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Dynamique des écoulements fuide-particules cisaillés : simulations numériques locales

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OATAO

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OATAO is an open access repository that collects the work of Toulouse researchers and

makes it freely available over the web where possible.

This is an author-deposited version published in:

http://oatao.univ-toulouse.fr/

Eprints ID : 15889

To cite this version

:

Bouteloup, Joris and Lacaze, Laurent and Bonometti, Thomas and

Charru, François Dynamique des écoulements fuide-particules

cisaillés : simulations numériques locales. (2015) In: 22e Congrès

Français de Mécanique, 24 August 2015 - 28 August 2015 (Lyon,

France)

Any correspondence concerning this service should be sent to the repository

administrator:

[email protected]

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✷✷ ♠❡ ❈♦♥❣%&' ❋%❛♥*❛✐' ❞❡ ▼/❝❛♥✐1✉❡ ▲②♦♥✱ ✷✹ ❛✉ ✷✽ ❆♦9: ✷✵✶✺

❉②♥❛♠✐&✉❡ ❞❡* +❝♦✉❧❡♠❡♥/*

✢✉✐❞❡✲♣❛3/✐❝✉❧❡* ❝✐*❛✐❧❧+* ✿

*✐♠✉❧❛/✐♦♥* ♥✉♠+3✐&✉❡* ❧♦❝❛❧❡*

❏✳ ❇❖❯❚❊▲❖❯(

✱ ▲✳ ▲❆❈❆❩❊✱ ❚✳ ❇❖◆❖▼❊❚❚■✱

❋✳ ❈❍❆❘❘❯

■♥"#✐#✉# ❞❡ ▼)❝❛♥✐,✉❡ ❞❡" ❋❧✉✐❞❡" ❞❡ ❚♦✉❧♦✉"❡✱ ❯▼❘ ❈◆❘❙✴■◆8❚✴❯8❙ ✺✺✵✷✱ ✷✱ ❆❧❧)❡ ❈❛♠✐❧❧❡ ❙♦✉❧❛✱ ✸✶✹✵✵ ❚♦✉❧♦✉"❡✱ ❋A❛♥❝❡

❘!"✉♠! ✿

❉❡' '✐♠✉❧❛:✐♦♥' ♥✉♠/%✐1✉❡' ❧♦❝❛❧❡' ❞✬/❝♦✉❧❡♠❡♥:' ❧❛♠✐♥❛✐%❡' ❝✐'❛✐❧❧/' /%♦✲ ❞❛♥: ✉♥ ❧✐: ❞❡ ♣❛%:✐❝✉❧❡' ♦♥: /:/ %/❛❧✐'/❡'✳ ▲❛ ♠/:❤♦❞❡ ♥✉♠/%✐1✉❡ ❡♠♣❧♦②/❡ ❡': ✉♥ ♠♦❞&❧❡ ❞❡ :②♣❡ ❊✉❧❡%✲▲❛❣%❛♥❣❡✱ %❡♣♦'❛♥: ♣♦✉% ❧❛ ♣❤❛'❡ ✢✉✐❞❡ '✉% ❧❡' /1✉❛:✐♦♥' ❞❡ ◆❛✈✐❡%✲❙:♦❦❡' ♠♦②❡♥♥/❡' ♣❛% ✉♥ :❡%♠❡ ❞❡ ♣♦%♦'✐:/ ε ❡: ♣♦✉% ❧❛ ♣❤❛'❡ '♦❧✐❞❡ '✉% ❧❡' /1✉❛:✐♦♥' ❞❡ ◆❡✇:♦♥ %/'♦❧✉❡' ♣♦✉% ❝❤❛1✉❡ ♣❛%:✐❝✉❧❡✳ ❯♥ :❡%♠❡ ❞✬✐♥:❡%❛❝:✐♦♥ ✢✉✐❞❡✲♣❛%:✐❝✉❧❡ ♣❡%♠❡: ❞❡ ❝♦✉♣❧❡% ❧❡' ❞❡✉① ♣❤❛'❡'✳ ▲❡' '✐♠✉❧❛:✐♦♥'✱ ❡✛❡❝:✉/❡' P Rep ❡: ρp/ρf ❝♦♥':❛♥:' ❡: ❜❛❧❛②❛♥: ✉♥❡ ❣❛♠♠❡ ❞❡ ♥♦♠❜%❡ ❞❡ ❙❤✐❡❧❞' θ ❛❧❧❛♥: ❞❡ ✵✳✶ P ✵✳✺✱ ♦♥: ♣❡%♠✐' ❞❡ :%♦✉✈❡% ✉♥ ❙❤✐❡❧❞' ❝%✐:✐1✉❡ θc✱ ❝❛%❛❝:/%✐'❛♥: ❧❡ '❡✉✐❧ ❞❡ ♠✐'❡ ❡♥ ♠♦✉✈❡♠❡♥: ❞✉ ❧✐:✱ ❡♥:%❡ ✵✳✶ ❡: ✵✳✶✷✳ ❯♥❡ /:✉❞❡ ❞✉ ❞/❜✐: ':❛:✐♦♥♥❛✐%❡ ❞✉ ♠✐❧✐❡✉ ❣%❛♥✉❧❛✐%❡ ♣❡%♠❡: ❞❡ ✈❛❧✐❞❡% ❧❛ ♠/:❤♦❞❡ ♥✉♠/%✐1✉❡ ✉:✐❧✐'/❡✳

❆❜"()❛❝( ✿

▲♦❝❛❧ ♥✉♠❡%✐❝❛❧ '✐♠✉❧❛:✐♦♥' ♦❢ ❧❛♠✐♥❛% '❤❡❛% ✢♦✇ ❡%♦❞✐♥❣ ❛ ❜❡❞ ♦❢ ♣❛%:✐❝❧❡' ❛%❡ ♣❡%❢♦%♠❡❞✳ ❚❤❡ ♥✉♠❡%✐❝❛❧ ♠❡:❤♦❞ ✉'❡❞ ✐' ❛♥ ❊✉❧❡%✲▲❛❣%❛♥❣❡ ♠♦❞❡❧✱ '♦❧✈✐♥❣ :❤❡ ◆❛✈✐❡%✲❙:♦❦❡' ❡1✉❛:✐♦♥' ❛✈❡%❛❣❡❞ ❜② ❛ ♣♦%♦'✐:② :❡%♠ ε ❢♦% :❤❡ ✢✉✐❞ ♣❤❛'❡ ❛♥❞ ◆❡✇:♦♥✬' ❡1✉❛:✐♦♥' ❢♦% :❤❡ '♦❧✐❞ ♣❤❛'❡✳ ❆ ✢✉✐❞✲♣❛%:✐❝❧❡ ✐♥:❡%❛❝:✐♦♥ :❡%♠ ❡♥❛❜❧❡' ❛ :✇♦✲✇❛② ❝♦✉♣❧✐♥❣✳ ❚❤❡ '✐♠✉❧❛:✐♦♥'✱ ♣❡%❢♦%♠❡❞ ✇✐:❤ Rep ❛♥❞ ρp/ρf %❡♠❛✐♥✐♥❣ ❝♦♥':❛♥: ❛♥❞ ❝♦✈❡%✐♥❣ ❛ %❛♥❣❡ ♦❢ ❙❤✐❡❧❞' ♥✉♠❜❡%' θ ❢%♦♠ ✵✳✶ :♦ ✵✳✺✱ ❛❧❧♦✇ :♦ ✜♥❞ ❛ ❝%✐:✐❝❛❧ ❙❤✐❡❧❞' ♥✉♠❜❡% θc✱ ✇❤✐❝❤ %❡♣%❡'❡♥:' :❤❡ ❧✐♠✐: ❜❡:✇❡❡♥ ':❛:✐❝ ❛♥❞ ❞②♥❛♠✐❝ ❜❡❞✱ ❜❡:✇❡❡♥ ✵✳✶ ❛♥❞ ✵✳✶✷✳ ❆ ':✉❞② ♦❢ :❤❡ '❛:✉%❡❞ ♣❛%:✐❝❧❡ ✢♦✇ %❛:❡ ✈❛❧✐❞❛:❡' :❤❡ ♥✉♠❡%✐❝❛❧ ♠❡:❤♦❞ ✉'❡❞✳

▼♦45 ❝❧❡❢5 ✿ ❚;❛♥5♣♦;4 5?❞✐♠❡♥4❛✐;❡✱ ❈❋❉✲❉❊▼✱ ♥♦♠❜;❡

❞❡ ❙❤✐❡❧❞5✱ ?❝♦✉❧❡♠❡♥4 ❧❛♠✐♥❛✐;❡

❛✳ ❊✲♠❛✐❧ ✿ ❥♦*✐+✳❜♦✉.❡❧♦✉♣❅✐♠❢.✳❢*

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✶ ■♥#$♦❞✉❝#✐♦♥

▲❡ "#❛♥&♣♦#" &)❞✐♠❡♥"❛✐#❡ ❡♥ ♠✐❧✐❡✉ ♥❛"✉#❡❧ ❢❛✐" ✐♥"❡#✈❡♥✐# ❞❡& ♠)❝❛♥✐&♠❡& ❝♦♠♣❧❡①❡& 3 ❞✐✛)#❡♥"❡& )❝❤❡❧❧❡& ❞❡ "❡♠♣& ❡" ❞✬❡&♣❛❝❡✳ ❆✜♥ ❞❡ ❞)❝#✐#❡ ❝❡& &②&✲ "<♠❡&✱ ✐❧ ❡&" ❡♥ ♣❛#"✐❝✉❧✐❡# ♥)❝❡&&❛✐#❡ ❞❡ ♠✐❡✉① ❝♦♠♣#❡♥❞#❡ ❧✬✐♥✢✉❡♥❝❡ ❞❡& ✐♥"❡#✲ ❛❝"✐♦♥& ❧♦❝❛❧❡&✱ ❡♥"#❡ ❧❡ ✢✉✐❞❡ ❡" ❧❡& ❣#❛✐♥& ❡" ❡♥"#❡ ❧❡& ❣#❛✐♥& ❡✉①✲♠@♠❡✱ &✉# ❧❡& &"#✉❝"✉#❡& ❞❡ ❣#❛♥❞❡& )❝❤❡❧❧❡& ✭#✐❞❡&✱ ❞✉♥❡&✱ ❡"❝ ❬✶❪✮✳ F♦✉# ❝❡❧❛✱ ♥♦✉& ✉"✐❧✐&♦♥& ❞❛♥& ❝❡""❡ )"✉❞❡ ✉♥❡ ♠)"❤♦❞❡ ♥✉♠)#✐G✉❡ ❜❛&)❡ &✉# ✉♥❡ ❛♣♣#♦❝❤❡ ❞❡ "②♣❡ ❊✉❧❡#✲ ▲❛❣#❛♥❣❡ ❬✷❪✳ ❈❡""❡ ♠)"❤♦❞❡ ♣❡#♠❡" ❞❡ #❡♠♦♥"❡# ❛✉① ❝❛#❛❝")#✐&"✐G✉❡& ❧♦❝❛❧❡& ❞❡ ❧✬)❝♦✉❧❡♠❡♥" ✢✉✐❞❡✲❣#❛✐♥& ❞✬✉♥❡ ♣❛#" ❡" ❞✬❛❝❝)❞❡# ❛✉① ✐♥❢♦#♠❛"✐♦♥& ♠)❝❛♥✐G✉❡& ❧✐)❡& ❛✉① ❝♦❧❧✐&✐♦♥& ❡♥"#❡ ❣#❛✐♥& ❞✬❛✉"#❡ ♣❛#" "♦✉" ❡♥ ♣❡#♠❡""❛♥" ❧❛ &✐♠✉❧❛"✐♦♥ ❞✬✉♥ ❣#❛♥❞ ♥♦♠❜#❡ ❞❡ ❣#❛✐♥& ✭O(106)✮✳ ▲❡ ♠♦❞<❧❡ ❊✉❧❡#✲▲❛❣#❛♥❣❡ ❡&" ❜❛&) &✉#

❧❛ #)&♦❧✉"✐♦♥ ❞❡& )G✉❛"✐♦♥& ♠♦②❡♥♥)❡& ♣♦✉# ❧❛ ♣❤❛&❡ ✢✉✐❞❡ ❡♥ ♣#)&❡♥❝❡ ❞❡ ♣❛#✲ "✐❝✉❧❡& ✭❏❛❝❦&♦♥ ❬✸❪✮✱ ❝♦✉♣❧)❡ 3 ✉♥❡ ♠)"❤♦❞❡ ❛✉① )❧)♠❡♥"& ❞✐&❝#❡"& ✭❉❊▼✮ ♣♦✉# ❧❡ &✉✐✈✐ ❧❛❣#❛♥❣✐❡♥ ❞❡& ♣❛#"✐❝✉❧❡&✳

❈❡""❡ ♠)"❤♦❞❡ ❡&" ✉"✐❧✐&)❡ ✐❝✐ ♣♦✉# ❞)❝#✐#❡ ❧❛ ❞②♥❛♠✐G✉❡ ❝♦✉♣❧)❡ ❞✬✉♥ ❧✐" ❞❡ ❣#❛✐♥& ❝✐&❛✐❧❧)& ♣❛# ✉♥ )❝♦✉❧❡♠❡♥" ❤♦♠♦❣<♥❡ ❡" &"❛"✐♦♥♥❛✐#❡ ♣#♦❝❤❡ ❞✉ &❡✉✐❧ ❞❡ ♠✐&❡ ❡♥ ♠♦✉✈❡♠❡♥"✳ ❈❡& ♣#♦❜❧)♠❛"✐G✉❡& ♦♥"✱ ❛✉ ❝♦✉#& ❞❡& ❞❡#♥✐<#❡& ❞)❝❡♥♥✐❡&✱ ❞)❥3 )") ❛❜♦#❞)❡& ❞❛♥& ✉♥ ❝❡#"❛✐♥ ♥♦♠❜#❡ ❞✬)"✉❞❡& ♣♦✉# ❞❡& ❝❛& ❞✬)❝♦✉❧❡♠❡♥"& ❧❛♠✐♥❛✐#❡& ♦✉ "✉#❜✉❧❡♥"&✱ G✉❡ ❝❡ &♦✐" ❡①♣)#✐♠❡♥"❛❧❡♠❡♥" ❬✹✕✽❪ ♦✉ ♥✉♠)#✐G✉❡✲ ♠❡♥" ❬✾✕✶✶❪✳

◆♦✉& ♣#)&❡♥"♦♥& "♦✉" ❞✬❛❜♦#❞ ❧❡ ♠♦❞<❧❡ ♥✉♠)#✐G✉❡ ✉"✐❧✐&) ❛✐♥&✐ G✉❡ ❧❡& ♣❛✲ #❛♠<"#❡& ♣❤②&✐G✉❡& ❡" ♥✉♠)#✐G✉❡& ✐♠♣♦#"❛♥"& ♣✉✐& ❧❡& #)&✉❧"❛"& ❞❡ &✐♠✉❧❛"✐♦♥& &♦♥" ❛♥❛❧②&)&✳

✷ ▼,#❤♦❞❡ ♥✉♠,$✐0✉❡

▲❡ ❧♦❣✐❝✐❡❧ ✉"✐❧✐&) ✐❝✐ ♣♦✉# ❧❛ ♣❤❛&❡ ❣#❛♥✉❧❛✐#❡ ❡&" ❧❡ ❝♦❞❡ ●#❛❉②▼ ✭●#❛♥✉✲ ❧❛# ❉②♥❛♠✐❝ ▼♦❞❡❧❧✐♥❣ ❬✶✷❪✮ G✉✐ #❡♣♦&❡ &✉# ✉♥❡ ♠)"❤♦❞❡ ❉❊▼ ❬✶✸✱ ✶✹❪✳ ❈❡""❡ ♠)"❤♦❞❡ ❝♦♥&✐&"❡ ❡♥ ❧❛ ❞)"❡#♠✐♥❛"✐♦♥ ❞❡& "#❛❥❡❝"♦✐#❡& ❞✬✉♥ ❣#❛♥❞ ♥♦♠❜#❡ ❞❡ ♣❛#"✐❝✉❧❡& ❡♥ #)&♦❧✈❛♥" ♣♦✉# ❝❤❛❝✉♥❡ ❞✬❡♥"#❡ ❡❧❧❡& ❧❡& )G✉❛"✐♦♥& ❞❡ ◆❡✇"♦♥ ❡" ❡♥ ❝♦♥&✐❞)#❛♥" ❧❡✉#& ✐♥"❡#❛❝"✐♦♥& ❛✈❡❝ ❧❡ ✢✉✐❞❡ ❡♥✈✐#♦♥♥❛♥" ♦✉ ❛✈❡❝ ❞✬❛✉"#❡& ♣❛#"✐✲ ❝✉❧❡&✳

❖♥ ❝♦♥&✐❞<#❡ ❛✐♥&✐ ✉♥ ♥♦♠❜#❡ Np❞❡ ♣❛#"✐❝✉❧❡& &♣❤)#✐G✉❡& G✉✐ )✈♦❧✉❡♥"✱ &❡❧♦♥

❧❡ &❡❝♦♥❞ ♣#✐♥❝✐♣❡ ❞❡ ◆❡✇"♦♥ ✿ ρpVpd✉p dt = X j6=p Fpj+❋murs+ ρpVp❣ + ❋f p+❋lub ∀p = 1, .., Np ✭✶✮ Ipdωp dt = X j6=p τpj+ τmurs ∀p = 1, .., Np ✭✷✮

(4)

✷✷ ♠❡ ❈♦♥❣%&' ❋%❛♥*❛✐' ❞❡ ▼/❝❛♥✐1✉❡ ▲②♦♥✱ ✷✹ ❛✉ ✷✽ ❆♦9: ✷✵✶✺ ❛✈❡❝✱ ♣♦✉( ❝❤❛*✉❡ ♣❛(+✐❝✉❧❡✱ ρp ❧❛ ♠❛//❡ ✈♦❧✉♠✐*✉❡✱ Vp ❧❡ ✈♦❧✉♠❡✱ ✉p ❧❛ ✈✐+❡//❡ ❞❡ +(❛♥/❧❛+✐♦♥✱ ωp❧❛ ✈✐+❡//❡ ❞❡ (♦+❛+✐♦♥✱ Ip❧❡ ❝♦❡✣❝✐❡♥+ ❞✬✐♥❡(+✐❡✱ Fpj ❧❡/ ❢♦(❝❡/ ❞❡ ❝♦♥+❛❝+ ❡①❡(❝6❡/ ♣❛( ❧❡/ ❛✉+(❡/ ♣❛(+✐❝✉❧❡/ /✉( ❧❛ ♣❛(+✐❝✉❧❡ ♣✱ ❋murs❧❡/ ❢♦(❝❡/ ❡①❡(❝6❡/ ♣❛( ❧❡/ ♠✉(/✱ ❋lub❧❡/ ❢♦(❝❡/ ❞❡ ❧✉❜(✐✜❝❛+✐♦♥ ❡①❡(❝6❡/ ♣❛( ❧❡/ ❛✉+(❡/ ♣❛(✲ +✐❝✉❧❡/ ✭✈♦✐( ■③❛(❞ ❡+ ❛❧✳ ❬✶✷❪ ♣♦✉( ♣❧✉/ ❞❡ ❞6+❛✐❧/✮✱ ❡+ ❋f p❧❡/ ❢♦(❝❡/ ❡①❡(❝6❡/ ♣❛( ❧❡ ✢✉✐❞❡ /✉( ❧❛ ♣❛(+✐❝✉❧❡ ♣✳ τpj✱ τmurs /♦♥+ ❧❡/ ❝♦✉♣❧❡/ ❡①❡(❝6/ (❡/♣❡❝+✐✈❡♠❡♥+ ♣❛( ❧❡/ ❛✉+(❡/ ♣❛(+✐❝✉❧❡/ ❡+ ❧❡/ ♠✉(/✳ D♦✉( ❧❛ ♣❤❛/❡ ✢✉✐❞❡✱ ❧✬❛♣♣(♦❝❤❡ ❞6✈❡❧♦♣♣6❡ ♣❛( ❏❛❝❦/♦♥ ❬✸❪ ♣❡(♠❡+ ❞❡ ❞6✲ ❝(✐(❡ ❧❛ ❞②♥❛♠✐*✉❡ ❞✉ ✢✉✐❞❡ I ✉♥❡ 6❝❤❡❧❧❡ ♣❧✉/ ❣(❛♥❞❡ *✉❡ ❧❡/ ♣❛(+✐❝✉❧❡/✳ ❊❧❧❡ ❝♦♥/✐/+❡ I (66❝(✐(❡ ❧❡/ 6*✉❛+✐♦♥/ ❞❡ ◆❛✈✐❡(✲❙+♦❦❡/ ❡♥ ♠♦②❡♥♥❛♥+ /♣❛+✐❛❧❡♠❡♥+ ❧❡/ ❣(❛♥❞❡✉(/ ❧♦❝❛❧❡/ ❞❡ ❧✬6❝♦✉❧❡♠❡♥+ I +(❛✈❡(/ ✉♥❡ ❢♦♥❝+✐♦♥ ε✱ ❝❛(❛❝+6(✐/❛♥+ ❧❛ ❢(❛❝+✐♦♥ ✈♦❧✉♠✐*✉❡ ❞✬❡/♣❛❝❡ ♦❝❝✉♣6❡ ♣❛( ❧❡ ✢✉✐❞❡ ♣❛( (❛♣♣♦(+ ❛✉① ♣❛(+✐❝✉❧❡/ ❞❛♥/ ✉♥ ✈♦❧✉♠❡ 6❧6♠❡♥+❛✐(❡✳ ▲❡/ 6*✉❛+✐♦♥/ ❞❡ ❝♦♥/❡(✈❛+✐♦♥ ❞❡ ❧❛ ♠❛//❡ ❡+ ❞❡ ❧❛ *✉❛♥+✐+6 ❞❡ ♠♦✉✈❡♠❡♥+ ❞❡✈✐❡♥♥❡♥+ ✿ ∂ε ∂t+ ∇ · (ε✉f) = 0 ✭✸✮ ρfε Df✉f Dt = ∇ ·❙ f❢ + ρ fε❣ ✭✹✮ ❛✈❡❝ ✉f ❧❛ ✈✐+❡//❡ ❞✉ ✢✉✐❞❡✱ ρf /❛ ♠❛//❡ ✈♦❧✉♠✐*✉❡✱ ❢ ❧❛ ♠♦②❡♥♥❡ /✉( ❧❡ ✈♦❧✉♠❡ 6❧6♠❡♥+❛✐(❡ ❞❡/ ❢♦(❝❡/ ❡①❡(❝6❡/ ♣❛( ❧❡/ ♣❛(+✐❝✉❧❡/ /✉( ❧❡ ✢✉✐❞❡ ❡+ ❙f ❧❡ +❡♥/❡✉( ❞❡/ ❝♦♥+(❛✐♥+❡/✳ ❙♦♥ ❡①♣(❡//✐♦♥ ♥❡ ♣❡✉+ Q+(❡ ❡①♣❧✐❝✐+❡♠❡♥+ ❞6+❡(♠✐♥6❡ ❡+ ❞✐✛6✲ (❡♥+/ ♠♦❞S❧❡/ ♣❡✉✈❡♥+ Q+(❡ +(♦✉✈6/ ❞❛♥/ ❧❛ ❧✐++6(❛+✉(❡✳ ◆♦✉/ ❝❤♦✐/✐//♦♥/ ✐❝✐ ✉♥❡ ❡①♣(❡//✐♦♥ 6*✉✐✈❛❧❛♥+ I ✉♥ ✢✉✐❞❡ ❛♣♣❛(❡♥+ ❞♦♥+ ❧❡/ ♣(♦♣(✐6+6/ ❞6♣❡♥❞❡♥+ ❞❡ ❧❛ ♣♦(♦/✐+6 /♦✉/ ❧❛ ❢♦(♠❡ ✿ ❙f = −εpI + µ∗∇m+ ∇✉t m  ✭✺✮ ❛✈❡❝ p ❧❛ ♣(❡//✐♦♥ ❤②❞(♦❞②♥❛♠✐*✉❡✱ I ❧❛ ♠❛+(✐❝❡ ✐❞❡♥+✐+6✱ µ∗= µ fε−2.8 ❧❛ ✈✐/✲ ❝♦/✐+6 ❡✛❡❝+✐✈❡ ❞6✜♥✐❡ ♣❛( ●✐❜✐❧❛(♦ ❡+ ❛❧✳ ❬✶✺❪ ❛✜♥ ❞❡ ♣(❡♥❞(❡ ❡♥ ❝♦♠♣+❡ ❧❛ ♣(6/❡♥❝❡ ❞❡/ ❣(❛✐♥/ ❞❛♥/ ❧❡ ✢✉✐❞❡ ❡+ ✉m= ε✉f+ (1 − ε)✉p❧❛ ✈✐+❡//❡ ❞❡ ♠6❧❛♥❣❡ ✢✉✐❞❡✲♣❛(+✐❝✉❧❡/✳ ▲❡/ ❞❡✉① +❡(♠❡/ ❞✬✐♥+❡(❛❝+✐♦♥ ✢✉✐❞❡✲♣❛(+✐❝✉❧❡/ ❞❡/ 6*✉❛+✐♦♥/ ✭✶✮ ❡+ ✭✹✮ /♦♥+ (❡❧✐6❡/ ♣❛( ❧❛ (❡❧❛+✐♦♥ /✉✐✈❛♥+❡ ✿ ❢ = 1 Vc X p∈c ❋f p ❛✈❡❝ ❋f p= V p∇ ·❙f+❋drag ✭✻✮ ▲✬❡①♣(❡//✐♦♥ ❞❡ ❋f p❡/+ ✉♥❡ ❛♣♣(♦①✐♠❛+✐♦♥ ❢♦(♠✉❧6❡ ♣❛( ❈❛♣❡❝❡❧❛+(♦ ❡+ ❛❧✳ ❬✷❪✳ ❖♥ ✉+✐❧✐/❡ ❧❡ ♠♦❞S❧❡ ❞❡ +(❛Y♥6❡ ♣(♦♣♦/6 ♣❛( ❚❡♥♥❡+✐ ❡+ ❛❧✳ ❬✶✻❪ ♣♦✉( ❝❛❧❝✉❧❡( ❋drag✱ ♠♦❞S❧❡ ✈❛❧❛❜❧❡ ♣♦✉( ✉♥❡ ❧❛(❣❡ ❣❛♠♠❡ ❞❡ ♥♦♠❜(❡ ❞❡ ❘❡②♥♦❧❞/ ❡+ ❞❡ ❝♦♠♣❛❝✐+6✳ ❈❡ /♦♥+ ✐❝✐ ❧❡/ /❡✉❧❡/ ❝♦♥+(✐❜✉+✐♦♥/ ❤②❞(♦❞②♥❛♠✐*✉❡/ ✐♠♣❧6♠❡♥+6❡/ ♣♦✉( ❧❡ ❝♦✉♣❧❛❣❡ ✢✉✐❞❡✲❣(❛✐♥/✳

(5)

θ = µfγ (ρp−ρf)gdp = τ (ρp−ρf)gdp , Rep= γd2 pρf µf dp γ τ θc 0.12±0.03 Rep 1.5 × 10−5 0.76 θ = 0.4 0 ≤ up ≤ 0.1γdp θ Rep = 0.48 ρp/ρf = 4 g γ hbed≈10dp ≈ 0.7

(6)

U0 x z φinit ≈ 0.56 q = 1/(Aφbed)VpPpup(m2/s dp 5 × 10−4 m3 µf 10−3 P a.s ρf 103 kg.m−3 ρp 4 × 103 kg.m−3 e 0.8 µ 0.1 ∆x/dp= ∆y/dp= 2∆z/5dp 2 ∆ts 5 × 10−6 s ∆tf 10−4 s tsimu 400 s qsat θ q = a(θ − θc)3/2 θc≈0.1 Rep θ = 0.4 qsat θ

(7)

❘!❢!#❡♥❝❡'

❬✶❪ ❋✳ ❈❤❛((✉✱ ❇✳ ❆♥❞(❡♦11✐✱ ❛♥❞ 3✳ ❈❧❛✉❞✐♥✳ ❙❛♥❞ (✐♣♣❧❡7 ❛♥❞ ❞✉♥❡7✳ ❆♥♥✉❛❧ ❘❡✈✐❡✇ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝'✱ ✈♦❧✳ ✹✺✭✶✮ ✿✹✻✾✕✹✾✸✱ ✷✵✶✸✳ ❬✷❪ ❏✳ ❈❛♣❡❝❡❧❛1(♦ ❛♥❞ ❖✳ ❉❡7❥❛(❞✐♥7✳ ❆♥ ❡✉❧❡(✕❧❛❣(❛♥❣❡ 71(❛1❡❣② ❢♦( 7✐♠✉❧❛1✐♥❣ ♣❛(1✐❝❧❡✲❧❛❞❡♥ ✢♦✇7✳ ❏♦✉%♥❛❧ ♦❢ ❈♦♠♣✉:❛:✐♦♥❛❧ G❤②'✐❝'✱ ✈♦❧✳ ✷✸✽✭✵✮ ✿✶✕✸✶✱ ✷✵✶✸✳ ❬✸❪ ❘✳ ❏❛❝❦7♦♥✳ ❚❤❡ ❉②♥❛♠✐❝' ♦❢ ❋❧✉✐❞✐③❡❞ G❛%:✐❝❧❡'✳ ❈❛♠❜(✐❞❣❡ ▼♦♥♦❣(❛♣❤7 ♦♥ ▼❡❝❤❛♥✐❝7✳ ❈❛♠❜(✐❞❣❡ ❯♥✐✈❡(7✐1② 3(❡77✱ ✷✵✵✵✳ ❬✹❪ ❋✳ ❈❤❛((✉✱ ❍✳ ▼♦✉✐❧❧❡(♦♥✱ ❛♥❞ ❖✳ ❊✐✛✳ ❊(♦7✐♦♥ ❛♥❞ ❞❡♣♦7✐1✐♦♥ ♦❢ ♣❛(1✐❝❧❡7 ♦♥ ❛ ❜❡❞ 7❤❡❛(❡❞ ❜② ❛ ✈✐7❝♦✉7 ✢♦✇✳ ❏♦✉%♥❛❧ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝'✱ ✈♦❧✳ ✺✶✾ ✿✺✺✕ ✽✵✱ ✷✵✵✹✳ ❬✺❪ ❆✳ ❍♦♥❣ ❛♥❞ ❆✳ ❚❛♦✱ ▼✳❛♥❞ ❑✉❞(♦❧❧✐✳ ❖♥7❡1 ♦❢ ❡(♦7✐♦♥ ♦❢ ❛ ❣(❛♥✉❧❛( ❜❡❞ ✐♥ ❛ ❝❤❛♥♥❡❧ ❞(✐✈❡♥ ❜② ✢✉✐❞ ✢♦✇✳ G❤②'✐❝' ♦❢ ❋❧✉✐❞'✱ ✷✼✭✶✮ ✿✕✱ ✷✵✶✺✳ ❬✻❪ ❆✳ ❊✳ ▲♦❜❦♦✈7❦②✱ ❆✳ ❱✳ ❖(♣❡✱ ❘✳ ▼♦❧❧♦②✱ ❆✳ ❑✉❞(♦❧❧✐✱ ❛♥❞ ❉✳ ❍✳ ❘♦1❤♠❛♥✳ ❊(♦7✐♦♥ ♦❢ ❛ ❣(❛♥✉❧❛( ❜❡❞ ❞(✐✈❡♥ ❜② ❧❛♠✐♥❛( ✢✉✐❞ ✢♦✇✳ ❏♦✉%♥❛❧ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝'✱ ✻✵✺ ✿✹✼✕✺✽✱ ✻ ✷✵✵✽✳ ❬✼❪ ❚✳ ▲♦✐7❡❧❡✉①✱ 3✳ ●♦♥❞(❡1✱ ▼✳ ❘❛❜❛✉❞✱ ❛♥❞ ❉✳ ❉♦♣♣❧❡(✳ ❖♥7❡1 ♦❢ ❡(♦7✐♦♥ ❛♥❞ ❛✈❛❧❛♥❝❤❡ ❢♦( ❛♥ ✐♥❝❧✐♥❡❞ ❣(❛♥✉❧❛( ❜❡❞ 7❤❡❛(❡❞ ❜② ❛ ❝♦♥1✐♥✉♦✉7 ❧❛♠✐♥❛( ✢♦✇✳ G❤②'✐❝' ♦❢ ❋❧✉✐❞'✱ ✶✼✭✶✵✮ ✿✕✱ ✷✵✵✺✳ ❬✽❪ ▼✳ ❖✉(✐❡♠✐✱ 3✳ ❆✉77✐❧❧♦✉7✱ ▼✳ ▼❡❞❛❧❡✱ ❨✳ 3❡②77♦♥✱ ❛♥❞ ❊✳ ●✉❛③③❡❧❧✐✳ ❉❡✲ 1❡(♠✐♥❛1✐♦♥ ♦❢ 1❤❡ ❝(✐1✐❝❛❧ 7❤✐❡❧❞7 ♥✉♠❜❡( ❢♦( ♣❛(1✐❝❧❡ ❡(♦7✐♦♥ ✐♥ ❧❛♠✐♥❛( ✢♦✇✳ G❤②'✐❝' ♦❢ ❋❧✉✐❞'✱ ✶✾✭✻✮ ✿✕✱ ✷✵✵✼✳ ❬✾❪ ❏✳ ❏✳ ❉❡(❦7❡♥✳ ❙✐♠✉❧❛1✐♦♥7 ♦❢ ❣(❛♥✉❧❛( ❜❡❞ ❡(♦7✐♦♥ ❞✉❡ 1♦ ❧❛♠✐♥❛( 7❤❡❛( ✢♦✇ ♥❡❛( 1❤❡ ❝(✐1✐❝❛❧ 7❤✐❡❧❞7 ♥✉♠❜❡(✳ G❤②'✐❝' ♦❢ ❋❧✉✐❞'✱ ✈♦❧✳ ✷✸✭✶✶✮ ✿✶✕✶✷✱ ✷✵✶✶✳ ❬✶✵❪ ❖✳ ❉✉(b♥✱ ❇✳ ❆♥❞(❡♦11✐✱ ❛♥❞ 3✳ ❈❧❛✉❞✐♥✳ ◆✉♠❡(✐❝❛❧ 7✐♠✉❧❛1✐♦♥ ♦❢ 1✉(❜✉❧❡♥1 7❡❞✐♠❡♥1 1(❛♥7♣♦(1✱ ❢(♦♠ ❜❡❞ ❧♦❛❞ 1♦ 7❛❧1❛1✐♦♥✳ G❤②'✐❝' ♦❢ ❋❧✉✐❞'✱ ✷✹✭✶✵✮ ✿✶✕ ✷✺✱ ✷✵✶✷✳ ❬✶✶❪ ❊✳ 3❛♣✐71❛✱ ❉✳ ❉✐♠✐1(❛❦✐7✱ ❛♥❞ ❙✳ ●✳ ❨✐❛♥17✐♦7✳ ❉✐(❡❝1 ♥✉♠❡(✐❝❛❧ 7✐♠✉❧❛✲ 1✐♦♥ ♦❢ ✐♥❝✐♣✐❡♥1 7❡❞✐♠❡♥1 ♠♦1✐♦♥ ❛♥❞ ❤②❞(❛✉❧✐❝ ❝♦♥✈❡②✐♥❣✳ ■♥❞✉':%✐❛❧ ✫ ❊♥❣✐♥❡❡%✐♥❣ ❈❤❡♠✐':%② ❘❡'❡❛%❝❤✱ ✺✵✭✷✮ ✿✻✸✵✕✻✸✽✱ ✷✵✶✶✳ ❬✶✷❪ ❊✳ ■③❛(❞✱ ❚✳ ❇♦♥♦♠❡11✐✱ ❛♥❞ ▲✳ ▲❛❝❛③❡✳ ▼♦❞❡❧❧✐♥❣ 1❤❡ ❞②♥❛♠✐❝7 ♦❢ ❛ 7♣❤❡(❡ ❛♣♣(♦❛❝❤✐♥❣ ❛♥❞ ❜♦✉♥❝✐♥❣ ♦♥ ❛ ✇❛❧❧ ✐♥ ❛ ✈✐7❝♦✉7 ✢✉✐❞✳ ❏♦✉%♥❛❧ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝'✱ ✈♦❧✳ ✼✹✼ ✿✹✷✷✕✹✹✻✱ ✷✵✶✹✳ ❬✶✸❪ ❋✳ ❘❛❞❥❛e ❛♥❞ ❋✳ ❉✉❜♦✐7✳ ❉✐'❝%❡:❡✲❡❧❡♠❡♥: ♠♦❞❡❧✐♥❣ ♦❢ ❣%❛♥✉❧❛% ♠❛:❡%✐❛❧'✳ ❖①❢♦(❞ ✿ ■❙❚❊✱ ✷✵✶✶✳ ❬✶✹❪ 3✳ ❆✳ ❈✉♥❞❛❧❧ ❛♥❞ ❖✳ ❉✳ ▲✳ ❙1(❛❝❦✳ ❆ ❞✐7❝(❡1❡ ♥✉♠❡(✐❝❛❧ ♠♦❞❡❧ ❢♦( ❣(❛♥✉❧❛( ❛77❡♠❜❧✐❡7✳ ●/♦:❡❝❤♥✐1✉❡✱ ✈♦❧✳ ✷✾ ✿✹✼✕✻✺✱ ✶✾✼✾✳

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✷✷ ♠❡ ❈♦♥❣%&' ❋%❛♥*❛✐' ❞❡ ▼/❝❛♥✐1✉❡ ▲②♦♥✱ ✷✹ ❛✉ ✷✽ ❆♦9: ✷✵✶✺ ❬✶✺❪ ▲✳●✳ ●✐❜✐❧❛+♦✱ ❑✳ ●❛❧❧✉❝❝✐✱ ❘✳ ❉✐ ❋❡❧✐❝❡✱ ❛♥❞ 7✳ 7❛❣❧✐❛✐✳ ❖♥ :❤❡ ❛♣♣❛+❡♥: ✈✐>❝♦>✐:② ♦❢ ❛ ✢✉✐❞✐③❡❞ ❜❡❞✳ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡%✐♥❣ ❙❝✐❡♥❝❡✱ ✻✷✭✶✕✷✮ ✿✷✾✹ ✕ ✸✵✵✱ ✷✵✵✼✳ ❋❧✉✐❞✐③❡❞ ❇❡❞ ❆♣♣❧✐❝❛:✐♦♥>✳ ❬✶✻❪ ❙✳ ❚❡♥♥❡:✐✱ ❘✳ ●❛+❣✱ ❛♥❞ ❙✳ ❙✉❜+❛♠❛♥✐❛♠✳ ❉+❛❣ ❧❛✇ ❢♦+ ♠♦♥♦❞✐>♣❡+>❡ ❣❛>✕>♦❧✐❞ >②>:❡♠> ✉>✐♥❣ ♣❛+:✐❝❧❡✲+❡>♦❧✈❡❞ ❞✐+❡❝: ♥✉♠❡+✐❝❛❧ >✐♠✉❧❛:✐♦♥ ♦❢ ✢♦✇ ♣❛>: ✜①❡❞ ❛>>❡♠❜❧✐❡> ♦❢ >♣❤❡+❡>✳ ■♥:❡%♥❛:✐♦♥❛❧ ❏♦✉%♥❛❧ ♦❢ ▼✉❧:✐♣❤❛'❡ ❋❧♦✇✱ ✸✼✭✾✮ ✿✶✵✼✷ ✕ ✶✵✾✷✱ ✷✵✶✶✳ ❬✶✼❪ ❇✳ ❈❛♠❡♥❡♥ ❛♥❞ ▼✳ ▲❛+>♦♥✳ ❆ ❣❡♥❡+❛❧ ❢♦+♠✉❧❛ ❢♦+ ♥♦♥✲❝♦❤❡>✐✈❡ ❜❡❞ ❧♦❛❞ >❡❞✐♠❡♥: :+❛♥>♣♦+:✳ ❊':✉❛%✐♥❡✱ ❈♦❛':❛❧ ❛♥❞ ❙❤❡❧❢ ❙❝✐❡♥❝❡✱ ✻✸✭✶✕✷✮ ✿✷✹✾ ✕ ✷✻✵✱ ✷✵✵✺✳ ❬✶✽❪ ❊✳ ▲❛❥❡✉♥❡>>❡✱ ▲✳ ▼❛❧✈❡+:✐✱ ❛♥❞ ❋✳ ❈❤❛++✉✳ ❇❡❞ ❧♦❛❞ :+❛♥>♣♦+: ✐♥ :✉+❜✉❧❡♥: ✢♦✇ ❛: :❤❡ ❣+❛✐♥ >❝❛❧❡ ✿ ❊①♣❡+✐♠❡♥:> ❛♥❞ ♠♦❞❡❧✐♥❣✳ ❏♦✉%♥❛❧ ♦❢ ●❡♦♣❤②'✐❝❛❧ ❘❡'❡❛%❝❤ ✿ ❊❛%:❤ ❙✉%❢❛❝❡✱ ✶✶✺✭❋✹✮ ✿✕✱ ✷✵✶✵✳ ❬✶✾❪ ❏✳ ❙✳ ❘✐❜❜❡+✐♥❦✳ ❇❡❞✲❧♦❛❞ :+❛♥>♣♦+: ❢♦+ >:❡❛❞② ✢♦✇> ❛♥❞ ✉♥>:❡❛❞② ♦>❝✐❧❧❛✲ :♦+② ✢♦✇>✳ ❈♦❛':❛❧ ❊♥❣✐♥❡❡%✐♥❣✱ ✸✹✭✶✕✷✮ ✿✺✾ ✕ ✽✷✱ ✶✾✾✽✳

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