HAL Id: hal-00461261
https://hal.archives-ouvertes.fr/hal-00461261
Submitted on 4 Mar 2010
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.
Application of a two-point closure to the dispersion of particles in isotropic and sheared turbulence
Wouter J.T. Bos, Jean-Pierre Bertoglio
To cite this version:
Wouter J.T. Bos, Jean-Pierre Bertoglio. Application of a two-point closure to the dispersion of particles
in isotropic and sheared turbulence. iTi Conference on Turbulence 2005, Sep 2005, Bad Zwischenahn,
Germany. pp.217-220, �10.1007/978-3-540-32603-8_44�. �hal-00461261�
) *$+"%!#,"!$-./!"%0/ ) (#$,"%$ )
#"21 $3# $
4*57698;:=<>?5A@ BDCFE#G7:=BHC9IKJMLN:=<;<O:3>P:=<Q8O5HRHSTLT5
UWV=X9YZQV[\Y^]_Z\`+a7`+bc`OdeV=f7]Ngih7`a7`kjMlnmohA]pa7`OjM`;[MaFqr?dkY=hAjK[\]pgihA`
stdkY=mp`3u!`kfi[vZQVmp`aH`UFwiY=fWx7stdkhAmom_wixylHZQV=fAdO`
z|{A}M~9 3+~ \M}
>PBD8Ok:=SN57<Ne.BH@8O:/W<O@Q88;55H<OBHSNSTE:=@O<OLN:,8;F:/SNLTCW:8\?:=:CFBH<Q8OLTSN:E9LT@QI
W:=<O@;LN57C*BHCFE8O:Lp9LTCR#5DPBFB7@;@;LN7:@OBHSTBH<=tyLTCF:8O:Ctt@;:7:<kBDSBD698O5H<k@FBH:
6F@Q:^E8;L @%STLNCF8O5BHEE<;:^@;@!8O:<O5HST:¡5DtE9L @Q:<k@;LN57CtH¢57<2:BHFSN:7H£¤BH<Ok:=H:^¥76F:
BDCFE¦£t:^@QLT:6F<§¨i6F@;:=E©8O:«ªMEE©¬BD:=E|36FB7@QLNIK®+5H<O¯BDS,°BD<OH5iyL BDC©ST57@;6<;:
±
ªM¬,®+°*²%57<M8O:,@;=BDS BD<?@;:=e8O<;6F¡8O5@Q8;6WE98;:<O:S Bi8;LTH:E9L @Q:<k@QLT5HC³5DtFBH<Q8OLTSN:
FBDLT<O@=F´µ<O:=:CA8<O:yLN:=5H2E9LT@;:<k@QLT5HCBDCWE#LN9LNCFRBDC:,576CFE#LTC¢FBDSTH5iyL k·¶¸
¹iºN»
¼K.½vC«8;:<;:^@Q:=CA8/?57<;*Lp8LT@@QF5i C*8OFBi8B<O:=:=CA8:y8;:=CF@;LN57C¾5D+ªM¬,®+°
8;58O:BH@;:5H25H57RH:CF:5H6W@ 8;6F<;6FSN:=CF: Lp8O*B6FCLp57<;:^BDC¿@OBHSTBH<+RH<kBHELN:=C78
BDC*W:BDFSNLT:=E8;5@;LNCR7SN:IÀWBD<;8;L ST:E9L @Q:<k@QLT5HC*LNC*W5H8;LT@;5D8O<;57L ,8;6F<;6FSN:=CF:BDCFE
5H5HR7:C576F@@QF:=BD<PÁF5i,
ÂÄà Q}Å?Æ\ÇÉÈ3Ê9~ QËÆ\Ç©+ ;Ì7È,ǤFÌy \M}Ä Q}Í QÌA%~9MÈ3 QÎ~9+Ï33Æ\Ǥ}3nÇ
4*:F<k@\8/5HCF@;LTE:<BDCLT@;5D8O<;57LT/8;6<O6ST:CA8,H:=SN59Lp8\#F:=STE*5HC* L k*P:LTW5A@Q:¯B
5HCW@\8kBDCA8M6CLN5H<O:=BDC8;::<kBi8O6<;:3RH<kBHE9LT:CA8?ÐLTC¯8;:ÑHIKE9LT<O:=e8OLN57CtHÒ:@;=BDS BD<
LT@BH@O@;6:=ECF5HC9IvE9LNÓ6F@;LTH:3@;58;FBD8?8O:3:^¥A6FBi8OLN57C5H<P8O:@OBHSTBH<2ÁF6We8;6WBi8;LT5HC<O:=BHEF@Ô
ÕHÖ
ÕHרÚÙ Ð+Û+ÜÞÝ
±
^²
Lp8O¯Û
Ü
8O:+ÑHIK5H5HC:=C78%5HW8O: ÁF6F8;6FBD8;LTCR/7:ST5yLp8\/F:S EËH½K8PBDC::=B7@QLTST@;::C
ALNCA8O:RH<kBi8OLNCFR8;:@OBHSTBH< :=¥A6FBD8;LT5HC#5iH:=<BFBD<;8;L ST:8;<kBiß\:^e8;57<;8;FBD8+LNC#8OL @+B7@Q:
8;:@;=BDS BD< ÁF6We8;6WBi8;LT5HC Ö LT@3E9LT<;:^e8;ST<;:=STBD8;:=E8O58O:7:<;8;L BDS.@Q:=FBD<kBi8OLN57C#ELT@Q8OBHCF:
Ñ Ü
5H.8O:,FBD<;8;L ST:<;57àLN8O@ LTCLN8;L BDSt57@;Lp8OLN57CtÔ
Ö
ئÙ
ÐÑ Ü ±
¨H²
è Ñ Üé
è
×êØ
¨ Û3Ñ
Ü
ØÚÙ
¨DÐëËì Û Ö Ý ±
¼7²
LNC LTk/8;:P5HCST6CyC5i C¥A6FBHCA8;LN8\L @¤8;:@;=BDS BD<;IÀ7:ST59LN8\+<O57@O@\Iv57<;<O:S Bi8OLN57C Û Ö
í+@;LNCR³8;:ªM¬,®+°0SN5A@Q6F<;:57<+8O:@OBHSTBH<QIKH:=SN59LN8\³<;5A@;@QIK5H<O<;:=STBD8;LT5HC³@;:=e8O<;6F
îW¤ïK Û Ö
BHC*:¯:iBDST6FBD8;:=Et¤Ò:<;:^@Q6FSp8OLNCR#<O:=ELT8;LT5HCF@57<8;:¯:=BHC¾@;¥A6FBH<;:^E
E9LT@;S BH::CA82BD<O:M576CFELTCR75y5yEBDR7<;:=::CA8 Lp8O8O:S BH@O@;LT=BDS9E9L @Q:<k@QLT5HC8;:=5H<O
5D%Ò.ByST5H<ðDÀFÒF:<O:=@;6SN8O@BH<;:@;5i CLTCFRH6F<;:H
10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1
0.01 0.1 1 10 100
ñMòNóôõyô.öM÷^ø øù ú\ûüùý+÷=þ9ÿötþAù÷=þ =ù;úú\ù þ ^øH÷Qûüùýkøÿ=ùTúQú\ù=þ
;ÌAÈ,ÇtFÌy \M}Í Q}! +#" 2Å2Ǥ}+ÇtM+Ì Ì$ ÇtÊ&%3('
Ò.ByST5H<<O:=E9L e8O:=E B ×*) :WBALT5H6F<5H<8;F:5H<OL,+=5HCA8OBHS E9LT@;:<k@QLT5HC LNCÎ@Q8OBi8OLN57CFBD<O
5H5HR7:C:=5H6F@@QF:=BD<ÁF5i,!4ÎLNST:#¬3®¾L @7:<O*STLTLN8;:^E¾8O5*¯BD7:S BD<ORH:8OLN:5H<
STBH<;R7:.- :yC57STE@?Cy6:<?<O:=E9L e8OLN57CF@yB,<O:=:CA8M:y8O:CF@;LN57C5Dn8O:ª?¬®3°Í8;:=5H<O
8;5@;=BDS BD<PLN9LNCFRLTC³575HRH:=C:576F@P@Q:^BD<?ÁF5iÚ/^tBDSTSN5i@M8;5@Q8;6FE98;:E9L @Q:<k@QLT5HC
LNC5H<OL+5HCA8kBDSËBDCFE³H:=<Q8OLT=BDSËE9LT<;:^e8;LT5HC5H<ST5HCR8;LT:=@ BHCFE³yLT<Q8O6FBDSTST Lp8O5H68STLTLN8
LNC0- :=ACF5HS E@Cy6:<^
4*:57CF@QL E9:=<F5H5HR7:C:=5H6F@@QF:=BD<3ÁF5iÚ LN8;·B:^BDC*H:ST59LN8\RH<kBHE9LT:CA821
Ø
è435iè ÑF^Ò:M7:<;8;L BHS7E9L @QFSTB7::=C78!5H:9@¤8;:@;BH:?BDCWBDST5HRH3 Lp8OB+C5HC9IvE9LNÓ6W@QLTH:
@;=BDS BD<+B7@ LNC¿LT@;5D8O<;57LT8O6<O6ST:CF:HWÒF:/5H<OL+5HCA8kBDS¤E9L @;STB7:=:=CA8+LT@+5iP:H:=< :yI
SNL LN8;STBDÓ:^e8O:=Ey¯8;:,:^BDCH:ST59LN8\nÔ
Õ76
Ü
Ü;: Ñ Ü1%Ý
±
îA²
Lp8O
8 Ü
8;: 6 IK5H5HC:=C78 5H¤8;F:3ÁW6Fe8O6FBi8OLNCR7:ST5yLp8\W:S EËF´C#BHEFE9Lp8OLN57CFBDSn8O:<O
FBH@8O5¯W:BHEE9:^E8;58;F:/@;=BDS BD< :=¥A6FBD8;LT5HC8;5¯5798OBHLNCB¯E9LT<O:=e8+BHCFBDST5HR7H9Ò:/:^¥A6FBiI
8;LT5HCF@5H<8O:*@OBHSTBH<O@ Ö BDCFECB|5H<O<;:^@Q5HCWE9LNCFR*8O5¾8;F:¿H:=<Q8OLT=BDS+BHCFE 57<;L+57C78kBDS
E9LT@;S BH::CA8O@ BH<;:C5i<;:^@Q:=8;LTH:=SNnÔ
ÕHÖ
ÕH× ØÚÙ
ÐEDeÛ Ü ±
ïH²
Õ B
Õ7× ØµÙ ÐGF
8 Ü Ù Ö 1 ±
ð7²
½vC³8O:=@;:3:^¥A6FBi8OLN57CF@ÐED3LT@B:^BDC#@;=BDS BD<?RH<kBHE9LT:CA8LNC³8;:,ÑHIKE9LT<O:=e8OLN57C³BDCWE³ÐGFLT@B
:^BDC@OBDS BD< R7<OB7E9LT:CA8 LNC#8O: 6 IvE9LT<;:^e8OLN57CtFÒ:,:^¥76WBi8;LT5HCW@5H< 8O:/:=BHC@;¥A6FBH<;:^E
E9LT@;S BH::CA8E9L @\8kBDCF:,<;:^BHEËÔ
Õ Ñ Üé
Õ7×êØ
¨ Û ÜÑ Ü ±
/D²
Õ 6 ÜÑ Ü
Õ7× Ø 8 ÜÑ Ü : Û Ü6 Ü : Ñ Üé 1
±IH
²
Õ 6 Üé
ÕH× Ø ¨ 8 Ü6 Ü : ¨ Ñ Ü6 Ü1
±IJ
²
±
K7²
Ò:<O57@O@QIK5H<O<;:=STBD8;LT5HCF@
8 ÜÑ Ü
BDCFE Û ÜÑ Ü
BD<O:,BDR7BHLNC8O<;:^Bi8;:^E#BH@@OBHSTBH<?ÁW699:=@=FLTC8O:
BH@;:,5D2B¯7:<;8;L BHSt:=BDC@OBHSTBH< RH<kBHE9LT:CA8 LN8;iBDST6:
Ù
7WÒ:<O57@O@\Iv57<;<O:S Bi8OLN57CF@
8 Ü6 Ü
BDCFE Û Ü6 Ü
BD<O: 8;<O:=BD8;:=E¯B7@M@OBHSTBH<%ÁF699:=@?LTC¯8;:3<O:=@;:CF:5DtB/F5H<OL,+=5HCA8OBHS:=BDC
@;=BDS BD<MR7<OB7E9LT:CA8=78OBHALTCRLNCA8;5BH=576CA8M8;F:5H6STLTCR8O:<O
± Ù Ö 1M²2LTC³:^¥76WBi8;LT5HC
±
ð7²
Ò:LNC9ÁW6:CF:,5D.8;L @576STLNCFR8O:<OàLT@C:R7SN:^e8;:^E³LTC8;:,:^¥76WBi8;LT5HC³5H<8O:8;<OLNFSN:
5H<O<O:S Bi8;LT5HCW@Ò:E9:8kBDLTST@5H+8O:ST57@;6<;:6F@;:=E§8;5·5H8OBDLTC ¥76WBDCA8;LN8OBD8;LTH:<;:^@Q6SN8O@=
FBH@;:=E5HC8;:/ª?¬®3° 8;:=5H<OHBHC:5H6FCFELNC>P57@,/=K
Ò:<O:=@;6SN83L @+@;5i CLTCFRH6F<;:¨ÒF:FBHSNSTLT@Q8;L /<O:R7LN:LT@+<;:^@Q:=CA8BDRABDLTCBDCFE
Biü8;:=<3B8O<OBHCF@;Lp8OLN57CFBDStWBD<;8+B ×*) :FByLT5H6<+LT@+5HF@;:<OH:^EË´P8+8O:/ST5HCR7:=@Q88;LT:=@ 8O:
:9W57C:CA8OLTBHSFRH<O5i8;5H8O:+yLTC:8OLT+:C:=<;R7LTC¯5H57RH:CF5H6F@2@;:^BD<%ÁF5iÎ=BD6F@;:=@8O:
E9LT@;:<k@QLT5HC8;5¯E9:=AL Bi8O:<;57 8;FLT@@OBHSNLTCRBDCFE8O5R7<;5i BH@Q8;:<8OFBDC ×*)
LNM*?}+ËÆQ3Ìy \M}
Ò:¿BDCWBDST5HRH¾5H8;::^¥A6FBi8OLN57C 57<¯8;F:¿@;=BDS BD<ÁF6We8;6WBi8;LT5HC LTC B8;6<O6ST:CA8ÁW5i
Lp8O·B:=BHC·FBH@O@QLTH:@OBDS BD<R7<OB7E9LN:=CA8BHCFE8O:E9:=STB7:=:=CA8/5DPBÁF6FLTEFBH<Q8OLTSN:
BDSTSN5i@8O5/6F@;:+ST57@;6<O:=@2E:H:=SN57W:^E5H<MWBH@O@QLTH:@OBHSTBH<2Lp9LTCR,8;55H98kBDLTC¯<;:^@Q6SN8O@%5H<
ÁF6L EFBH<Q8OLTSN:E9L @Q:<k@;LN57Ct^ÒF:ª?¬®3° SN5A@Q6<O:HE9:=H:ST5H:=E/5H<!8;F:@OBHSTBH<tÁF6niL @
6F@Q:^E LN8;5769859E9LNW=Bi8;LT5HC³LNC³8;L @MP5H<O8;5<;:^E9L e8 @QLTCRHST:+WBD<;8;L ST:3E9L @Q:<k@;LN57C¯LNC
LT@;5D8O<;57LT!8O6<O6ST:CF:?BDCWE5H5HR7:C:=5H6F@Ë@;:^BD<ËÁF5i,^Ò:2<O:=@;6SN8O@¤BH<;:%LTCBHRH<O:::=C78
Lp8O#STB7@;@;L BDSË@OBHSNLTCRSTB@=
10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4
0.01 0.1 1 10 100
< x’ 2 >
St t 2
t 3
ñMòNóô(PFô(QRQùST=þAû÷^ønÿ=ù ú;ú\ùþUV+÷,ú\ù þ ^øWH÷Qûüùýkøù þYX72* Z;þ[*?\HúúX;v÷(]W
^ ÇE_OǤǤ}+nÇ.Ì
`a.bcadea ã!V[\df@A`kmoY=Z a <]ghAjv]oY^f]pf·VUhy`kmpaY?>.@AYi,Y=ç^`kf7`kY^h7j,[\h7Z\XAh7mo`kfAdk` a ]a `OhAmo`OZ\]pV=f
VfyV=m_wHjv]ojaj \Dú\ûIk4l7knmyýOùkpo#;úqkrmstk j x áuOvAwTxvOZyzDx `T{ O {7a
áZa b a UVZ\d@7`t|[}`egihA`3V=fAab a UF`kjv]o`kh7Z ac~ @A`V==Amo]odeV[\]oY^fY>.`eaAaDwDåÀa7Vi2=9`eai/VZ*Y|D]NVf
dkmoY=jvh7Z\`kj[\Y/[*@A`c=7Z\Y^X7mo`ti Y?>%a7]oj=9`OZ\jv]oY=fY?>=yVZv[\]odkmo`e=yV]oZ\j
a
l7kGkýQ÷þAù q\;;x
á z u``
ATx
` AO7x `T{7`a
A a.bca lAV=mY|D]pdf@Wx deasb Vt`katD]üxFb a[ `OZ\çVjvjvY^mpV a æËVZv[\]odkmo`kjV=fyaYhA`kmpa7jM]ofAhA]pa[\hHZ\XAh7å
mo`kf7dk` a o#kpeÿvk(Xútkox[w?AOZu{` Ax { wyDx á zz `a
O a.[a `OZvZexAU aæ a7â V=fAçHxyVfyaU aa u!Y^momo]ofAja sE<R bi,YDa7`OmFY?>tV=AV=jvjv]|` jvdeVmNVZ2]_[*@
Vh7fA]> YZ*ii,`eV=fç=ZQV=a7]o`kfi[ a Xúqk!ø\DùÿúQx u` y x `t z x `T{{ 7a
y aPâCaä a~a ã!Y=jkx caG~ Y=hA]omüx¤V=fAa¿ä aåÞæ a ã!`OZv[\Y=ç^mo]oY a0 `OwHfAY=mpa7jfDhiX9`OZ3a7`q=9`kfyaH`kfAdOwY>
[*@7`MjvdkV=mpVZWAh¡3j=9`kdO[vZ\h;iÉ]of,]ojvY=[vZ\Y=7]od?[\h7Z\X7hAmo`kf7dk`W2]o[*@V hAf7]> Y=Z*ijvdeV=mpVZ%ç=ZQV=a7]o`kfi[
a
(Xútkp!ø\Dùÿ=ú\x ` w7`eá ;x á zzy a
a.bca¢e~ VewHmoY=Z a <]ghAjv]oY^f*XDwdOY^fi[\]ofHh7Y^h7j2i,Y?|^`ti,`kfi[\j a Weýk#£=þ9ÿvþ¤÷=ûX;km$eýkpx
á á zv u`t{ x á`eá x `T{^á`a
w aPâCaä a~a ã!Y=j a M÷úQú\ù¥m9ý\÷^øo÷?ù¦=ùTþ ?ù þ3û\7q§q\Hø;þAûv]W a æG@;<[*@A`kjv]ojkxstdkY^mo`2u!`kfi[vZQVmp`
aH` UFwiY=fWx á zzvy a