Recent advances in modeling and simulation of nanofluid flows—Part II Applications

(1)

HAL Id: hal-02051317

https://hal-univ-rennes1.archives-ouvertes.fr/hal-02051317

Submitted on 4 Mar 2019

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.

Recent advances in modeling and simulation of nanofluid flows-Part II Applications

O. Mahian, L. Kolsi, M. Amani, Patrice Estellé, G. Ahmadi, C. Kleinstreuer, J.S. Marshall, R.A. Taylor, E. Abu-Nada, S. Rashidi, et al.

To cite this version:

O. Mahian, L. Kolsi, M. Amani, Patrice Estellé, G. Ahmadi, et al.. Recent advances in modeling and simulation of nanofluid flows-Part II Applications. Physics Reports, Elsevier, 2019, 791, pp.1-59.

�10.1016/j.physrep.2018.11.003�. �hal-02051317�

(2)

Recent Advances in Modeling and Simulation of Nanofluid Flows-Part II: Applications

$! !%

⁾

!&+&#)!

^,-

& $$$%!

^.

(!)##/

^/

&&(. $!

⁰

#$%*

#!%)*(+(

¹

(-() ##

²

&(*-#&(

³

!-+

⁴

$%) !!

⁵

$!

!.$%

⁵

&$ !&%,!))

⁺⁶

),(-*

⁷⁸

#!" ) )!%

⁹

&%&'

^:

&(()'&%!%+* &($!#&$!$ !%$!#&$

)"+07747.0-51+)46/16--:16/)6,#-+06747/A'1)61)7<76/$61>-:;1<A'1)6"0))6@1 016)

*-6<-:.7:,>)6+-,#-+06747/1-;-:,7?;1$61>-:;1<A7.);00),);00),:)6

+4=1,-+0)61+;#0-:5)46/16--:16/)6,=4<180);-47?!-;-):+0)*7:)<7:A$#$!)*

-8):<5-6<7.-+0)61+)46/16--:16/)+=4<A7.6/16--:16/16/76/3=<;$61>-:;1<A7.#-+06747/A

#076*=:1)6/57,)6/373 #0)14)6,

,-8<-+06/16--:16/744-/-7.6/16--:16/)14$61>-:;1<A)141<A")=,1:)*1)

-)*7:)<7:A7.5-<:747/A)6,-6-:/A;A;<-5)<176)4-6/16--:16/;+077476);<1:$61>-:;1<A7.76);<1:

#=61;1)

.-+0)61+)4)6,6-:/A6/16--:16/-8):<5-6<"0)01,-0-;0<1$61>-:;1<A#-0:)6:)6

/$61>!-66-; !-66-;:)6+-

0-8):<5-6<7.-:76)=<1+)4)6,-+0)61+)46/16--:16/4):3;76$61>-:;1<A($"

1-8):<5-6<7.-+0)61+)4)6,-:7;8)+-6/16--:16/7:<0):7416)"<)<-$61>-:;1<A!)4-1/0$"

2-8):<5-6<7.-+0)61+)46/16--:16/$61>-:;1<A7.%-:576<=:416/<76%#$"

3"+07747.-+0)61+)4)6,)6=.)+<=:16/6/16--:16/"+07747.07<7>74<)1+;)6,!-6-?)*4-6-:/A 6/16--:16/$61>-:;1<A7.-?"7=<0&)4-;-6;16/<76-?"7=<0&)4-;=;<:)41)

4-+0)61+)46/16--:16/-8):<5-6<0)41.)$61>-:;1<A*=0)*1$61<-,:)*51:)<-;

5-8):<5-6<7.-+0)61+)46/16--:16/-:,7?;1$61>-:;1<A7.);00),);00),:)6

6#0-+),-5A7."+1-6+-#0-!7A)46;<1<=<-7.#0)14)6,")6)5"=-))=;1<)6/373 #0)14)6,

776416-):6)4A;1;)6,8841-,)<0-5)<1+;!-;-):+0:7=8-8):<5-6<7.)<0-5)<1+;)+=4<A 7."+1-6+-16/*,=4)B1B$61>-:;1<A-,,)0 ")=,1:)*1)

8-8):<5-6<7.)<0-5)<1+; =)1,1B)5$61>-:;1<A;4)5)*),)31;<)6

9)+=4<A7.-?"+1-6+-;)6,#-+06747/1-;$61>-:;1<A7.#-0:)6#-0:)6:)6

:)+=4<A7.)<0-5)<1+;$61>-:;1<A7.4=2!4=2!75)61)

(3)

Abstract

# " " '!) (#" # ""# ) #+' ' &) #& $$ (#"' &"" &#! ( ## " # (&#"*'(#'# &+(&("'-'(!'$&() & -+"#!$&(#(,$"'#

,$&!"( '()')&('!) (#"#(&! )'-'(!&%)&'$)"&'(""#

()"& -"$-' $"#!"#)&&""('-'(!"('##!$ ,""# )' '-'(!')( '!$ -"$$&#,!(#"'!)'(#'"(#'(& "(+"(""#'

" !&#' $"#!" ' #" (' #' ( #!$)((#" $$&##& '( #

$$&#'(#'# (!(!( !# ""(!$ !"(" ("$&(#

('&++$&'"((( '#&#)'$$&#'((&)'#&!# """# ) #+'+" ''"(#'" $'"(+#$'$$&#'#+"$&((!"

#!$)((#" !(#'#&'# "((&"'$#&(%)(#"'''#(+(""# ) #+&& - ')!!&." )"("(&""(# )!"( !"( ((# (.!""!(#'

" &"" !(#' ') ' ''$(* $&( -"!' " !# ) & -"!' ,( ( ('('()'#" '!) (#"#""# ) #+"&#)'&!'"#")&(#"'&&+

")!& '()'"( (&()&!#'( -#)'#"*&#)'#&!'#(,"&'')''# &

# (#&' ( $ ( " $&# '# & # (#&' !&#"" ' & &(#&' " '( )&"

'(## &' #"+(+#(&!$#&("(""# ) #+$$ (#"'(("(#"'"(#(

&"(+"" '!) (#"'((#)'"&"(#!$)((#" $$&#'#"

( #+"(&! $&#&!"$&(#"'"(" )"#(' ($-' !# #"(

#!$)((#" &') (' " - ( "#+ $' " (' & ')'' " ( #" +(

'#!')'(#"'#&(",('($'"(' $&'"(&*+$&$&"(+#$&(''"("

(##!$&"'*&&"#&&'&&'"$&((#"&'"(&'("""# )'""(

!"-$$ (#"'#""# ) #+'

Keywords:"# )'$-' !# ' !# "("%)'

(4)

)#,$,+1#+10

Introduction CFD techniques for nanofluid flow solution

!/,0!)# 0#"1#!&+'.2#0 '+'1#"'$$#/#+!#*#1&,"

'+'1#3,)2*#*#1&,"

,**,+ 0#",$14/#

'+'1##)#*#+1*#1&,"

,**,+ 0#",$14/#

1&#/*!/,0!)# 0#"1#!&+'.2#0 ,+1/,),)2*#'+'1#)#*#+1#1&,"

,2+"/6)#*#+1#1&,"

-#!1/)#1&,"

#0&)#00*#1&,"0

#0,0!)# 0#"1#!&+'.2#0 11'!#,)17*++*#1&,"

'00'-1'3#-/1'!)#"6+*'!0*#1&,"

'!/,0!)# 0#"1#!&+'.2#0,)#!2)/"6+*'!0'*2)1',+

Three dimensional computational modeling of nanofluid flows 12/)!,+3#!1',+,$++,$)2'"$),4 4,"'*#+0',+)3#/0201&/##"'*#+0',+)*,"#)'+%

&1/,)#",#01&#1#!&+'.2#-)6,+1&#*,"#)'+%/#02)10 &1'01&##$$#!1,$++,$)2'"16-#,+1&#&#11/+0$#//1#

&1'01&##$$#!1,$*%+#1'!$'#)"--)'#"1,+12/)!,+3#!1',+++,$)2'"$),4 &1'01&##$$#!1,$!3'16!,+$'%2/1',+,++12/)!,+3#!1',+

,/!#"!,+3#!1',+,$++,$)2'"$),4 12"'#020'+%1&#0'+%)#-&0#--/,!&

12"'#020'+%1&#14,-&0#--/,!&

'5#"!,+3#!1',+,$++,$)2'"$),4 12"'#020'+%1&#0'+%)#-&0#--/,!&

12"'#020'+%1&#14,-&0#--/,!&

Conclusions Gaps and future work

!(+,4)#"%*#+1 References

(5)

1- Introduction

'"('&'(,"'% &,&' &'%"(+#% !'#,&&&*,&)(##%"'

&&" ''"!&!&)!'&!"'%*& ''(',!'% !!

' "!'%('"! " )%"(& #,& #!" ! '" ' &,&' , "%")% "&'& "

+#% !'&%"'!%%%"%"!!#'+#% !'&'(&"!!& ('"!"

'&%&,&' !%&!,"%&(''%!')'"+#% !'& & !'"!!%'"

'& %)* +#% !'& "! !!"(& % &" ' & #%"') ( '" ' +#!& "

!!"( #%#%'"! ! !"% '"! ! %" &( +#% !'& & ' , ('& ! ! ' ' !!"& ! "*! &,&' & & !& "')' %&%%& '" (&

" #(''"! ( ,! & '!$(& '" & (' !!"(&& &,&' & * !

"'! "! ' % "*% +#!& ! , , ' ' "! &,&' #%"% ! ! '&

%'"!&# '" %"& #,&& "% "! " !!"(&& &,&' & ' & %' '' '

" !!'#,&#!" !'!'#%"% !"'&,&' %%&''% !""*!

* &(' " '" &% '& #!" ! ! &' %' " '& %)* )

" #%!&)&(&&"!")%"(&#,&#!" !%&#"!&"%''%!&%!! !'!

!!"("&"!'&&"'#,& "&!#%'"'%)*' &'"&( %- ' "&' #"#(% " #(''"! ( ,! & '!$(& '' % (& "% & ('"! "

!!"(&""*!'&%!'&'(&"! "!"'% &,&' & #",!!!"(

&&'"%!(%%)!,#&!!&! "!"!!"("*&%

#%&!'!&" #%"#"&&%)!"%('(%*"%

2. CFD techniques for nanofluid flow solution

!%, ' & (' '" &") ' !!"( "* #%" & !,', (& " ' !"!!%

!'(% " ' ")%!! $('"!& %"% '" ! ' )"', ! ' #%'(% &

" #(''"!(,! &'!$(& ,(&!'&#%'' !'!$(&

'' % (& '" '% ! ' "* ! ' '%!&% %'%&'& ! !!"( "*& % %,

%)*

(6)

"# $"""##!!$""&"

!""# $"

"""# $"

!""# $"

$!!"#"!# # $"!"$#$&"

##!"#&!"'#$"'""#"$&"#

&$"#!"!# # $"!"#

2.1. Macroscale based techniques

!" " # $" "$ " # ! # %$ # # &#

%!# !# !# $#" # ! $#" #!$ "!#(# ! #

!""!"!!'!%&

2. 1.1. Finite difference method

(7)

2)#/ 40 -/, )6 1&# $'/01 ,+# 4&, '+1/,"2!#" 8$'+'1# "'$$#/#+!# *#1&,"9 1, 0,)3# "'$$#/#+1')

#.21',+0 '+ ,/# 1&+ 6#/0 %, '+'1# "'$$#/#+!# #1&," '0 1&# ,)"#01 +" #0'#01 +2#/'!)--/,!&1,0,)3#1&#$),4+"&#11/+0$#/-/, )#04&#/#1&#",*'+,$0,)21',+'0+,1

!,-)'!1#",20#1&'0#1&,"$'/011&#%,3#/+'+%#.21',+00&,2)" #!,+3#/1#""'0!/#1'7#"1, )%# /'!$,/20'+%1&#6),/0#/'#0$,/#5-)#

#!,*#0

4&#/#'+"#5,$'01+"0

$,/ 1&# +,"# +2* #/ '+ 1&# 5 "'/#!1',+ #51 1&# !,-211',+) ",'+ 0&,2)" # "'3'"#" '+1, 0#/'#0 ,$ +,"#0 '+ #!& !,,/"'+1# "'/#!1',+ '+))6 1&# )'+#/ )%# /'! #.21',+0 /# 0,)3#" 6 '1#/1'3#1#!&+'.2#0+"1&#3)2#,$"#0'/#"-/#1#/0/#, 1'+#"1#!&+,"#&#'0#061, 20#+"#$$#!1'3#,+01/2!12/#"#0&#0+1&#,1&#/&+"'1'0+,10&#)-$2)+--/,!&$,/!,*-)#5

$),40+"%#,*#1/'#04&#/#'1'0%#+#/))6/#.2'/#"1,"'0!/#1'7#1&#$),4$'#)"20'+% ,"6$'11#"

%/'"'+,/"#/1,20#,/#5-)#01+'+#1)20#"1,0'2)1#1&#*'5#"!,+3#!1',+

,$ ++,$)2'"0 '+ )'""/'3#+ -,/,20 !3'16 &#/##1 #1 ) #-),6#" 1, 0'*2)1# +12/)

!,+3#!1',+ ,$ ++,$)2'"0 '+ )'""/'3#+ !3'16 4&#/# ,2+"/6 !,+"'1',+0 /# 0 $2+!1',+ ,$

1#*-#/12/# 01+'+ #1 ) 012"'#" $),4 '+ 0.2/# !3'16 4'1& 3/' )# -/,-#/1'#0 ,$

++,$)2'"020'+%0'+%0#)#1)'+3#01'%1#"1&#'5#"!,+3#!1',+,$++,$)2'"0 '+0.2/#!3'164&#/#&#1#/40'+01))#"'+,+#,$'10!,/+#/0)0 #/6#1)--)'#"1, 0,)3#1&#!,+(2%1#+12/)!,+3#!1',+'+0.2/#!3'164&#/#1&#1#-#/12/# ,2+"/6!,+"'1',+0 '+ 1,- +" ,11, 4))0 !&+%# 0 0'+20,'") $2+!1',+ 00+ +" /+" 20#" 1, +)60'01&#$),4,$++,$)2'"0'+&#1-'-#2+"#/-#/',"'!&#1),"'%2/# "'0-)600!&#1'!,$

0,#-/, )#01&1&3# ##+0,)3#" 6

(8)

2. $

%

&#$##!

1.2. Finite v

%' ( $ '

%# #

!#%(%!

volume me

&%

$#'(

& (!

# )

!#$$ #

thod

( ( %%#

!# $

!

# !&$

($%# &

%&#$ ($

%$!!#

#

$

$%%

#$%%

%# %

%#%'%

$% $%

#%

%#%

%

%"&%

'# * '

%%#

!# &

$# ' ' &

!)

&#&$

'#%%

$

(9)

^{2# -}

$. /# 0)$/ 0/2- )-(' ) $. /# )/-' .0-! 3/ /#

($)!.'0/$)$.$1$ $)/**)/$)00.)/-'1'0( .)/# "1 -)$)" ,0/$)$.++'$

/* #!/# (# 1'0 .! .$- +-( / -.- '0'/ //# )/ -! #)/-'1'0(

/# ) / /# .0-! . ! )/-' 1'0( . /# +-( / - 1'0 . - ./$(/ /#-0"# $)/ -+'/$*)

$)''4!- #)/-'1'0( /# #$ 1 $)/ "-'/ -(.- )1 -/ /'" -$ ,0/$).4 0.$)",0-/0- !-(0' # !$)$/ 1'0( ++-#$. .4/*0) -./) 0. 1 -4/ -(#.

' -+#4.$' .-$+/$)# 1)/" !$.$/.!' 3$$'$/4!-0. $)(+' 3!'2.) /# /# -) !$/.($)$.1)/" $./# $!!$0'/4! 3/ )$)"/#$.( /#!*-#$"# -/#)

⁾

- -0-4$)( '$)"

. ( )/$) -'$ - (./ )0( -$' ./0$ . ) ))!'0$. #1 0. . 1 -' 3(+' . *!

2#$#- "$1 )$)/# !''2$)"#* /' (+'4 /(* '/# )/0-')1 /$)*!

))!'0$.$)1$/42$/#2142''..( ' /' 0. /.$(0'/ /# ))!'0$!'2$) 1$/40) -(") /$!$ '4).$ -$)"+-/$'.'$+)# /!'03).$ 2''.#(($) /' )0( -$''4 .$(0'/ )%0"/ # / /-).! - ! ))!'0$. $) ($-#)) ' # / 3#)" - 0) -/# '($)-- "$( # 4 (+'4 )/# .$)"' +#. (* '$)/# $-./04)" /' (+'4 /(* '!- )1 /$)*!))!'0$.$)1$/42$/#0' .$)0.$'214 2''.2# - /# 0' -$)"-)"$)/ #)$,0 2.++'$ $) )'"#')4.'1 /#

)/0-' )1 /$) ! ) $))!'0$ ($3/0- ! $)$ '$,0$. ) ))+-/$' . $).$ .,0- 1$/4 0) - $!! - )/ . /. ! 0)-4 )$/$). # 4 (+'*4 ) .$)"' +#.

#(" )0. ++-# (* ' 2$/# / (+ -/0- + ) )/ +-+ -/$ . $) /# .$(0'/$) ) )/# - 2-&$) )'"#')4 (+'4 !-(* '$)"!- $1 -!+-'$/-0"#

'' /-$"0- .#2./# .# (/$.!1 ( )/$) +-' (./#/#1 (+'4

(10)

/

"

,"

!

" $#

!

/( (&&(' '

"''% -+#'

*&()"1+#

!

#!-*

' +

!

%)(),#

*+**)*#',

#

+ +( ,/*

(*

+('," (*

# ,/#,")*&

* )$!+

0&)% + ' ," +#

*('.

"!

" $#

&#++#(' *(&

+ -+ (* + + "'# , %

#'!%)"+

,#(',-*-%

!!

&)-%#+"*

+#&-%,#('

% +#&- ))*("

', %(/#'

$

#"$#

( ''( %-#

-%, ''(

,+ ( ,-/+#'

#

# %(/+ * %-# %(/ # .*#(-+ &

'.+,#!,

!

#' /.1

&(%+ (*

1#'

(11)

$ 73 1 6 - 89' 789 (3 (3

;- 73 74 75

2. 2 3

0

97 % ;-6

30:* 2&896&0 3)*0 +36 2&

6.2/1&2 29 -6(-78 '.2'38-0 9)= &039) 300(836 '=

327.)6)'=

-.(- 463:.) 3+8;&6* .7 ' 4(.+.(463'0 59&6(&:.8=

!

"$

1.3. Finite e

28-*+.2.8**

3+8*2927869(

1287*+

7*3+;-.(-

"

(32:*(8.32

&23+09.) +03 91'*6 .2 &

&0$%*14 0&1.2&6&2)

).7 8 &0 $ 8;34-&7

=$%

)7 & 8 '38- 8- 6*)

01736

97.2,'38-

##

$%

element me

012818- (896)17- +36.28,6&8 .7323+8-*

;&7 2 3+ #2;

3; .2*&

4&68.&00= -*

403=) )896'90286

% 97*)

&4463&(-

7&6(-67 8330'3< +36 )9() (378

*<&140**

7.2,0*4-&7

"$

+.,96*7&6*6

ethod

-3) -7&2)8-

8.2,8-*,3:

1&.2).++

97) .2

&8*62&23+09

$ % 98.0.>

&8) 89'* .

""

6,.17;-.

" ! 90*6.&29

&6.2(6&7.

730:.2, 2&2

&7 ;*00 &7

2,8&0$

7*&2)8;34

"$

6*46.28*);.8

8-)31&.2 ,3:62.2,*

:62.2,59&

62(7'8;

& &2) 36

9.).2 & (&:

>*) "

.2 ;-.(- 837.19 .08-+03;

! 83 13)0 906.&2 . .2,0=97.2, 23+09.)'&7*

8-* &7 3+

%98.0.>) 4-&7&4463

'

8-4*61.77.32

23+73098.32

*59&8.327&6

&8.327*&(-

;**2&

*2>.2. $%

:.8= '=&779

0 ;&86 190&88-*+03

;&7&7791*

0 8-* 2&23+

.,96* 7-3 34*27396(

) 463'01 + &)&48.2, )4*2

&(-*7

$%

2+63149'0.7

2.7).:.)) 6730:)&7 861.7190

&2)

1403=) 91.2, 7.2,0 83 &2&0=7

2&23+09.) ; 3;3+&2.32

)83'7.2 +09.) +03; .2 3;7 7(-*1&

73+8;&6

1 -* &):&2

&2) 13).+=

8313)*

# 7-*6

.283&76.

7;.,-8).2 08.40.*)'=&

-+.2&059

"

04-&7 -31 7.7 8-* :&6

;&7 97*) &

2&23+09.).2 2,04-&7

2 & 4&6&'3

&8.(7 3+ 8-*

79(-&74*

28&,* 3+ 34

=.2, 8-* 73+

*02&23+09.)

!

73++.2.8**0 28,6&073:

&;.,-8+92 9&8.327&6.

83 13,*2397 .&8.327 3+

&7 (330&28 2&(.6(90&6 2&238-6 0.( 8639,- 463'017

2 427396(*

+8;&6* +36 +03;.2&

01287

68-7*

2(8.328-*

.28-*

(12)

'0

$0

"$

8 0 /6 80 . 80

$0

"

#:

3.0'/0/- 0.1-*$"5&%(

$$6.6-"5*0/

80#"

6-51):4$4 03&9".1-&

4:45&.

6.&3$"- 4.

03,*"/$0 8)

&-5/("/%4 03,/('-6*%

0/4%&3/(5)

"5".*"/%

:/"/0'-6*%4

-/&"3"-(&#3 (&0.&53&4 /

"4&%40'58"3

"/%-&9

"43*/&5"

. 64*/( /"

.6-"50/ 8

0&5"- !*

)&3&/"/0'-6 40-%'$"50

%8"4".*9 )&5&.1&3"56

/( ! 44&&(

3"$&26"50 !/&0'%

3&1"$,"(&4

"-5)06()

"- !64&

"/0'-6%4 8

5) &91&3.

/7&45("5&%

6*%'-088"

0/*/"4)&-- 9563&0'1"3"

63&%&1&/%&

64&%-&9

#

0/4)&."*

%*4"%7"/5"(

45)"5)"7&#

)5)&'03.&3

&%

5) 5)& 4/(

.&/5"- %"5"

%5)&3."-&''

"4$0/4*%&3&

-"/%56#&3

"''*/"/%/"

&/50'1301&

50&7"-6

*/"%7"/5"(

&40'.

#&&/64&%' 31"$,"(&*4 50.0%&- (-&1)"4& "1

"/% '06/%

''$&/$:"/%

&%"4"4/(- 3&(&/&3"57&

"/01"35*$-&4

&35*&44&&

6"5&5)&1&3

#

&0'*4 .":#&3&-"

'03/"/0'-6*%

4.03&$0..

5)&1&3'03.

1130"$) ) (00% "(3&&

%&/5301:(&

-&1)"4&'-6

&5:1&-"5&/

4)&:64&%

*("

3'03."/$&0

454'-&9#*- 5&%50306/%

%'-084*.6 .0/-:64&%

."/$&0'"

)&: $0.1"3

&.&/5 #&58&

&/&3"50//

6*%6/+0&

/5)&"54503"

%

0'%*3&$5"#4

5:'03%&"-

%0''&33034

6-"5*0/"3&

1)05070-5"*

3&% 5)& 3&46

&&/ 5)&.

/"4:4 5"- !*/

"(&4:45&.

'035)&.0

403#&340-"3

/(85)

"/%&3303

*$5)&3."- 6-54 0' 5)&

/ "/05)&3 45&.64*/(

7&45*("5&%

8)&3&5)&

0%&-*/(#:

$0--&$5034

(13)

$! & " #$ !!#% #( ## # "#! "("# "% (

$# (""(""!#"!"! ##!('" .+,++)+"'-0"-!

)+&",,"('+(&).%",!+

2.1.4. Other macroscale based techniques

!+ + ,(& (-!+ &+( ,% , -!'"*., -!- !/ ' ., -( ,(%/ ''(%." %(0, 0!"!++"%2+/"0!+

#!$###

('-+(%(%.&"'"-%&'--!(",)(0+.%-!'"*.-(,(%/%."%(0,"' (&)%1 (&-+",'",(&"'-"('(',(-!-"-!,-!'"-,(

-!, -0( -!'"*., "' "-,% !"$!(%,%&" ", )"('+ +,+!+ "' -! ., ( -(

,(%/ ''(%." )+(%&, (+ 1&)% , 3 !"$!(%,%&" ' !", (%%( ., !/ ., -(,(%/%(0'!--+',+(''(%.","'(&)%1 (&-+",2(-!-!,"' %)!, ' -0()!, ))+(!, (+ 1&)% !"$!(%,%&" ' '#" (',"+ -! (&-+2 ,!(0' "' " '"'/,-" - -! -( %-+" "%('''(%." %(0 "' -! )+,' ( )(+(., &" ' +"-"(' ' '(-!+ 0(+$ !"$!(%,%&" ,"&.%- -! - ( & '-"

"%(''-.+%('/-"('"',")(+(.,.+//"-22,!&-"",,!(0'"'"

(14)

"

+ ,.

(

&

#, -"+

( (

"(

"

/

*

#

$"% #!

#

"-+&4(.

+', .'-#

.+ -2)#

(.'+2#'- -+#1,2,-&

',,+2(

+(.!"(.--"

('/-#('(

(',#+#'!' (0/+#,%#

"

",)-+%&

+#%,#'

*.-#('-"

"!

!.+,++)+

%

.'+2%&

#(' (+&.%

%%2-"(.

!+%*.-#

&"')) ('%2-(,(%/

"(&#' 0-+, ''( %.#,,

#&#--()+(

" "

&-"(#,

#!' .'-#(

(.'+2

!!$

+#'-0#-")

""

&'-&-"(

-#('-(('/

.'#'!,.+

#('"

)%#%

(+-"/%

(+1&)%

''( %.#

,,#'!%)"

(%&, (+0

!%(%)) ('1)',#(

('#-#(','

% +&#,,#(' +(

5#, /+-) ( (

"#,(.'+2 '-#%,/

%.( -".'

%$$2 ,#',#'

",0(+$#' 0"#"+

)+(".'%#

(',0"#"&

'-"((+

# (&).%#,"+

,.,-(

+-#%# + 2#&&+, 2#'-!+%*

/+2,#!'# # '$'(0'/+

2'(3$

#'%#',&

'! %.#

', .'-#(

#$-"

&.,-,%

#'-,2,-

#! "#

+

,+#'.&

'-#%*.-#

#'-" %(0

*.-#('' '-+.-#(

+#%,('-"

2&

&#''.%.,

#!

(',(%.-#('

"-#,, --(

&., (+

!#

&+#%))+

#('-('#'- 0 #%-2)#

-"'#, ('( (&).

"(.'#'!

&)%(2 /#-2.' ))%##

1#,-,

('1)', (&)-#%

'.&+#%,

+("-"-.

-!+%*.-#

%%2%%

+-#3-((

.--#('-#&

,.+ +- -(,(%/' +&!'-#

#%#-2(

,#('( -"

0#-"-"

,(%.-#(''

#!# $

.,,

#('(/+

-"

(-#' ,#'#-

-"+-"' '-.+%

# #%2

)''-

# +'-#%

,)-+%

(15)

.'50& 5' 7#-6' 0( # 1#3#.'5'3 #5 '#%* 10+/5 +4 0$5#+/'& $: +/7'34+0/ 0( 5*' '+)'/(6/%5+0/

'91#/4+0/4#/&40+5&'1'/&40/+/(03.#5+0/0$5#+/'&(30.5'80-'%0.165#5+0/#-&0.#+/#/&/05 0/-: /'+)$03+/) /0&'4 '/ #11-+%#$-' 50 # )+7'/ 130$-'. 41'%53#- .'50&4 #3' '953'.'-: (#45 '#4:50(03.6-#5'#/&'9+$+5'910/'/5+#-#%%63#%:8+%1307+&'4%0/7'3)'/%'(#45'35#/#/:05'3 /6.'3+%#- .'50& 8+5 #-)'$3#+% #%%63#%: !" 6' 50 5' 3'453+%5+0/4 0/ '9+45'/%' 0( 5' '+)'/(6/%5+0/ '91#/4+0/ 08'7'3 41'%53#- .'50&4 #3' -+.+5'& 50 7'3: 4+.1-' )'0.'53+'4 46%* #4

%#//'-(-08$'58''/+/(+/+5'(-#58#--403(-08+/#53+1-:1'3+0&+%426#3'('8456&+'4#7'64'&

41'%53#-.'50&45040-7'/#/0(-6+&130$-'.403'9#.1-' #,+('5#-!"65+-+;'&#41'%53#-.'50&

50456&:5''(('%50(#.#)/'5+%(+'-&0/5'0/4'50(%0/7'%5+0/0(/#/0(-6+&4$'58''/580+/(+/+5' 1-#5'48'3'5'41#%'$'58''/5'5801-#5'48#44#563#5'&8+5103064.'&+#4''+)$*'

#65034#7'64'&5'60/)+03/0.0&'-+/5'+3456&:

0 64' %0..0/ 5'%/+26'4 46% #4 #/& # .'4* .645 (+345 $' )'/'3#5'&

8+% %#/ 40.'5+.'4 $' # %#--'/)' 0 40-7' 5+4 130$-'. .'4-'44 #-40 %#--'& .'4(3'' 03 )3+&-'44.'50&4#7'$''/130104'&/.'4-'44.'50&4#4'50(/0&'48+5*065#/:%0//'%5+0/

#3'4%#55'3'&5306)0655'40-65+0/&0.#+/! "'.045%0..0/.'4-'44.'50&4#3'!"

'4*(3''-0%#-'5307#-'3,+/

+/+5'10+/5.'5*0&4

#&+#-$#4+4(6/%5+0/#1130#%'4 '4(3''$06/&#3:4%*'.'4

/%0.1#3+40/8+5#45#/&#3&#.'4-'44#1130#%*+4'#4+'35064'(03.0&'-+/)130$-'.4

#/&%#/#7'+)'3#%%63#%:/5'05'3#/&5'%0.165#5+0/#-%0450(#.'4-'44#1130#%+4 0(5'/)3'#5'35#/#45#/&#3&! "

4#/#-5'3/#5+7'#)3#/)+#/40-7'34#3'#/05'3'9#.1-'0(.'4-'44.'50&48'3'/0.'4*+464'&

+/ 5' 40-65+0/ '/'3#--: # 4'5 /6.$'3 0( (-6+& .0-'%6-'4 #3' &+453+$65'& 3#/&0.-: +/ 5' (-08

&0.#+/ 03 '9#.1-' +/ 0-'%6-#3 :/#.+%4 #/& +44+1#5+7' #35+%-' :/#.+%4 5' .05+0/0(5'4'.0-'%6-'4+4)07'3/'&$:'850/<44'%0/&-#80(.05+0/8*'3'#4'50((03%'4#3'

#%5+/)0/5'(-6+&.0-'%6-'4!"/5'(03%'4$'58''/5'.0-'%6-'4#3'%#-%6-#5'&$:64+/) 5' +/5'31#35+%-' 105'/5+#-4 46%* #4 5' '//#3&0/'4 105'/5+#- / 5' #1130#%* 5*' (03%'4

%0/4+450((3+%5+0/(03%'4%0/4'37#5+7'(03%'4#/&3#/&0.(03%'4#4'&0/,/08-'&)'0(5*'(03%'4

(16)

#%5+/) 0/ .0-'%6-'4 5' #%%'-'3#5+0/ 7'-0%+5: #/& 104+5+0/ 0( 5' 1#35+%-'4 %#/ $' %#-%6-#5'& *+4

#--08445#5+45+%#-#7'3#)+/)50$'+.1-'.'/5'&50%#-%6-#5'5'$6-,1301'35+'40(5'(-08&0.#+/46%*

#4&'/4+5:7+4%04+5:.#44&+((64+7+5:7'-0%+5:13'4463'453'#.(6/%5+0/#/&7035+%+5:035'3.#- 53#/41035130$-'.45'(+345-#80( '3.0&:/#.+%4+4#11-+'&505'(-6+&.0-'%6-'48+%%#/5*'/

$' 40-7'& (03 5' .0-'%6-'4 5'.1'3#563' +/ 5' (-08 &0.#+/ +.+-#3-: 5' $6-, 1301'35+'4 46% #4 5'3.#- %0/&6%5+7+5: 5'3.#- &+((64+7+5: +/5'3/#- '/'3): #/& '/5#-1: 0( 5' (-08 &0.#+/ %#/ $'

%#-%6-#5'& 64+/) 45#5+45+%#- #7'3#)+/) !" /05'3 #)3#/)+#/ .'50& +4 .005* #35+%-' :&30&:/#.+%4 !" 8+% +4 #-40 # .'4-'44 .'50& / 5+4 .'50& 5*' %0/5+/66.

%0/4'37#5+0/ -#84 )07'3/+/) 5*' (-08 +' #7+'350,'4 '26#5+0/4 #3' 53#/4(03.'& (30. 1#35+#-

&+(('3'/5+#- (03. +/50 +/5')3#- 64+/) #/ +/5'310-#5+0/ (6/%5+0/ 5#5 %#-%6-#5' 5' ,'3/'- 0( 5' (-08 7#3+#$-'4 #5 #/: )+7'/ 10+/5 ' .#+/ &+(('3'/%' $'58''/ #/& .'50&4 +4 5#5 5' +/5'3 1#35+%-'(03%'4+/#3'&'3+7'&(30.5'%0/5+/66.%0/4'37#5+0/-#84#401104'&5064+/)+/5'3 1#35+%-'105'/5+#-4+/!" '+4#113013+#5'(03456&:+/)-#3)'&+4%0/5+/6+5+'4+/5'(-08

#/&#45'#&7#/5#)'5#5$6-,(-6+&1301'35+'4#3'#/+/165505'40-65+0/6/-+,'8'3'5':#3' 0651654 08'7'3 +54 #11-+%#5+0/ 50 .+.+% .'404%#-' #/& /#/04%#-' #11-+%#5+0/4 +4 /05 #4 8'-- '45#$-+4'& #4 5'3 .'4-'44 .'50&4 #3' 5' &+4%3'5' '-'.'/5 .'50& !" #/& 5' .005

+44+1#5+7'#35+%-':/#.+%4!" '8#4+/530&6%'&50%0.$+/'5'#&7#/5#)'40(

$05* #/& 8'3' +5 3'5#+/4 5' 5'3.#- (-6%56#5+0/ ('#563'4 (06/& +/ #/& #4 5' (-'9+$+-+5:+/ &+4%3'5+;+/) 5' #7+'3<50,'4 < '26#5+0/)07'3/+/)0( 5' 4:45'. 64+/)5'

#11309+.#5+0/4!"

/ 5' -+5'3#563' 5'3' #3' # -+.+5'& /6.$'3 0( 1#1'34 0/ /#/0(-6+&4 5#5 #7' '.1-0:'& .'4-'44 .'50&403+/45#/%'+,(#3#/&#.00&+! "64'&5'.'4-'44-0%#-'5307<#-'3,+/#1130#%

5040-7'/#563#-%0/7'%5+0/+/#426#3'%#7+5:(+--'&8+5#-6.+/#8#5'3/#/0(-6+&+)63'%4084 5' &0.#+/ $06/&#3: %0/&+5+0/4 #/& /0&'4 &+453+$65+0/ +/ 5' 130$-'. %0/4+&'3'& $: +,(#3 #/&

#.00&+! "=#4%#/$'4''/5'3'+4/0%0//'%5+0/$'58''/)3+&40+#/&0107!"456&+'&5' '(('%50(4#1'0(6/#/01#35+%-'44641'/&'&+/8#5'3+/5'3.40(/#563#-%0/7'%5+0/*'#553#/4('3+/#

%#7+5: 65+-+;+/) # 3#&+#- $#4+4 +/5')3#- '26#5+0/ .'50& 063#/5#4 '5 #- !" '.1-0:'& # .'4-'44

#1130#%%0.$+/#5+0/0(10+/5%0--0%#5+0/.'50&8+55'.07+/)-'#45426#3'4#11309+.#5+0/50

40-7'5*'/#563#-%0/7'%5+0/0(/#/0(-6+&4+/#103064%#7+5:

(17)

"

2.2

%#"

2. Mesosca

##$

"#

##

##%

"$

"

le based tec

!%#$$

#

$!%

%$(

$

chniques

$ $$#

$$$)

%#$$"%

(#(#

! !

%$#(#$

)$

%#$#%

#$$$

#$#

$

%$$

#'

!

$'

## $&

%'

$&"

!

%"# " "

%"

"$(

'#$

$$&

!

! "$ '$

"#$

#$

$#$

&"$#

"##

$

"$

#"

!%$#

(18)

2.2.1. Lattice Boltzmann method

"-* 0&88.(* 308>1&22 18-3) ;&7 +.678 .2863)9() .2 8-* 7 "-* &440.(&8.32 3+ 8-.7 18-3) .2 7.190&8.2, -&8 &2) 1&77 86&27+6 3+ +09.)7 74(.&00= .2 (3140< ,3186.7 &2) 1908.(3143228 +03;7 -&7 .2(6&7) .2 .286:2.2, =&67 )9 83 .87 ++(8.:277 +0<.'.0.8= &2) 7.140.(.8= .2 (314&6.732 83 8- 86&).8.32&0 (31498&8.32&0 +09.) )=2&1.(7 18-3)7 "- +92)&128&0 .)&3+.7'&7)328-).7(6813:1288-36=3+&783+&68.+.(.&0+09.)4&68.(0740&()32 0&88.(7 "- 138.32 3+ 8-7 +09.) 4&68.(07 .7 7.190&8) '= +3003;.2, 8-* :3098.32 3+ & 467(6.') 308>1&22 59&8.32 "-* 1378 7728.&0 &):&28&,* 3+ 8-* .7 8-* .2(36436&8.32 3+ 1.(637(34.(

4-=7.(&0 .286&(8.327 3+ 8- +09.) 4&68.(07 .2 8- 2916.(&0 7.190&8.32 &2) 6:&0 8- 1737(&0

1(-&2.71 3+ -=)63)=2&1.(7 "&/.2, 8- 308>1&22 59&8.32 .283 (327.)6&8.32 8-* )*27.8=

).786.'98.32 +92(8.32 .7 .2863)9() 83 646728 & )27.8= 291'6 3+ 8-* 4&68.(07 ;.8- 8-

1.(637(34.(:03(.8=&88-437.8.32&2)8-8.1&2)8-4&68.(07).786.'98.32(&2'**<4677)&7

$%"-1378(3113213)0797)+367.190&8.327&68-&2)0&88.(7

463:.).2,

^8-

&2)

^8-

36)*6 .738634= .2 8-6** ).127.327 674(8.:0= ;-.(- .7 .009786&8*) .2 .,

$%"-78&2)&6)238&8.3221+36).7(68:03(.8.7.274&8.&0).127.327.797)8-639,-398 236)6837.190&88-2&23+09.)+03;'=8-8-)=2&1.(4&6&1867&++(8.2,8-2&23+09.) +03; '-&:.36 7-390) ' &2&0=>) +631 8- 1737(34.( 43.28 3+ :.; )9* 83 :&6.397 +36(7 32 8-

2&234&68.(07!.)./&2) &>&0.$%463:.))&(32(.76:.;3+67&6(-789).7328-973+

+36 &2&0=>.2, 8-* (32:(8.: -&8 86&27+6 3+ 2&23+09.)7 2 &238-*6 789)= !.)./ &2) &1&8 $%

7911&6.>) 6(28 &):&2(7 3+ .2 78.1&8.2, 8- -=)63)=2&1.( &2) 8-61&0 '-&:.36 3+

2&23+09.)7&2)).7(9777:6&0(-&002,.2,463'0178-&8-&:*'**22(39286*).28-.7.779&08-

&2) &7&2. $% 789).) 8- 2&23+09.) +6** (32:(8.32 .2 &2 7-&4) (&:.8= 97.2, &2) 8-*

.2+092(73+2&234&68.(0(32(286&8.32&2)).&186&7;00&7 &=0.,-291'6328-8*146&896

&2)+03;+.0)7;6**:&09&8)0./.8&0$%&440.)83:&09&88-2&896&0(32:(8.32 3+2&23+09.).2&2.2(0.2)7-&4)(&:.8=&2)8-.14&(83+60:&284&6&1867328--=)638-61&0 (-&6&(86.78.(7 3+ 8- 2&23+09.) ;&7 .2:78.,&8) #-39 8 &0 $ % ):*034) & 13)0 83

&2&0=>8-1.<)(32:(8.32-&886&27+6&2)+03;)=2&1.(73+&091.2&2&23+09.).2&(9'.((&:.8=

92)6 1&,28.( +.0) 3-''. 8 &0 $% 1403=) 83 7.190&8 8-* +36() (32:(8.32 3+

".

;&8*6 0

;&86 &2) 9;&86 2&23+09.)7 .2 &2 <82)) 796+&(7 (-&220 -&71. &2)

(19)

#

2.2 '#

( - &/

) , )

&-

0 .' -.

-#' .#'

#0-"#

/1.,#(

2.2. Dissipa

")

#,)(()- ()*,.#&

--/'*.#)(#- /#.)."- ),,) .(

,%)1(

/,#(&#+/#

--

"' 1",

&)#.#- ',./,

.#-.#&-' '/&.-3- '-&-1

-./#."

(*),)/-

ative partic )(.#(//'

&"..,(

,)-)&-#-1 --.#&&**&#

&) ."*

(-) (' 0( ),'#

(() &/#

')-. '#&#, ."*,)--

&,.#)(-

*,--/,0#

'*&#(!/"

-.')10 1"#",()

#,-.(- 0#.3'*&)

cle dynamic 0#,.)%

(- ,**&#

1".",."

&"#-#-

*,)&'/(

!."'(

,)-&-3- -3-.'- ),

,**,)"

--.,.- ,)' ( ),- -)-#.3(

"(&3-#-!#

0,."*,#' ).**,)*,#.

)(&1-) 3#(!

#!/,#-,

cs method -+/.#)(-

.#)(--/"

-&) ((

--/(

,)(-#, ( ,*.") -.'--#'

,*,)&'-#

')(!#-, '.,%#(!#(

,)'."- (-#.3'--

#0-)'*, ',3,1

. ),.,%#

) .",')3

,*,#(.1#."

-, -#((() ()-&-") ,-)&03 .#)(), ) #,#-(' '#&,,%

#(0)&0#(!&

,.*,.#&

(#0#/&.) '#,)-)*#

# /-#0#.3

"(-#0.

%) ."#-'

#(!&,!,.

3('#- ),

"*,'#--#)(

!(,&&3/

&/#-"' /&-#'/

3)'*,#(!

,)-)&-3-.

'."()-&

)1() ."

(!."-&-

-#'/&.#)(

)'-) ."

#.#&- / ( &)1,.

.)/.'#

'.")#-."

'*),&(

(./,&)(

( ,)'*/&#-"

/-.)-./3 '$),)(

/&.),# .

!."'(

'- ),1"#

&#--/'.

)(.#(//' -)(."),

('.")-#- &/#(&

/,.",/&%*

.-(, ,)-)*#

"2.,'&3 -*.#&-&

(0.#)() (

",

3(,!3.,(

,(*,.#/&

.")(.#(//

,*.")

"'( ,

#)('3!

--/'*.#)(

,) )/.

-')&/&,

&/&.#(!."

,),.#--/

)0,/- .#&-) ."

3-'&&*"3

&-.".)/

(() &/#-

(-*),.#(

,&31#."

/' ."-

*."#-

!#(.) (-(

('),

3('#-

#, /"- -#(!

-#&(

/,#(."

(20)

-$#!,,&+(($","'&+& &*$,!%,!'"+'%(-,,"'&$$10(&+".&,"%

'&+-%"& .&'-,,"& +-('%(-,+!,"%&+(,"$+$+',!!,,&+, ,!&&'+$$* ,!&1,+%$$,!&,!'&.&,"'&$'&,"&--%+$+!-+

"&,%",+(,"$&,%('$+$+%-+,(,-!++$+&(,--+"&

%+'+'("(,"$+%,!'++'$$3'+ "&"& 4%,!'+/!!+"%-$, (,"$(+&,+ '-(',-$$-"%'$-$+!+%,!'+('++++,!-&")-(",1,'

%'$$,".$1-$#(!1+"$+1+,%+&""&,$1(,-,!++&,"$,"$+',!(*,"&&,

"&,,"'&+/",!"&$-"%'$-$+$,".$1&,'+ *"&,!&")-',!"+,1("+,!

"++"(,".(*,"$1&%"+%,!'

!"++"(,".(,"$1&%"+%,!'"+'+ "&.+"'&'/!*!

(,"$(+&,+ '-('(#,',-$%'$-$+!(,"$+**&'%$1

"+,"-,"&,!$'/'%"&&(,"$"&,,"'&'1+'&+.,"'&'%++%'%&,-%&

&* 1+"++!'/&" !" -+!'/++!%,"',!$-"(,"$+&/$$(,"$+

/!!(,"$(+&,+ '-('$-"%'$-$+'".,!$'/"&,!'%"&'1 '+"+-+&("'"'-&1'&","'&"+(($",,!'%"&"&$,&'-,$,,' -&,

%++'&+.,"'&-,!'-&#'-&1'&","'&"+(($",,!/$$+,'(*'-,!

&'+$"('-&1'&","'&!%,!'&(",'%($0!1'1&%"+,%"'&&'+$

/",!%-!!" !'%(-,,"'&$""&1,!&,","'&$-"& &,1+,!%,!' /+(($",'+.$!,,&+(($","'&+"&$-"& '&-,"'& &'&.,".!, ,&+(($","'&+'(-$-"+&&,-$'&.,"'&2&!,,&+*/",!

,%(,-(&&,('(,"+

(21)

,

"

$

"

"' ,(

!'#*

,%#*""

#%"

"#!%#&

$$ '#"&

"""# (

")&#&',

"# (&(&

$%' &

"'!%#

"# (&-&

(

*& '#!

!#'#" &

($%' $$ '

''%"&$#

#("%&'

"'

&(

#&#$#%

( '%!

)"

!!"%

&#'!'

&

'#"&"'

#%'$$ '#

'"'

&'*#$%#

"'%'#"

#"(')

' ,'%"&%!

#")&'

%&( '&"

"(%'

#"&'!#

''%"&%!

#$%'&!(

&%'

"'*"

)',"'!

$$%#'

!"&!&

''"' ''''

%+'"

&'" ("'

!"&!'

&' "

"$%#

$%'

!%#&#$

'#!#

&,&'(,"

'%$%' '!

'##'%$

$%#$%'&

''&$

'#'!%

#(%'*

& &#

'#"(

''%"&$#%'

"'%!

"%,'%"

'#&*

%'

&%''

!%#&

%#&#$"'

*"""#

&%' ('#" (+

*'"""#

('('#"&

&$#%''*

&('#%

''%"&%

%! #"(

#$ ,*

%'#"&

(-&( ) "#

&#(%%"

# (&

"(

"

')', '"

'*"

)&#&',

#

'*"

(22)

,.0%(!/+0$!.3+. /$!0./"+.)! 0$!).+/+,%1('2(1!/+"2%/+/%05* 0$!.)(

+* 10%2%05%0+)%.+/+,%,.0%(!,.0%(!"+.! $!0"(14%0!.0%+/!03!!%0!.0%*#

,.0%(!/

$!%0!.0%+!03!!,.0%(!//$+3%%#%/#+2!.! 50$!"+((+3%#!-10%+/

^%& ^%& ^%&

%&^2%/ %&^+* %&

3$!.!0$!+*/!.20%2!"+.!

%//%,0%2!"+.!

* .* +)"+.!

.!#%2!%(/+

+10/"+.5!40!.("+.!,,(%! 0+0$!,.0%(!//1$/+ 5"+.!,.!//1.!#. %!0

!0$!0!.)/

.!+* 10%2!2%/+1/* .* +)$!0"(14!/.!/,!0%2!(5/#%2!*

%$!+/!.20%2!%0!.0%+!03!!0$!,.0%(!/3$%$.!#1(0!/0$!,+0!0%(!*!.#5

!03!!0$!,.0%(!/%/ !0!.)%! ".+)0$!/00!!-10%++"0$!/5/0!)(/+0$!/0.!#0$+"

%//%,0%2!* .* +)"+.!/3%0$%0$!/5/0!)+""+((+3/0$!"(1010%+ %//%,0%+*0$!+.!)0+

3..05+/!.20%++"0$!0+0()+)!01)%*/5/0!)

!!.((53!%#$0%#"10%+/.!1/! %0$!+/!.20%2!* %//%,0%2!"+.!/3$!.!0$!5 !(%*!

)++0+%((53%0$0$!%0!.,.0%(!/!,.0%+ %/0!!+)%#6!.+0.%0%(10+"". %1/

$!+/!.20%2!"+.!/$2!"%!.!,1(/%2!01.!+!3$%$ +!/+0+10"+.00.0%+*

!03!!,.0%(!/$!)+/0+))+(51/! 3!%#$0%#"10%+"+.0$!+/!.20%2!"+.!%/0$!

.++0* ..!/+"0.!,1(/%+)+ !(

/"+.0$! %//%,0%2!* .* +)"+.!/3!%#$0%#"10%+/7/!0( #!!.(%6! 3!%#$0%#

"10%+3$%$%/+!+"0$!)+/0")+1/)+ !(/$/,.+2!1.0!"+.$%#$2%/+/%05"(1% /3$%$

)%)%$%#$$)% 01)!.)//0./,+.0,,(%0%+/ $%#$.* 0(*1)!.0$!.)(

,,(%0%+/7/)+ !(%/#%2!*/

(23)

%&/3"&$%0&$#1 0&+* )&)& $."0".2&/ +/&055"&0%"..&/&$0%""4,+"0+.5 )+!&#5&*$0%"2(1"+#

/ 10+##.!&1/%&/3"&$%0&$#1 0&+,(5/'"5.+("&)&)& '&*$

0%".)(0./,+.0&30".3%"."7*!

%"&* +.,+.0&++#"4,".&)"0(!0#+.0%"1('**+#(1&!2&/ +/&05&0%")+!"(

"!+"51,!0&$0%"!&//&,0&2"#+. "/,.+,+/"!51! 1!6/0" %*&-1"

&),.+2"/6/"0(3"&$%0&$#1* 0&+0+ +10#+.0%"**+#(1&!6/2&/ +/&052.&0&+*3&0%

0%"!+)&6/ + "*0.0&+**!0"),".01."1!,.+,+/"!0%"#+((+3&$(+$5

%"." &/0%""4,".&)"0((5)"/1."!2&/ +/&05/#1 0&+*+#**+,.0& ("/ +* "0.0&+

*!0%"(+ (0"),".01."+#0%"**+#(1&!+0"0%00%",.)"0".

&/)+!&#&"!

3"&$%0&$#1 0&+**!0%",.)"0".

&/3"&$%0&$#1 0&+*3%&("

&/0%",1."#(1&!

2&/ +/&05

+/"-1"0(5!&//&,0&2"!.!+)#+. "/ "+0&"!5

%"&* +.,+.0&++#0%"1('0%".)( +!1 0&2&05+#"4,".&)"0((5 +((" 0"!!0#+.$&2"

**+#(1&! "& (1!"!5!,0&$0%" +!1 0&2"%"0#(14"/%1/0%"05,& ( +*!1 0&2"

%"0#(14)+!"( *"!"/ .&"!/

%"."2&/"!2(1"+#0+0(%"0#(14&/ %&"2"!5(0".0&+*+#0%"

,.)"0".$&2"&0%"#+.)".

."(0&+*%&/,.)"0".&/ ( 1(0"!/

(24)

$)2#$#$.($!$ # /!-$/$)!0)/$))/# /4+$')./)/+-*+ -/$ .

!-(0'/$)*)./)/1'0

'' # /!-/$)$.0. $)./ !# (0)/!# /

!-/$*)

$.0.0''4 ./$(/ 4/0)$)"0)/$'$/(/# ./# 1'0 *!/# -('$!!0.$1$/4 - $/$.

..0( /#//# !0)/$)#)" .2$/#($)6./ (+ -/0- )))!'0$) )/-/$) )). ,0 )/'4- '/$*).#$+2..0"" ./ .

2# - $. 3+ -$( )/'( .0- 0'&/# -(')0/$1$/4!))!'0$.$)"/#$.!0)/$) /# !$)'. /! ,0/$)!-/# # /!'03 .- "$1 )4/# !''*2$)"

*)

$%

# - '/$*) /2 )+-( / -.

)

$..

$)''4/# !/-$)/# -)(# /!'03) 3+- .. .

2# -

$./# '/5()))./)/

2.3. Microscale based techniques (Molecular dynamic simulation)

# !'0$ !'2 $. 0.0''4 .'1 !- ))!'0$. 0.$)" /# )/$)00( /# -4 2#$# $)1*'1 . /#

#4+/# .$.!*)/$)00(( $0(( )/$) $)/# +- 1$0.. /$)/# )/$)00(++-3$(/$)

(25)

#!."%').4/"2%!+$/7.&/202/",%-3!44(%3-!,,%2%.$/&4(%.!./-%4%23)9%2!.'%%'&,/7!2/5.$

#!2"/..!./45"%3%4#$5%4/#/-0!2!",%#(!2!#4%2)34)#$)-%.3)/.3/&4(%&,5)$#/.&).%$30!#%!.$

4(%-/,%#5,!2&2%%0!4(.4()32%'!2$-%4(/$34/$)2%#4,83/,6%&/24(%&,5)$-/4)/.!44(%-/,%#5,!2 3#!,% 7%2% ).42/$5#%$ (% -/,%#5,!2 $8.!-)# 3)-5,!4)/. -%4(/$ 7!3 &)234 ).42/$5#%$ "8 ,$%2!.$!).72)'(4!44(%%.$/& .4()3-%4(/$4(%-/,%#5,%3!.$!4/-3!2%2%'!2$%$!3

&,5)$0!24)#,%37(/3%$8.!-)#!.$).4%2!#4)/.3!2%#/-054%$/6%2!4)-%0%2)/$(%0%2)/$/&4)-%

&/2 4(% #!,#5,!4)/. )3 $%&).%$ !##/2$).' 4/ 4(% 4)-% .%%$%$ 4/ 2%!#( ! $8.!-)# %6/,54)/. "%47%%.

3834%- #/-0/.%.43 3/,6%3 ! 3834%- /& ).4%2!#4).' 0!24)#,%3 53).' 4(% %74/.;3 %15!4)/.3 /&

-/4)/. !.$ 30%#)&)%3 4(% 42!*%#4/2)%3 /& -/,%#5,%3 !.$ !4/-3 . !$$)4)/. -/,%#5,!2 -%#(!.)#3 /2 ).4%2!4/-)#0/4%.4)!,&/2#%&)%,$3!2%53%$4/#!,#5,!4%$&/2#%3"%47%%.4(%0!24)#,%3!.$4(%)20/4%.4)!,

%.%2')%3

%#%.4!$6!.#%!.$02/'2%33)..!./&,5)$)#3(!3-!$%!02/-)3).'3)-5,!4)/.!002/!#().4()3

&)%,$&/202/",%-3/.!002/02)!4%,83-!,,,%.'4(!.$4)-%3#!,%33#(%-!4)#/&!7!4%2"!3%$

.!./&,5)$#/.4!).).'#!2"/..!./45"%3)3')6%.).)' (%-%4(/$)3+./7.!3!

6)!",%-%4(/$4/).6%34)'!4%4(%&,/70(83)#3)..!./3#!,%02/",%-33).#%)4)3!",%4/02/0%2,8

$%4%2-).%4(%-/,%#5,!23425#452%/&4(%&,5)$!.$352&!#%3!.$4/02%$)#44(%!4/-)34)#).4%2!#4)/.3

!""!2)%4!, !.$%%%4!, 15!.4)4!4)6%,82%6)%7%$2%#%.4!$6!.#%3/&)..!./&,5)$)#

3834%-3).#,5$).'%.(!.#%-%.43).-%!352%-%.4/&.!./&,5)$4(%2-/0(83)#!,02/0%24)%34(%2-!, 7!,,-/$%,).'4(%#54/&&2!$)53!.$).)4)!,#/.$)4)/.3%4#(%3%2%6)%70!0%2302/6)$%$6%28

#2)4)#!,$)3#/6%2)%3!.$02/0/3%$)-0/24!.4!2%!3.%%$%$4/"%4!+%.).4/#/.3)$%2!4)/.).&5452%

2%3%!2#(!.$4(534(%2%!$%2)32%#/--%.$%$4/2%&%24/4(%3%!24)#,%3&/2&524(%2$%4!),%$).&/2-!4)/.

/.4()3-!44%2(%-%4(/$7!3(%2%4/53%$).6!2)/53).6%34)'!4)/.3/..!./&,5)$34)353%$&/2 -%!352).'4(%6)3#/3)48!.$4(%2-!,#/.$5#4)6)48/&.!./&,5)$!4$)&&%2%.44%-0%2!452%3!.$53).'

$)&&%2%.40!24)#,%3)9%3 ).6%34).'(%!442!.3&%2#(!2!#4%2)34)#3/&.!./&,5)$ !.$4(%)2352&!#%

.!./3425#452%3 /)3%5),,%&,/7/&!.!./&,5)$ #(!/4)#-/6%-%.43/&.!./0!24)#,%3 %4#

##/2$).'4/%)%4!, -/,%#5,!2-/4)/.7)4(/544%-0%2!452%'2!$)%.4!2%&/5.$4/"%

2%30/.3)",%&/2(%!442!.3&%2).4(%!2%!/&,/76)3#/53(%!453).'!./.%15),)"2)5-/'(2!)%%4

!, 53%$4/3)-5,!4%4(%/)3%5),,%&,/7/&.!./&,5)$3#/.4!).).'0,!4).5-!.$#/00%2 0!24)#,%3!.$7%2%!",%4/35##%33&5,,8%-0,/84(%!2'%3#!,%4/-)#/,%#5,!2!33)6%,8!2!,,%, )-5,!4/2#/$%&/24()30520/3%.4()3-%4(/$4(%%..!2$:/.%30/4%.4)!,'/6%2.34(%

!4/-)#).4%2!#4)/.!",%35--!2)9%33/-%!$6!.4!'%3!.$,)-)4!4)/.3/&$)&&%2%.4!002/!#(%3

(26)

(27)

#1&," "3+1%#0 '*'11',+0

Finite Difference

-

'*-)'!'16

-

#)1'3#)60))!,+3#/%#+!#1'#

-

2*#/'!)#3)21',+,$1&#"'$$#/#+!#

#.21',+'00'*-)#

-

'*-)#!,"#01/2!12/#0

-

+12'1'3#+"#061,'-)##+1$,/0'*-)#

-/, )#*0

-

+)/%#0!)#-/, )#00))1'*#

01#-0/#+##"#"

-

,*-/#"1,+##"0&'%&

+2* #/,$%/'"01,%'3#%,,",21-21

-

'$$'!2)11,#-),6$,/!,-)'!1#"

01/2!12/#0 Finite Volume

-

,-)#5#0%#,#1/'#0!+ #!,+0'"#/#"

-

#2*++!+ ##0')6#01 )'0&#"

-

2'1 )#$,/+01/2!12/#"/'"0

-

,3#/+'+%#.21',+0/#4/'11#+0'+1#%/)

$,/*

-

//#%2)/%#,*#1/'#0/#.2'/#$/

*,/##$$,/11&+

Finite Element

-

#/6%#+#/)*#1&,"+"20#"$,/0,)3'+%

3/'#16,$-/, )#*0

-

&#01'$$+#00*1/'5+"1&#),"3#!1,/!+ #

!,-21#"$,/#!&#)##+1

-

)),4'+%11#+1',+1, #!,+$'+#"1,0'+%)#

16-'!)#)#*#+1

-

-1')3/'1',+,$-/,-#/1'#0!+ #&+")#"

4'1&/#)1'3##0#

-

//#%2)//#%',+0!+ #*,"#))#"4'1&%/#1#/

!!2/!6

-

2'1 ')'16$,/+,+)'+#/#.21',+0

-

'%&$)#5' ')'16'+!&+%'+%#)#*#+10'7#0

-

'5#" ,2+"/63)2#-/, )#*0/##0'#/1,

&+")#

-

+60&-#"%#,#1/6!+ #,"#))#"4'1&

1&#0*##$$,/1

-

.21',+0/#2!&,/#

1&#1'!))6!,*-)#5

-

2*#/'!))6'+1#+0'3#+"!+1(#

,/#1'#+"#,/601,/%#

0-!#

-

,/#1&#1'!0'+3,)3#")#00 -&60'!)0'%+'$'!+!#

-

,*-)#5!,"#01/2!12/#0

Boundary Element

-

#00"1'0/#.2'/#"1,/2+-/,%/*

#$$'!'#+1)6

-

))0601#,$#.21',+0"'0!/#1'71',+1(#0 -)!#,+)6,+1&#02/$!#,$ ,"6

-

#))02'1#"1,-/, )#04'1&,-#+",'+0

-

,,"%/##*#+1 #14##++"

-

,/##$$'!'#+1'+*,"#))'+%"2#1,1&#

/#"2!1',+,$"'*#+0',+0

-

"3+1%#,20'+1&#!0#,$-/, )#*04'1&

'+$'+'1#,/0#''+$'+'1#",'+0

-

#00"1-/#-/1',+1'*#

-

/,%/**'+%!,*-)#5'16

-

'0+,102!!#00$2))6--)'#"1, +,+)'+#/$)2'"$),4-/, )#*0

-

1/'!#0/#06**#1/'!"#+0#+"

+,1#061,0,)3#

-

,)21',+1'#'0),+%+"#*,/6 0'7#'0)/%#

-

,,/$,/1&'+01/2!12/#0

-

++,+)'+#/-/, )#*01&#'+1#/',/

201 #,"#))#"

(28)

Lattice Boltzmann

-

$###$#%$"

" %$

-

%$" $$"#

-

%$"%$ #'#%$#

-

%$"## $##%#

#'# "#%$"

"%#

-

%$" )"# $"$#

$'%%#

-

&"$" )#

" "$##%#&##$)

$"%$&$)#$

#$"$"'"

-

$#$$%"

"'"$&")

"$#%$"$%"#

"!%"#$"$$'#

$$ %$$) ( ##"

)"#%"))"#

-

%&##$)#$"$ %$

"$"$$$%$$#

"$$$"($$$

$

-

(#)$"'#

$$#"$

#$"$"'"

-

%$ #'#'$"

&"$#$)# "$

Dissipative

Particle Dynamics

-

"##$'$"$%"

-

"'$#$"#$"%$

$$

-

%$" $$"#

-

%$""%# $##$

$#$"%$%$#

-

"#" "$$#%$"

# $$ "##$$"""$

$%")###%

#"$$$%%##

&"$#

-

%$"#%# #'##

-

%$"%$ #'#

-

$"$"'"($#"$'

##$$"##

-

%$"$" "$$"$#$

"# &)%$$"$

" %#"#$'%%#

$#%""%

-

"#"$$

$' "$"#$

)# "$"#$#)#$

%"#%$

$" )# " "$#"$

%$ "$"#

-

%$$)'

"'$$"

$&%#

Recent advances in modeling and simulation of nanofluid flows—Part II Applications

HAL Id: hal-02051317

https://hal-univ-rennes1.archives-ouvertes.fr/hal-02051317