O
pen
A
rchive
T
OULOUSE
A
rchive
O
uverte (
OATAO
)
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 : 14503
To link to this article: URL:
http://hdl.handle.net/2042/57596
To cite this version : Luminari, Nicola and Airiau, Christophe
Sensitivity and stability analysis of a rigid canopy flow. (2015) In:
22ème congrès Français de Mécanique, 24 August 2015 - 28 August
2015 (Lyon, France).
Any correspondance concerning this service should be sent to the repository
administrator:
staff-oatao@listes-diff.inp-toulouse.fr
!"#$%$&$%' (") #%(*$+$%' ("(+'#$#
,-( .$/$) 0,-(",1' 2,3
! "#$%&'(%
) *! +%(%'#
! "#$%& '#( ))*+ ,-(./"-01'0.& '234567389 :5 %;<=;<75& >==95 :< 06! , ?3==5 .;<= & @AB** %;<=;<75& $6 2C5&
23C;= !=<?32 63D3?E8!E6&CF6378;GF5! 363 <D3?E8!E6
!"#$%&#'
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
,-./0(123 2-&2%4%5%4.) '160%&4) 24'7%8%4.) 9'&0:.
( )*#$+,-&#.+*
H<6 6575 6CF 32856578 C;4567 8F5 ?;:5==32I ;E J;K ;456 C;?G=5L 7<6E C57 MC;?G=3 28& G;6;<7& 2;8 7?;;8F !!!N 8; 3245783I 85 G;773O=5 :6 I 65:<C83;2! > 7<6E C5 C;4565: OP 2 66 P ;E 63I3: CP=32:567 755?7 8; O5 I;;: 78 6832I G6;O=5?
! "#$"%#$" "& '()(*& &)+ ,"#-" *#$" "(.& "& %()$&) &/&- + !0 "& 1!, !.&) +2-" ( )#$#34 '!)!2+ (03 (0#+! )!'#- *&3#2*5
6# " "& '2)'!+& !7 "& -!2'%&3 1!, ")!2$" (03 !.&) '&)*&(8%& %(9&)+ "& *( "&*( #-(% *!3&% ')!'!+&3 89 :"#+(%8&) # (03 ;&'7 <=> "(+ 8&&0 (0(%9+&35 ?)!* ( +& !7 &@'&)#*&0 + !.&) ( )#$#3 -(0!'9 <=> ( '()(*& )#- -!*'%&@ *!3&% "(+ 8&&0 82#% 2+#0$ ( 3)($ 720- #!04 ( 2)82%&0-& *#@#0$ %&0$ " *!3&% (03 "& *!*&0 2* 12#3 1!, &A2( #!0+5
B0 <C> ( +#*'%&) *!3&% #+ ')!'!+&3 82 "& -!*'()#+!0 ,# " &@'&)#*&0 (% 3( ( (''&()+ ! 8& !7 %!,&) A2(%# 9 "(0 "& #0# #(% *!3&%5 D!*& *!)& -!*'%&@ ('')!(-"&+ -(0 8& 7!203 #0 <E4 F> 82 ,& #0.&+ #$( & "&)& "& G)+ '()(*& )#-!)#$#0(% -(0!'9 *!3&% <=> 8&-(2+& # #+ 82#%3 7)!* "&!)& #-(% &A2( #!0+ (03 &@'&)#*&0 (% 3( ( G #0$ (03 8&-(2+& "& ')!G%& -(0 8& )('#3# %9 -!*'2 &3 ,# " +!*& A2# & $!!3 ($)&&*&0 ,# " &@'&)#*&0 (% 3( (5
D#0-& "& *!3&% #+ %()$&%9 8(+&3 !0 &*'#)#-(% 720- #!0+ &@ )('!%( &3 7)!* &@'&)#*&0 (% 3( ( ( +&0+# #.# 9 (0(%9+#+ "(+ ! 8& '&)7!)*&35 H"& *(#0 #3&( #+ ! 3&G0& ( A2(0 # 9 ,"#-" A2(%#G&+ "& ')!G%&5 I+ #0 8!203()9 %(9&)+4 (0 #0 &$)(% "#-J0&++4 +2-" (+ "& 3#+'%(-&*&0 "#-J0&++ δ1 *(9 8& ('')!')#( & (03 # #+
"2+ +& (+ !8K&- #.& 720- #!0(%5 H"& +&0+# #.# 9 (0(%9+#+ (%%!,+ ! 3& &)*#0& "& $)(3#&0 !7 "#+ 720- #!0(% ,# " )&+'&- ! (09 .()#(8%& !7 "& *!3&% ! +"&3 %#$" !0 "& 3&'&03&0-#&+ (03 "& #*'!) (0-& !7 "& -"!#-&+ ')!'!+&3 #0 <=>5
I7 &) "#+ + 239 ,& (%+! (0(%9L& "& %#0&() + (8#%# 9 !7 "& ')!G%& 8(+&3 !0 ( *!3(% ('')!(-"4 (03 ,& #0# #( & "& + 239 !7 "& +&0+# #.# 9 !7 "& &#$&0.(%2&+ 32& ! *!3#G-( #!0 !7 "& 8(+& 1!,5
!"#$%& %"'"()*'+, ($-).
H"& 1!, -!0G$2)( #!0 (03 0!M ( #!0+ ()& 3&G0&3 !0 G$2)& = N)#$" O5 H"& ')!8%&* #+ (++2*&3 ! 8& !0&M 3#*&0+#!0(% (03 '()(%%&%4 "& #0-!*')&++#8%& 8!203()9 %(9&) 1!, #+ +& #0 *! #!0 89 "& +%!'& !7 "& $)!203 +2)7(-& >?@5 H"& -!0 )#82 #!0 !7 "& -(0!'9 ! "& 8(+& 1!, #+ *!3&%%&3 89 ( 3)($ -!&P-#&0 (+ ( 720- #!0 !7 0!)*(% -!!)3#0( & z ,# " (0 &A2#%#8)#2* &A2( #!0+ ,)# &0 (+ QgS = ∂u ′w′ ∂z + 1 2CD(z)aU (z) 2 N=O
g #+ "& $)(.# 9 (--&%&)( #!05 u′w′ #+ "& 2)82%&0 + )&++ (03 U(z) #+ "& *&(0
+ )&(*,#+& .&%!-# 95 ?)!* "#+ &A2( #!0 ( -!*'%& & *( "&*( #-(% *!3&% #+ 3&)#.&3 (03 3& (#%&3 #0 <=>5 R&)& ,& +#*'%9 )&'!) "& *(#0 &A2( #!0+ ! +!%.& ,"#-" -!0+ # 2 & "& -!0+ )(#0 + !7 "& +&0+# #.# 9 (0(%9+#+Q
CDC = 1 + 10Re −2/3, R e= Uhd/ν !" U2− U1 Uh = (h − z1)ΩCDAη¯1a 1 − U1 2 Uh2 #" ¯ η1 = f ( ¯β1); β¯1= z1 h, U1= u1(z1), U2= u2(z2) $" ∆U Uh = 1 + 16ad %" CDA = CDC 1.16 (1.16 − 9.31ad + 38.6ad 2 − 59.8ad3) &" CD(z) = η(β)CDAχ(z) '" d dz du1 dz 2 = 1 lc 1 2CDau 2 1(z) − gS z1≤ z ≤ h (" tml = α(h − z1) )" lc = 0.22(h − z1) *+" lac = 0.22 ξ tml **" u2(z) = Uh+ 2√gS 3lac (z2− h)2− (z2− z)2 h ≤ z ≤ z2 *!" U1 = s 2gS CD(z1)a = u1(z1), Uh= u1(h) = u2(h) *#" ,- ./0 12030-. 4526 40 3107897:;;< 8-.25=>70 :- :==8.85-:; 0?>:.85- .5 8@A 1530 ./0 75-.8->8.< 5B ./0 =028C:.8C0 48./ 203107. .5 z 128@0 0D15-0-." :. ./0 .51 5B ./0 7:-51< z = h" u′ (h− ) = u′ (h+)48./ : χ B>-7.85-E χ(z) = 1 − e−σ(z−h) z1−h *$" F/0 7:-51< 83 @:=0 5B 3@:;; 7<;8-=023 5B /08G/. hH 5B =8:@0.02 d :-= a B25-.:; :20: 102 20B020-70 C5;>@0 =0-38.< 5B ./0 7<;8-=02"I F/0 125J;0@ 83 =8C8=0= 8-.5 .45 20G85-3 300 9GI * 28G/."H ./0 923. 5-0 8-=0D *" 8-38=0 ./0 7:-51< :-= 48./ ./0 3.20:@4830 C0;578.< u1(z) :-= ./0 3075-= 5-0 :J5C0 8. 8-=0D !" 48./ ./0 C0;578.< u2(z)I F/0 =2:G 750K780-. CD 83 30. .5 L025 :J5C0 z = h :-= 83 @5=0;;0= 48./ 30C02:; B>-7.85-3 8-38=0 ./0 7:-51<I F/0 7:-51< @5=0; J0G8-3 :. z = z1H >-=02 ./83 /08G/. ./0 C0;578.< =503 -5. 7/:-G0I F/0 .>2J>;0-70 @5=0; 83 :33578:.0= 48./ ./200 ;0-G./ tmlH lc :-= lacI ,- :==8.85-H ./0 0?>:.85- * 7:-J0 8-.0G2:.0= :-:;<.87:;;< :J5C0 ./0 7:-51< .5 <80;= u2(z)I M8-:;;<H : -5-;8-0:2 51.8@8L:.85- 125J;0@ 48./ # -03.0= N04.5- 8.02:.85-3 83 20?>820= .5 35;C0 ./0 B>;; 30. 5B 0?>:.85-3I M8G>20 * 3/543 ./0 1259;0 5J.:8-0= B52 7:30 O 5B ./0 20B020-70 1:102 :-= 8. 83 75@1:20= .5 ./0 0D1028@0-.:; =:.: 48./ 02252 J:23"I P5@0 3@:;; =837201:-7803 :20 5J302C0= =>0 .5 ./0 752207.85- @:=0 ./25>G/ ./0 χ B>-7.85- .5 30. 75-.8->8.<
χ n1 = 20 qi n2− n1= 3 Qi(z) qi= [a, d, h, α, ξ, z1, z2, Uh, U1, U2, lc, lac, ¯η1, β1, tml, CDC, CDA, Uh ′ , σ, β] i = 1..n1 Qi= [χ, CD, u1] i = 1 + n1..n2 L = E + L E E = ǫ0E + n1 X i=1 ǫi q2 i 2 + n2 X i=1+n1 ǫi Z h 0 Q2 i 2 dz E
E = δ1= Z z2 0 [U2− u(z)] dz = Z h 0 [U2− u1(z)] dz + Z z2 h [U2− u2(z)] dz
!" #"$%!&' ǫi()" *'")+,"-".,".& (., &(/" $.&0 (110*.& &!" ,$2".'$0. 03 &!"
-()(2"&")'4 5(.6 0&!") 1!0$1"' 30) E ()" -0''$78" .(&*)(8869 (., 10))"'-0.,$.% 2(&!"2(&$1(8 ":-)"''$0.' 1(. 7"102" ;*$&" 80.% (., 102-8":4
L ,"'1)$7"' &!" 10.'&)($.&' 03 &!" 0-&$2$<(&$0. -)078"24 =. L ()" $.18*,", &!" 10.'&)($.& #$&! 10.'&(.& -()(2"&")' )"3"))", (' ci (., 10.'&)($.&' #$&!
3*.1&$0.' )"3"))", (' Ci(z) 4 >"'-"1&$?"869 &!" @(%)(.%" 2*8&$-8$")' (''01$(&",
&0 10.'&(.& (., 3*.1&$0.' ()" )"3"))", (' pi 0) Pi(z)A
L = n1 X i=1 pici+ Z h 0 n2 X i=1+n1 Pi(z)Ci(z) dz BCDE
!" '".'$&$?$&6 sqi 0) SQi ()" ,"&")2$.", 3)02 &!" ,")$?(&$?"' 03 &!" 8(+
%)(.%$(. 3*.1&$0.(8 #$&! )"'-"1& &0 &!" 1!0'". -()(2"&")' (' δL = δE = n1 X i=1 sqiδqi+ Z h 0 n2 X i=1+n1 SQi(z)δQi(z) dz
!" ?()$(&$0. 03 &!" @(%)(.%$(. 3*.1&$0.(8 #$&! )"'-"1& &0 (88 &!" -()(2"&")' 8"(,' &0 ( 8$."() '6'&"2 30) &!" *./.0#. @(%)(.%" 2*8&$-8$")'4 !" 3*88 '6'&"2 $' 2(," 76 FG )0#' B.*27") 03 )"2($.$.% -()(2"&")'E (., CH 108*2.' B.*27") 03 )"2($.$.% 10.'&)($.&'E #!$1! 1(. 7" '08?", 76 )",*1&$0.4 !" z+,"-".,".& '".'$&$?$&6 3*.1&$0.' 1(. 7" I)'& '08?", (.(86&$1(886 (., I.(886 ( 8$."() '6'&"2 $' 8"3&9 $4"4A
∂L
∂qi = Ai,jpj+
∂E ∂qi
#!")"9 30) ":(2-8"9 &!" 2(&)$: 0. &!" )$%!& !(., '$,"9 30) i = j = 19 )"(,'A A1,1 = ∂L ∂a = n1 X i=1 2 p12gS a2C DAη¯1 − p1d16Uh − p8CDC !3 a 2d3(59.8) + 2 ad2 (38.6) + d(−9.3) + + p10 1 − U1 2 Uh2 (h − z1)ΩCDAη¯1− 2 (ad16 + 1)d16 + + Z h 0 p17(z)CD(z)u12(z) lc2 dz BCJE BFKE =. &!" -)"?$0*' ";*(&$0. &!" 3*.1&$0. p17(z) $' &!" (,L0$.& 03 &!" ?"801$&6
-)0I8"9 (., $& $' &!" '08*&$0. 03 &!" (,L0$.& ";*(&$0. #$&! !020%"."0*' 70*.,()6 10.,$&$0.A
−2 d dz du1(z) dz dp17(z) dz −p17(z)CD(z) au1(z) l2 c = h !"# $% &'( ) *'++ ,%-'. ,/%0+'1 *2(3 ) -45)/' 267'/(20+' 1)(/289 -%1' ,)/)1'('/-3)7' (% 0' )--51'. :%6-()6(9 )6. (3' :%//'-,%6.26& -'6-2(272(2'- 7)62-3; <6 (3' =%++%*26& *' =%:5- %6 (3' -'6-2(272(2'- )--%:2)('. (% (3' =%++%*26& =%5/ ,)/)1'('/- > a (3' =/%6()+ )/') %= (3' :)6%,? ,'/ 562( 7%+51'9 d (3' :?+26.'/ .2)1'('/9 h (3' 3'2&3( %= (3' :)6%,? )6. α (3' =/):(2%6 %= (3' -3')/ +)?'/ (3)( +2'- *2(326 (3' :)6%,?; @ -':%6. )6)+?-2- *)- .%6' )--5126& α :%6-()6( )6. 0? )..26& (3' -'6-2(272(? (% (3' ,)/)1'('/- ξ; <( )++%*- (% 7'/2=? 3%* (3' -'6-2(272(? %= a9 d )6. h :)6 :3)6&' *3'6 *' -'+':( .2A'/'6( ,)/)1'('/- (% -%+7' (3' -?-('1; $% :%1,)/' -'6-2(272(? *32:3 )/' .'B6'. *2(3 .2A'/'6( 562(9 *' .'B6' (3' /'+)(27' '//%/ )-> ∆E E = sqi qi E ∆qi qi C' )/02(/)/2+? -'( ∆qi/qi= 1 = ǫrqi; ∆E E ǫra DE;!! F ǫrd G;H"F ǫrh DI;IGJK F ǫrα K;JE F ǫrξ I ∆E E ǫra DE;K! F ǫrd G;HE F ǫrh DI;IGE F ǫrα I ǫrξ DI;"LM F $)0+' "> N'+)(27' '//%/ ()0+' =/%1 (*% )6)+?-2-; @- B/-( :%6:+5-2%69 (3' /'-5+(- -3%* (3)( (3' ,)/)1'(/2: 1%.'+ 2- -'6-2(27' (% (3' 26,5(- a )6. d *32:3 )/' &'%1'(/2:)+ ,)/)1'('/- %= (3' :)6%,?; $3' 1%.'+ ),,')/- (% 6%( 0' 7'/? -'6-2(27' (% α )6. ξ *32:3 )/' ,)/)1'('/- /'+)('. )( (3' 12826& +'6&(3 (3'%/?; $32- )6)+?-2- :)6 0' 21,/%7'. 0? %(3'/ ('-(- )6. 0? ,'/=%/126& =5/(3'/ -'6-2(272(? )6)+?-'-;
!"#$%& '(%)"*"(+
,#%*+-"-$3' ('1,%/)+ +26')/ -()02+2(? )6)+?-2- 2- 0)-'. %6 (3' :+)--2:)+ =%/15+)(2%6 26 ('/1- %= 6%/1)+ 7'+%:2(?9 )6. ) .':%1,%-2(2%6 26 6%/1)+ 1%.'- (3)( /').-> w(x, y, z, t) = ˜w(z)ei(kxx−ω t) *2(3 kx (3' /')+ -(/')1*2-' *)7' 6510'/ )6. ω ) :%1,+'8 6510'/ *3'/' (3' /')+ ,)/( 2- ) =/'45'6:? )6. (3' 21)&26)/? ,)/( ) &/%*(3 /)(';9 $3' .'D :%1,%-2(2%6 %= (3' O%* B'+. *2(3 ) 1')6 O%* (3' ,/'72%5- ,/%B+' U(z)# )6. (3' ,'/(5/0)(2%6 w(z) +').- (% (3' *'++ P6%* Q//DR%11'/='+. '45)(2%69 *3'/'!"# $ D2 %$ "# &%'() * !+&!,-" "# . ,%+& & $!/-"!/ 0!"# $ .1 ," "% "# +%$2-) &!$ ,"!%+ z3 (−iω + ikxU )(D2− k2x) − ikxU′′− 1 Re (D2 − kx2) 2 ˜ w = 0, Re = Uhd ν ω !. -+ !4 +/-)' 5%$ " 21%$-) ."-(!)!"6 -+& ˜w !. -+ !4 +5'+,"!%+7 8# !4 +/-)' 1$%() 2 !. .%)/ & 0!"# 9-")-( 0# $ "# :; <'-"!%+ !. &!.,$ "!= & '.!+4 - .1 ,"$-) .,# 2 0!"# ># (6.# / 1%)6+%2!-). ?.!= N@ -+& - 2-11!+4 ( "0 + "# . 2!A!+B+!" -+& "# .1 ,"$-) &%2-!+ CDE7
8# .1 ,"$'2 !. .#%0+ !+ B4'$ F7 8# 2%$ '+."() 2%& !. .#%0+ 0!"# -$ & &%" !+ "# B4'-$ -+& "# ,%-$-$ .1%+&!+4 !4 +5'+,"!%+. --$ 1)%"" & !+ B4'-$ G7
!"#$% &' u ()* w +%$,#$-(,!.) /%0.1!,2 +$.30%4 .5 ,6% 7.4, #)4,(-0% %!"%)7.*%8 9.,6 ,6% /%$,!1(0 ()* ,6% 6.$!:.),(0 /%0.1!,2 6(/% ( 7(;!7#7 !) 1.$$%4+.)< *%)1% .5 ,6% !)=%1,!.) +.!), .5 ,6% 1().+2 =.>? ,2+!1(0 .5 () !)4,(-!0!,2 .5 !)=%1,!.)(0 )(,#$%8
!"#$%&%'%&( &) '*+%*&%)#$ %# &," -"*# .)/
@!)1% >% 6(/% 4%%) ,6(, 4.7% /($!(,!.)4 .$ %$$.$4 .5 ,6% !)+#, +($(7%,%$4 1() 16()"% ,6% 46(+% .5 ,6% 7%() =.>? !, !4 !),%$%4,!)" ,. 1.7+#,% ,6% /($!(,!.) .5 ,6% 4+%1,$#7 *#% ,. 16()"%4 .5 ,6% 7%() =.>8 .00.>!)" ,6% >.$A +$%4%),%* !) B&C >% *%,%$7!)% ,6% 5#)1,!.) )(7%* Gu >6!16 7%(4#$%4 ,6% 4%)4!,!/!,2 .5 ,6% %!"%)/(0#%4 ,. 16()"%4 !) ,6% 7%() =.> +$.30%8 c = ω α, δc = Z ∞ 0 Gu K δU dz Gu= w∗(D2− kx2) ˜w − (w∗w)˜ ′′ D6% E#(),!,2 w∗ !4 ,6% (*F.!), %!"%)5#)1,!.) $%0(,%* ,. ˜w? ()* ,6% ./%$0!)% 7%()4 1.7+0%; 1.)F#"(,%8 D6% 1.)4,(), K 1() -% 1.7+#,%* 5$.7 !),%"$(,!.) .5 ,6% %!"%)5#)1,!.) ˜wG4%% B&CH8 D6% (*F.!), 5#)1,!.) !4 ,6% 4.0#,!.) .5 ,6% (*F.!), I$$<@.77%$5%0* %E#(,!.)? 4%% BJC ()* B&C 5.$ *%,(!048
D6% $%(0 ()* !7("!)($2 +($, .5 ,6% 5#)1,!.) GU ($% +0.,,%* !) 3"#$% K ()*
!, 1() -% 4%%) ,6(, ,6% /($!(,!.) .5 GU ($% 1.$$%0(,%* ,. ,6% *%$!/(,!/% .5 ,6%
7%() =.>? ()* ,6%$%5.$% >!,6 !,4 4,(-!0!,2 +$.+%$,!%48 L%, #4 5#$,6%$ ).,% ,6(, ,6% 4%1.)* *%$!/(,!/% !4 30,%$%* -2 ( "(#44!() 5#)1,!.) -%5.$% !,4 #4% !) ,6% I@ %E#(,!.)8
!"!#!$%!&
!" #$%&'()*+,%- ./ '01 2*34- 5/./- 6$* (%7%,*1 8+9:,$ 94 ;*8*,',*1 &$*'+ ('<*+&/ >):-% ?-'"2%0-' ?-'-)%0@ =9(/ >?- @??>/
@" AB$7%1- C/D/ '01 5*00%08&90- E/A/- A,')%(%,< '01 6+'0&%,%90 %0 A$*'+ F(9:& A3+%08*+ G33(%*1 .',$*7',%B'( AB%*0B*& @??!
H" I9,,'+9- G/- J9+)*,,- C/ '01 KLB$%0% C/- 6$* *M*B, 94 )'&* N9: ;'+%',%90 90 N9: &,')%(%,< AB (C2+, .-0@B =9(/ >OP- @??H- 33/ @QHRH?@/
>" 2*34- 5/./- F(9: '01 ,+'0&39+, %0 +*8%90& :%,$ 'SL',%B ;*8*,',%90 8##2)C ?-D+-E "F (C2+, .-0@)#+0' =9(/ >>- @?!@- 33/!@HR!>@/
T" EL390,- A/- #9&&*(%0- F/- C<- J/- E* K'08+*- U/- 5*790- C/- '01 I+L0*,-V/- .91*((%08 :';%08 B+93& L&%08 ('+8*R*11< &%7L(',%90W B973'+%&90 :%,$ *X3*+%7*0,& '01 ' (%0*'+ &,')%(%,< '0'(<&%& AB (C2+, .-0@B ;9(/ PT@- @?!?-33/ TY>>/
P" I*(B$*+- A/U/- 5'+7'0- Z/2/- '01 F%00%8'0- D/D/- 6$* :%01 %0 ,$* :%((9:&W N9:& %0 49+*&, B'093%*& %0 B973(*X ,*++'%0 8##2)C ?-D+-E "F (C2+, .-0@)#+0' =9(/ >>- @?!@- 33/>OQRT?>/
O" G0,[9:%'[- G/- E<0'7%SL* 'LX ,*73& B9L+,& 1\L0 ,9L+)%((90 %&9(] C$/E/ ,$*&%& ^_0%;*+&%,] C'L( A')',%*+ 1* 69L(9L&*- F+'0B*- @??T`/
