HAL Id: hal-00691603 https://hal-enpc.archives-ouvertes.fr/hal-00691603 Submitted on 26 Apr 2012 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Unified semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method Martin Ferrand, Dominique Laurence, Benedict Rogers, Damien Violeau, Christophe Kassiotis To cite this version: Martin Ferrand, Dominique Laurence, Benedict Rogers, Damien Violeau, Christophe Kassiotis. Uni- fied semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method. International Journal for Numerical Methods in Fluids, Wiley, 2013, 71 (476-472), pp.Online. 10.1002/fld.3666. hal-00691603
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HAL Id: hal-00691603https://hal-enpc.archives-ouvertes.fr/hal-00691603
Submitted on 26 Apr 2012
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Unified semi-analytical wall boundary conditions forinviscid, laminar or turbulent flows in the meshless SPH
To cite this version:Martin Ferrand, Dominique Laurence, Benedict Rogers, Damien Violeau, Christophe Kassiotis. Uni-fied semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshlessSPH method. International Journal for Numerical Methods in Fluids, Wiley, 2013, 71 (476-472),pp.Online. 10.1002/fld.3666. hal-00691603
rt♥ ♦♥r② ♦♥t♦♥s s ♦♥ ♦ t ♠♦st ♥♥ ♣rts ♦ t♠♦♦t Prt ②r♦②♥♠s P ♠t♦ ♥ ♠♥② r♥t ♣♣r♦s ♥ r♥t② ♦♣ s ❬ ❪ rt ♦♥r② ♦♥t♦♥s r ss♥t s♥ ♥ ♠♥② ♣♣t♦♥s r ♣rs ♦♥ ♦♥ ss rqr s s ♦rs ♦♥ ♦t♥ ♦s ♦r s♦r♥ strtrs t♥ s♥ ①♥s strtr ♥trt♦♥s ♥ ♣♦r♣♥ts t t s ♥♦♦s ♣rrqst t♦ ♠♣r♦ tr♥ ♠♦♥ ♥r s
♥② ♠t♦s ♦r ♠♣♠♥t♥ s♦ s ♥ P ♥ ♦♣♦r t ♣st t♦ s ♠♦♥ t ♠♦st ♣♦♣r ♥ ♦♠♠♦♥② s ♥tr r♦ t♦rs
♣s ♦rs s s ♥♥r♦♥s ♣♦t♥t ❬❪
tt♦s ♣rts ♦r tr♥t② ♦st ♣rts t ♠♣t②r ♦ t r♥ s♣♣♦rt ♥ ♦♥r② t rt ♣rts t♣rsr ♣②s q♥tts s s ♣rssr ♥ ♦t② t♦ ♥♦r♥♦ s♣ ♦r r s♣ ♦♥t♦♥
♠♥②t ♦♥r② ♦♥t♦♥s s ♦♥ rt♦♥ ♦r♠t♦♥♥tr♦ ② sr♠ t ❬❪ r r♥♦r♠③t♦♥ ♦ tqt♦♥s s ♠ t rs♣t t♦ t ♠ss♥ r ♦ t r♥ s♣♣♦rt❲ sr r♥t ♦ ts ♠t♦ ♥ ➓ r ♥tr♥s r♥t
♥ r♥ ♦♣rt♦rs r ♠♣♦② tt ♥sr ♦♥srt♦♥ ♣r♦♣rts
♦ ts ♠t♦s ♥ts ♥ rs ♥♥r♦♥s♣♦t♥t ♦r ♦r♥② srs t ♥trt♦♥ t♥ ♣rs ♦ t♦♠s ♥s s t♦ ♠♦ t r♣s♦♥ t♥ ♣rt ♥ ♦♥r② ♣rts ♠t♦ s s② t♦ ♠♣♠♥t ♥ ♦r ♦♠♣① ♦♠trs ♥ ♦♠♣tt♦♥② ♣ ♦r t s t♦ s♣r♦s ♦r ♦r ♥st♥ t s
♠♣♦ss t♦ ♠♥t♥ ♣rts ① ♦♥ rt ♥ t ♣rs♥ ♦rt②
tt♦s ♣rts r♦♠♠♥ ♥ ❬❪ ♣r♥ts s ♥♦♥♣②s♦r ♦r t ♣♦st♦♥♥ ♦ ♦st ♣rts ♥ ♦♠♣① ♦♠trs♥ ♣rtr② ♥② ♣rtr② ♥ 3D ♦r♦r t ♦♠♣tt♦♥♦rt rqr s ♥♦t ♥ ♥ tt ♥rs t ♥♠r ♦ ♣rtst♦ t ♥t♦ ♦♥t ♥ t srt s♠♠t♦♥s ♥
♥② t s♠♥②t ♣♣r♦ s ttrt t♥s t♦ ts rt♦♥rt♦♥ ♠♥s tt s♦♠ ♣②s q♥tts s s ♠♦♠♥t♠ t♦♠t② ♦♥sr ❯♥♦rt♥t② t ♦r♥ tt♠♣t ♥♦t ♣rs♥t r ♥ s♠♣ ② t♦ ♦♠♣t r♥♦r♠③t♦♥ tr♠s ♥tr♦ rtr♠♦r t ♦r♠t♦♥ ♣r♦♣♦s s ♥♦t t♦ r♣r♦ ②r♦stt ♣rssrs ♦r t♦ t ♥t♦ ♦♥t t sr strss ♦♥
s ♣♣r tr♦r ts t s♠♥t② ♣♣r♦ ♦ ❬❪ ♥ ①t♥st s♦ tt t r② ♦ t ♣②s s s t ♣rssr ♥①t t♦ ss ♦♥sr② ♠♣r♦ ♥ t ♦♥sst♥t ♠♥♥r ♦♣ ♦r ♦rrt♦♣rt♦rs ♦s s t♦ ♣r♦r♠ s♠t♦♥s t tr♥ ♠♦s s ♦r ♣rs♥t tr ② ♥s
• t♠ ♥trt♦♥ s♠ s ♦r t ♦♥t♥t② qt♦♥ rqrs ♣rtr tt♥t♦♥ ♥ s r② ♠♥t♦♥ ② ❱ ❬❪ ♣r♦ tr s♥♦ ♣♦♥t ♥ s♥ ♣♥♥ ♥ t♠ ♦ t ♣rts ♥st② ♥♦ r♥r♥t ♦rrt♦♥s r s ② s♥ ♥r♦♥r② r♥♦rrt rs♦♥ ♦ t t♠ ♥trt♦♥ s♠ ♦ t ♦r♠ ♣r♦♣♦s ♥❬❪ ♦♥t♠ s♠t♦♥s ② st ♦r tr♥t ♦s ♥ t ♦♥t①t♦ rt ♦♥r② ♦♥t♦♥s r ♣♦ss
• ♦ ♦♠♣t t r♥ ♦rrt♦♥ ♠♥ ♥ ♦♥t ❬❪ s ♥ ♥②t s ♦♠♣tt♦♥② ①♣♥s rs sr♠ t
❬❪ ♥ t ❬❪ s ♣♦②♥♦♠ ♣♣r♦①♠t♦♥ ♥ t t♦ ♥ ♦r ♦♠♣① ♦♠trs ❲ ♣r♦♣♦s r t♦ ♦♠♣tt r♥♦r♠st♦♥ tr♠ ♦ t r♥ s♣♣♦rt ♥r s♦ t ♥♦t♠ ♥trt♦♥ s♠ ♦♥ s ♥② s♣ ♦r t ♦♥r②
• ♦♥r② tr♠s ss r♦♠ t ♦♥t♥♦s ♣♣r♦①♠t♦♥ r ♥② sr s♠♠t♦♥s ♦♥② rqr ♥♦r♠t♦♥ r♦♠ ♠s ♦ t ♦♥r② t♥q ♦♣ r ♦s s t♦ ♦rrt t♣rssr r♥t ♥ s♦s tr♠s ♥ ♥ ♣r♦ ♣②s② ♦rrtsr strss s♦ tt ♥ t s♦♥ qt♦♥ ♦ sr q♥tt②♥ s♦ rt② s♥ P s s t tr♥t ♥t ♥r②♦r ts ss♣t♦♥ ♥ k − ǫ ♠♦ ♦ tr♥
s ♣♣r s ♦r♥s s ♦♦s ♥ t ♥①t t♦♥ ♥tr♦ t ♦r♠t♦♥s ♦r ② ♦♠♣rss P ♥tr♦♥ t s srtst♦♥s ♦r♦♣rt♦rs s♦s ♦rs ♥ tr♥ ♠♦♥ ♥ t t♦♥ ♦♦♥ t♥ ♦♣ t ♦♥sst♥t ♦♥r② ♦♥t♦♥s t ♠♣r♦ t♠ ♥trt♦♥ ♥ ♣♦st♦♥♣♥♥t t♥q t♦ ♦♠♣t t ♥st② ♣♣rt♥ ♣rs♥ts t ♦♠♣tt♦♥ ♦ t r♥♦r♠③t♦♥ tr♠s s♥ ♥♦ t♠♥trt♦♥ s♠ ♦r ♣rs♥t♥ t ♥♥ ♦r ♥ t t♦♥ ♦♥♥♠r rsts
s P ♦r♠t♦♥s ♦r ② ♦♠♣rss
t♦♥♥
♦♥srt ♦r♥♥ qt♦♥s
st② ♦♠♣rss t♦♥♥ s ♠♦ ② st ♦ ♣rts ♥♦t ② t ssr♣ts (.)a ♥ (.)b ♥ ♦♠♥ Ω st ♦ t ♣rts s ♥♦t ② F r ♣rt a ∈ F ♣♦sssss ♥♦r♠t♦♥ ss ts ♠ss ma ss♠ ♦♥st♥t ts ♣♦st♦♥ ra ts ♦t② ua t r♥♥ rt ♦ t ♣♦st♦♥ ts ♥st② ρa ts ♦♠ Va = ma
ρa
♥ ts♣rssr pa s♣t srt③t♦♥ s s ♦♥ t♥ ♥tr♣♦t♦♥ ♦rr♥ ♥t♦♥ w, t ♦♠♣t s♣♣♦rt Ωa t♥ rrs t♦ t s♣♣♦rt ♦ tr♥ ♥t♦♥ ♥tr ♦r ra ♦ rs R ❲ ♥r② ♥♦t ② t ssr♣ts(.)ab t r♥ ♦ q♥tt② t♥ t ♣♦st♦♥s a ♥ b ♦r ♥st♥uab ≡ ua − ub ♥ rab ≡ ra − rb ♦ ①♣t♦♥s r ♠ t t ♦♦♥♥♦tt♦♥s wab ≡ w (rab) ♥ ∇wab ≡ ∇aw (rab) r t s②♠♦ ∇a ♥♦tst r♥t t t ♣♦♥t ra
❲t ts ♥♦tt♦♥s ♦♠♠♦♥② s ♦r♠ ♦t ♦♥t♥t② qt♦♥ s s ❬❪
dρa
dt=∑
b∈Fmb∇wab.uab
r ddt
♥♦ts t r♥♥ rt tt s t♦ s② t rt ♦♥t ♣rt ♣t t ♥ r r♦♠ t ♦♦♥ ♥t♦♥ ♦ t ♥st②
ρa =∑
b∈Fmbwab
♥s ♠♦♠♥t♠ qt♦♥ ♥ rtt♥ s ♦♦s
dua
dt= −
∑
b∈Fmb
(pa
ρ2a
+pb
ρ2b
)∇wab + g
r g s rt② qt♦♥ ♦ stt ♥s ♥st② ♥ ♣rssr
pa =ρ0c
20
γ
[(ρa
ρ0
)γ
− 1
]
r ρ0 s t rr♥ ♥st② ♦ t c0 s t s♣ ♦ s♦♥ ♥ γ = 7s ♥r② ♦s♥ ♦r tr
❱s♦s ♦rs
s♦s tr♠ s s ♥ ts ♦r ♥ P ♦r ♦♠♣t♥ t s♦s tr♠1
ρ∇. (µ∇u) s t♦ ♦rrs t s ❬❪
1
ρa
∇. (µ∇u)a =∑
b∈Fmb
µa + µb
ρaρb
uab
r2ab
rab.∇wab
t r♥t ♦♣rt♦r s ♥ ② ∇a ≡ ex∂
∂xa
+ ey∂
∂ya
+ ez∂
∂za
(ex, ey , ez) ♥
t ss t♦r tr ♦ t rts♥ ♦♦r♥t s②st♠ ♥ 3D
r t ②♥♠ s♦st② µ s ♥ ②
µ ≡ νρ
♥ ν s t ♥♠t ♠♦r s♦st②
♣rt♦r ♥t♦♥s ♦r tr ♥②ss ♥ srt ♦♣rt♦rs r♥t GradaAb r♥ DivaAb ♥ ♣♥ Lapa (Bb, Ab)♦ rtrr② srt sr Ab ♥ Bb ♦r t♦r s Ab s
GradaAb ≡ ρa
∑
b∈Fmb
(Aa
ρ2a
+Ab
ρ2b
)∇wab
DivaAb ≡ − 1
ρa
∑
b∈FmbAab.∇wab
Lapa (Bb, Ab) ≡ ρa
∑
b∈Fmb
Ba + Bb
ρaρb
Aab
r2ab
rab.∇wab
r Aab ≡ Aa − Ab ② r ♣♣r♦①♠t♦♥s ♦ t ♦♥t♥♦s r♥tr♥ ♥ ♣♥ ♦♣rt♦rs rs♣t② ♥♦t ② ∇a ∇. ♥ ∇.∇s t qt♦♥ ♦ ♦♥t♥t② ♥ t ♠♦♠♥t♠ qt♦♥ ♥ rrtt♥ s ♦♦s
dρa
dt= −ρaDivaub
dua
dt= − 1
ρa
Gradapb + g +1
ρa
Lapa (µb, ub)
♦♣rt♦rs Grada ♥ Diva r s t♦ s♦♥t t s ♣♦ss t♦♥ r♥ts ♦ ts tr ♦♣rt♦rs ♦♥sr♥ ts ♣r♦♣rt② ♦ ♥t♦♥s ❬❪
r♥ ♠♦♥ ♥ P
②♥♦s ♣♣r♦ ♦♥ssts ♥ ♦♥sr♥ ♦♥② t ♠♥ ♣rt ♥♦t ②u ♦ t ♦t② u ♥ t qt♦♥ ♦ r t♦s t♥ ♠♦♥ tts ♦ t tt♥ ♣rt ♦ t ♦t② ♦♥ t ♠♥ ♦t②
k− ǫ tr♥ ♠♦ ♦♣ ② ♥r t ❬❪ t♦ tr♥s♣♦rt qt♦♥s ♦ k t tr♥t ♥t ♥r② ♥ ǫ ts ss♣t♦♥ t♦ t♠♦♠♠t♠ qt♦♥ s ♠♦ s ♦♦s
Dρa
Dt= −ρaDivaub
Dua
Dt=
1
ρa
Gradapb + 23ρkb + Lapa (µb + µTb, ub) + g
Dka
Dt=
1
ρa
Lapa
(µb +
µTb
σk
, kb)
+ Pa − ǫa
Dǫa
Dt=
1
ρa
Lapa
(µb +
µTb
σǫ
, ǫb)
+ǫa
ka
(Cǫ1Pa − Cǫ2ǫa)
♥ t st♦st ♣♦♥t ♦
r t rtD
Dt≡ ∂
∂t+ u.∇ s t r♥♥ rt ♦♥ t
②♥♦s r ub k − ǫ ♠♦ ♥s t tr♥t ♥ts♦st② νT ≡ µT
ρt♦ t tr♥t ♥t ♥r② k ♥ ts ss♣t♦♥ ǫ ②
νTa = Cµ
k2a
ǫa
t ♦♥st♥ts σk σǫ Cǫ1 ♥ Cǫ2 ♥ ② ❬❪
♣r♦t♦♥ tr♠ ♦ k Pa s ♥ ②
Pa = νTaS2a
r S2a ≡ 2Sa : Sa s t sr ♠♥ rt♦str♥ t♥s♦r str♥ rt
♦ t ♠♥ ♦t② s ♥ ② Sa ≡ 12
(∇au + ∇au
T) ❱♦ ♥ ss
❬❪ srt③ ♥ t P ♦r♠ t ♦t② r♥t
Gradaub ≡ − 1
ρa
∑
b∈Fmbuab ⊗ ∇wab
♥ t ♦♦♥ ♦r s ♦ s♠♣t② r♦♣ ♦rrs t♦ ♥♦t
♦t② ♥ ♣rssr ♥ ts ♥♦t r♥♥ rtsd
dt rr
s♦ ♣ ♥ ♠♥ tt ♥r tr♥t ♦♥t♦♥s t ttr q♥tts r♦♥sr s ②♥♦sr ♥ t r♥♥ rt s ♦♥ t②♥♦s r ♦t②
♦♥r② ♦♥t♦♥s ♥ rt t♠
st♣♣♥
rt♦♥ ♦ ♦♥r② tr♠s s♥ ♦♥t♥♦s
♥tr♣♦t♦♥
sr♠ t s r♥♦r♠st♦♥
♥st ♦ ss♠♥ tt ρa ≃∑
b∈Fmbwab ♥rst♠ts ρa ♥ t
♣rt a s ♦s t♦ ♦♥r② s r sr♠ t ❬❪ r♥♦r♠s t st♠t♦♥ s♥ ♥t♦♥ γa
ρa ≃ 1
γa
∑
b∈Fmbwab
r γa s ♥ ②
γa ≡∫
Ω∩Ωa
w (r′ − ra) dV ′
♥ ♥r γa s ♥ r♥ ♣♥♥ ♦♥② ♦♥ t ♣♦st♦♥ ♦ t♣rt a t rs♣t t♦ ♦♥rs ♦ Ω r r♦♠ s♦ ♦♥r② γa = 1.
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b ab R
Ω
Ωa
γa
∂Ω
r r♥♦♥r② ♥trt♦♥
♥tr♦t♦♥ ♦ γa ♥t♦ t rt♦♥ ♦ t ♦r♥♥ qt♦♥s st♦ s♦♠ ② r♥s s s t ♥ ♦♥t♥t② qt♦♥ ❬❪ r strt♥r♦♠ s rrtt♥
dρa
dt=
1
γa
∑
b∈Fmb∇wab.uab −
ρa
γa
∇γa.ua
s ♥ ♦♠♣r t ♦r♥② t r♥t ∇γa s ♥ ②
∇γa ≡∫
Ω∩Ωa
∇aw (r′ − ra) dV ′ =
∫
∂Ω∩Ωa
w (r′ − ra)ndS′
r t s♦♥ ♥tr s ♦t♥ s♥ t ss t♦r♠ ♥ r n st ♥r ♦♥r② ♥♦r♠
♥ ♦rr t♦ t t ♥ ♥tr♥ ♦rs ♥ ♦♥tt ♦rs sr♠t ❬❪ r t ♥tr♥ ♥r② s♥ t qt♦♥ ♦ r♥ ② ♦t♥ ♥ ♥tr♥ ♦r t♦ t ♣rssr ♦rrt♦♥ ♦ sr♠ t
s t ♦♣rt♦rs r② t s♦♥t♦♥ ♣r♦♣rt② ♥ rt♦♥s st ♦s ts ♣r♦♥ t ♦♥sst♥② t♥ t ♠♦♠♥t♠ qt♦♥ ♥ t ♦♥t♥t② qt♦♥ ♥ tr♠ ♦ GradK
a Ab s s② sr♠ t s ♦♥r② ♦r ♥ t ♦♦♥ ♣r♦♣♦s ♥♥♥ ♦r♠ ♦ t ttr ♠♦ ♥ ♠♦r rt r♣rs♥tt♦♥ ♦r♥ts ♦♥ t
♥r s♣ ♦ t ♦♥r②
♦r ♦♥rs ♦ rtrr② s♣ tr s♣ ∂Ω ♦ t ♦♠♥ Ω, s ♣♣r♦①♠t t strt s♠♥ts ♥ 2D ♥♦t ② t ssr♣t (.)s
b aR
e1
e2Ss
ns
♦♥r② t s♠♣rts e ♥ r♥ ♥ ts♠♥ts s sr Ss ♥ ♥ ♥r ♥♦r♠ns
e
θ
e e
t ♦ t ♦♠ ♦ ♥ ♣rt
r ♦♥r② ♣r♦♣rt② ♥t♦♥s
♥♦r♠ ns ♥ sr r Ss s r st ♦♥t♥♥ ts♠♥ts s ♥♦t ② S s♠♥t s ♥ ② t♦ ♣♦♥ts ♥♦t② t ssr♣t (.)e1 ♥ (.)e2 ♥ ♥t ♦♠ Ve ♥ ② Ve = me
ρ0
r ρ0 s rr♥ ♥st② ♥t ♦♠ ♦ ♣rts s rt♦♥♦ t ♥t ♦♠ ♦ ♣rts Vf ♦r ♥st♥ ♦r ♣♥ Ve = 1
2Vf
♦r ♥r② ♦r ♥ ♣rt ♦♥ t ♥ ♥ θ s♣② ♦♥t r Ve = θ
2πVf
st ♦♥t♥♥ t ♣rts s ♥♦t ② E s ♣rtss♦ s♠ ♣rts ♥ ts rt r ♦ ♣rtr ♥trst ♦r r♦r♥t ♣rssr t t s♦ ♦♥r② ♥ ♥ ♦r ♥ strtr ♦♣♥ ♦r ①♠♣ ② r s♦ s t♦ ♠♣r♦ r② ♦ t ♦♥t♥t②qt♦♥ s t② ♠♠ t t s ♠♣♦rt♥t t♦ ♥♦t tt t② rt♥ ♥t♦ ♦♥t ♥ t ♦♥t♥t② qt♦♥ ♥ ♥ t ♠♦♠♥t♠ qt♦♥ E ⊂ F ♥ t② r r♥ ♣rts tt s t♦ s② t② r ① t s ♠♦t♦♥ss ♥ ♦s ♥♦t ♣♥ ♦♥ t ♠♦♠♥t♠ qt♦♥
①t ♥ t ♦♥trt♦♥ ♦ t s♠♥t s ♥ t ♦ ∇γa t♦
∇γas ≡(∫
re2
re1
w (r) dl
)ns
♥ t♥ ∇γa ♥ ♦♠♣♦s ♥
∇γa =∑
s∈S∇γas
sr♣t♦♥ ♦ t ♦♥r② ♦♠tr② ♥ ①t♥ t♦ 3D ② ssttt♥ t s♠♥ts ② tr♥s ♥ ts s ts ♦♠ Ve ♦ ♣rts♦ s t s♦ ♥ ♦ t
❲ ♦rrt r♥ts
♠♥ s♥t ♦ ♣r♦s t♥qs t♦ t ∇γa s tt t r
♥t ♦♣rt♦r ♥ ② s ♥♦t rt ♥r ♦♥r② ♦r t rsr♥ ♥ s tt r♥ts ♦ ♦♥st♥ts r ♥♦♥ ③r♦ s♠t ♥♦r♠ ♦r♣rssr ♥ ♣r♦ ♣♣ t♦t ♥② ♦② ♦r t r♥t
♦ t ♣rssr s ♥♦t ③r♦ r②r ♥ ♣rts rrr♥ t♠ss sr ♦ ♦rrt tt ♥ ♦ t♦ t ♦♥t♥♦s ♥tr♣♦t♦♥♦ ♥ rtrr② ♥tr ♥t♦♥ f t ♣♦♥t r
〈f〉 (r) =1
γ (r)
∫
Ω∩Ωr
f (r′) w (r) dV ′
r r ≡ |r − r′| ♥ Ωr s t r♥ s♣♣♦rt ♥tr ♥ r ② ♥tr♣♦t♥t r♥t ♦ t ♥t♦♥ f ♥ t s♠ ② t ♦♠s
〈∇f〉 (r) = − 1
γ (r)
∫
Ω∩Ωr
f (r′) ∇w (r) dV ′
− 1
γ (r)
∫
∂Ω∩Ωr
f (r′) w (r)ndS′
r t rt♥s s ♦t♥ ② ♥ ♥trt♦♥ ② ♣rts ♥ n s t♥r ♥♦r♠ ♦ t ♦♠♥ t t ♣♦st♦♥ r′ ♥ ♥ s tt t♦♥r② ♦♥t♦♥s ♣♣r ♥tr② tr♦ t s♦♥ ♥tr ♦
♦r♦r ♦♥sr tt t r♥t ♥ ② s srt ♣♣r♦①♠t♦♥ ♦ t ♦♥t♥♦s r♥t ∇f ≡ ρ∇
fρ
+ fρ∇ρ t♦ ♦t♥ s②♠♠tr
♦r♠t♦♥ ♦t♥ s s♦ ❬❪
〈∇f〉 (r) ≃⟨
ρ(r)∇f
ρ+
f
ρ(r)∇ρ
⟩(r)
= − 1
γ (r)
∫
Ω∩Ωr
[f
ρ(r′) ρ (r) +
f
ρ(r) ρ (r′)
]∇w (r) dV ′
− 1
γ (r)
∫
∂Ω∩Ωr
[f
ρ(r′) ρ (r) +
f
ρ(r) ρ (r′)
]w (r)ndS′
♥ ♥ t ♦♥r② ♦♥t♦♥s ♣♣r ♥tr② ♥ ♥ ♥♦ stt ♦♦♥ srt ♦♣rt♦r r♥t ♦r ♥ rtrr② Ab s
GradaAb ≡ ρa
γa
∑
b∈Fmb
(Aa
ρ2a
+Ab
ρ2b
)∇wab −
ρa
γa
∑
s∈S
(Aa
ρ2a
+As
ρ2s
)ρs∇γas
♠t♦ t♦ ♦♠♣t ∇γas sss ♥ ➓ rs t ♦♠♣tt♦♥ ♦ ρs ♥ As ♥stt ♥ t ♣rr♣ ♦♥ ②♥♠ ♦♥r②♦♥t♦♥s ♦t tt t srt r♥t rs r♦♠ sr♠t s ♥ ② ♦♥② ♥①t t♦ ♦♥r② ♠♥s tt ♦♥srt♦♥♣r♦♣rts r st r r♦♠ t s
t s ♥♦ ♦rrt s♥ t s♠ t P r♥t ♦ t♦r ♣♣rs ♥ t ♦t② r♥t ♦r ♥st♥ ttr q♥tt②♣②s ② r♦ ♥ t k − ǫ tr♥ ♠♦ s t s rs♣♦♥s ♦r t ♣r♦t♦♥ ♦ ♥t ♥r② ♥ t t tt ♥ ♥♥ ♦ t str♥ rts t rst ♥ t ♥t② ♦ t ♦♥r② t s ♠♣♦rt♥t t♦ rt ♥ts r s ♦r♠ ♥s t♦ ♦rrt t rs♣t t♦ t ♦♥rs♥ tt t t♥s t♦ ♥rst♠t t str♥ rt ♥①t t♦ ♦ ♦rrttt ♣r♦♣♦s ♥ s♠r ② s
s ♥ ➓ t ♦♥r② tr♠s ♣♣r ♥tr② ♥ t s♦♥ ♥ ♦ r♦♠ ♥ ♥trt♦♥ ② ♣rts ♥ n s t ♥r ♥♦r♠ ♦ t ♦♠♥t t ♣♦st♦♥ r′ ❲ ♦♥trt♦♥s ♥s ♥ t ♣♦st♦♥ r s r r♦♠t ♦♥r② s♥ t r♥ w s ♦♠♣t s♣♣♦rt s t ♣r♦♣♦s♦rrt ♣♥ ♦♣rt♦r s
1
ρa
Lapa (Bb, Ab) =1
γa
∑
b∈Fmb
Ba + Bb
ρaρb
Aab
r2ab
rab.∇wab−1
γaρa
∑
s∈S(Bs∇As + Ba∇Aa) .∇γas
s♠ s st ♦s t ♥ rtrr② t♦r Ab ♥ s s
t♦ ♦rrt t s♦♥ tr♠ ♥ t rt♦s qt♦♥
❲ ♦rrt ♦♣rt♦rs ♥ t srt rt♦s
qt♦♥s
②♥♠ ♦♥r② ♦♥t♦♥s ♦♥ t ♣rssr
r♥t ♦♣rt♦r ♣♣ t♦ t ♣rssr s
Gradapb ≡ ρa
γa
∑
b∈Fmb
(pa
ρ2a
+pb
ρ2b
)∇wab −
ρa
γa
∑
s∈S
(pa
ρ2a
+ps
ρ2s
)ρs∇γas
♥ r t t♦ ♦♠♣t t ♣rssr ps ♥ t ♥st② ρs ♦♥ t ♥♦s ♥ ♥ r r♦st ② t rst ♦rr ②♥♠ ♦♥t♦♥ s∂ρ∂n
= 0 s ♦♥t♦♥ s ♦♥sst♥t t t r♥♦r♠③t♦♥ ♦♥ ♥ qt♦♥ ♥ t ♣rs♥ ♦ rt② ♥ ♠♦t♦♥ t ♥♦♦s ♦♥t♦♥ ♦♥ t♣rssr s rtt♥ s
∂
∂n
(p⋆
ρ+
u2
2
)= 0
r p⋆ ≡ p − ρg.r ♥ u s t ♠♥t ♦ t ♦t②
♦ ♦♠♣t t ♣rssr ♥ t ♥st② t t ♥ P ♥tr♣♦t♦♥♥ s ♦r t ♣rts ♥ E t ♦ ♠ s t♦ r ♥ s♣t♦
ρe =1
αe
∑
b∈F\EVbρbwbe
pe
ρe
=1
αe
∑
b∈F\EVb
(pb
ρb
− g.rbe +u2
b − u2e
2
)wbe
r t st F \ E ♥♦ts ♣rts F ①♥ ♣rts ♥ E ♥r αe s ♥ ②
αe ≡∑
b∈F\EVbwbe
sr♣t♦♥ ♦ t ♣r tr αe s ♥ ♥ ➓ s qt♦♥
r t s ♠♣♦rt♥t t♦ ♥♦t tt t ♥tr♣♦t♦♥ ♦♥ s s ♦♥ ♣rts ♥ F ♦ ♥♦t ♦♥ t♦ t st ♦ ♣rts E ♦r s♠♣t② r♠♥ ♥ 2D ♥ ♥ t ♥st② ♥ t ♣rssr t t ♠♥ts st♦
ρs =ρe1 + ρe2
2ps
ρs
=pe1/ρe1 + pe2/ρe2
2
r t ♥♦s ♦r ♣rts e1 ♥ e2 r ♥ ♥ ➓ sstrt② t♦ t q♥tts t t rs t♦ ♦♠♣t t sr strss ♥ ➓ ♥ s♦ t ♦ sr tr♥s♣♦rt ② t ♦♥ ♣♣♥① r s ♥♦ t♦rt rstrt♦♥ t♦ t s♣♥ ♦ t ♥♦s
❲ sr strss
♦rrt ♦r♠ ♣♣ t♦ t u s
1
ρa
Lapa (µb, ub) =1
γa
∑
b∈Fmb
µa + µb
ρaρb
uab
r2ab
rab.∇wab−1
γaρa
∑
s∈S|∇γas| (µa∇ua + µs∇us) .ns
♥ s♦ s ♥ ♣ ♦ Lapa ♥
♦♥r② tr♠s r t♥ trt s♥ t rt♦♥ ♦t② uτ ♥②
µ∂u
∂n
∣∣∣∣wall
≡ ρuτuτ
r♣rs♥ts t sr strss t t ♥ t ♣rs♥t rt ② ♦♥♥t♦♥ uτ s ♦s♥ t♦ t s♠ rt♦♥ s t ♦ ♦t② s t♥r♣s (µ∇u)s .ns ♥ t ♦♥r② tr♠ ♦ s ♥ ss ♥t ♠♥t♦r ♥t ♦♠ rt♦s ♦s
(µ∇u)a .ns ≃ ρuτsuτs
♦♠♣tt♦♥ ♦ t rt♦♥ ♦t② ♥ ♠♥r s rt♦♥♦t② uτ s q♥tt② ♥ t t ♦♥r② ♦ ♦♠♣t t ♥ ♦♠♣tt♦♥ ②♥♠ ♦ ♥st ♦ s♥ ts ♥t♦♥ s②t t ♥t ♦ ♥♦♥ t ♣②s ♦r ♦ t ♦t② ♥ t♥t② ♦ t ♦♥r② ♦r ①♠♣ ♥ ♠♥r tst s t ♦t② ♣r♦ s ①♣t t♦ ♥r ♦s t♦ t ♥ t♥ t ♦♦♥ rt♦♥s♣t♥ st♥ t♦ t z ♥ ♦t② ♦♥ t u ♦s
uτuτ = limz→0
νu
z
♠♥ ♥t s tt ♦ ♥♦t ♥ t♦ st♠t t rt ♦t ♦t② ♥①t t♦ t r t s t t♦ ♦♠♣t ♥♦tr♥t s tt ♥ ①t♥ t ♥t♦♥ ♦ t rt♦♥ ♦t② ♥ t rr ♣rts ♥trr t t ♦♥r② tt s ♥ t r♥ s♣♣♦rt♥trsts t s ♥ ♥
uτauτa =νua
za
r za s t st♥ t♦ t ♦r ♣rt a
♥t② t♦ t ρsuτsuτs s ♥ t ♦♥t♥t② ♦ strsss t♦ sts
ρeuτeuτe =1
αe
∑
b∈F\EVbρbuτbuτbwbe
♥ rhosuτsuτs s t r t♥ t ♣rts e1 ♥ e2 ♥♥ ➓ s ♦r♠ r s♠r t♦
s♠ trt♠♥t s ①t♥ t♦ tr♥s♣♦rt qt♦♥ ♦ sr ss k ♦r ǫ ♥ t k − ǫ tr♥ ♠♦ ♥ t ♣♣♥①
♦♥srt♦♥ sss t♠ ♥trt♦♥ ♦r t ♦♥t♥
t② qt♦♥
♦r♥ t♠ ♥trt♦♥ s♠ s ♥ ♣r♦s ♦r ❬❪ ❬❪ s s♠♣rst♦rr s②♠♣t s♠ r ♥ ♠♣t ♦t② s s ♥ t♣t s ♦ t ♣♦st♦♥ ♥ ♥st②
un+1a = un
a − δt
ρna
Gradnapn
b + g
rn+1a = rn
a + δtun+1a
ρn+1a = ρn
a + δt∑
b∈Fmb∇
nwab.un+1ab
r t s♣rsr♣t (.)n rrs t♦ t t♠ st♣ n ♥ t♦ t t♠ t =
n∑
i=1
δt
♥ ts s♠♠♣t s♠ t ♦ts r ①♣t rs t ♣♦st♦♥sr ♠♣t ♥ t ♦♥t♥t② qt♦♥ ♣♦st♦♥s r ①♣t rs t ♦ts r ♠♣t ♦r ts rs♦♥ ♦ ♥♦t rt t rs ♦ qt♦♥ s ρaDiv ub
♠♣r♦♥ t t♠ ♥trt♦♥ ♦ t ♦♥t♥t② q
t♦♥
♦ ♣t t ♣r♦s t♠ ♥trt♦♥ s♠ t♦ t ♠t♦ ♦ sr♠t ♥ t ♣rs♥t ♠♦ ♦♥ t ♦♦♥ s♠ s ♣♦ss
un+1a = un
a − δt
ρna
Gradn
apnb + g
rn+1a = rn
a + δtun+1a
ρn+1a = ρn
a +δt
γna
[∑
b∈Fmb∇
nwab.un+1ab − ρn
a∇nγa.un+1
a
]
r t ♦♣rt♦r Grada s tr ♦r ①♣r♥ ♦ t t♦rss s♦♥ ts ♣♣r♦ s♠s t♦ stst♦r② rsts ♦r ♠ r s♥ s♥ r t s♦ s r ♣t ♠♣r♠ t rt② s♠t♠ st♣ ♦r ♥ r♥♥♥ ♦♥t♠ s♠t♦♥s ♥ ♥♥ t rt② r t♠ st♣ ♣rts ♥r t ♠♦ rt② ♦♥rss♦② ♥ ♥t② ♣ss tr♦ t ♦♥r② s s ♥ r rt② t♠ st♣ s st ② rs♥ t ♥♠r s♣ ♦ s♦♥ c0
♣r♦♠ s s ② t ♦♥t♥t② qt♦♥ ♥ ♣rts ♥r t♦♥r② r ♦st♥ ♠♦♥ ♥ ♦rt tr ♥sts rs♥ t♥ t ♣rssr rt t♦ t ♥st② ② t qt♦♥ ♦ stt ♦♠s♥s♥t t♦ rt r♣s ♦r t♦ ♥ t ♦tr ♦rs
♦r♥ ♦ ts ♣♥♦♠♥♦♥ s t tr♠ δtγn
a
ρna∇
nγa.un+1a ♥ t t♠
srt③ ♦♥t♥t② qt♦♥ ♥ ♦♥sr s♥ ♣rt♠♦♥ t♦rs t t♥ t t♠s tn ♥ tn+1 r♦♠ t st♥ zn
t♦ zn+1 t ①t rt♦♥ ♦ t ♥st② s ♥♦t r♣r♦ ② t srt
X(m)
Z(m
)
0 0.10
0.2
0.4
0.6
0.8
1
P(Pa)
100009000800070006000500040003000200010000
Pr♦s t♠s♠ tc0 = 20m.s−1
X(m)
Z(m
)
0 0.10
0.2
0.4
0.6
0.8
1
P(Pa)
100009000800070006000500040003000200010000
Pr♦s t♠s♠ tc0 = 100m.s−1
X(m)
Z(m
)
0 0.10
0.2
0.4
0.6
0.8
1
P(Pa)
100009000800070006000500040003000200010000
t♠s♠ tc0 = 20m.s−1
r ♦♠♣rs♦♥ ♦ t ♣rssr ♥ t tr ♣t ♥ ♣r♦♦♣♥ ♥♥ ♦♥ r♦♠ t t♦ rt ♦r t♦ r♥t t♠ s♠s tr ts♠ ♣②s t♠
♦r♠ ❬❪ s s②st♠t rr♦r ♦ t t♠ srt③t♦♥ ♦ t ♦♥t♥t②qt♦♥ s ♥sr
♥② ♦tr ♥trt♦♥ t♠ s♠s ♥ ♦♥sr s s ♣r♦t♠ s♠ ♦ r t rr♦rs ♥ t ♥trt♦♥ ♦ t ♦♥t♥t②qt♦♥ ♦ ♠ ♥ t ♣rs♥t ♦r s t♦ ♦♥sr ♥ ♣♣r♦ tts t ♥st② ①♣t② s ♥t♦♥ ♦ t ♣rts ♣♦st♦♥s s s ♠♦r r♦st ♣♣r♦ s♣② tr s r♣s ♦r ♥t♦♥ ♦ t♣rssr ♥ ♥ ♦ t ♥st② ♥ ♦♠♣rss ♦s
♦♠♣t② ♣♦st♦♥♣♥♥t ② t♦ ♦♠♣t t ♥st②
rtr♥ t♦ t ♠♥ ♦ ♦rrt♥ t ♥♦♠♣t r♥ s♣♣♦rt ♥ s tt t ♦rrt ♦♥t♥t② qt♦♥ ♦♠s r♦♠
t t s♠ t♠ ts ♣r♦♣rt② t♦tr t s♠♣t t♠st♣♣♥ ♥srst ♦♥srt♦♥ ♦ ♥ ♥r② s ❬❪ ♦r ts
s s t♦ t ♦♦♥ t♠ ♥trt♦♥ s♠
un+1a =un
a − δt
ρna
Gradn
apnb + g
rn+1a = rn
a + δtun+1a
(γaρa)n+1
=(γaρa)n
+∑
b∈Fmb
(wn+1
ab − wnab
)
r s♦s t rst ♦t♥ ② r♥ t t♠st♣ ♥ ♥t s②st♠t rr♦r ♥ t ♥st② qt♦♥ ♥ t t♠ s♠ t trst ♦t♥ t t t♠s♠ t rr t♠ st♣ ② ♦♥s♦ s♠ ♦ s ♦r t♦ st ♥ t♦ ♦♥r t t s r tt ♦st ♥st② r♥ t stst♦♥ t♠ t tr ♣t s rss rs t s ♥♦t t s ♦♥ t r t t ♥ s♠
♥t③t♦♥ ♦ t ♥st② t♠s♠ rqrs ♥ts ♦r t ♥st② ♥② ♦s r ♣♦ss rst ♦ ♣rt a♥ t rr♥ ♥st② ρ0 s ♥t
ρ0a = ρ0
s s ♦♥ ♣r♦s② t t ♦♥t♥t② qt♦♥ ♦ ♥ts tt t ♦♥t♥t② qt♦♥ ♦♥② ♠srs t rt♦♥ ♦ ♥st② ♥ ♥♦tt ♥t s♦rr ♦ t ♣rts ♦r t ♠♥ r s ♥ ♥♦♠♦♥t② t♥ ♣rts ♦ t♦ ♥♦♥♣②s ♦r s s ♣rts♦r♥② t t rsr rt♥ r♦♥ ♦ r♣s♦♥ ♥ srr♦♥ ②♦trs tr ♥ t s♠t♦♥ ♥ t ♥t ♥st②
ρ0
a
s ♥t③
♦r♥ t♦
ρ0a =
1
γ0a
∑
b∈F0
mbw0ab
s ♥t③t♦♥ s t ♥t t♦ ♠♥t♥ ♦♠♦♥t② t♥ ♣rtst t♥ rqrs rsr ♦rrt♦♥
rsr ♦rrt♦♥ γ ♦rrt♦♥ ♣rs♥t s♦ r ♦s ♥♦t t
♥t♦ ♦♥t ♥② rsr ♦rrt♦♥ r ρa ≡∑
b∈Fmbwab ♣♥s
♦♥② ♦♥ t ♣rts ♣♦st♦♥s s s ♦r ♦♥srt♦♥ ♣r♦♣rts ♣r♦♠ s tt ρa ♠srs t♦ r♥t q♥tts
t r♥ ♦ t ♣rts s q♥tt② ♦ ♥trst ♥
t ♣rs♥ ♦ ♦s t♥ t r♥ s♣♣♦rt ♦ ♣rt
❲ r t t♠ st♣ ② stt♥ t s♣ ♦ s♦♥ t 100m.s−1 ♥st ♦20m.s−1
t s rqr tt t ♦ s ♦rrt t γa ♥①t t♦ t ♥♦t ♥①tt♦ t rsr ♦ ts s t ♦♦♥ ♣r tr ♥②
α (r) =∑
b∈F
mb
ρb
w (r − rb)
s♦ tt ♦r ♣rt a ∈ F \ E
αa ≡∑
b∈F
mb
ρb
wab
♦r ♥ ♣rt e ∈ E ♥ ♦r t ♠ ♦ s♠♥t s ∈ S t♦ r♥t♥t♦♥s ♦ α r s
αe ≡∑
b∈F\E
mb
ρb
web
s♥ αe s s t♦ t q♥tts s s t ♥st② ♦r t ♣rssr tt ♥ ♦s♥ ♥♦t t♦ t ♥t♦ ♦♥t ♣rts ♥ ♥tr♣♦t♦♥ s♥ ♦♥② t ♣rts ♦ t ♣②s q♥tts s st ♥st② ρ ♦r t ♣rssr p t t s s ➓
♠ s t♦ ♣♣② t ♣r tr ♦♥ t ♥st② t t♠st♣ t ♦♥② ♥①t t♦ t rsr s tt t ♥st② s ♥♦t ♦rrtr②r t αa s ♥ t ♥t② ♦ rsr tr s ♦♥t♥♦s♠① t♦ ♦rrt t ♦♥t♥t② qt♦♥
ρa [βγa + (1 − β) αa] = ρa =∑
b∈Fmbwab
r
β = exp
[−K
(min
αa
γa
; 1
− 1
)2]
♥ K s t♥ t♦ ♥ rtrr② ♦ − ln(0.05)
0.012≃ 30000 s♦ tt
β ≤ 0.05 ♥ αa
γa
≤ 0.99 ♦t β s ♥ t sr♠rr ♥s t ts s ♠♦st ♦♥ rs t t♥s t♦ ③r♦ s ♣♣r♦ t rsr
♦♠♣tt♦♥ ♦ t r♥♦r♠③t♦♥ tr♠s
♦r♠ ♥t♦♥s ♦ t ♦♠tr q♥tts γa ♥ ∇γa ♦r ♣rta r
γa ≡∫
Ω∩Ωa
w (r − ra) dr
∇γa ≡∫
Ω∩Ωa
∇aw (r − ra) dr =
∫
∂Ω∩Ωa
w (r − ra)ndS
❲ r tt ∇γa r♣rs♥ts ♥ ♣♣r♦①♠t♦♥ ♦ t ♥♦r♠ t♦ t ♦r ♣rt ♦t t t ♣♦st♦♥ ra t ♣r♦s ♣♣r♦s s♥ ♣♦②♥♦♠♣♣r♦①♠t♦♥ ❬ ❪ ♥ ♥②t s♦t♦♥ ❬❪ ♥ srt s♠♠t♦♥ ♦r♦♥r② ♣♦♥ts ❬❪
s ♣♣r♦s ♥ts ♥ s♥ts sss rrr♥ ♦♠♣tt♦♥ ♦ t r♥♦r♠③t♦♥ tr♠ ♦ t r♥ s♣♣♦rt ♥r s♦ s ♦t♥ t t♠ ♥trt♦♥ s♠ tr② ♠♦r s②♦♥t♥ ♦r ♥② s♣ ♦ ♦♥rs ♣rs♥t ♥ ➓
♥②t ♦ ∇γa
❲t t ♦♥r② ♦ t ♦♠♥ ♦♠♣♦s ♦ s♠♥ts ♥♦t t t ssr♣t (.)s s♠♥t s ♥ ♥r ♥♦r♠ ns ♥♥♥ ♣♦♥t re1 ♥ ♥♥♥ ♣♦♥t re2 s r ♥ ♥ ♦♠♣t t ♥②t ♦ t ♦♥trt♦♥ ∇γas ♥ ②
∇γas ≡(∫
re2
re1
w (r) dl
)ns
r ♠ s♦♥ t ♥t♦♥s ♦ t ♦♠tr ♣r♠trs s t♦♦♠♣t t ♥②t ♦ ∇γas t
s ♦r t q♥t ❲♥♥ r♥ s ♥ ts ♦r
h
∫re2
re1
w (r) dl =(q2 cos α2)
πPq0
(q2) −(q1 cos α1)
πPq0
(q1)
+q40
π
(105
64+
35
512q20
)
sign (q2 cos α2) ln
(q2 + |q2 cos α2|
|q0|
)
−sign (q1 cos α1) ln
(q1 + |q1 cos α1|
|q0|
)
r t ♣♦②♥♦♠ ♥t♦♥ Pq0
s ♥ ②
Pq0(X) =
7
192X5 − 21
64X4 +
35
32X3 − 35
24X2 +
7
4
+q20
(35
768X3 − 7
16X2 +
105
64X − 35
12
)
+q40
(35
512X − 7
8
)
r q0 ≡ |raei.ns|h
qi ≡ |raei|h
♥ qi cos αi i ∈ 1, 2 r s♣② ♥ r
s ♥②t s ♥ st♠ts ♦ t rr♦r t♦ t ♣♣r♦①♠t♦♥s ♥ r ♦♠♣r t ♥②t ♥ ♣♣r♦①♠t s ♦ ∇γa
♥st t st♥ t♦ ♣♥ srt ♣♣r♦①♠t♦♥ s s ♥②
∇γas ≃ wasSsns
r t s ss♠ tt t r♥ s ♦♥st♥t ♦♥ s♠♥t s
r♥ s s t q♥t r♥ s ❬❪ ♥ t rt♦δr
h= 2 r δr
s t ♥t st♥ t♥ t♦ ♣rts ♥ h s t s♠♦♦t♥ ♥t s♦t rr♦rs r ♥ s
ǫ∇γa=
∣∣∇γ♥②ta −∇γsrt
a
∣∣∇γ♥②t
a
❲ ♦♥sr ♦♥② t ♦♠♣♦♥♥t ♦ ∇γa ♦rt♦♦♥ t♦ t ♥ r t rr♦r ♦r t srt③t♦♥ ♦ ∇γa s r② ♦♦ ♦r s ♣♦♦r srt③
t♦♥ rt♦δr
h♦r ♣♥ ss t♥ 0.1% rs t srt③t♦♥ s
s②st♠t rr♦r ♦r t ♣♣r♦①♠t♦♥ ♦ t ♦ γa ♦♥ t ♦rr ♦ 3%
❱s ♦ t ∇γa ♥t♦♥ ♥st t st♥ ♦t
❱s ♦ t rr♦r ♥t♦♥ ǫ∇γa♥st t s
t♥ ♦ t
r ♥②t ♥ r ♥ ♦♠♣t s ♥ r♥ ♦ t ♥t♦♥s ♥stt st♥ t♦ ♣♥
t srt③t♦♥ rr♦r ♦ ∇γa ♥ t ♣rs♥ ♦ ♦♠♣① ♦♥r②s s t s ♦♥ tt t rr♦r s rr ♦r♦r t rr♦r ss②st♠t s♦ tt t ♠♥t ♦ ∇γa s ②s ♥rst♠t ❲t♥ s♠t♦♥ ts s t♦ ♥♦♥♣②s ♦r ♣rts s t♦rs♦♥ ♦♥r② s♥ t rt② s ♥♦t ♥ ♦♠♣t② ② t r♣s ♦r ♣r♦♣♦rt♦♥ t♦ ∇γa
♦r♥♥ qt♦♥ ♦r γa
r♥ ♠t♦ t♦ ♦♠♣t γa ♦r ♣rt a ♥r s♦ ♦♥r② s sst t♦t t ♥ ♦r tt♦s ♣rts ♥ s tr♦r s♠♣r t♥ ♥♥②t ♦♠♣tt♦♥ ♠♥ ♦ t ♣rs♥t ♠t♦ s t♦ s ♦r♥♥qt♦♥ ♦ γa
dγa
dt= ∇γa.ua
γa = 1 ∂Ω ∩ Ωa = ∅
r t ♥t♦♥ ♦ t r♥t s ♦♠♥ t t t ttdra
dt= ua
♥♦tr ② t♦ ♦♥sr ts qt♦♥ s t♦ r♠r tt s q♥t t♦
∂γa
∂t= 0
γa = 1 ∂Ω ∩ Ωa = ∅
s ♠♥s tt t γa ♦s ♥♦t ♣♥ ♦♥ t t♠ t ♦♥② ♦♥ ♣♦st♦♥♥ s tr♦r ♥ r♥ s t s s ♠♥s t♦ ♦♠♣t γa t♦ ♦r♥t t ∇γa s sr t♦ ♦♠♣t s♥ t ♥ ①♣rss s sr ♥tr
rt♦♥ ♥ ①t♥ ♦r ♠♦♥ ♦♥rs ♥ t♦♥r② ♥ ② r ♠♦♥ ♦r♠ ♥ ts♥s tt s♠♥t ♦r tr♥ ♦♠♣♦s♥ t s ♠♦♥ t ts ♦t② t ♦♦♥ ♦r♠ s ♦t♥
dγa
dt=
∑
s∈S∇γas.u
Rs
a
γa = 1 ∂Ω ∩ Ωa = ∅
r uRs
a s t ♦t② ♦ t ♣rt a ♥ rr♥ r♠ Rs r ts♠♥t s s ① γa s ♥♦ ♦♠♣t ② s♦♥ t ♦ qt♦♥s ♦♥ t ♥♦ ♦ ∇γas ♦♠♣t r♦♠
♥t③t♦♥ ♦ t γa
♥t③t♦♥ st♣ ♦ γ0a s ♦♥ ② ♠♥ rt tr♥s♦r♠t♦♥ ♦r
♣rt ♥t② ♥①t t♦ s♦ rtr♦♥ s∣∣∇γ0
a
∣∣ > 0 ♠♦t r♦♠ ts strt♥ ♣♦st♦♥ r0
a t♦ ♥ r r t ♥t♦♥ γ (r) ≡ 1 ♦r♥st♥
ra = r0a + l
∇γ0a
|∇γ0a|
r t ♥t l s t♥ t♦ 2R R s t rs ♦ t ♦♠♣t r♥s♣♣♦rt
st ♦ t ♣r♦♣♦s ♠t♦ s s♣② ♥ r ♦r ①♠♣ tr ♣rt ♥ r s ♣ ♥ t s ♥ r t ♦
t ♦s ♥♦t ♠♦
γ s 1 ♥ s ♠♦ t♦ ts ♥t ♣♦st♦♥ ♦♥ t ♣t ♦ t r rr♦ ♣t♥ t ♦ γa t rs♣t t♦ t ♦r♥♥ qt♦♥
0.82 0.84 0.86 0.88
0
0.01
0.02
0.03
0.04
0.05
10.950.90.850.80.750.70.650.60.550.5
γa
γ (r) = 1
γ (r) < 1
r t ♦ t ♥t③t♦♥ ♦ t γ ♥①t t♦ s♦
♦t tt t qt♦♥ ♦ γa s ♥trt ♥ t♠ t s♦♥♦rr t♠♥trt♦♥ s♠ t♦ ♣r♥t s②st♠t ♥trt♦♥ rr♦rs s ➓ ♥t♦
γn+1
a = γna +
1
2
(∇
nγa + ∇n+1γa
).(rn+1
a − rna
)
t s♦ ♦♥r② s ♠♦t♦♥ss ♥r ♦r♠ ♦r ♠♦♥ ♦r♠ s
γn+1
a = γna +
δt
2
∑
s∈S
(∇
nγas + ∇n+1γas
).(uRs
a
)n+1
♦♥t♦♥ ♦♥ t t♠ st♣ s rqr t♦ ♣ t ♥trt♦♥ ♦ γa st
δt ≤ Ct,γ
1
maxa∈F ; s∈S
∣∣∣∇nγas.(uRs
a
)n∣∣∣
r Ct,γ = 0.005 r♦♠ ♥♠r ①♣r♥ s s ♥tr ♦♥t♦♥ ♥t s♥s tt t t♠ st♣ rss ♥ ♣rts ♣♦sss st ♦t② ♥♣♣r♦♥ ♦♥r② s s s♣s♥ ♥st ttr ♦♥t♦♥s ♦♥sr ♥ t♦♥ t♦ t s t♠st♣♣♥ ♦♥t♦♥s ♥ P
s ❬❪
♠r rsts
♠♥r ♥♥ ♦ tst s
♦ tst t rt♦♥ tr♠s ♥ t t sr strss ♦r♠t♦♥ ♠♥rP♦s ♦ ♥ ♦s♥♥ t ♣r♦ ♦♣♥ ♦♥rs s s♠t ♥♥ s ♠tr ♦ 1m t s♦st② ν s st t 10−1m2.s−1
s♦ tt t ②♥♦s ♥♠r s 10 s♦s tr♠ s ♠♦ t t♦rrt ♠♦ ♦ ♦rrs ♦♠♥ t t♦ ♦♠♣t t rt♦♥♦t② r s♦s tt t ♦r③♦♥t ♦t② ♣r♦ s ♥ ♦♦ r♠♥t t t ♥②t s♦t♦♥ ♥ ♥ t ♥t② ♦ t ts♠♦♥strt♥ tt t sr strss ♦rrt② ♥s t ♦② ♦r
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.20
0.2
0.4
0.6
0.8
1
10.90.80.70.60.50.40.30.20.10
ux
uxz
(m)
r P♦s ♦ ♥ ♣r♦ ♣♣ t ②♥♦s ♥♠r ♦ 10♦♦r ♦ts r♣rs♥t t ♦t② ♦ ♣rts t t st② stt rs t ♦ts • r t ♥②t ♣r♦
♦♠♣tt♦♥ ♦ t str♥ ♠♥r s ♦s s t♦ t ♦r♠ s♥ tr s ♥♦ ♥♥ ♦ t str♥ rt ♦♥ t ♦ ts s ts♦st② s ♦♥st♥t s s tst s ♦s s t♦ ♦♠♣r t ♦ t ♦♠♣t str♥ rt t♦ ts ♥②t ♥②t ♣r♦ ♦♦t② s
ux (z) = 4 Reν z
D2
(1 − z
D
)
s t♦ t ♦♦♥ ♥②t ♦ S
S (z) = 4 Reν
D2
∣∣∣2 z
D− 1∣∣∣
S(s-1)
Z(m
)
0 1 2 3 40
0.2
0.4
0.6
0.8
1
t♥r ♠♦
S(s-1)
Z(m
)
0 1 2 3 40
0.2
0.4
0.6
0.8
1
♦rrt♦♥ ♦ t♥r ♠♦
r ♦♠♣rs♦♥ ♦ t str♥ rt ♦r r♥t ♠♦s ♥ ♠♥r ♥♥♦
❲ ♥♦t ♥ r tt ♥ ts t♦rt tst s t ♦rrt ♠t♦
s stst♦r② r♣r♦t♦♥ ♦ t sr strss ♥①t t♦ t ♥t ♣r♦s ♠t♦s
t tr ♥ ♠ r ♥ t♥ t
♥ ts st♦♥ t ♣r♦♣♦s s♠ s tst ♦♥ ♠♦r ♦♠♣① ♦♠tr② ♦♥ssts ♦ s♥ ♦ ♣♣r♦①♠t② 2m ♥t ♥ 1m t t
♦ π2
rad ♥ ♥√
28
m ♦ t ♥ t ♦tt♦♠ ♠ ♦ t t♥ s♦♠tr② s ♦s♥ s♣② s♥ t trs ♦t s♦♥t♥♦s♣♦♥t ♥ s♦♣♥ ♣r♦ tsts t ♦rrt♦♥ ♦ t r♥ ♥ t ♣rs♥♦ rt② ♦♠♣rs♦♥ s ♠ t♥ r♥t ♠♦s ♥ st tr s♥ ②♥♠ s r t s♦st② ν s st t 10−2m2.s−1
t tr s
♦♠ trt♠♥ts ♦r s♦ ♦♥rs sr r♦♠ ♥ ♥t② t♦ r♣r♦ ♦rrt② st tr s r ♦♠♣r t rsts ♦t♥ ♥ t s♥s t 0.5m ♦ tr ♦r tr ss t ♥♥r♦♥s r♣s ♦rss ❬❪ t tt♦s ♣rts ♠t♦ s ❬❪ ♥ t♥ ♥ ♣r♦♣♦s ♠t♦s ①♣t t r♣s ♦rs ♣r♦ ♣♦♦r rsts s t rs ♥ t s♥s tt ♣rts ♣ s♥ ♦♥ rt t s t♦ tt tt t ♠ss♥ r ♥ t r♥ s♣♣♦rt s ♥♦t ♦♠♣♥st ♥ tst rt② s ♥♦t ♥ s♥t② ♣♦t ♦ t ♣rssr ♦ ♣rts♥st t ♣t s tr♦r ♥♦s② ♥ ② r♣r♦ ♥①t t♦ t ♦tt♦♠ tt♦s ♣rts ♠t♦ s r s ttr rsts t t♦♥t♦♥ s ♥♦t ♥sr ♥ s♦ t ♣rssr ♣r♦ s st ♥♦s② ♦r♦rts ♣♣r♦ s ♣r♦♠t t♦ sr ♥ ♦♠♣① ♦♠trs ♥ rqrst ♣rts t♦ ♠♠ t ♦♥r② ♥rs t ♦♠♣tt♦♥ ♦st ♣rs♥t ♠t♦ s s♣r♦r rsts ♥r ♣rssr ♣r♦ ♥ ♥rt ♦tt♦♠ ♥ ③r♦ ♦t② s s♦♥ ♥ r
②♥♠ s
s♠t♦♥ ♦ ♠ r t t s♠ ♦♠tr② s ♥ ♣r♦r♠ ♦r tt♦ ♦♥r② t♥qs ♣r♦s② sr ♥ t ♣rs♥t ♦♥ trs ♥t② ♦♠♥ ♦ 1m t ♥ 0.5m t ♦♥ t t♥ s ♦ ts♥ ♥ t rsts s♦♥ ♥ r ♣♣r♦s ♥sr ♠♣r♠♦♥rs t ♦t ♦ r♣s ♦rs ♠t♦ ♥ tt♦s ♣rts ♠t♦ ♥♦sr ♣rssr rtr♠♦r s♠t♦♥ t ♥r rs♦t♦♥s ♥ ♦♠♣t ② ♦♥ rs♦t♦♥ ♥ ♥ t ♥♠r ♦ ♣rts♥♣s♦ts ♦ t ♣rssr t t s♠ ♣②s t♠ r ♣♦tt ♦♥ tr
♦♠♣rs♦♥ t ❱ s♠t♦♥ ♦♥ t ♥t ❱♦♠ ♦♣♥s♦r ♦ ♣♥♦♠ ♦ t ♣rssr ♦♥ t t s ♦ t s ♣r♦♠♥ s♣② ♦♥ r
tt♦s ♣rts
♥♥r♦♥s t②♣ r♣s ♦r
Prs♥t ♠t♦
r ♦♠♣rs♦♥ ♦ t rt ♦t② ♦r st tr ♥ t♥ t ♦r r♥t ♦♥r② ♦♥t♦♥s tr 20s
♦♠♣rs♦♥ ♦ t k − ǫ ♠♦ t P ♥ ♥t
❱♦♠s s♠t♦♥ ♦ s ♣ss
♥ ♠ s t ♦♥ rr t ♦♥t♥t② ♦ t ♦ s sr♣t ♥ t♠rt♦♥ ♦ s s ♥trr♣t ♦r s♣s s s s♦♠ s♠♦♥ t ②rqrs t ss t♦ ♠rt r ♣rr ♦ rst♦r t ♠rt♦♥ ♣r♦sss ♣sss r ♥st ♦♥sst ♦ ♠♥② r♣t♥ ♠♥ts ♥ ♥ ♦♥sr s ♣r♦ ♦ ♠♥s♦♥♥ ♦ ts ♦♠♣♦♥♥ts rqrs t ♥♦ ♦ t tr♥t ♦ t♥ st♦♥ s♥ t s③ ♦t r s ts t t② ♦ t s t♦ s♠ ♣str♠
♥♥r ♥ ♦♥s r♣s ♦rs tt♦s ♣rts
Prs♥t ♠t♦ Prs♥t ♠t♦ t t s♠rs♣ srt③t♦♥
r ♦♠♣rs♦♥ ♦ t ♣rssr ♦r ♠ r tst s ♥ t♥t ♦r r♥t ♦♥r② ♦♥t♦♥s
3 parti lesSPH 2 × 104 parti lesSPH 8 × 104 parti les
r ♦♠♣rs♦♥ ♦ t t♠ ♦t♦♥ ♦ t ♣rssr ♦♥ t t s♦ t t♥ t ♣rs♥t P ♦r♠t♦♥ ♥ t ♥t ♦♠ ♦♣♥♦♠ t r♥t s♣ srt③t♦♥
t♦ t ♦r ♦ t ♦ t♦ s ♣ss s 3D rsr♦ ♦♥sr r 2D s♠t♦♥s r t rt rt♦♥ r ss♠t♦ ♥ ❲ r♣t r t s♠t♦♥s ♣rs♥t ♥ ❱♦ t ❬❪ ♦♠tr② ♦ t x−♣r♦ s♠t♦♥ s ♣rs♥t ♥ r rsts ♦t♥ ② P r ♦♠♣r t♦ s♠t♦♥s ♦♥ t ♦❴tr♥ ② t ♥t ❱♦♠ ♦ ♦♣ ② s ❬❪ ♠ ♦ s ♦♠♣rs♦♥ s t♦ t t ♣r♦r♠♥ ♦ P ♦r tr♥ts♠t♦♥ ♦♠♣tt♦♥ s ♦♠♣r t② s♦♥ t s♠ qt♦♥s
t ②♥♦sr r t♦s t t k − ǫ ♠♦ t t s♠
♣rssr r♥t rs♣♦♥s r♥ t ♦ ∆p
ρ∆x= 1.885m.s−2 t t t♦
r♥t ♣♣r♦s r♥♥ ♥ r♥ t t♦ r♥t srt③t♦♥♣♣r♦s P ♥ ♥t ❱♦♠
♦t② ♣r♦s t ♦t♦♥s P1 P2 ♥ P3 ♥ ♥ r r♣♦tt ♥ r rsts s♦ tt t ♠ss ♦ s ♣r♦s♦s t♦ t ♦♥s ♦t♥ t t ♣rs♥t P s♠ t s♦ ♥♦t ttt t st♥r P ♠t♦ ❬❪ t ♣rt ♦t② ♥ ② s♦st②strt♦♥s ♥♦t t t ♥t ❱♦♠ ♦♥s ❲t t ♣rs♥t ♠♦ ♦♥♥ s tt t r♠♥t s r② stst♦r②
♦♥s♦♥
♣rs♥t rt s ♣rs♥t ♥ ♣♣r♦ t♦ t s♦ ♦♥r②♦♥t♦♥ s ♦t s♠♣ ♥ r♦st s♠♣t② s ♥ t ♠♥♥r ♦♠♣t t r♥ r♥♦r♠③t♦♥ tr♠ γa t ♥trt♦♥ ♥ t♠ ♦♥② rqrs t ♦♠♣tt♦♥ ♦ ts r♥t ∇γa r♦st♥ss s t♦ t ♥trt♦♥ ♥ t♠ ♦ t ♦♥t♥t② qt♦♥ ♠s t ♥st② ♣♥ ♦♥② ♦♥ t ♣rts ♣♦st♦♥s s ♦s ♦♥ t♠ s♠t♦♥t rt② r t♠ st♣ ♥ s ♠♦r ♥t ♦r ♦♥srt♦♥♣r♦♣rts
♥t♦♥ ♦ ♥ ♦♥r② ♦rrt r♥t ♥ ♣♥ ♦♣rt♦rss s t ♦♣♣♦rt♥t② t♦ ① ♦♥r② ♦♥t♦♥s ♥ ①s ♦♥ t ♣rssr t sr strss ♥ ♥ t sr s s s k ♥ ǫ ♥ ♠♦♦ tr♥
♦r ♥♠r♦s sss st rqr ♥stt♦♥ ♥ ♦♣♠♥t ♥♠②
❱t t ♣rs♥t ♦r♠t♦♥ ♦♥ r♥t tst s s s ♣r♦tr♥t ♠♣
♣t t r♥♦r♠③t♦♥ t♦ 3D t ♠♥ ♥ s t♦ ♥ ♥♥②t ♦r♠ ♦r t ♦♠♣tt♦♥ ♦ t ♦♥trt♦♥ ♦ sr♠♥t s ♦r t ♦ ∇γa ♦ ♣rt a s ♥♦t ②∇γas ♦r ② t♦ ♦♠♣t rt② ♥ ♣♣r♦①♠t ♦ t
t② t t♦rt ♦♥srt♦♥ ♦r ♥♦♥♦♥srt♦♥ ♦ ♠♦♠♥t♠♥ ♥r ♠♦♠♥t♠ ♥ s♣② ♥ ♣r♦ ss
♦♠♥ t ♣rs♥t ♣♣r♦ t ♥♦♥♣r♦ ♥tr♥ ♦♥t♦♥s
r♥t ♦♥t♦♥s ♥ P
♦♠♣tt♦♥ ♦ t rt♦♥ ♦t② ♥ tr♥t
s
❲ ♥ ♣♣② t s♠ ♦rrt♦♥ ♦ t s♦♥ tr♠ ♦ t ♠♦♠♥t♠qt♦♥ s t ♠♥r ♦♥ ♥♦t♥ tt (µ + µT )S.n ≃ ρuτuτ ♥ t ♥t②
♥t ♦ t ♦t② tP❯
r♥t s♦st② νT t ♦❴tr♥
r♥t s♦st② νT t P❯
r♥t s♦st② νT t ♦❴tr♥
♥t ♥r② k t P❯ ♥t ♥r② k t ♦❴tr♥
r ♦♠♣rs♦♥ ♦ t k − ǫ ♠♦ t r♥♥ P ♣♣r♦♥ ♥ r♥ ♥t ❱♦♠ ♠t♦ ♥ s♠t s ♣ss
♦ ❯♥ t qt♦♥ ♥s rt♦♥ ♦t② t♦ t ♠♥♦t② ♥ t tr♥t s ♥ ♥♦tr ♥t♦♥ s♦st② s♥♦t ♦♥st♥t ♥②♠♦r ♥ s s♣♣♦s t♦ ♥r ♥ t ♥t② ♦ ♥ t ♥ s♦♥ tt t ♦t② ♣r♦ ♥ tt r s ♦rt♠s♣ ts ③♦♥ s t ♦ ②r ♦♥sr t ♣rt a t♦ ♥
♦r♠② t k = 0 ♥ t♥ νT = 0 s♦ tt r♦r t ♠♥r s ♦rt s♦s s②r r t ♠♥r s♦st② s ♠♦r ♠♣♦rt♥t t♥ t tr♥t ♦♥ ss② ♦r ♥r♦♥♠♥t ♦s r② t♥ s♦ tt ♦ ♥♦t ♥♦r k t♦ ③r♦ t
♥ ♥♥ ♦ ts ss♠♣t♦♥ s r ♥ 10% ♦ t ♥♥ ♣t
Pr♦s ♥ P1 Pr♦s ♥ P2 Pr♦s ♥ P3
r Pr♦s ♦ t ♦t② ♠♥t ♥ tr r♥t ♣♥s ♥ t s♣ss k − ǫ ♠♦ ♥ r ♥ t k − ǫ ♠♦ t ♦❴tr♥ ♥r♥
t ♦ ②r ♦ s♠♦♦t uτ ♥ ♦t♥ r♦♠ t ♦♦♥ t ♥ trt ♦rt♠
|ua|uτa
=1
κln(zauτa
ν
)+ 5.2
♦ qt♦♥ ♠st r② tt t ♥♦♥♠♥s♦♥ st♥ t♦ t zauτa
νs rtr t♥ 11 ❲ ♦ s♦ s ♦ s ♦r r♦ s ♦r s
s ♦ ♦t ♥ t ♠♥r ♥ t ♦ ②r s s rs t t♥ t k − ǫ ♠♦ ♠st s♦ ♠♦ ♦r ♦ ②♥♦s ts♦r ♠♦r ♥♦r♠t♦♥ s ❬❪
❱♦t② t t
❲ ♦sr ♥ t ♦r♠ tt t ♦t② t t us s ♦♥sr♥ t ♦♥r② tr♠ ♦r♠② t ♥♦s♣ ♦♥t♦♥ ♦ ♠♣♦s tt t♦t② t t s t ♦t② ♦ t ts 0 ♦r ♠♦t♦♥ss s s ♠♣♦s ♦r ♠♥r ♦ ♥ t tr♥t s t s ♣rr t♦ ♥♦t♦ s♦ t s♦♣ ♦ t ♦t② ♣r♦ s ♠ rr t t t♥ ♥ t♦ ②r r ♣rts r ss♠ t♦ s ♥t t♦ trt② Sa ♥①t t♦ t ♥ t♦ ♥tr♣♦t t ♦t② t t ♦ ♦ s♦ t t ♦t② ue ♦ ♣rts ♥ t s♦s ♥ rt♦♥tr♠s
due
dt=
1
γe
∑
b∈Fmb
µTe + µTb
ρeρb
ueb
r2eb
reb.∇web
︸ ︷︷ ︸s♦s tr♠
− 2uτeuτe
γe
∑
s∈S|∇γes|
︸ ︷︷ ︸rt♦♥ tr♠
❲ ♥♦t r tt qt♦♥ s t ♠♦♠♥t♠ qt♦♥ ♣♣ t♦ ♥ ♣rt t ♥tr rt② ♥♦r ♣rssr r♥t ♥ ♥ us t♦ t r t♥ ♣rts e1 ♥ e2 ♥ ♥ ➓ stt t♦ ♦ ♥ ♦ s♣ ♦t② t t ♥ ②♥♦s ♥♠rs♠t♦♥ s s♦ s ♥ ♠♥② ♦s s s ♥ ♥t♠♥ts s r♦t ❬❪ ♥t② t♦ r ♥ ♠♥ tt t ♣rts ♥ E r ♥ t r♥ ♣♦♥ts ♥ ♦ ♥♦t ♠♦ t t ♦t② ue
t t t ♦t② t s t♦ s② ♦♥② s t ♦t② ue t♦ ♣ts♦s ♦rs ♦ ♣rts ♥trt♥ t t ♥ t♦ ♦♠♣t tstr♥ rt S
① ♦♥t♦♥s ♦♥ t ♥t ♥r②
♣♥ ♦♣rt♦r ♣♣ t♦ t tr♥t ♥t ♥r② rs t♦
1
ρa
Lapa
(µb +
µTb
σk
, kb)
=1
γa
∑
b∈Fmb
2µ + µTa/σk + µTb/σk
ρaρb
kab
r2ab
rab.∇wab
s t s ss♠ tt tr s ♥♦ ① ♦ k r♦♠ t ♦♥r② ∂k
∂n=
0 t t qt♦♥ s♦ s ♥ ♣ ♦ Lapa ♥ ♣②s ♠♥♥ s tt t tr♥t ♥t ♥r② s ♦♥② rt ② t♠♥ ♦ ♦r♦r ♥st ♦ s♣②♥ ♦♥r② ♦♥t♦♥s t t t ♥t♦♥ ♣♣r♦ srs t ♥ ♦ tr♠s ♥ t ♥t② ♦
t r t s ss♠ tt P = ǫ ts ♠♣s tt t ♦♥t♦♥∂k
∂n= 0
s ♥♦t ♦♥② t t t ♥ t ♦ ♥t② ♦ t s♦ ♦♥r② ss♠ t ♦ t♦ ② tr♥t tt s t♦ s② t t♥
s♦s s②r ♥ s ②♥♦s ♥♠r s ♦r k − ǫ ♥ t ♦ ♥♦t s♦ t k− ǫ ♠♦ ♣ t♦ t r k s t♦rt② ①♣t t♦ 0 t ♣ t♦ s♠ st♥ δ r♦♠ t r t tr♥ s ②sts νT ≫ ν ♠♥ ♥t ♦ t ♣rs♥t r♥♥ ♣♣r♦♦♠♣r t♦ ♥ r♥ ♦♥ s tt t r♥♥ ♣rts ♥ F \ E t st t st♥ ♦ t ♦rr ♦ δr r♦♠ ♥② t s ♦♥ ♦t ♠♥ ♥ts ♦♠♣r t♦ t ♠ss ♠t♦s r t ♦♥r②♠♥ts r s♣♣♦s t♦ t rt st♥ t♦ t t r ♦♥②♦s ♦♥ ♣rts r t② t ♥♦♥③r♦ st♥ r♦♠ t s
♦ st♠t k t t ♥ t s ♦r t ♥st② t♣rssr ♦r t sr strss
ke =1
αe
∑
b∈F\EVbkbwbe
❲ ♥♦t tt ts ♣♣r♦①♠t♦♥ s ♦♥sst♥t t t ss♠♣t♦♥∂k
∂n= 0
♥②
ks =ke1 + ke2
2
① ♦♥t♦♥s ♦♥ t ss♣t♦♥ ♦ ♥t ♥r②
♣♥ ♦♣rt♦r ♣♣ t♦ ǫ rqrs t ♦r ∂ǫ/∂n ♥♥ ss♠ tt t ♦ s ② tr♥t t♥ r② s♥ ♣rt♥ t r ♦ ♥♥ ♦ ∃s ∈ S/ |∇γas| > 0 s ♥ t ♦ ②rr
k ≃ u⋆2
√Cµ
ǫ =u⋆3
κz
νT = κu⋆z
t qt♦♥s ♥ r r♦♠ t qr♠ P = ǫ
r z s t st♥ t♦ t ♣rt a s ♥trt♥ t sr s stt z = max (ras.ns; δr) r δr s t ♣rt r ♥t s♣♥κ s t ❱♦♥ r♠♥ ♦♥st♥t t t ♦ 0.41 ♥ u⋆ s rt♦♥ ♦t②♠sr♥ t tr♥
u⋆s =
√ks
C1
4
µ
♥ r♦♠ qt♦♥ ♥ ♦r t ① ♦ ǫ
νTa
σǫ
∂ǫa
∂ns
= − 2u⋆4s
σǫκδras
t♦r 2 s ♣r♦ ② rst♦rr ♣♣r♦①♠t♦♥ t ① s tt t st♥ z
2 s r② s ♥ s♣② r s ǫ s s♣♣♦s
t♦ r② s 1zr z s t st♥ t♦ t
s t ♣♥ ♦♠s
1
ρa
Lapa
(µb +
µTb
σǫ
, ǫb)
=1
γa
∑
b∈Fmb
2µ + µTa/σǫ + µTb/σǫ
ρaρb
ǫab
r2ab
rab.∇wab
+4
γaρa
∑s∈S |∇γas| ρs
u⋆4s
σǫκδras
s s ♥ ♣ ♦ Lapa ♥
r♥s
❬❪ sr♠ ♦♥t s ❲ Pr♦t rt♦♥ ♦r♠t♦♥s ♦♥tt ♦rt♠ ♦r r ♦♥rs ♥ t♦♠♥s♦♥ s♣ ♣♣t♦♥s
❬❪ r ♦r♥ ss♥r♥ rr♥t P ♥ ♠♣r♦ s♣ ♠t♦♦rs r ♦rr ♦♥r♥ ♦r♥ ♦ ♦♠♣tt♦♥ P②ss
❬❪ ♦♥♦ ♥♥t t t ♥♥ ♦ s♠♥②t ♣♣r♦ ♦r s♣ ♠♦♥ ♦ s♦ ♦♥rs4th P ♦rs♦♣ ♥ts r♥