Lecture-20_(11-24-09)

Lecture-20_(11-24-09) - APPH 4200 Physics of Fluids Fluid...

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Unformatted text preview: APPH 4200 Physics of Fluids Fluid Instabilities (Ch. 12) November 24, 2009 1.! ! 2.! Kelvin Helmholtz Instability Centrifugal Instability Tuesday, November 24, 2009 1 Kelvin-Helmholtz Instability Tuesday, November 24, 2009 2 More Beauty Tuesday, November 24, 2009 3 More Beauty Tuesday, November 24, 2009 4 Nature, vol. 430, pp 755-758 (2004) Also plasma! Tuesday, November 24, 2009 5 Kelvin-Helmholtz Instability ¡. Ii l V, AJ f-( II úA ,. iJ i. -1 l.. S' TV i .. , 1' éJ "" ~ U" .P, ~ ii :. t: J( ~ U-z p , (. k e ,-V.N - H i!h-~Hó(.n- I N ,n+fZIi..., T" DlffT~J L.'~/""'7 t:rc'f.a t:F -l" ç'7Yßt '" i i-K -, l=tÁo(-q a l!t! I ~ ~MIFA. (Hit S" tl ¡SM 0'" E4C'ò""&.( Sr/lAr l ¡:~c.+7ì. ~. Tuesday, November 24, 2009 6 (f . -t- ,u. ___Ls~ i) (. cI)1.... ~6. 7'ù 0 \ -~---------- -~----- ---------- -- --------- ------ ~---- ~- -- --- -- --~-------- - -~----- -~~ -- ----- -----,----..--~~- t1_~__~ =-~--- 6. -((tT .. --I/-;~-----(ltÆ--f-'-lr~--J---~--H---- .-.----------------. _.__., ___._..__..., ___._rtl_J..J_~~~.._. ____.__.__.__...u.._____.._._._______~___._u__._ _ _...___. _.uu__ _..____.__uu___.____ __.___ _.__.___.______..__._..___ Potential Flow Away from Interface -t---- _H Eo N'-7~--------LiTu£--r:-f,----------e: :oj:? ~ê---- ---- -- ------------------- ---.r-u.-----.-..---..--.-------~~-.--------------~--.---- _ _._______________..____.__.._______._u__.._._u________..------ _..u___ .____u___u__._ ! L~-- if - CJ)! l Jl ¡ ==---=-- fl~~EO ..~=: ?L-~_Ö. ----=-~~--=- ~= =I=~==:-~~=-;~;=;~~i-i~-=~-~=-~~-~::;;£-=~-=-..--~.. ____.___________________________________U___H________ ___O_tJ_,,_____-l_.._r-_d:~,iJ-.J'-~t.í1~~"'~ i ø(' i ~ _L. ~. _. ._n" _ ____________'_____._'.'_____.___u.. ....'__.._______.____.__ _____..__._____________.___._________________________._,..______________~_____.__ .0________.__________ _____________,_.._____.______ i i ì ..~~-~-i~~:~~~:~-~;-:;-=--==~==- ~--=~==--~~-Tuesday, November 24, 2009 7 i~J1g¿~-il-t--l-Q ~, il~ -J __~ ~ ~~-eA--CJ1------------.. .___mu_____m_________.____ T..~~-~-i~~:~~~:~-~;-:;-=--==~==- ~--=~==--~~--;_ _. ._n" _ ____________'_____._'.'_____.___u.. ....'__.._______.____.__ _____..__._____________.___._________________________._,..______________~_____.__ .0________.__________ _____________,_.._____.______ i i ì ~i~J1g¿~-il-t--l-Q ~, il~ -J __~ ~ ~~-eA--CJ1------------.. .___mu_____m_________.____ --------t---------..--.~.----.----.I ! ! I i ...__._____._._.__.__ ......_______._.__.__..___________.n._______________. ___.____.__..~._.m__.__._.______.___ ......_._..___..____._._.._____ I Dynamic Boundary Condition - ~- +_._...._.._._---.-----...--~--~---..--.~--- .__.._._____m.___._______________. .. _._m.__~___..u_ .._. ..._..____..__ ._. .__.__._...___ ..__~----- c. l_____~_____..____~____~~-1"-----u--_-----~-----u---..___________...__..___~_______ ----i-------- ~ ,------- .. - .. .. -.--- ------~-l'a,-f~-----*O--~-(..iC---~-l--- ! ~ ~ ,i i _ t-~¡_-..~~----------.. --------------------.---------~~---i7----:l..fl...-Â-~-).-..--.l------ ______________________._.__.______c._.___________ __.._._u_ ...----_.,--~----~-------------_.~-~-_.._-~-~--, ------------_.~------~--_._--------unn..'t--O'~ e I ....7i Q~__ ¡J l- ~-- D ~--f% . _ 71-j--=---Q..--------------u--------. _-~~---~-~-c.c-..~__----ílc-..~~--~-r-------___~ -£._____..._________________n______._______________ I .1 ~---- -- C.1i~n.J_6J vl~--..-----.-------!-\)~.------&4~d-.¿'----t:---A.â-k9l-. -------- -------I J---~~----------.-----_____________ ______L_~I"d_e~c._f---u--~------------ _______.___u__u_ i ! --_._--_. ----_.__.--._-_._._---- -T---.----R-c.u' ê¡------ -' -----..~- ________ __~/_~!"!'! c:¡-l _ u------T------------- J-tfl~-C)-71----2"--------=--~----- . ) ï. .+.___ _ u _ ..... u... __... ......_...__ __ .. ___._..._____ ____.__.u___...__m._.__.____ _._ j -------~-= E--~ ~_li ~.:.t7~ut~..-ò-== Tuesday, November 24, 2009 8 i I )i I ! ! i I i i A.711l -fA-C.if Dynamic Condition (cont.) Oø.; ~ A" ~ (~~_7. ) -I ~ I: Q tv Ól. l. '- 0", i ! ! ~ i ~ 'fi. -;1 ¡ Q '~f ~ .. ~ ;; ~ _ :i 11 .2 -t -.: '" :i ~ -l q~ñ C~~o T I:2A \ :211 .. L( "l ';.l -: f 1' iW (~~D) a~.Jlla~i ~.o ti7... s "Ç(~ v4.c. ~~, 7" (~ i(\7tf) -L; -l'~) = 0 P!.ïU.l,& So ~ ~ lA.",'S';4 E ..oJ'r úÆ.,-i)C ere ;I,)i-IwTlA-t. O-ii (!,)IwiiAJ4Jo.JJ I r i 1' il.-A L A-c.lltlst í~~A"'l Wf f EQ.u~ &-m Tuesday, November 24, 2009 ç~, c: ""-. D~"-A....C 1l.4'-.-LI . 9 , ----- ._------,----------- . ____.l_.tj:..~~__l?_kJ.L_________u__u___.._________ __.___u________.____u_.__________.u_..__ u ___ ...u_.. ... ..________.._____u_ ~ i9 Bernoulli’s Condition ------_.._-----t-- ---- ,. i , .____..____¡___$.-'_Ç._~lLLa.~-S-ùA._f___a__l_~~~!r__________ ____..________.___.______.~____..___n______ . .~--i==:---=~=--¡;¡l ¡j/-:~t_l=-r~~-:-lxr-y:~=---)=_:-- :--=~~ .- .... ---.... -.. ------...~---....---..-~-.---1'.--.-..-_.--..---.-m..-. ~-----.--.---.-.-.... .--.---l-..----.-.- u_..________.__.___ ____r-u.. _______~______~____.____________________.Qc)N$~T.------n_________..___ __ ~c)..~_~.._1'_m___________ ..__.___ . .. .____L__Cl~ A-~_Lc_lA1-rf.y_~__O_d__t.~_~_i'_._'___m___u______________________._____________________ n_=~ ._=~~~==~-~~~~~~=~~n--~=~===-===--nn_ ~__~~=~-~~~~~~~~ - ----~_¡-~-----l---,.I,e,¡..4---:n..u.-s----ç.f-J)-II-l;iI---fJ-t1-- . ___.______________________________n.______u_________ ..... --.. _ ..-1-... .. .--e.-u.A..-c.ai""t"..-----~----.-~-~-o.-....----------------- --.--~------..------~-------.---- ...- ..-.- .=~~~~=~:'~~-==1l~-eS~~-c:~1~~u. ~__1";~-:tg-_;~:L;.=-= - - - -1 ~ - -- Tuesday, November 24, 2009 -~ -~i --~-___ _ __ _____ ____ _ ______ ____ 10 A-l).,..~ "' OAG 's-oLvTIJ-l I)e¡ çA-1l ç ~'r(tf~ OC).,,..4r Q~""/)i"Pd_J Setting up Normal Mode Solutions - oi iw+- J,'f) ~ l4l,~ HA-..,f ~== f, (./t-':L "' -ø - .1 L . -) ( -~ Iff ('l,.:,-l) oR ~ (~)Q) n/~ .()"" '1øO 7)6-v we Ilus, .. ,1 ., )¡ -l - j ((, -t - e..i' (~¿ d) J, ) tfz ('fr +,,+) = 1; o cf, A " S f' t1 0/, ': t7 ~ 'i 0 '?~ ( ~ Tuesday, November 24, 2009 11 &J -Ol._~_~t.___a_~_~~lo..~'f ____~q~_~_1I:n.._~________ Applying Boundary Conditions r== m~-~m_:L-.c.~-.. ... L~-~_ù~=--.---..=----=--- -"f i --- ----r---t .-..i-.-------.. ~~¡~-~--.:~~~~~~--:~;~~~~~===~~:Ç~~=~l!f= , ---..---'-.-t----~.-------------------.--------.---~-..---_.-----~~-----.-.----,~--------------.-.-------~-.-.--.---_._,------~--~ ---.JJ'1M-~!lJL,l.~¿---'ld-.JlW!Ø-~-L.--n-~~.Q-l-î!-l~---______~~________ ....._~~____..________~_ J ..-1-.-...---..... .-.-- _.__n_.._._.~._.._~.__.______.___.____________.__.____.~_________.____._..._______._._____..._._.___...._._,____._______.._.,.__.._.._u_.m i j----~Prt~~1.4~--~~.. .1mi..~-=---__=______=~=m==___=_ .==-- ... ... .--=--~-.p2--Ç-~-=-+AÙ~;~--l i~J. ..-= ---n.-.-------~- -----~-.--._---------------------.------,,--.--~\-------.---.-1 --:iwO--.:ciknrJ ~~---~~4-7WJd-JJ~IC~~NJ'~-----~-L~~7--~----)----------------- ---_.~-------_.__._._-_._-------------~-._-----------~----~--_._. , Tuesday, November 24, 2009 ,--------~_.__._._.__._- 12 (i .--.-.L-~..d-~_._~~'.__~El.níh.ßS_.______..________. _._.___.____~____._._.______ Normal Modes -- ---~--- .-___._.___~_____.__..__ ._____.___.______.'.___.__ ___._._. _._______..___..___0__.__.._________ ____. '.n_.._.__..__..~______...______________,,______________._,_. ~__' ___._ .__ ______._______ _____._.__.~_ .-----..----.. ._.-- _._...._____..._______n_..__.____...____..._.____...____.-l"-..----.....--.--.--..-. ..---.------... ---..-....---~.-.-- __.;.___... _.__.______________..____.________.___.~______,_____._____..___ _ _.. ____._____ ..__._.n.___.__......_.._.._.___ __ __.._,__,_,,_____ ________n____'________________ _____..__J.._ _____ ____ .__..__.u__ .-.. f ~-+,L '- Æ/'-2.l. ...7 J. Pi Ú. --.4y~)ta~-~~--.el1J~-( _.~~~~~~~~-~~~)~=.n-=~~.~=.~-n--- ------- .-.--- --~------ ____un __ _~____ __.. _ _~_ ~____ un _________.____ -- --~~.. ----.----T -~-~- -- ----- -- ---- ~ n ---------.It--l.---~-- ---- ..- _ ---- ---- n___ ----~------ -____~__.______._.~__._~__.____.__..._______._.._"_._.~____._ .~___ __~,. __._.~____~___ ___________________...___...._..__.___.._.______._____._.__ _.___~__.____ I i ~ :. -J'(l.-A,tt)'ho ~~:~_=I~_~=~-_T.l~_ç--~-___A-..__~ù.~ _(l~1!_C__~~_~.u ((.~_~..uJ~~..~_.____~__!-:-_-..n....n--u--... ...~_~~_~_::- . -- ~~~-~ . l---?~-h~)~~~l '-..lla t~i1:_lu~~J -- .......... Tuesday, November 24, 2009 13 --~------~------------.------.-~--~----------.----._-- -'-.-.---.---------------~----.---~--------.-------...------------. ...-..---. --,--------..~---------~---..---- -t--.-- ---------------------.--.------------------------------------.-- ._.._______.._u~__.. ---....__... ---- ..- -_., -------------------r c !-((£ c(' : I Ñ Ç. 7Y 'ltt.. '7' t:o,.",_ l?iJ-l _ __ ~_irt-l__ oJ.,_ __ft_d Check: Condition for Gravitational Rayleigh-Taylor Instability _~_~___G./J~ ~ëill~_______-ll A-~ L Æ ( LtL..___C:.A't _~~_~.______ ._..~------------- --- --------------------------.- -te----l-t--~--i.-? '=-0--7 ____1'-( !~------.------- --.-.-.-..-.-._-.-- .-.-------------- - - -t--------~~---~-------n~--.-.--~~~ -Q----.-----------~---.._--- =_I-===-n=-~~zi-~=~-U;i. ~....::.;~Zi~~p'=)-==--m. i .....ni ....---b-S ~ir+-_:-~Pi--!-----.. , ¡ i F=- JJA _=m m~~~~~~:_,,;,~~--~m m_:==-:::~~~ ~~*¿~--f~~_:--~i~ I~_ r~ ~_~_;Tlt~----~~~V~---n~-~~=-:-:-:_~n= Tuesday, November 24, 2009 14 IIU S ~, L. i ìT / r i-~ I l./ Il l!r.".~ ,171 l::V ~ Instability/Stability irat (~~ -l ll~= .l - )L( 4J - )¡ c.,J + .. i) r (R,--lT w:: ~(p, £/1 +~ Pc + (Ji. Condition :! y.4, (Ml.) - J.ip'p~ ri.4- ~ + Pl. I At ~ r It 4,( I. C 7' ~H.' øV I i~~r.. ~L.' rr ~ \. -- \r ¡Ot f 15 lJy s r kG' c.c 1-1 A- S I-Æ~ Tuesday, November 24, 2009 tl-c ~ rlløt71 f&.. c ~7(.,,J Q3 Vortex Sheet .J......_.i!l).~T~-~n._-.Stif¿nii'e.-..~...~___....__ ....._.._ ! ._.__._._---~- --~fi -- 1-- ---i i i ; . r-- . ---i-.. .. --- --- - - ---- -- - --- __ _ ____ __ ______ n ____ __ _ __ ~_____ ___ _ . __n________.._..____..__.._...._______.. ..--.-......- ---.._..._.___._._.__. ---~--_I-£-~-= ~--7~ VO_1.:r--_'i__.Ç~££r------~------A~~.s-----c.~.S'-~ t.1L..___. .n._._.._ ._1 ------- ..-.---..-----.-..~-~-------------.---.---~----.----.------.-.---------.-..-----.-..---.--------.-------I i - .. It ~4 i~ , -. -.-..---.-.__~________~______________ __...___.n.____n_ _._._._.___.____._.___.~__._._._________.. ______..__._~__~________.....____ -----=--~... Jl-,_______ - - --~ --; -'VX ..-:;-0 - ~-r -- -lkô ~n,lø Aa. - Q-£-T"-al'-c -- ----- -~-~- - --- -- _n. __n_ _ __ _ _ _ --E .. U.l O-s _ --- _ n - _n __ --i------..--~ - - -. "" ---- -"L-n - m ----.----------~-.~..tD'Z--.-.--- -------...------------------- ________! _~ - - ~ __~_ ___ __ __ _~~~_~_ _~ ___~____ _____ ____ - ____.__~______.~_____._________________"...___________u_____.u..___._.___ .__ ---f-----:;-:t ~ +-4 ---~G-) ------------- ..... . ..:l-(~i~~--y ~~ ø-~ ~~-~()-"~-"-~=J~~""T~.I. _--== i ---------t-i -------~-i-~---..-..-------------.-.----~-------------------~.-~--------.--------..-.--..---~----.-----.-~ .--~--------- .__ _._¡.______._..__.__.__________.._.____._____._________~.._______.____n______._ __ _n_.________.._____.___.____._.__...._.____ Tuesday, November 24, 2009 16 E 'ie (l c ~ ---~I~lJ61 nergetics i of a Velocity Shear Layer V lL c." c ,7., c~,irs i W (-l) ., /l ". i ~;: Ì" Â , l' i it " I J.u S 7'1b~1 ¿ 1"1 l( t; i -l.. (; I ;V & T ( ,.~ f!.-l! f 'f (__ø ~ 0 (.1 - r f Ú 1,1(2. c. CJ c 71-" .. -h ~ i c ~ Fi~ ê Ñ.., l' "" t l UC h ~ r A s I AJ ST-:u , iT O.VlrL d~J ¡ rL C)~ \ So f2 ! ~ e di-iS/. '" A f ~ f, ~ )tr IE ,¡EJf'r &F S:H~ A-/J.. I ~-11",, "1 -l 14d~. Tuesday, November 24, 2009 17 Centrifugal Instability ~ "" ") ~ '" t ~ i~ ~ ~ ~ \( .0 ~ E ~ tI ~ l .. ~ ~ c..... -----. "" ~~ ~ (J IL. ~ t) ~ \t -J 0 .. ~~ () "~ 'l ) ~ J.. ;) t \l \i .l " ~ Q ~ \J t: F ~ l- -l '-( .l= 0 -i "\ ,, ~ ~ .. i () .. ~ ). ~ ~ ~ l\ i t: ~- ......" ~ l V ~ (' \-c_:~ ~ ¿\~ "' 1~ ~ " lo t \t ~ ~ .. ~ (~ .. .. '" "" ~ " c. ¡. ). ~ " II ~ .. ') ~ 'a c: L 't 0 .. ~ ':t . ~ OL "" fW ,. H ~ t: ~ '" fl ~ ,. ~ .. ~ /" ~ V\ ~ L ~ w t) ~ .i l - c. t' \' \J t. "" \i i: VI ~ ( \l " .. \0 .. VI V' \l iqO ') ') \l ~ l:_ ~ ~ 0" ~ V\ .. 3 (; ~ a \. ~ "3 V\ .i " ,). . ~ "' \L - CN /' u C- ~,. 1: ~ ~ "~ \- ~ . ~ -. c. U \U Tuesday, November 24, 2009 ù. ~ "" \A .V CJ v ~ "" ~ 'o 0 - ~ ~- 18 ~ f(,q '( LE ,.,tf 's F ,.EIl' Y CO;.OC ï/vv .~' \. , i=l~ I rit , .~: Rayleigh’s Energy Condition ~l ~~ VllAT iS TI-f~ ChA~4fr ¿)P E,veR~'r , (l LJHe,u 'TWo flc)TI4Tlilf rL.()l.O It ¡ii,¡,s" / ..TEIi G.i- "' ~4 e l L£T VOL.&J""E' IJF T~o fCl-Vitl ,'2,""f r 0. t' E &l c.A- (. (, A, (!O-l,L1I if( r l- CA) . r¿ /) V=-:2 'f rat J l- ~ l :: 2'f 'fi. cP r d-l E#1M1Y :: f l tJ; c5v , ~( , "" c~ trl11. R 1. i'Zi ) I'll i I" i. ::L(-)JV ('I. (, l-1l=Ac r:: 211 n tl f) l)VI'SCIO l.,.u:, ,""t:"..¡"..ISSI/J..L FLØ.. lZ!,jUl-TS''' kEtV,,,.s iHe~_ u. l ..,T.A L .ENe,a.,..y :: .. c: r'l -l l.. \ ( V ~ 171. cui c c) c.A-1 D Ñ l CS t'o,.tT A-,.ï lJi FL v i oD Tk QVIIS. F/i"itL E".e 4~ '( " !-l.,3 \J C L t + rl."t) oU a: if c Tuesday, November 24, 2009 / -i _ ~1í i' c i A F '" F,_A" - 1/V1l iAL :: -- ¿¡ v r. ã. - ã'\ + i: flL æ L _ t) r t. ( I ( ) 1 (i - J. )J 19 e Inviscid Condition for Centrifugal Instability ~ .. C. f\ \l l" :) .. l.l ~ . .. ~ ~ l. tv ~ t: " '" ~ .. 1: 14 t \J L ~ VI ~ \f ~ " ~ :i ~' i .. ol i: ~ r \U ~ () Vl t¡ ~ oJ .. .. \11 J 1I "" Q ~ ç .. \J \r -l .. ~ \l ~ -l~ r! ... ~- -".. ,. .. . ,. .. (. .. ê 'u ~ oJ lJj ~ '" '2 (\ \I ~ Co ~ .. ~ ~ .. ~ l; \U .. ~ 'il '~ ~ ~ .. ) ~ ~ .. .) .. .. (I _ ~ ~ u. ~ ~ - , 3 1" .. Q " '/' t. oA . I (V ~ LNc. "' t: /\ , c. c. '0 -~ \'" - ~ _ ,. 'l.. J .. '... ~ \~ - f'~ "" v; \l J () :) ~ - Q v ~ \L l( \1 \l ~ ~ Tuesday, November 24, 2009 ~ ~ () () ..v ~\~ 20 ~ \QI Centrifugal Instability: Cylindrical Coordinates ~ '" '" "J (L ,. \ '" ') ~ 'I \) .. , .. "" l ": ti ~ rl\~ (': (J . u .a ~ ~ ~ ~(l tj ') l. .. ~ II 1 . /' ~ ~ i. rL r t .~ \) ~ ~ l~ ~~ i ~ ~ ., f ~ ~ t ~ .. :r Q li ;j ~ ~~ it t, .1 ~ i 1. iJ vi 1 C3 '"" . ~ ~ q: t ~ o .l " "l VI t: .. .'l "' tl J ~ó ~ IJ " ~~ t u.tì" .. ~ ~ ~.+ " ~ Ii )( ~ ., .. , , c, e ~i ~~ ~ ~ ~ ~r ~ t; .. ~ .. 'U ~ t '0 t + U (~ ,. .. \J ~ \I .. 'i~ c: CL '')( t3t. ). \I \I ~ l Cl ~~ ~~ .. o t c: .. '0 ~ f i ¥ .. v \. \' \l t .. Q lJ t. ,. ).. .. L . -- VI ~ ~ (~l) II \t .i (~'.. ~ l 0. .. ~ ~ t ~ '" ) l '3 :) lU i~ ~ ~~ ~~ ,~ \ \I .. l .. . t "' L ~ ,= .) .. .. ,'i .. l'j ~ ~ "' '" ~ (' -i '3 l~ .J ~ ~ .. ~ '-i V\ 't ) J l"" ~ ~ i l: ) '0 ~ -r " Q. -( -J~ ~ ~ ~ ~ ,. .. " ß~ ~ .. ~ iu ~ ~ u. .. (~ q .. .. .. . Q ~~ \i, '" -i i: Jo ui t. 1 t q: l " - n ~ (-= '" l. '; ,~ ~ . ct ) 0\0 l. 1= R '- l o ~ . . ~ ~ Q ~ ~ q au ') ~ J, .. ~ . -i ~ o ~ "l ~ '1 ~ ~ Vi .. o Tuesday, November 24, 2009 ~ ct 1 '- t. \J 'i ~ "' ~ 1: ~ '0 ~ .l ~ ~ ~ F ~ uJ & 21 ~ c: ~ '" I') tv Axisymmetric Flows q: ~ l , ~ + ,~ :I .. ~ t. ~ ~~ l ~ , u l .. \I ,. l -J Q. " " .. '~\ .L (~I~ l "' "' ID E J. .. VJ ~ IA l l a0 (":"( Ci .. ~ .. \J J~ () ~ )- Yo \) \J ,',, tv Q" ~( ~ i. ~ "" I.. ... S ~~ \" "t ~~ , I.. " i. " ~ l~1\ ~ ~ IJ c1 C' (il~ o 1-l -l ci ,. '" T ~ d ~ ~ ( Q.\ ~ ~\t i ~'. l. ,. 0 t) 'r /' ~~\ t. , l~~ + t ~ it -l '1\ " 'i~\ l"~ l 4. \ .. .. ~ T ~ C' t\ U. .. ~ l~ '" 3 .. ~ "" '0 1! .. l ~ ~ \n i -\~ l l~ -H \\I ~~ ~ " . \l ( Ñ\ ~ 1.~ ~ " l~\ t ~ C\ Ii . '" - '" Vl - ~ l ~ '- f'::::l 0 \) II 1.J w - ~ (~L ô\ ~ l': r \J d: :X ~ . . ( ~ C1 (\ .. ~ ~t.' l '''" 1\ ~ (" '" . ~~ .. .. ~na " \I \ .. Tuesday, November 24, 2009 ~ ~ .. .. .. .c ~\( ~ '¡: ~\~ , + .. \~ l~ .. \, 22 ~ ./ 'U ~ 'l -. ~ ~ .) ~ "V" "3 ~~ ~~ J~ b ,~ Q cI .. ~I t' C" f\ - Hydrostatics ~ ~ \. /" I, 'l -\ -I l. .. l- .. bI~ tv It ') i ,. ~ ~ tv ., .. , fV fI f' ~ Il '\ ):i r . c.i ~ fl C\ -I~ 0. \ .. "; ~ ~r '.. \i V\ r; t: .. "l ~ I, -Ir IV fl C! n" ~- ~tl ~AI ~ i /" ~ oJ ~ ~ .. )U N (\ :: ~l~ \i b .. " i' f' N C: \. a: tt .. ii '' ". () h " \u '- ~ ., P i L -. ,f .. . ('c. . (Çb t f' .. .. C1 r .. ~at V\ '1 ~ ~ " ~ " vl - \\ t '\ ~ .t -~ e: Jt Ii - 1 t r.l e:" \A .. .. II ~ .. . \U + d~ ~ \A V' t: 1 + \Ò OJ t") ~ . ct F ~ tI a ~ va . \J ~ \, ~ U 0 "J i; c: fã \~ Q: -0 \\. (l v I: ? \\A .. e a -i \1 .. C:- ß! ç l- '" ~ ~ ~ u. " Tuesday, November 24, 2009 ~ ~ /" Cl ~ - .. 23 el g ~ rJ ~ ~ i 1: Linear Dynamics t~'t i .. ~ Ç) Q l. -\ t' 1" ~i~ ltU .. ~l~ c. W Q i. C1 ~ ' '~ \ tv \I ~~t ~ Ì' ~ + t" A "\ t ~ '). ri ~. .. ") l" (~" ¡v ': N ~ ?' ~ + 1\t) JD (~~ ., l ~.. \) v ~ --~ ii '" i ii C\ () ~lti + ~~fb ~ (~f t) r 'i ~ .~ G Ii " ~~'i) ( -i ~ ~ritb + t lo:\ t r~ ii (\ ~~~ l\\ :: \) ~\ ~ i.. \ tl "l ~ C" "" -I Q. II -l \ ~ I .l l~. \ ~ f~ -l ~~ 1- (:s" l to ;. ~ pi ~ + ~~tb\ .u ~(á 't1 t\ C\ (' .. (' 'l ~ lJ. i a. - /' \ \ ti C\ ,~,~ ~ "' ~ (Ö ~ i\ ~\ t \\ P\ç C" t" -. (. -: " ~ -+ ~ N Vi .~\~ (, ~ (:1 \ \ t" f\ C' tt ,,~ ,:sf( i .. l~~\ .I (~~ (~A. Vl ~ \I 'i ¿ Q. \ Q. II t (~ ~ 1: ~ t'~ ~~ ",' "" u ~ 1\ ~ ~~ ~ l':. ~ (- ~ - II ~ 3 .J ~hl \~ V' . ~ .. ;i \t Q .. .. Tuesday, November 24, 2009 a t t . .. 'Í 0 ~~ ' .. .. 24 Ta - (1/2) (1 + Qi/Q¡) , er - (12.41) Sir G. I. Taylor’s Solution (1923) is attained. The nondimensional axial wavenumber at the onset of instabilty is found to be ker = 3.12, which implies that the wavelength at onset is Åer = 2nd/ kef ~ 2d. The height of one cell is therefore nearly equal to d, so that the cross-section of a cell is nearly a square. In the limit Qi/ Qi -+ 1, the critical Taylor number is identical “The closeness of numberagreement between hisintheoretical to the critical Rayleigh the for thermal convection discussed the preceding section, for which the solution was given by without precedent in the and experimental results was Jeffreys five years later. The agreement history of fluid mechanics.” p. 476 40 D.iRi v /I I I I II 1"" I II i I I Ii /I i I I D.¡ _Ri D.i - Rr 40 flow of water. The ratio of represents Rayleigh's o D.iRi v 300 Figure 12.11 Taylor's observation and narow-gap calculation of marginal stabilty in rotating Couette radii is Ri/ R¡ = 1.14. The region above the cure is unstable. The dashed line inviscid criterion, with the region to the left of the line representing instabilty. 25 Tuesday, November 24, 2009 Centrifugal Convection Tuesday, November 24, 2009 26 Taylor Vortex to Turbulent Vortex From: http:/ /www.staff.ncl.ac.uk/a.j.youd/tcf/tcf.html Tuesday, November 24, 2009 27 Summary • • • • Velocity shear is common place in natural systems with stratified layers. When the velocity shear exceeds a threshold set by mass stratification, the Kelvin-Helmholtz Instability develops. Instability threshold can be calculated with amazing accuracy (like the centrifugal instability). Mixing at the interface, reduces the free energy available and drives unstable growth. 28 Tuesday, November 24, 2009 ...
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