unified semi-analytical wall boundary conditions for
TRANSCRIPT
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
methodMartin 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 meshlessSPH method. International Journal for Numerical Methods in Fluids, Wiley, 2013, 71 (476-472),pp.Online. 10.1002/fld.3666. hal-00691603
❯♥ s♠♥②t ♦♥r②
♦♥t♦♥s ♦r ♥s ♠♥r ♦r tr♥t
♦s ♥ t ♠sss P ♠t♦
rr♥ r♥ ♦rs ❱♦ ss♦ts
♠ ♠rt♥rr♥r
②
♥tr♦t♦♥
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.
e
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
b
b
b
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
♥ ①♣rss ♥ tr♠s ♦ ♥ ♦♠♣t ♥ rt♦♥ s♥s ♦♣rt♦rsr♥t ♥ r♥ ♥ ♦♥ ♥ ♥
GradKa Ab ≡ ρa
∑
b∈Fmb
(Aa
γaρ2a
+Ab
γbρ2b
)∇wab −
Aa
γa
∇γa
DivKa Ab ≡ − 1
γaρa
∑
b∈FmbAab.∇wab +
1
γa
Aa.∇γa
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
Gradaub = − 1
γaρa
∑
b∈Fmbuab ⊗ ∇wab +
1
γaρa
∑
s∈Sρsuas ⊗ ∇γas
X(m)
Z(m
)
0 0.05 0.1 0.15
0
0.05
0.1
0.15
0.2
P(Pa)
5500050000450004000035000300002500020000150001000050000
sr♠ t s r♥t ♦♣rt♦r s ♥ t♦ ♠♥t♥ ♥♦r♠ ♣rssr Prts ♠♦ t♦♥ ♥♦♥♣②s qr♠
X(m)
Z(m
)
0 0.05 0.1 0.15
0
0.05
0.1
0.15
0.2
P(Pa)
5500050000450004000035000300002500020000150001000050000
♣rs♥t r♥t ♦♣rt♦r s t♦ ♠♦st ♠♥t♥ ♥♦r♠ ♣rssr s ♥ qr♠
r ♦♠♣rs♦♥ ♦ r♥t ♦♣rt♦rs ♥ ♥ ♥ ♦r♣rssr③♣r♦ ♣♣
♥ t t ♦rrt ♦♠♣♦♥♥ts t str♥ rt S s ♦♠♣t ♦r♥t♦ t ♥t♦♥ ♥ ②
♣♥
♣♥ rst♥ r♦♠ t ♦rrs ♠♦ ♥ s♥ s srtst♦♥ ♦ t ♦♥t♥♦s ♥tr♣♦t♦♥ 〈∇.µ∇f〉 (r) r f s t ♦♥t♥♦ssr ♥ µ t ♦♥t♥♦s s♦st②
〈∆ (µ, f)〉 (r) ≡ 〈∇.µ∇f〉 (r) = 〈∇r′ . [(µ∇f) (r) + (µ∇f) (r′)]〉 (r)
= − 1
γ (r)
∫
Ω∩Ωr
[(µ∇f) (r) + (µ∇f) (r′)]∇w (r − r′) dV ′
− 1
γ (r)
∫
∂Ω∩Ωr
[(µ∇f) (r) + (µ∇f) (r′)] .nw (r − r′) dS′
♦♠♥ t t ♥t r♥ ♣♣r♦①♠t♦♥
∇f(r). (r − r′) ≃ (f(r) − f(r′)) ≃ −∇f(r′). (r′ − r)
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♦♠
d (γaρa)
dt=
d
dt
(∑
b∈Fmbwab
)
γa
dρa
dt+ ρa
dγa
dt=
∑
b∈Fmb∇wab.uab
♥ s♠r ② ❱ ❬❪ stt tt t ♦♥t♥t② qt♦♥ s strt②
q♥t t♦dρa
dt=
d
dt
(∑
b∈Fmbwab
) t♠ s ♦♥sr ♦♥t♥♦s r
qt♦♥ s s ② t♦ ♥trt ①t② ♥ t♠ t q♥tt② γaρa
t ♣rts ♣♦st♦♥s ♠♦ r♦♠ rnb t♦
rn+1
b
s ♣r♥ts s②st♠t
t♠ ♥trt♦♥ rr♦rs ♥ ♠s ρna ♣♥ ♦♥② ♦♥ t ♣♦st♦♥s ♦ ♣rts
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
0500100015002000
0 0.5 1 1.5 2Pressurefor e(N)
Time (s)
VOF 7 × 103 ellsVOF 3 × 104 ellsVOF 1 × 106 ellsSPH 5 × 10
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♥
❬❪ ♦♥♥ tr ♣ ♣rt ♦♥r② ♦rs ♦r rtrr② ♦♥rs ♦♠♣tr P②ss ♦♠♠♥t♦♥s
❬❪ r♦♥ ♦ r♦ ♦ ♠ ♥♠r ♥♠r s♠s ♦rt s♣ ♠t♦ ♣♣t♦♥ ♥ r sr ♦s ♥ ♣t♦♥ tr♥s 4th
P ♦rs♦♣ ♥ts r♥
❬❪ ♦③é ss♥r♥ ♦r♠ ① ♠t♦ t t ♦♥r② ♦r s♣ 4th P ♦rs♦♣ ♥ts r♥
❬❪ ♦♥♥ ♠t♥ r sr ♦s t P ♦r♥ ♦ ♦♠♣
tt♦♥ P②ss
❬❪ ❱♦ ss ♠r ♠♦♥ ♦ ♦♠♣① tr♥t rsr♦s t t s♣ ♠t♦ ♥ ♦r ♥tr♥t♦♥ ♦r♥ ♦r
♠r t♦s ♥ s ♦ ❯tt♣①♦♦r
❬❪ ❱ P ❲t ♣rt ♣r♦s tr♠♠t♦s♥①t tr♠ ♥ s♠♦♦t♣rt ②r♦②♥♠s t ♦s t ♣♣
❬❪ ♠♥ ♦♥t ②♥♠ r♥♠♥t ♥ ♦♥r② ♦♥tt ♦rs ♥s♣ t ♣♣t♦♥s ♥ ♦ ♣r♦♠s ♥t ♦r♥ ♦r ♠r
t♦s ♥ ♥♥r♥ sr t♦s ♥t♥s ♥ ♣♣t♦♥s
❬❪ ♦♥♥ ♠♦♦t ♣rt ②r♦②♥♠s ♥♥ r ♦ str♦♥
♦♠② ♥ str♦♣②ss
❬❪ ♦rrs P ♦① P ❩ ❨ ♦♥ ♦ r②♥♦s ♥♠r ♥♦♠♣rss♦s s♥ s♣ ♦r♥ ♦ ♦♠♣tt♦♥ P②ss
❬❪ ❱♦ ♥s ♦ ② ♦♠♣rss ♦s ♥ ts ♣♣t♦♥
t♦ t P ♠t♦ t♦ ♣s
❬❪ ♥r ♣♥ t♠t ♠♦s ♦ tr♥ ♦♥♦♥♠ Prss
❬❪ rr♥ r♥ ♦rs ❱♦ ♠♣r♦ t♠ s♠ ♥trt♦♥ ♣♣r♦ ♦r ♥ t s♠ ♥②t ♦♥r② ♦♥t♦♥s♥ s♣rts Pr♦ V th P ♥tr♥t♦♥ ❲♦rs♦♣ ♥str❯
❬❪ ♦st♥ P♦♦ ♦ ♦♥srt♦♥ ♦ t ♠t♦♥♥ s♦♥❲s② ♥ r♥s♦ 3rdedition
❬❪ r♦♥ ♦ Pr♥s♦♥ ♠r s♠t♦♥ ♦ t ♦ ♥ Pt♦♥ tr♥ s♥ t ♠sss ♠t♦ P ♥ ♥ s♠♣ s♦♦♥r② trt♠♥t t r♦♣♥ ♦♥r♥ ♦♥ r♦♠♥r②
②♥♠s ♥ r♠♦②♥♠s t♥s r
❬❪ ❲♥♥ Ps ♣♦②♥♦♠ ♣♦st ♥t ♥ ♦♠♣t② s♣♣♦rt r ♥t♦♥s ♦ ♠♥♠ r ♥s ♥ ♦♠♣tt♦♥
t♠ts ♠r
❬❪ ❱♦ ss ♥♠♦ ♦r ♦rt ♦♥ s ♣ss t s♣ ♥ r♥ ♦s t ♥♥♦ tr♥t ♦sr Pr♦ IIIrd P ♥tr♥t♦♥ ❲♦rs♦♣ s♥♥ ss
❬❪ ♦❴tr♥ ♥t ❱♦♠ ♦ ♦r t ♦♠♣tt♦♥ ♦ r♥t
♥♦♠♣rss ♦s ♥str ♣♣t♦♥s ♦ ♥tr♥t♦♥ ♦r♥♦♥ ♥t ❱♦♠s
❬❪ P♦♣ r♥t ♦s ♠r ❯♥rst② Prss
❬❪ r♦t ②r♦②♥♠s ♦ r r ♦s ♦♥ t t
♥t ♠♥t t♦ ♦♥ ❲②