICD J4 2EKEJ
O KC K = 90m1/3 s−1 14DE 9IK2 2IK14DTQ4 HICD JG4K1460J4 941 2EJ2CJ1 4F 941
6I9QJ31EF3IK1 SC3 1C3T4KF!
!"!"!" #$%&'()*+,$% - .*)/*,%'0 1 2.)3
-41 HEDHE3KM1 CF3J31Q1 HICD FEH3114D J4 2EKEJ 51CD J4 RIK9 4F J41 0ID917 IKF HICD
9364K13IK 50 cm×20 cm×5 cm 5TI3D U3MCD4 !.?7! -41 HEDHE3KM1 KGQFEKF HE1 2EDa
DQ1 9EK1 JE 93D42F3IK 8ID3bIKFEJ4; J4 KI60D4 9G3KF4D142F3IK1 4KFD4 94CY HEDHE3KM1
HED CK3FQ 94 JIKMC4CD 9QH4K9 94 JE 6EK3VD4 9IKF 241 94DK34D1 1IKF 3K1FEJJQ1!
*41 3KF4D142F3IK1 H4CT4KF cFD4 O JGID3M3K4 94 FCD0CJ4K241 SC3 ECM64KF4KF JE DCa
MI13FQ EHHED4KF4 941 HEDHE3KM1! -41 2IKLMCDEF3IK1 CF3J31Q41 9EK1 JE 1C3F4 941
4YHQD34K241 H4CT4KF KQ24113F4D 3K93[QD4664KF 941 HEDHE3KM1 ID34KFQ1 4K JIKM IC
4K JEDM4! -4 2I4^234KF 94 DCMI13FQ E 9IK2 QFQ 2EJQ HICD J41 94CY 2IKLMCDEF3IK1!
!" #$%&'()* +, )-./0(%(.
0.03
0.05
0.07
0.09
0.11
0.13
-0.1 0.9 1.9 2.9 3.9 4.9 5.9 6.9 7.9
Abscisse (m)
Hau
teu
r d
'eau
(m
)
Mesure pour Q=15 l.s-1 ±5% Théorique Q=15 l.s-1 Critique - Q=15 l.s-1
Mesure pour Q=40 l.s-1 ±5% Théorique Q=40 l.s-1 Critique - Q=40 l.s-1
Mesure pour Q=60 l.s-1 ±5% Théorique Q=60 l.s-1 Critique - Q=60 l.s-1
!"#$% +, 1 2 #345676893: ;<9 =8>:<9 ;?<6@ ABC378D@<9 6E<F =<9 E6=<@79 <G5C78H
4<:A6=<9 53@7 =< F6=6>< ;@ F3<IF8<:A ;< .A78FJ=<7 53@7 =6 F3:K>@76A83: L #6:6=
M :@ N,
0<9 4<9@7<9 ;< B6@A<@7 ;?<6@ 7C6=89C<9 ;6:9 =<9 F6:6@G <: 567568:>9 93:A
9O:ABCA89C<9 ;6:9 =< A6P=<6@ +, Q =<9 4<9@7<9 P7@A<9 93:A 7<>73@5C<9 ;6:9 =<
A6P=<6@ %,! <: 6::<G< %, 06 4CAB3;3=3>8< ;< F6=F@= ;<9 =8>:<9 ;?<6@ ABC378D@<9
57C9<:AC< M =6 9<FA83: +,!,!, <9A F3:9<7EC<, '= 655676RA A3@A<S389 D@< 53@7 F<7H
A68:<9 4<9@7<9T =< 7C>84< ;?CF3@=<4<:A CA68A A377<:A8<=, &3@7 =<9 =8>:<9 ;?<6@
ABC378D@<9 <: 7C>84< A377<:A8<=T U+, VW <9A ;89F7CA89C< <A 7C93=@< ;< =?643:A E<79
=?6E6= M 567A87 ;< =6 E6=<@7 4<9@7C< <: 7C>84< A377<:A8<= <A =6 5=@9 <: 643:A
53998P=<, 06 F3:S73:A6A83: ;<9 =8>:<9 ;?<6@ ABC378D@<9 6E<F =<9 4<9@7<9 <G5C78H
4<:A6=<9 <9A 57C9<:AC< M =6 K>@7< +,!V <A <: K>@7<9 %,! <A %,X <: 6::<G< %, &3@7
=<9 4<9@7<9 <G5C784<:A6=<9 UA6:A <: F3:K>@76A83: Y<: =3:>Z D@?<: F3:K>@76A83:
Y<: =67><ZWT 8= 9<4P=< D@?8= O 68A 53@7 F<7A68:<9 4<9@7<9 @: ;CF6=6>< 9O9AC46A8D@<
UM =?6P9F899< x = 1, 20m 53@7 =6 F3:K>@76A83: Y<: =3:>Z <A M =?6P9F899< x = 0, 50m53@7 =6 F3:K>@76A83: Y<: =67><ZW, *: <[<AT F<9 4<9@7<9 93:A :<AA<4<:A 9@7C=<EC<9
567 765537A 6@ 7<9A< ;< =6 =8>:< ;?<6@ <A =<9 <77<@79 ;< 4<9@7< :< 5<74<AA<:A M
!"! #$%&'(#)*#+ '&,-(+&#+ ."
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Théorique Q=15l s-1
Critique - Q=15l s-1
Mesure pour Q=15 l.s-1 ±5%
Théorique Q=40l s-1
Critique - Q=40l s-1
Mesure pour Q=40 l.s-1 ±5%
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Théorique Q=15 l.s-1Critique - Q=15 l.s-1Mesure pour Q=15 l.s-1 ±5%Théorique Q=30 l.s-1Critique - Q=30 l.s-1Mesure pour Q=30 l.s-1 ±5%Théorique Q=45 l.s-1Critique - Q=45 l.s-1Mesure pour Q=45 l.s-1 ±5%
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Théorique Q=15l/s
Théorique Q=30l/s
Mesure pour Q=15 l.s-1 ±5%
Mesure pour Q=30 l.s-1 ±5%
Critique - Q=15l/s
Critique - Q=30l/s
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Hau
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Théorique Q=15 l.s-1
Critique - Q=15 l.s-1
Mesure pour Q=15 l.s-1 ±5%Théorique Q=30 l.s-1
Critique - Q=30 l.s-1
Mesure pour Q=30 l.s-1 ±5%
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Hauteur d'eau modélisée avec SW2D
Hauteur d'eau modélisée avec SW12D
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=56 <F;2I8=534 I8625 53 7>3942>3 05 =18D@92@@5! %>;6 4>;45@ =5@ 9>3H?;6842>3@E =8
7>6A;=842>3 Q !."R 8 :56A2@ 05 0<456A2356 =5 9>5B92534 05 +4629C=56 <F;2I8=534
:>;6 =5@ T8;45;6@ 0158; A232A8=5 54 A8V2A8=5 A5@;6<5@! -8 02W<65395 53465 =5@
I8=5;6@ 0<456A23<5@ :>;6 95@ 05;V T8;45;6@ 0158; <4834 46S@ 782D=5E =8 A>J5335
05@ T8;45;6@ A5@;6<5@ :>;6 9T8F;5 9>3H?;6842>3 8 <4< ;42=2@<5 :>;6 98=9;=56 =5
9>5B92534 05 +4629C=56 <F;2I8=534!
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;35 @<625 015V:<625395@ :6<@534834 ;3 =24 A235;6 D>60< :86 05;V :=8235@ 0123>3U
0842>3 083@ =5@F;5==5@ 4683@24534 ;35 :86425 0; 0<D24 8 <4< 6<8=2@<5 QI>26 Z2?;65
!"[R! +;6 ;35 =>3?;5;6 05 4mE ;3 =24 A235;6 659483?;=8265 8 <4< 8A<38?<! -5@
=86?5;6@ 0; =24 A235;6 @>34 7 cmE 17 cm 54 28 cm :>;6 05@ :6>7>305;6@ 65@:59U
42I5@ 05 10 cmE 10 cm 54 5 cm! %>;6 9T8F;5 9>3H?;6842>3E 02W<6534@ 0<D24@ >34
<4< 23G594<@ 53 8A>34 \ <I534;5==5A534 8I59 ;3 0<I56@>26 L =18I8= :>;6 ?8683426
;3 0<D>605A534 @;6 =153@5AD=5 0; =23<8265 0; =24 A235;6! -5@ T8;45;6@ 0158; >34
<4< A5@;6<5@ 53 02W<6534@ :>234@ 0; =23<8265! %86 82==5;6@E 05@ A5@;65@ 05 I245@@5
>34 <?8=5A534 <4< 6<8=2@<5@ :>;6 =5@ 9>3H?;6842>3 0<D>608345@!
*5@ A5@;65@ >34 <4< 6<8=2@<5@ @>24 8I59 ;3 4;D5 05 %24>4 Q:>;6 05@ T8;45;6@
0158; 782D=5@RE @>24 :86 A5@;65 8; ]>445;6! Y; 7824 05 =8 02B9;=4< :>;6 :>@242>3U
356 9>665945A534 =5 4;D5 05 %24>4 083@ =18V5 05 =1<9>;=5A534E 95@ A5@;65@ @>34
65=842I5A534 2A:6<92@5@! %86 82==5;6@E :>;6 =5@ A5@;65@ 8; ]>445;6E 2= <4824 65=8U
42I5A534 02B92=5 05 0<459456 9>665945A534 =5 :8@@8?5 0; ]>445;6 L ;35 8D@92@@5
0>33<5 54 05 65=5I56 =5 45A:@ 9>665@:>30834! -5@ A5@;65@ 05 I245@@5@ @>34 0>39
7>645A534 53489T<5@ 015665;6@ 02B92=5A534 F;8342H8D=5@!
%>;6 9T8F;5 9>3H?;6842>3E =8 9>A:8682@>3 05@ A>0<=2@842>3@ 8I59 =5@ =>?2U
925=@ +X."Y 54 +X"Y 5@4 6<8=2@<5! -5@ A>0<=2@842>3@ 9>A:86<5@ @>34 D8@<5@ @;6
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2" #$%&'()!*+$% , -+* .+%/() 28 cm 0
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#012345675 8294: *;6<4:4;6 =>=? K @m1/3 s−1A"BC7DBE 5 l s−1 %=F <5 <2>53F;43 GH
"BC7DBE < 5 l s−1 82>53F;43 357:=6IJ?=435 @h = 10 cmA GH
"BC7DBE.K< 10 l s−1 82>53F;43 357:=6IJ?=435 @h = 10 cmA GH
"BC7DBE"K< 20 l s−1 82>53F;43 357:=6IJ?=435 @h = 10 cmA GC
"BC7DBEGK 40 l s−1 %=F <5 <2>53F;43 GC
!"#$!% !. L #012345675F 32=?4F25F 1;J3 ?= 7;6MIJ3=:4;6 N 7 cm O!
1=3 3=11;3: =J0 >4:5FF5F 56 ?4: D=P5J3 7;335F1;6< Q 75?J4 ;9F53>2 156<=6: ?5F
5012345675F!
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<T2>=?J53 ?= 7=1=74:2 <J D;<U?5 Q 35FFJV53 ?5F 5=J0 <29;3<25F 1J4FWJ5 ?5F <29;3X
<5D56:F 2:=456: ?;7=?4F2F J64WJ5D56: FJ3 ?5F <5J0 135D453F DU:35F <J :3;6Y;6!
-5F <4Z2356:5F 5012345675F 32=?4F25F F;6: 35I3;J125F <=6F ?5 :=9?5=J !. !
-5F D5FJ35F <5 [=J:5J3F <T5=J 1;J3 ?T56F5D9?5 <5F 7;6MIJ3=:4;6F F;6: 132X
F56:25F Q ?T=66505 \!.! -5F 7;D1=3=4F;6F <5F ?4I65F <T5=J D;<2?4F25F =>57 ?5F
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132F56:25F Q ?= MIJ35 !" @5: Q ?= MIJ35 \!G 56 =66505 \ 1;J3 ?5F =J:35F 50X
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D;<2?4F25F 65 7;335F1;6<56: 1=F =J0 D5FJ35F 32=?4F25F! #6 5Z5:R ?= :J39J?5675
5F: ?= 7=JF5 1346741=?5 <5 ?= 153:5 <5 7[=3I5 ;9F53>25 <=6F ?= 1=3:45 32:32745 <5
?T27;J?5D56:! -5F ?;I4745?F <5 7=?7J? 65 1356=6: 1=F 56 7;D1:5 75 1[26;DU65R 4?
5F: ?;I4WJ5 WJ5 ?T;6 65 3513;<J4F5 1=F ?= 156:5 <5 ?= ?4I65 <T5=J ;9F53>25!
%;J3 ?5F 5012345675F <29;3<=6:5F @"BC7DBE <R "BC7DBE.K<R "BC7DBE"K<
5: "BC7DBEGKAR ?5F ?4I65F <T5=J F4DJ?25F F;6: 35?=:4>5D56: F4D4?=435F 1;J3 ?5F
<5J0 ?;I4745?F 5: 7;335F1;6<56: =J0 D5FJ35F 32=?4F25F @=J0 4D13274F4;6F 13UFA!
8=6F :;JF ?5F 7=FR ;6 35:3;J>5 WJ5 ?= 7;:5 <5 ?= FJ3]=75 ?4935 D;<2?4F25 =>57
+^"8 5F: 1?JF 9=FF5 WJ5 75??5 13;<J4:5 1=3 +^."8! -T=113;04D=:4;6 FJ3 ?5
3=V;6 [V<3=J?4WJ5 15J: _:35 46734D4625! %=3 =4??5J3FR ?T=9=4FF5D56: <5 ?= ?4I65
<T5=J D;<2?4F25 1=3 +^"8 FJ3 ?5F 50 135D453F 756:4DU:35F 5F: ?42 Q ?= 7;6<4:4;6
Q ?= ?4D4:5 =D;6:! #6 5Z5:R ?5 <294: J64:=435 Q ?= 7;6<4:4;6 ?4D4:5 =D;6: 5F:
FJ11;F2 J64];3D5 FJ3 :;J:5 ?= F57:4;6! -= [=J:5J3 <T5=J 2:=6: 1?JF 4D1;3:=6:5
<=6F ?5 ?4: D465J3R 75?= F5 :3=<J4: 1=3 J65 >4:5FF5 <T27;J?5D56: 1?JF ]=49?5 <=6F
?5 ?4: D465J3 WJ5 <=6F ?5 ?4: D=P5J3!
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<2?4F25F 65 7;335F1;6<56: 1=F =J0 D5FJ35F 32=?4F25F! -5F `J7:J=:4;6F <5 ?= FJ3X
]=75 ?4935 ;9F53>25F <J3=6: ?5F 5012345675F 15J>56: _:35 46734D4625F @>;43 a4IJ35
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7;D1:5!
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0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
Abscisse (m)
Co
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(m
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Cote de la surface libre mesurée (m)
Cote de la surface libre modélisée avec SW2D
Cote de la surface libre modélisée avec SW12D
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0.155
0.160
0.165
0.170
0.175
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Abscisse (m)
Co
te d
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a s
urf
ace l
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(m
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Cote de la surface libre mesurée (m)
Cote de la surface libre modélisée avec SW2D
Cote de la surface libre modélisée avec SW12D
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4<9@7A<9 53@7 =6 D3:E>@76B83: F 08B 48:<@7 7 cm G,
!"! #$%&'(#)*#+ '&,-(+&#+ ./0
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
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Q (l/s)
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Vmin/Vmaj - Mes
Vmin/Vmaj - SW2D
Vmin/Vmaj - SW12D
!"#$% !"1 2 '3456 789 :5489989 8; <54 =5;8>? 3>@ :5489989 8; <54 =3A8>? B6>?
<3 C6;DE>?3456; F -54 =5;8>? 7 cm G!
%6>? H:3<>8? <3 C3B3C54H 789 C6789 I ?8B?67>5?8 <3 :548998 78 <JHC6><8=8;4 73;9
<8 <54 =5;8>? 84 73;9 <8 <54 =3A8>?K <8 ?3456 789 :5489989 8; <54 =5;8>? 3>@ :5489989
8; <54 =3A8>? 599>9 789 =89>?89 3 H4H C6=B3?H I C8<>5 599> 78 <3 =67H<593456;
L:65? M5E>?8 !"1N! &43;4 76;;H89 <89 5=B?HC5956;9 9>? <89 =89>?89 78 :548998K
<89 ?34569 =67H<59H9 96;4 8; 37HO>3456; 3:8C C8>@ 6P98?:H9! (< 3BB3?3Q4 46>48R659
O>8 <3 :548998 78 <JHC6><8=8;4 8; <54 =5;8>? =67H<59H8 3:8C +S."T 894 46>A6>?9
9>BH?58>?8 I C8<<8 8; <54 =3A8>? L<8 ?3456 ?8948 9>BH?58>? I .N! -3 U3>48>? 7J83>
H43;4 ;HC89935?8=8;4 B<>9 5=B6?43;48 8; <54 =5;8>?K <8 48?=8 78 R?6448=8;4 V 894
=65;9 5=B6?43;4 L<3 U3>48>? 5;48?:8;3;4 3> 7H;6=5;348>? 78 LW!0XNN! *8 ?3456
48;7 :8?9 . O>3;7 <8 7HP54 3>E=8;48 73;9 <3 =89>?8 6Y <3 75ZH?8;C8 78 U3>48>?
7J83> 8;4?8 <8 <54 =5;8>? 84 <8 <54 =3A8>? 894 ?8<345:8=8;4 =65;9 5=B6?43;48 B6>?
<89 R6?489 U3>48>?9 7J83> LC6??89B6;73;4 I >; 7HP54 5=B6?43;4N! #; ?8:3;CU8K
+S"T C3<C><8 789 :5489989 8; <54 =5;8>? B<>9 R35P<89 O>J8; <54 =3A8>?K 3:8C >;
?3456 O>5 48;7 :8?9 0, 85 B6>? 789 7HP549 5=B6?43;49!
%6>? 789 7HP549 5=B6?43;49 L84 76;C B6>? 78 R6?489 U3>48>?9 7J83>NK <J8Z84 78
<3 B?H98;C8 7> <54 =5;8>? 9>? <3 :548998 789 HC6><8=8;49 894 75=5;>H ;643==8;4
O>3;7 <3 ?>E6954H 7> <54 =5;8>? 894 98;95P<8=8;4 <3 =[=8 O>8 B6>? <8 <54 =3A8>?!
(< 98=P<8 76;C O>8 <8 C6=B6?48=8;4 78 +S."T 9654 B<>9 C6UH?8;4 B>59O>8 <8 ?3456
48;7 :8?9 . B6>? <89 7HP549 R6?49! \6>48R659K <8 C6=B6?48=8;4 F 5;C6UH?8;4 G 78
+S"T B8>4 [4?8 8@B<5O>H B3? <3 =3;5]?8 76;4 894 5=B69H8 <3 C6;75456; I <3 <5=548
3=6;4 L7HP54 >;5435?8 >;5R6?=8 9>? <3 98C456;N!
-89 :5489989 =67H<59H89 84 =89>?H89 96;4 B?H98;4H89 73;9 <8 43P<83> ^!" 84
!" #$%&'()* +, )-./0(%(.
#12345678912 :;<98 #12=98912 7>7? K @m1/3 s−1ABC DEFCGB+= 25 l s−1 :;>H6I196 6HE87245?796H @h = 5 cmA +B
BC DEFCG!J= 30 l s−1 :;>H6I196 6HE87245?796H @h = 5 cmA +
!"#$!% +, K L *MN;69H2EHI 6;7?9I;HI N156 ?7 E12345678912 O 098 F92H56 17 cm P,
Q ?7 3456H R,+ H2 722HMH R, 0HI >98HIIHI F1=;?9I;HI I128 N?5I S79<?HI T5H EH??HI
FHI56;HIU H8 EH N156 ?VH2IHF<?H =HI =;<98I 8728 H2 ?98 F92H56 T5VH2 ?98 F7WH56 H8
N156 ?HI =H5M E1=HI =H E7?E5?, 0HI =9X;6H2EHI 1<IH6>;HI NH5>H28 Y86H HMN?9T5;HI
N76 ?H S798 T5H ?HI E1=HI =H E7?E5? S15629IIH28 ?7 >98HIIH F1ZH22;H I56 ?7 >H689[
E7?H =H ?V;E15?HFH28 7?16I T5H ?HI >7?H56I FHI56;HI I128 =HI >98HIIHI =H I56S7EH
@N59IT5H FHI56;HI 75 \188H56A, 0HI >98HIIHI HMN;69FH287?HI 128 ;8; 6HE7?E5?;HI,
0H =;<98 8187? 8672I98728 =72I ?H E727? H8 ?V]ZN18]^IH T5H ?H 67891 =HI >98HIIHI
F1ZH22HI I56 E]7T5H IHE8912 =V;E15?HFH28 HI8 ;47? 75 67891 =HI >98HIIHI FHI5[
6;HI @H2 I56S7EHA H2 ?98 F92H56 N76 67NN168 75M >98HIIHI H2 ?98 F7WH56 NH6FH88H28
=V;87<?96 ?H IZI8^FH I59>728 _
Q = V1D,recalcS1D + V2D,recalcS2DV1D,recalc
V2D,recalc=V1D,mes
V2D,mes
1` Q HI8 ?H =;<98 8187?U Sk ?7 IHE8912 =H ?V;E15?HFH28 k 7>HE k = 1D ou 2D aVk,mes 6HN6;IH28H ?7 >98HIIH FHI56;H H2 I56S7EH =H ?V;E15?HFH28 k H8 Vk,recalc
?7 >98HIIH F1ZH22H I56 ?7 IHE8912 =V;E15?HFH28, 07 6;I1?58912 =H EH IZI8^FH
NH6FH8 =H 6HE7?E5?H6 ?7 >98HIIH F1ZH22H @V1D,recalc H8 V2D,recalcA N156 E]7T5H
;E15?HFH28, 0HI >98HIIHI F1=;?9I;HI I128 7?16I H2 7=;T578912 7>HE ?HI >98HIIHI
F1ZH22HI HMN;69FH287?HI @>196 b9456H R,+ H2 722HMH RA,
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0HI HMN;69H2EHI I56 ?H ?98 F92H56 =H DEF =H ?764H 128 ;8; 6;7?9I;HI 7>HE 52
=;>H6I196 H2 N7619 F92EH =H +EF E1FFH E1286c?H 7>7? H8 N156 =HI =;<98I =H
25 l s−1 H8 30 l s−1, &76 79??H56IU ?H ?98 F92H56 2V;8798 N7I EH286; N76 67NN168 75
E727? =H FHI56H =5 S798 =HI =9FH2I912I =HI <69T5HI H8 N76N7924I, 0H ?98 F7WH56
H2 69>H 475E]H ;8798 H2 HXH8 F192I ?764H T5H ?H ?98 F7WH56 H2 69>H =6198H, 0HI
=9X;6H28HI HMN;69H2EHI 6;7?9I;HI I128 6H4615N;HI =72I ?H 87<?H75 +, K,
&156 ?HI F1=;?9I78912IU ?H =;<98 =VH286;H =5 F1=^?H E166HIN12=798 75 =;<98
HMN;69FH287? F7W16; =H +d @E166HIN12=728 Q ?7 F764H =VH66H56 =5 F1=5?H Q
F7IT5HIA, *2 HXH8U 52H N6HF9^6H I;69H =H F1=;?9I78912 7 ;8; 6;7?9I;H N156 ?H
=;<98 HMN;69FH287? F79I ?HI ?942HI =VH75 ;879H28 IZI8;F789T5HFH28 861N <7IIHI
N76 67NN168 75M ]758H56I =VH75 FHI56;HI,
0HI FHI56HI =H ]758H56I =VH75 I128 6H4615N;HI =72I ?H 87<?H75 R,! H2 72[
2HMH R, 07 E1FN7679I12 =HI ?942HI =VH75 F1=;?9I;HI 7>HE ?HI ]758H56I =VH75
FHI56;HI HI8 N6;IH28;H Q ?7 3456H +,BD N156 ?VHMN;69H2EH BC DEFCGB+= @H8 Q
?7 3456H R,K H2 722HMH R N156 ?VHMN;69H2EH BC DEFCG!J=A, '? 7NN767e8 T5H ?HI
!"! #$%&'(#)*#+ '&,-(+&#+ ./0
0.125
0.130
0.135
0.140
0.145
0.150
0.155
0.160
0.165
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Abscisse (m)
Co
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a s
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(m
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Cote de la surface libre mesurée (m)
Cote de la surface libre modélisée avec SW2D en lit majeur
Cote de la surface libre modélisée avec SW2D en lit mineur
Cote de la surface libre modélisée avec SW12D
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;54?9156 >=9126=56E F51156 A><7:2?56 A9> .G HI =?94? ?<:?5J<26 A1:6 A><FK56 756
L915:>6 5CA=>2;54?9156 B:5 F51156 A><7:2?56 A9> .GHI, &9> 92115:>6E 156 F<?56 75
19 6:>J9F5 12@>5 ;<7=126=56 A9> .GHI 45 6<4? A96 156 ;M;56 A<:> :45 9@6F2665
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Abscisse (m)
Cote
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m)
Cote de la surface libre mesurée (m)
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0.110
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cote de la surface libre (m)
0.13
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l’écoulement
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Cote
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m)
Cote de la surface libre mesurée (m)
Cote de la surface libre (m) - SW2D
Cote de la surface libre (m) - SW12D
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O2
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Abscisse (m)
Cote
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m)
Cote de la surface libre mesurée (m)
Cote de la surface libre (m) - SW2D
Cote de la surface libre (m) - SW12D
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Rati
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lit
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Exp
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Mesure pour Q=15 l.s-1 ±5% Théorique Q=14.25 l s-1 Critique - Q=14.25 l s-1
Mesure pour Q=40 l.s-1 ±5% Théorique Q=38 l s-1 Critique - Q=38 l s-1
Mesure pour Q=60 l.s-1 ±5% Théorique Q=57 l s-1 Critique - Q=57 l s-1
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0.03
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Mesure pour Q=15 l.s-1 ±5% Théorique Q=15.8 l s-1 Critique - Q=15.8 l s-1
Mesure pour Q=40 l.s-1 ±5% Théorique Q=42 l s-1 Critique - Q=42 l s-1
Mesure pour Q=60 l.s-1 ±5% Théorique Q=61.2 l s-1 Critique - Q=61.2 l s-1
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Théorique Q=14.25 l.s-1Critique - Q=14.25 l.s-1Mesure pour Q=15 l.s-1 ±5%Théorique Q=28.5 l.s-1Critique - Q=28.5 l.s-1Mesure pour Q=30 l.s-1 ±5%Théorique Q=42.75 l.s-1Critique - Q=42.75 l.s-1Mesure pour Q=45 l.s-1 ±5%
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teu
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Théorique Q=14.25l s-1
Critique - Q=14.25l s-1
Mesure pour Q=15 l.s-1 ±5%
Théorique Q=38l s-1
Critique - Q=38l s-1
Mesure pour Q=40 l.s-1 ±5%
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Abscisse (m)
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)Théorique Q=15.3 l.s-1Critique - Q=15.3 l.s-1Mesure pour Q=15 l.s-1 ±5%Théorique Q=31.5 l.s-1Critique - Q=31.5 l.s-1Mesure pour Q=30 l.s-1 ±5%Théorique Q=47.25 l.s-1Critique - Q=47.25 l.s-1Mesure pour Q=45 l.s-1 ±5%
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Théorique Q=15.75l s-1
Critique - Q=15.75l s-1
Mesure pour Q=15 l.s-1 ±5%
Théorique Q=42l s-1
Critique - Q=42l s-1
Mesure pour Q=40 l.s-1 ±5%
0" #$%&'()!*+$% ,-% .!)'-/
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Théorique Q=14.25 l.s-1
Critique - Q=14.25 l.s-1
Mesure pour Q=15 l.s-1 ±5%Théorique Q=28.5 l.s-1
Critique - Q=28.5 l.s-1
Mesure pour Q=30 l.s-1 ±5%
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0.03
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Hau
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Théorique Q=14.25l/s
Théorique Q=28.5l/s
Mesure pour Q=15 l.s-1 ±5%
Mesure pour Q=30 l.s-1 ±5%
Critique - Q=14.25l/s
Critique - Q=28.5l/s
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Hau
teu
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Théorique Q=15.3 l.s-1
Critique - Q=15.3 l.s-1
Mesure pour Q=15 l.s-1 ±5%Théorique Q=31.5 l.s-1
Critique - Q=31.5 l.s-1
Mesure pour Q=30 l.s-1 ±5%
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Hau
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Théorique Q=15.75l/s
Théorique Q=31.5l/s
Mesure pour Q=15 l.s-1 ±5%
Mesure pour Q=30 l.s-1 ±5%
Critique - Q=15.75l/s
Critique - Q=31.5l/s
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Hau
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Hauteur d'eau modélisée avec SW2D
Hauteur d'eau modélisée avec SW12D
Hauteur d'eau mesurée pour Q=15 l s-1 ±5%
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Hauteur d'eau modélisée avec SW2D
Hauteur d'eau modélisée avec SW12D
Hauteur d'eau mesurée pour Q=60 l s-1 ±5%
2" #$%&'()*+) ,-./-034
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Hauteur d'eau modélisée avec SW2D
Hauteur d'eau modélisée avec SW12D en faisant l'approximation Rh=h
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Hau
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Hauteur d'eau modélisée avec SW2D
Hauteur d'eau modélisée avec SW12D
Hauteur d'eau mesurée pour Q=15 l s-1 ±5%
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Hau
teu
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(m
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Hauteur d'eau modélisée avec SW2D
Hauteur d'eau modélisée avec SW12D
Hauteur d'eau mesurée pour Q=40 l s-1 ±5%
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