P a rt icle Colli s ion s in Tur b u len t Flow s Michel Vok u hle - PowerPoint PPT Presentation

ÉCOLE NORMALE SUP ÉRIEURE DE LYON P a rt icle Colli s ion s in Tur b u len t Flow s Michel Voßk u hle Labo r a t oi r e de P hy s i qu e 13 Decembe r 2 0 13

ÉC O LE NORMALE SUP É R IE UR E DE LYON M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 2

ÉC O LE NORMALE SUP É R IE UR E DE LYON M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 2

ÉC O LE NORMALE SUP É R IE UR E DE LYON M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 2

ÉC O LE NORMALE SUP É R IE UR E DE LYON M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 2

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE NORMALE SUP É R IE UR E DE LYON P a rt icle Colli s ion s in Tur b u len t Flow s Out line In tr od u c t ion Pr evalence of s ling / ca ust ic s/ RU M effec t M u l t i p le colli s ion s K S v s. DN S M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 3

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE NORMALE SUP É R IE UR E DE LYON R ain fo r ma t ion and t he d r o p le t s ize di str ib ut ion t ime [ min ] r ela t ive ma ss 3 0 2 0 1 0 1 0 2 1 0 3 1 0 4 1 1 0 d r o p r adi us [ µm ] Lamb Meteorol. Monogr. (2 00 1);S haw ARFM (2 00 3) ∂ f ( a ) = + n u clea t ion / collision/ ∂t conden s a t ion c oale s cence M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 4

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N R ain fo r ma t ion and t he d r o p le t s ize di str ib ut ion t ime [ min ] r ela t ive ma ss 3 0 2 0 1 0 1 0 2 1 0 3 1 0 4 1 1 0 d r o p r adi us [ µm ] Lamb Meteorol. Monogr. (2 00 1);S haw ARFM (2 00 3) ∂ f ( a ) = + 1 a ′′ 2 Γ ( a ′′ , a ′ ) f ( a ′′ ) f ( a ′ ) d a ′ a 2 a n u clea t ion / 2 ∫ ∂t conden s a t ion 0 − ∫ Γ ( a , a ′ ) f ( a ) f ( a ′ ) d a ′ ∞ 0 a ′′ 3 = a 3 − a ′ 3 M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 4

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N R ain fo r ma t ion and t he d r o p le t s ize di str ib ut ion t ime [ min ] r ela t ive ma ss 3 0 Many Ot he r A pp lica t ion s 2 0 1 0 1 0 2 1 0 3 1 0 4 › p lane t g r ow t h in 1 1 0 d r o p r adi us [ µm ] pr o t o p lane t a r y di s k s Lamb Meteorol. Monogr. (2 00 1);S haw ARFM (2 00 3) › die s el spr ay s › ... ∂ f ( a ) = + 1 a ′′ 2 Γ ( a ′′ , a ′ ) f ( a ′′ ) f ( a ′ ) d a ′ a 2 a n u clea t ion / 2 ∫ ∂t conden s a t ion 0 − ∫ Γ ( a , a ′ ) f ( a ) f ( a ′ ) d a ′ ∞ 0 a ′′ 3 = a 3 − a ′ 3 M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 4

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N Kine t ic t heo r y In tr od u c t o r y exam p le: › s im p lifica t ion: only one p a rt icle s ize Colli s ion cylinde r › colli s ion r a t e fo r one p a rt icle R c = n π ( 2 a ) 2 ⟨ w ⟩ › ove r all colli s ion r a t e a N c = 1 2 n 2 π ( 2 a ) 2 ⟨ w ⟩ 2 a ������������������������������������������������������� Γ kin ( a ) ⟨ w ⟩ ∆ t Colli s ion ke r nel Γ kin ( a ) = π ( 2 a ) 2 ⟨ w ⟩ M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 5

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N ( Ine rt ial ) P a rt icle colli s ion s in Tur b u len t Flow s DN S › fini t e den s i t y ρ p > ρ f › Navie r–Stokes equ a t ion s › fini t e s ize 0 < a ≪ η › p e r iodic box › e qu a t ion s of mo t ion: › 3 84 3 g r id p oin ts d t = u ( X , t ) − V d X d V d t = V , › R e λ = 13 0 + G τ p Kinema t ic S im u la t ion s Maxey & R iley P hy s. Fl u id s (1 98 3) Ga t ignol J . méc . t héo r. a pp l . (1 98 3) › s yn t he t ic tur b u lence › dimen s ionle ss qu an t i t y: St oke s n u mbe r › efficien t F u ng e t al . JFM (1 99 2) a 2 τ p = 2 ρ p St = τ K ρ f 9 η 2 M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 6

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N De t e r mining t he colli s ion r a t e › p a rt of s im u la t ion box wi t h p a rt icle s M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 7

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N De t e r mining t he colli s ion r a t e › p a rt of s im u la t ion box wi t h p a rt icle s › divide in t o s egmen ts › know which p a rt icle s in which cell M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 7

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N De t e r mining t he colli s ion r a t e › p a rt of s im u la t ion box wi t h p a rt icle s › divide in t o s egmen ts › know which p a rt icle s in which cell › con s ide r only surr o u nding cell s M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 7

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N De t e r mining t he colli s ion r a t e ex tr a p ola t ion i s inexac t r a t he r us e in t e rp ola t ion Colli s ion ke r nel N c ( T ) = N c = 1 2 n 2 Γ ( a ) T V s y s M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 8

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N Pr evalence of t he s ling / ca ust ic s/ RU M effec t M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 9

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N S affman & Tur ne r JFM (1 95 6) › St → 0: p a rt icle s follow flow R c = n ∫ − w r ( 2 a , Ω ) Θ [ − w r ( 2 a , Ω )] d Ω 2 a › ave r age t o ob t ain t o t al colli s ion r a t e 2 ∫ 1 N c = 1 2 ⟨∣ w r ( 2 a )∣⟩ d Ω 2 n a › a ppr oxima t e ⟨∣ w x ( 2 a )∣⟩ = 2 a ⟨∣ ∂u x / ∂x ∣⟩ › a ssu me Ga uss ian st a t i st ic s wi t h ⟨( ∂u x / ∂x ) 2 ⟩ = ε / 1 5 ν 1 / 2 ( 2 a ) 3 Γ ST = ( 8 π 1 5 ) τ K M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 1 0

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N Colli s ion ke r nel (s) St → 0 S affman & Tur ne r JFM (1 95 6) 1 / 2 ( 2 a ) 3 Γ ST = ( 8 π 1 5 ) τ K St → ∞ Ab r aham s on Chem. Eng. Sci. (1 9 7 5 ) Γ A = Γ kin wi t h V r m s = ( η / τ K ) f ( St , R e λ ) √ π ( 2 a ) 2 η τ K f ( St , R e λ ) Γ A = 4 M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 11

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N Colli s ion ke r nel (s) St → 0 S affman & Tur ne r JFM (1 95 6) 1 / 2 ( 2 a ) 3 Γ ST = ( 8 π 1 5 ) τ K 1 00 80 Γ τ K /( 2 a ) 3 6 0 Pr efe r en t ial S ling / ca ust ic s/ RU M 40 concen tr a t ion ? effec t ? 20 0 0 1 2 3 4 5 St St → ∞ Ab r aham s on Chem. Eng. Sci. (1 9 7 5 ) Γ A = Γ kin wi t h V r m s = ( η / τ K ) f ( St , R e λ ) √ π ( 2 a ) 2 η τ K f ( St , R e λ ) Γ A = 4 M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 11

In tr od u c t ion Pr evalence of s ling M u l t i p le colli s ion s K S v s. DN S ÉC O LE N OR MALE SUP É R IE UR E DE LY O N Pr efe r en t ial concen tr a t ion 700 70 › ma t hema t ically exac t 600 y (pixels) 60 y (mm) Γ SC = 2π ( 2 a ) 2 g ( 2 a )⟨∣ w r ∣⟩ 500 50 40 400 › r adial di str ib ut ion f u nc t ion g ( r ) 100 200 300 400 10 20 30 40 50 x (pixels) x (mm) Moncha u x e t al . Phys. Fluids (2 0 1 0 ) ( D 2 − 3 ) , g ( r ) ∼ ( r / η ) r / η ≪ 1 Su nda r am & Collin s JFM (1 99 7) 3 5 3 0 R e λ = 13 0 25 g ( 2 a ) 20 15 10 5 0 0 1 2 3 4 5 6 St M . Voßk u hle › P a rt icle Colli s ion s in Tur b u len t Flow s 12