Shock Structure in a Two-phase Isothermal Euler Model

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Submitted on 31 Aug 2007

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Shock Structure in a Two-phase Isothermal Euler Model

Hervé Guillard, Vincent Perrier

To cite this version:

Hervé Guillard, Vincent Perrier. Shock Structure in a Two-phase Isothermal Euler Model. [Research Report] RR-6274, INRIA. 2007, pp.23. �inria-00168787v2�

a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Shock Structure in a Two-phase Isothermal Euler Model

Hervé Guillard — Vincent Perrier

N° 6274

August 2007

Unité de recherche INRIA Sophia Antipolis

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∂

∂t(ρ) + ∂

∂x(ρu) = 0 (6.1)

∂

∂t(ρY) + ∂

∂x(ρY u)−ε ∂

∂x(ρY(1−Y)ur) = 0 (6.2)

∂

∂t(ρu) + ∂

∂x(ρ(u)²+p) = 0 (6.3)

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αk

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αkρk =ρY2

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ρk

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α1+α2= 1^¨ ShšcSrv˜š˜š]p¬K`bQeUœuegiU;^i^adSgbUF^

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pk =pk(ρk)^¨

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









p1(ρ1) =p2(ρ2) (1−Y)

ρ1

+(Y) ρ2

= 1 ρ

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ur= (Y −α)∂p

∂x

¦

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`iU0cS^bnpgF¨©ªc`iQeUWgiU0TrvhšcSw!U;gžnp•`iQehk^t^aUFx£`ih™n9c›žUW^adeTTrvgihš®0Uœ`bQeUWTrv`bQeU;Trv`bhkx0rp˜Õuegbn9u‰U;ga`ih™UF^žnv•`bQehk^5Tn!w!U0˜C¨

! "

#U`tdS^t^bU`

U = (ρ, ρY, ρu)^t rpcSw›5gbh™`bU`iQeU$^a]!^a`bU;T Ã} ždecSw!U;g(`bQeU•–n9gbT

∂U

∂t +∂F(U)

∂x = ∂

∂x(D(U)∂U

∂x) ^E

›5h™`bQ.n9¡K—_hšnpdS^cSnv`irv`bhšnpcS^;¨­P(QeUTr‘`iQeU0Trv`bhkx0rv˜uegbn9u‰U;ga`ih™UF^nv•5^a]!^a`bU;T Ã} QSr—pU¡'U0U0c«^a`bd‰w!h™UFwh™c O rpcSw›UgiU•–U;g`bn

`iQehš^…›žnpgi¤Œ•–n9gt`bQSU,uegin_nv•ånv•å`iQeU,giU;^bde˜™`i^O`bQSrv`…›žU$gbUFx0rp˜™˜šU;w"hšc¸`bQShš^œ^aUFx£`ih™n9cM¨ Sh™gy^a`œ^bh™cSx0U$`iQeUˆ^b]!^q`iU0T Ã}thš^…¡Srpgbnp`bginpuehkxv¬

›žU$x0rpcw!U SceUœ`bQSU$´tU0˜šTQenp˜™`b®•–giU0UWU;ceU0gi„p]

fk(ρk), k= 1,2np•U;r9xyQuSQSrp^bU¡K]

f_k⁰(ρk) =pk(ρk)/ρ²_k

”

h™`bQ"`iQeU;^bUWw!UScSh¢`ih™n9cS^0¬S›Uhšc9`igbn!w!d‰xUœ`bQeU•–n9˜™˜šn‘›5hšce„•–decSx£`ih™n9c

η(U) =ρu²

2 +ρY f2(ρ2) +ρ(1−Y)f1(ρ1) ^O

” U`bQeU;c"QSr—pU

 ¼ Z ¼ <–Ž<¼ ; >K8

η(U) ^{$ , "%! ( $I$ %) 1 , % * (} , #"I$!$

Y ∈]0,1[

4, X$K! $!

D(U) ^{$ 4!$ $"! %,B $ % "}

©ªcSweU0U;w•–n9gt^aTn_nv`iQ^bnp˜šd!`bhšnpc‰^0¬

η(U)^—9U0gih‰U;^ž`bQSUWU;jKdSr‘`ih™n9c

∂η(U)

∂t +∇Uη(U)· ∂F(U)

∂x = ∂

∂x(∇Uη(U)·D(U)∂U

∂x)− ∇²_Uη(U)∂U

∂x ·D(U)∂U

∂x

DF†

¡ede`Oh¢`Ox;rvc¡‰U$^bQen‘›5c Ã^bU0UO `iQSr‘`

∇Uη(U)·∂F(U)

∂x = ∂(η(U) +p)u

∂x

›5Qehš˜šU

∇²Uη(U)∂U

∂x ·D(U)∂U

∂x = (ur)²

rpcSw`bQeU;cuSgbn9u‰nK^ah™`bhšnpc"~!¨#Dœhk^(uegin‘—pUFw

! "

©¦•›UWcSn‘› xn9cSxU;c9`igirv`bUnpc`bQSU Sgi^a`5npgyw!U0g5x0npc_—pUFx£`bhš—pUœuSrpga`5nv•å^b]_^a`bU;T Ã}

∂U

∂t +∂F(U)

∂x = 0 ^D6D

›žUWQSr—pU Ã^bU0U O M•–n9g(`bQeUWuSgbn_nv•

 ¼ Z ¼ <–Ž<¼ ; >>

. $I$ )2$ ""!!+K ( ,) ".+0% $ W %$# % , $ I$ $5K! 4B 2 ( $# % ,+$

λ1(U) =u−a , λ2(U) =u , λ3(U) =u+a ^DF~

^%

a ^{$' '$" 4 2$!.4 4R&% . 4'!" (%"M%$ $ K,} 1

ρa² =X

k

αk

ρka²_k

DF‡

13! (,R %$ K)%/%+04!$ !$ $%K! 4 ( $

λ1(U) =u−a ^,M4 λ3(U) =u+a ^{,*# %M++}

.!3+M !'^+0 ,R $ K%/%+#4 I$ $%K, 4 B

λ2(U) =u ^{$2+M !++ 4R,#6M,}

- . (/…$ ² )*45+

©ªc²`bQehk^^aUFx£`bhšnpc¬m›U^aQSn‘› `bQSrv`ˆ`iQeUTn!w!U;˜ B}ˆx0rpc«¡‰U"w!U;gbhš—pUFwµ•–gbn9T r?¡Srpgbnp`bginpuehkx—9U0gy^ahšnpc.np•t`bQeU10(rvU;ga¥ªVtdecS®0hkr‘`bn

Tn!w!U0˜ #D ¦¨ SSnpg`bQehk^ž›Ut›5hš˜š˜'r9^b^bdeTUt`iQSr‘``bQeU…uegbUF^b^bdegiUOrvc‰w—pU;˜™n!xh™`q]gbU;˜šrv !rv`bhšnpc`bhšTU;^`bU;cSw`bnˆ®0U;gbn,rpcSw­x0npcS^bhšweU0gm`iQeU

Sgy^a`a¥¦npgyw!U0gW¾QSrvueTrpc!¥CŸcS^a¤9np„U !uSrvc‰^ahšnpc?np•m`bQSU¡‰rvginv`bginpuShšx20(rpU0gb¥¦VOdece®0hkr‘`in­Tn!w!U0˜åhšc?`iQehš^˜šhšTh™`;¨ˆVtnv`iU,`iQSr‘`Whšc O ¦¬

`iQeUTn_weU0˜ Ã} œ›žr9^rv˜k^an"w!U0gihš—pU;w?•–gbn9T `bQSU­x˜krp^i^ahkx0rp˜`q›žnv¥¦—pU;˜™n!xh™`q]p¬Õn9ceUuegiU;^i^adegiU$`q›žn§Sdehkw Tn!w!U;˜ Ã^bU0U•–npgWhšcS^a`irpcSxU

€S¬ DF}!¬DF† "žh™c"`iQeUW˜™hšTh¢`…nv•å˜šrpgb„9Uœwegirp„x0n_U ­xhšU0cK`i^;¨P(QShš^t^bU;x`bhšnpc"uegiU;^bU0cK`i^5rpc¸rv˜™`bU0gicSrv`bUWw!U0gihš—rv`bhšnpc¸rvc‰wŒ`iQeU0giU•–n9gbUhk^

cenp`œUF^b^bU0cK`ihšrp˜`inŒ`iQeUgbU;Trph™c‰w!U0gœnv•m`iQeUuSrvu'U0gF¨W©¦`x;rvc‚¡‰U^a¤_hšueu'U;w?¡_]¸giU;r9w!U0gy^tcenp`œhšcK`bU;gbUF^q`iU;w?¡_]"`iQeUweU`irph™˜k^œnv•m`iQeU

w!U;gbhš—‘r‘`ih™n9cnp•B}£¨

" !)

P(QeUˆcenpce¥CUFj9dSh™˜šh™¡SgbhšdeTQD5ORE9}0(rvU0gb¥ªVtdece®;hšrv`bnTn!w!U0˜ 0D n9g…^bnpTU$nv•mh™`i^O—‘rvgihšrv`bhšnpcS^5Q‰r—pU$¡'U0U;c?gbUFxU;c9`i˜™]`bQSUˆ^bde¡!oqU;x`Onv•

^bU0—9U0gyrv˜›n9gb¤!^O`bnTn!w!U;˜`q›np¥CueQ‰rp^bU$§Sn‘›t^ 0D!¬Õ‡e¬ DF‡ t¨$©ªc‚`bQehk^^aUFx£`bhšnpc¬Õ›Uw!U;^ixgihš¡‰Urvc‚hš^bnv`iQeU0giTrv˜—pU;gi^bh™n9c?nv•`bQehk^

Tn!w!U0˜C¨ ” U(`iQ_dS^åxn9cS^ahkw!U;gmrO§‰n‘›±xn9Tu‰nK^aUFw,np•‰`q›žnh™TThš^ixhš¡e˜šUž§Sdehkwe^0¬

αk

w!U;cenv`iU(`bQeU5—9np˜šdeTUž•–gir9x£`bhšnpc‰^np•'UFrpxyQueQSr9^aU

α1+α2= 1^¬ ρk

`iQeU0hšgOueQ‰rp^bU,w!U0c‰^ah™`bhšU;^…›5Qehš˜™U

u_k rvgiUW`bQSU,—pUFx£`inpgt—9U0˜šn_x0h¢`ih™UF^0¨ ShšcSrv˜š˜š]

pk

rvgiU`bQeUˆuegiU;^i^adegiU;^tgbU;˜šrv`bU;w

`in`bQeUWueQ‰rp^bUweU0cS^bh¢`ih™UF^

ρk

¡_]­¡Srpgbnp`bginpuehkx…U;jKdSr‘`ih™n9cS^žnv•å^q`yr‘`iU

pk =pk(ρk) ^D;€

`iQSr‘`thšc¸rpx0x0npgywervc‰xUO›5h™`bQ?^a`irpcSwerpgiw­`bQeU;gbTn!w!]_cSrvThkxv¬!—9U0gih¢•–]

∂pk

∂ρk

>0

rpcSw"›UWcSnv`bU

ak(ρk) =p

∂pk/∂ρk

`iQeU,^bu‰U;U;w"np•å^bnpdScSwh™c"`iQeU$ueQSr9^aU

k^{¨ ”} h™`bQ¸`iQeU;^bU$cenp`irv`bhšnpcS^;¬!`bQeU,Tn!w!U;˜M`iQSr‘`O›žU x0npcS^bhšweU0g5hk^

∂α1ρ1

∂t + ^w!hš— (α1ρ1u₁) = 0 (15.1)

∂α1ρ1u₁

∂t + ^w!hš— (α1ρ1u₁⊗u₁) +∇α1p1 = p^© ∇α1+λ(u₂−u₁) (15.2)

∂α2ρ2

∂t + ^w!hš— (α2ρ2u₂) = 0 (15.3)

∂α2ρ2u₂

∂t + ^w!hš— (α2ρ2u₂⊗u₂) +∇α2p2 = p^© ∇α2−λ(u₂−u₁) (15.4)

∂α2

∂t + u^© .∇α2 = µ(p2−p1) (15.5)

©ªc `bQeUF^aUU;jKdSrv`bhšnpcS^;¬

p^{© rvc‰w} u^© ^a`irpcSw?giU;^bu‰UFx£`ih™—9U0˜š]Œ•–n9g…`iQeUh™cK`bU;ga¥C•Ãrpx0hšrp˜uegiU;^i^bdegbUrvcSw?—pU0˜šn!xh™`q]p¨ #MU0`œd‰^œweUSceU

Yk = (αkρk)/ρ `iQeU,Trp^i^5•–gyrpx`bhšnpc"nv•`iQeU,ueQ‰rp^bU

k ^›5QeU0giU ρ=P²

k=1αkρk

hš^t`bQeUˆTh¢ _`bdSgbU,weU0cS^bh¢`q]9¨

u^© hš^OQeU0giU,xyQenK^aU;c¸rp^

`iQeU$xU;c9`iU0g5np•Trp^i^(—pU;˜™n!x0h¢`q]

u^© =

2

X

k=1

Yku_k ^DF}

¾n9cS^ahkw!U;girv`bhšnpcS^Onpc?`iQeUU0cK`bginpu_]"w!hš^i^bh™uSrv`bhšnpc?np•m`bQSUˆTn!w!U0˜ –`bn¡‰U„phš—pU;c?˜šrv`bU;g…hšc?`iQehš^^bU;x`bhšnpc …^aQSn‘› `bQSrv`œ›5h™`bQ `bQehk^

xyQen9hšx0Unv•`bQeUWhšcK`bU0gb¥C•Ãrpxhkrv˜|—9U0˜šn_x0h¢`q]9¬_`bQeUhšcK`bU0gb¥C•Ãrpxhkrv˜ÕuSgbUF^b^bdegiU…TˆdS^q`tQ‰r—pU…`bQeU•–n9˜™˜šn‘›5h™cS„ˆ—‘rv˜šdeU

p^© =

2

X

k=1

Ykp¯ ^k

D‘

›5QeU;gbU

¯k= (2,1) ^›5QSU0c k= (1,2)¨5P(QeU,Tn!w!U;˜ Dtx0npcK`irph™c‰^5gbU;˜šrv er‘`bhšnpc"uSrpgirpTU`bU;gi^

λ^rvc‰w µ >0`bQ‰r‘`…w!U0`bU;gbThšceU

`iQeU…gyr‘`iU;^žr‘`ž›5QehšxyQŒ`bQeUœ—pU;˜™n!x0h¢`ih™UF^rvc‰w­uegiU;^i^adSgbUF^mnv•Õ`iQeU…`q›np¥CueQ‰rp^bU;^ågiU;r9xyQ­U;jKdehš˜™hš¡egih™dST¨©ªcTrpc_]^bh¢`idSr‘`ih™n9cS^ –¡ede¡S¡e˜™]

uehšu'U$§Sn‘›t^;¬S¡edS¡e¡e˜šUˆxn9˜™dSTc¬‰U0`ix¬‰`iQeU$uegiU;^i^bdegbUWUFjKdeh™˜šhš¡egbhšdeT2`bhšTU,hk^tU0 K`igbU;TU0˜š]^bTrv˜š˜›5Qehš˜™U$`bQeU,`bhšTUˆceU;x0U;^i^brpgb]­`bn

giU;r9xyQ—9U0˜šn!xh™`q] U;jKdSrp˜™h™`q] hk^$TˆdSxyQ.˜krvgi„pU0gF¨ŒP(Qehk^$˜šU;r9we^W`in?xn9cS^ahkw!U;gW`bQeUx˜krp^i^ahkx0rp˜å`q›np¥C—9U0˜šn!xh™`q]p¬Mn9ceU­uegiU;^i^adSgbUTn!w!U0˜k^

€‰¬ D;}e¬ DF† ›5QeU0giU?U;jKdSrv`bhšnpc D _¨hk^­giU0uS˜šr9xU;w ¡_]±`bQeUrp^i^adeTu!`ih™n9c¯nv•WuegiU;^i^adegiU¸U;jKdehš˜šh™¡egihšdeT

p1(ρ1) = p2(ρ2)^¨

´On‘›U;—pU0gF¬`bQehk^,˜krp^a`ˆTn!w!U;˜•Ãrph™˜k^W`in ¡'UŒQ_]_u‰U;gb¡'np˜šhšxŒrvcSw.hšc.`iQeU^bU;jKdeU;˜B¬›žUŒuegiU•–U;g$`bnw!U0gihš—pU B},weh™giU;x`b˜š] •–ginpT D

hšcS^a`bU;r9wnv•r9^b^bdeThšce„rˆuSgbhšnpgih|uegbUF^b^bdegiUU;jKdSrv˜šh™`q]¡‰U0`q›U;U0c`bQeU`q›žnueQSr9^aUF^0¨

F K8K8 e5<º <; · e L¦¼  ¼ML Ž A · ¼ ½ ·

©ªc¸`iQeUˆ^bU;jKdeU;˜B¬‰`bQSU,Tr‘`iU0gihšrp˜w!U;gbhš—‘r‘`bhš—pU,nv•r­jKdSrpc9`ih¢`q]

φ <sup>›5h¢`iQ</sup> giU;^bu‰UFx£`t`in­`iQeU,—9U0˜šn_x0h¢`ih™UF^

u_k

nv•mU;r9xyQ¸ueQSr9^aU

k r9^t›žU0˜š˜

r9^ž`bQeUWhšcK`bU0gb¥C•Ãrpxhkrv˜|—9U0˜šn_x0h¢`q]

u^©

›5hš˜™˜¡‰U$w!U;cenv`iU;w¡_]

Dkφ Dt = ∂φ

∂t +u_k.∇φ ^•–npg k= 1,2 ^rvc‰w k=^©

DE

X…^ahšce„`bQShš^5cSnv`irv`bhšnpcM¬_`iQeU$TnpTU;c9`ideT UFjKdSr‘`ih™n9cS^ D e¨~¦¥ DK¨(rvcSw`bQSUWTr9^b^5x0npcS^bU0gi—‘r‘`ih™n9cŒU;jKdSrv`bhšnpcS^ D _¨#D ª¥ D _¨€ ¬

h™`thš^tU;rp^bhš˜™]Œ^bU0U;cŒ`iQSr‘`5`iQeUW—pU;˜™n!x0h¢`ih™UF^

u_k

n9¡‰U;]­`iQeU•–np˜š˜™n‘›5hšce„U;jKdSrv`bhšnpcS^

α1ρ1

D1u₁

Dt +∇α1p1 = p^© ∇α1+λ(u₂−u₁) (19.1) α2ρ2

D2u₂

Dt +∇α2p2 = p^© ∇α2−λ(u₂−u₁) (19.2)

VOU _`;¬|dS^bh™cS„`bQeUˆ—pn9˜™deTUW•–gyrpx`bhšnpc¸UFj9d‰r‘`bhšnpc D e¨É£¬'›U,x;rvc¸giU0›5gih¢`iUW`bQSU,Trp^i^Ox0npcS^bU0gi—‘r‘`bhšnpc"UFj9d‰r‘`bhšnpc‰^ D_¨#D ¦¥ D_¨€

hšc`bU0giT np•ueQ‰rp^bUweU0cS^bh¢`ih™UF^

ρk

dec‰w!U0g5`iQeU•–npgiT

α1

D1ρ1

Dt +α1ρ1

w!h™—

u₁ = ρ1(u^© −u₁).∇α1+µρ1(p2−p1) (20.1)

α2

D2ρ2

Dt +α2ρ2

w!h™—

u₂ = ρ2(u^© −u₂).∇α2−µρ2(p2−p1) (20.2)

ShšcSrv˜š˜™]9¬!`bn„9U`5`iQeUWU;jKdSrv`bhšnpcS^ž•–n9g5`bQeUWuSgbUF^b^bdegiU;^

pk

nv•U;r9xyQuSQSrp^bUp¬!›žUW›5gbh™`bU`iQSr‘`O^bhšcSxU

pk=pk(ρk)^¬!`iQeU0c Dkpk

Dt = a²_kDkρk

Dt

rpcSw›UWn9¡!`irph™c

α1

D1p1

Dt +α1ρ1a²1

w!hš—

u₁=ρ1a²1(u^© −u₁).∇α1+µρ1a²1(p2−p1) (21.1)

α2

D2p2

Dt +α2ρ2a²2

w!hš—

u₂=ρ2a²2(u^© −u₂).∇α2−µρ2a²2(p2−p1) (21.2)

#U`,dS^,^adeTTrvgihš®0U`bQeUF^aUŒgbUF^ade˜™`i^;¨­©ªcµ`iU0giT nv•ž`bQSU­—‘rvgihšrp¡e˜™UF^

t(u_k, pk, α2)¬`bQeUŒhš^bnv`iQeU0giTrv˜åcen9c!¥CUFjKdeh™˜šhš¡egbhšdeT&Tn!w!U;˜

D (x;rvc¡‰UW›5gih™`a`bU;cdScSw!U0gt`bQeU$jKdSr9^ah™¥C˜šhšceU;rpg•–n9gbT

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 α1ρ1

D1u₁

Dt +∇α1p1 = p^© ∇α1+λ(u₂−u₁) (22.1) α2ρ2

D2u₂

Dt +∇α2p2 = p^© ∇α2−λ(u₂−u₁) (22.2) α1

D1p1

Dt +α1ρ1a²1

w!hš—

u₁ = ρ1a²1(u^© −u₁).∇α1+µρ1a²1(p2−p1) (22.3)

α2

D2p2

Dt +α2ρ2a²2

w!hš—

u₂ = ρ2a²2(u^© −u₂).∇α2−µρ2a²2(p2−p1) (22.4) D^©α2

Dt = µ(p2−p1) (22.5)

©ªc`bQSU$^aUFj9dSU0˜C¬!›U›5hš˜™˜weU0cenp`bU

Ck=ρka²_k `iQeUW¡ede˜š¤­Tn!w!de˜šdS^;¨

F K8>

<B0Ž · ; ’ ·¸¼ML  ; · ;

Ž; ¼ Z ¹

 ¼ Z ¼ <–Ž<¼ ; F K8

%

η(U), G(U) ^{! 4R&% . 4'!"}

η=α1ρ1(u²₁

2 +f1(ρ1)) +α2ρ2(u²₂

2 +f2(ρ2)) G(U) =u₁(α1ρ1(u²₁

2 +f1(ρ1)) +α1p1) +u₂(α2ρ2(u²₂

2 +f2(ρ2)) +α2p2)