Lecture-21_(12-1-09)

Lecture-21_(12-1-09) - APPH 4200 Physics of Fluids A Few...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: APPH 4200 Physics of Fluids A Few More Fluid Instabilities (Ch. 12) Turbulence (Ch. 13) CD December 1, 2009 i-.~ .~~__~__~~~m_ ..----.D_~~~(c._~...--~.E.----~-l-s.-c:-~~--,,-----/jQ-lL~a.lJ-l'--- C-~,çF. 1.! ! Viscous boundary layer and waves Stability of Parallel Flows 2.! t- 3.!._~ ~-l"-!.~___tl,, l.~r-t~_~Ii _m '-,.'_~'~~"" Model Introduction to Turbulence: Lorenz -- -------- --r _m___ I .. _______1 ! -----------t------- ....--.-~..----------.-----.; ~.. .._...._---_..._-_.~._-~-_..__.....__..__._-------_....-.__._-------_._--~-_.-.-------------~----_._------_.~-----_._-_.._---._---_._--.-- ._-¡ 1 -- -- ---.~(-- -___.~_____._. _____--__.__.__..____________________._.___n_______..n_....____ "___.__...____ ..____._.__.________ ___._..___ _,____ _____.________._.__..._~__.__._.. _,_______________._____.__.__.__~. ; -------.-_.l . u _________1 . ..__1 .. _____!.. __EIlù.~___c.ß_~t;;)li__c.__.. L, F' H 17 e-_~_~_.C=:l, Vl_oQ~_~~_~tL ~~LÇl~~________ (From Landau and Lifshitz, “Fluid Mechanics”, 2nd Edition, Elsevier, 2004.) .. _ --¡.-------';__~ ~ e.._' T(l!~ .,___E.c._r &.._L#-d-,___2 ~o ~.~_)______________.________n________~___ --~--~--------.--¡-E-~-cJ-L't-n----.--~---------l-aC----__n__.W__~_~T LS~Zt: ~__Q"'.~-l"" ~_~__ ==~--~ru~~=---==:-:-- l__u .. .......==--- =--=:--:.=:=--= _.._.._._.....u__._______~-..-..------~-----------------~--___~_____ _____u_ ~...____4-E--..~Lf.l-Q---.-8-i-ó ..L6________~______ _____..___~____ ____... .....___~________ò._______________--- _____~__tJ_s:_(..L'_C.__~~__---- ~_~~~A~_.____ ___nn_ _________I_n__._______,,~ C it.' -"'71 ""--r¿-ei-ii----------..---..---------~---.._----------..~.------~- -.-----.-....-----t----.~-..-.--~---F-4-c)-"---~-----.---.___________________~.n~~_..._..___..___..______._..__ I D" I..&..~,.. AI' í)c)J~tt¡a" ~------..-.---.-T..----- __._____._._________._.______...._.____~_u_____.__.__.~-.-.---.----.--.--.--------.~-~--.-..---.--..-- --l--~---~----------------ZQ~------ --::-~-.-------_T--~--------?~~---- _+/V A",. eA - '" ( tJ /' S : _~~-l U · c/ L'L:: _-"j r:--Y 17 .Y . _ . ._ .._ i i i --; ~ ~ I. L ¿ . " *",,/ 11' I /r-*,'/ .,U i I_ ¡ ì I I , u .V-l_~_,-~~~-- 'n_d .J .". 1._~_.,~____L4~..4..s----L v.___Ø.._~"4 L. ~7_l__l'___El._~.!_lø___~ í( Viscous Boundary Layer above an Oscillating Wall -l -----------~~--~--_._--------------~-----...._-~-------.____.____~-l-w ... M 6____ti_t-ø.l-/ r-nf.4 ~ SI cJN £__~.,,Lß4 "~n_£ t/ II r-1~ c"'!â .oLJ_ __________--..--_IÙ____I' is_.r__":..__'?!.l¿ Æ g &I. J -----~w----=-~--1---í IL___L/I lÎ í.~~.'1l_____LS__t~~gtl.i i I . .----.-.--..-- .-.-----.-.---.-.-------~--p--.---.-.-...--.- ----...----..---.....----.--.-----.----. ..---..---- --------1--n-.------ ------~-+-i i i 2 ---~_._.._~-----T-----_..._~-~--~--~..._-----_.__.__._-.~-~---------. ---~-------_._--~-._--"._..~_.._----------_.----~ I I i VfSCOùS. f"L~l. I~ L i i I I Equation for Viscous Flow Dynamics O'(N~ C c:~ . t:A /~~ I ~ t" I )l - (! cJlilJij ~S.iT " F ~Av i ei- ''(die e.s: ,-l i. ,. ~ l= I. ~ I~ i i I -; ~ .: .. .? ~ O'(N~ C c:~ . .; t .l ~ I N G 6 ..U.. 17 .J U .: c: (- -\- i VfSCOùS. f"L~l. ... Le "' I ßC)..l-~"1t 'r L' I I i t. '1 (It r:ù\ )l - (! cJlilJij ~S.iT ~ ttfl e " F ~Av i ei- ''(die e.s: ,-l i. ,. S C, L uï Ilh.J L t:A i .s /~~ ~ t" I l= I. ~ q~ ~ .:-E) :: -; (Y, .. .? ~ .; t .l ~ I N G 6 ..U.. 17 .J U .: c: (- -\- ---r '-I 17H ßC)..l-~"1t ! ¡ i i i i i 'r L' i .s CJ'W=YÅ~~ e t. '1 (It r:ù\ ~ ttfl ... Le "' Vi :: I I i ! ¡ i i i i i S C, L uï Ilh.J d¡ II., ¿ ) q~ (Y, -E) :: (i'" i) ~ ",7~+,) $ -fcil1 Jz c . t*-hl I i i I i : ---r '- 17H ~i' l Y, -t)=- "0 t()S (fJt - t'/J) 12- j CJ'W=YÅ~~ d¡ II., ¿ ) cil1 Jz c (i'" i) ~ ",7~+,) $ -f- Vi :: ~ . t*-hl 3 P S-'cï Q. c4" l) ~ ~i' U Y, -t)=- i"0 t()S Q ,..AJ,tf.. ò F l c S ( () cJ Dc ! 7ü (fJt - t'/J) I i i i : I 12- j tJi Penetration of Viscous Disturbances ~ tG..J I c.ie L. t Acu A- 't F wi ù i () !. c tl.." -+71 oJ - J b .l"" ì I ~~",.( tJi ti ì I I i i I i ! P S-'cï Q. c4" l) ~ ò F U c S ( () cJ i Dc ! 7ü Q ,..AJ,tf.. .. ri c) oJ IV ~"'l( ~ tl.i r I i i I i ! 'i( .A~ i1 : i i i i i ~~",.( 'i( .A~ i1 : i ; I føn ï11($ ! í ; ! í '( _ ../1 '( _ ../1 .. tl.i S. ( c) li 4. t .ucJ.M4~ -ie '( AL.S~4--I-~7 ù ~ G..J I c.ie L. t Acu A- 't F wi () !. c tl.." -+71 oJ - J b .l"" øt~ "".)',0£,7,,1' c.AvøJ (bTl-~ ,'\4,1 STIl~~ -ie '( AL.S~4--I-~7 ri c) oJ IV ~"'l( r é ': L'l + S. ( c) li 4. t .ucJ.M4~ OSc i "..øfi'"-l STIl~~ (bTl-~ ,'\4,1 FI' ~61c4~Cl ç "".)',0£,7,,1' c.AvøJ øt~V,!. (O~ i l) A-..p, -,4-lfìf ç V,!. ..~ i l) A-..p, -,4-lfìf (O~ -. g -t Y' L'l + r il t) c.i' A ,é ':lJ to: 2'''-r ~ I ~ I ()oI'" wA"'!'-i;."~~ føn ï11($ r il t) c.i' A , lJ to: ~ , , Ii A. OSc i.. "..øfi'"-l FI' ~61c4~Cl ..~ Y' '" S '1 2''' -r Ii A. --. .. 77 g -t 77 ()¡: 1M If l),t c/I-t.ATI"A.$. '" S '1 I ~ ()oI'" wA"'!'-i;."~~ ()¡: 1M If l),t c/I-t.ATI"A.$. 4 (¿ -- --- -- -----f---- --.- ~-- ----------.---..-.--.--.~--------- - __ ~~____ __ - ___ __._._______n"__ , ____~__.._.._______u_¡- f$.~-~_m_b¿ZA_~.'_.~---.~-'f~l"¡---.- __.. Example: Gravity Waves - - ........+ . H--- .________m_ .__u ... -- ------. .. . ..... Co TT'... ..-- ---- ,,________m__.____~.~------ ~-.Jl-Lf-----~-----l,-U-(..~j----kd~-eO-.---~-1-.-----"-~-~ v ... ____..~ ~~l'i_._~_____ ! ==---..-=-I-==-=wT.: ~l_~ .~-~:~. ... _~_.._=-=-.===-_-= ----.-.~.-...--¡--.-~-..--........--Ó.-.-t--..---.i-¥--.---___nn..._ ...____.n._________~___.._.~._____..____..______._n_____...--.----.-.--------__..__u__. .._.___.. ...._l . -..--------.---------.-v-...n-.-------.-----------.---.-___n__.______.__.___....__.._____....____..__.______.._ ____._~.___n___...... ____un..__.____. ¡:~ ."_____n____.__.__________... ---------------- . ..- .-...--.---...-----1-..---...----.----.....-----------n.--..---.----..---.----------..-..-.--~-----------------_.-..--.-.-- --..-.-- - i--r-.-------..-----n-.------I~-------I-f---ìJ~--I_O.~--T---c-~4-ir- = a_!L~ H.*-___.fl ""Cl____Å ~_.~. __LP.___._~____ ---------~--i---~--Ti- fA f rø:l'-ft I! -()c ( (b-:,--fl4 oJ ~~-A;lir--t..+~-6?-(S--c:.eì---n---- _____..__._-~-~.:~~--.--¡=~~=--= ili~_:;~~1 __~~~~~~~~_~~~=~~======~~-:==~:=~=_====-===n-=_._=~=---._~~= í _______________________ .__._l.u_~________.________._._____~__________~_. __.____________,__,_,____.__.__.____..____._______.________________.__.___~______..___~_~__~__ ___~.______"._ on _____.___ __ ___~________j.________ :~ 5 ~ ---~-------------~-- G) _~~- ~--- ___ ~__~__ ________~______________ ': -- --- --~ ~----j¡-- -- -----~- - ~ ¡ ! Viscous Effects on Parallel Shear Flow vi LL ... -i..- .- .- - u_._. -... ~ I. ~ ~ ! ~ 3 ~\ ~t j L (L '" LL '\/ 1- "' r "' ': G \. ~ ic ... !.! J ,j \J VI r ¡ '" "Ó V\ ., "- lt.: J "- ~ t: ~ )~ l1 i''- ~'J .i +. li "" y1 ~ "- ¿~ + 0. \ P- I) \. ~ .. .. ~ \ ...~ ~ \(J 0 .J v- lJ ~ t VI \.. 0 " J - ~ t vi ).. - IJ \.i .. .J \) .. J IL \ ~ .~ VI r- lol l '" t3 c. u. t ~ ~~ J .j " \, V: '" 1- .J '- ~ L \ ~ .j '\ 1/ 't " I- \. ,J '- lL "" \. \i I:: t i o )- ~ ~ 6 Orr-Sommerfeld Equation for 2D Dynamics in Shear Flow ~ ~'" ~ + Solution using Normal Modes ~ r'~ I ~ -'y. 7 tl ~ , \~ .J 1 ~ j /' ~o I ~ -. \. ;) .. I: J "- .. ~~ !L l~ \I ~b l -l~ ~ ~ \\ r .) C' \\ ¡ J(V \' \~- ~ & ¿': "' ~ ~ .. x "~ \) ~ ~5 '~cl (~ 0~ .i .J ~J (' 'l J I~ .. \.~ 1' i:~ ~ ~ ~. cv ~ (Vj) l~ ~~ ). ~ :) ~t' 0 I. It: q- \L 'ï .. a. 't \: J ~ 't ( II \J .. ~ ~ Ç\ (~ ~ I " to" 1j: + ''' ). (¡ (I ci 0 .¿i J~ ~- ~ i ~\l' -1 cl --¿ t~ \ Y. ~ ï~ ~ ~ a I' ~ )" l~ L II ~I ~ ci ~ ~ llJ III + UJ J ~ ~a. ~ t~\~ ~ tt í) 'X '~ J ~ I, rl: V) ': ~ ~ ~ " "- V) '0 1,- \' o ¿~ ~ \' ri ~ (~I ). \' \\' ': ~\~ Ih ~~ (~ . (~ ~ +i-i ~.. .. NI '" \~ + (J \1 ~n\~ Ip ~ Dr). t:: )- \I .. -IJ II (~ ~ +w \1. ,. ¿~ ~ ¿ l '~ j t~\~ · c~ .. VI (t 'r ~ Q t f \) ~ ~ i t '\ o ( -l\.J ~ "+ 0; -t ~ I~:G J ¿~I 'Á ~ =l -J -t l~ p D \' (': \ ~ .J "Á (-: -l \ 'l .: ~ r- ~ p + t "- ~ (= ': I' :: \1 Nto l.; \\ ti VI t~ t: -t l~..J\~ tJ\.. t~ "i .J~ 1'1 FlI .. ~ .J ~ a ~ VI ) ~ () II uJ ÇI tJj J lL b . i~ ~ ~ -J "' \ .(1 )~ l Ii r,:i.\.4 + I~ L + ~ IiJ -r II -iJ Q- \ ') ~ ~., V" -- l£\ "- g V" )v VI 0 ("0 J-i .~ 'i ¿ I' Cl rl () ~ l. .. ~ J~ 't \' lè '- t ~ ~ ~~ t t ~ \t ,. ) :s ~ \i a r' a ~ ~ ~ ~ V' ~ 1 ~ " .. ~ '- t~-t ~ ~ \i J o )- () 0 o i. \,U lc- I' 'YJ ~ í) ~ VI ~ IJ .~ vi f\ t ~ J \U ,c; t " J. t~ V' . ~ Ç! i ~ lU l .i ~ ~ ..; ~ - t F ~ \U ~ \- u l- ¡. J l'j t u. ~ (J ~ \/ :2 ~ .. -J ~ g ~ 8 (9 Orr-Sommerfeld Equation Q Orr-Sommerfeld Equation ~ (V ~ Q /' N QJ , 9 ~ , i ~ ~ () ~\ J.. \ll (Ç. Q' "\ .. N\ ~ ~Ill ~ 1'1 ~ "- ~ "' (V~ l '\ l~ ~ ii ~ ll. Ò). v .b !. '\ ti j. l" ~~ lL t l~x \L t \ Q .. l! ') '~'l ~ ~ ~ ~ -. "" ~ -r i- ~ -¿~ (\ (\ ~ ~¡ f ,J ~ o Q \J V\ 10 4 (~ l~ lj ~ 't ~ l~ i ~ , .'\ ~ ~F t~ \W " ~ j'~ ''l -'+ ,. -i~~ l Cl ~ ~ () ~\~ +- t' \ " ¿~ h 'C)\~ ~ ~ .+ ~ li ~ i~("\''1 "- Ii ~ f). e I rv l""). ~ J~ ~ () "- ~ Ii Nh j I- ~ ,a r. f' ~. '"~IJ tL J I ). ~ l~ \/ V¡ ~ \ r/ '" Ij l ~VI J (~ \l ~ ,,~ '- ~ '-l~ t ( c; ~CL Ç'\~ II ~ ~ (\ * VI 1-- ii (' ~ I' .j ~~\ ~ \6\~ b\ rJ )~ VI ~ "t 1 '0 oJ 1 Vl I l - C' ci ~\"j Cf C1 ~ Ç) t ~\). 'i \) VI )( ~ \A l~ ~ .. ~ ~ t ~ \" ~ \I bl ~ J ~ \ ). r- ~ Ç) J II (-L .¿~ -f ~ t~;( t Li l r6 't ~u \J ~ ~ (':). ~ .. , ~1i 11 \' ~ ~ ~ .. Ii ~ ~ -. \' t" \i l- '\I) LU ..j ~ ~ VI \~ ~\)- \t 8- c: VI ~ I ~ t~ ." p -t I ~ '- \i C: ~ ~~ .. ,. \;: .J ~ ~ ~ ~ l~ t '-. i ,... . ~ ¡~ ') '0 ~ \ ~ I. ~ ~ .. "' (,( V\Y.rI~ ~ ~ Ili If t (ti Jt ~ ~ '\ ~ r- L "0 ~ ~ ~ t:t '- lj C' ' f IJ. \J VI \ i VI '" .. .. ~ V\ ~ ~ "...: )- IcIt ~ I- -- 'ù " ,~ Q.3 t ~t ~ ~ I: ~ ~ ~ VI") ':t _ 't o ~ ~ rJ \\ '- l' V' - \. ') '- ~ /" ~ ~~ j J i ~ ~ -- l e ! II ~ .. ~ ) ~ " I:: c: ") I t. i ~ IIJ l. ~ ~ ~ " t .. ¿~ ~ lL li ~ ,J ~ Q. çl Ul ~ t ',. \l ".~ ~ J Q -. ~ Ii \J I~ (~~ :i ¿ ~L) lù l: C: ll ~ ~ ~ ~ tt ~'- (- .L ''I :i ç: "t I- .. J ~ \~ .. ~ i ~~ \~ .; ~ :J ~ 8 t ~ -. t '" ~ .. .. \/ ~ ~ a i () t J: ~ I. -.x o . ~)- ~ (- .3 i v- ~ t \, .. ~ ~ ~ ~ ~ ~, (J -. LU i ':s i t .. . ~ ~ ~ ~ F V\ c: V' ~ J 4J 10 11 Rayleigh’s Inflection Point Criterion ~ t t") Rayleigh’s Inviscid Criterion 7 :J ~ i"J , "- \a l~ ) ,~ \, .. .. \. '\ .. ~rl ,; Jl -j ttj f ~a tJ \\ \L ~i~ ~ V\ t i b rv .§ Ii ~ l. .t ~ cV ~ -~ iJ IV \i t) i .Q cl a f' ~J - il bi). ~ ~ ~it J 'rI -~ () tu 't ~ ~ ~ lU 1 -. r~ ~ ~ t ~ ~ It¡ \, \ 'í t \ \ ~ ) .. ~ LL (; \) t ~ t1 r- ~~-~ il ~ .. .~ ~ C ') ~ UJ "" ~ ~N \) ~ ~ .. "- ~\~ ~ ~ J '" ~ .,) rr ~ I ~ II IU ,;) tJ ~ I (" ~ \j ~ if I- ~ ~ ti~ ) .. 1-- .. ~ -~ I1 '" + ~ lc ~~ ~ ~ 0 (U "2 t; ~ I '0 ¡:oJ 1,l l, J ~ 'L~ ('-~ i ~ i. ~ ~ ~ ~J% J .j '\i\~ ~ ~V\~~ '- J t~i lJ ~ ~ l. i- ~ -~ -j ~~ :1 \. -. () j 'i \ '; "j(V~ t' ~ ~ £1 ~ r~ l .~ '- ~ ~ '- ~ ) ¡: .. ::~ ~ Ò l/ \, ~ \tJ "" ~ () , '~ " ~ t 't VI Q. ~ .ç 0 -';.. lJ . ct -vi t \( 'I ~( ~ '; r ~ (IJ tJ ~ \ \ " ~ ~ i ~ )-- ~ t ~ '" J \t " 9 J " " ~\ () tL li; ) - ~ (t r J-~ '- , r- 1- ~ l~ \A l- ~ a \\ \I - ). ~ .J ~ (- ~ 0 VI ~ ~ .. '~ \c ~ \' rI ~ ~ c: v \J -l ! V" ) \ \, VI 1 o " c. ~ ~ VI t ~ J 't ~ N.) ') ~\~ ~ tJ i l' .~ )? \~)- () ~ V) ~ J- ~ \J ,- UJ t ~ .. ~ '- ~ t .). a )~ \I ~ '- ~ ~ ~ b '- ~ \\ \ \. ~ ... Ql ~ )- L \ 3 -- \l. )- -. -i ~ \t ~ 't V\ "" "" '\ J\ ~ ~ '- c: ~ '- \l J ~ ~ \. L v " re ~ 12 G) 13 Energetics of Fluctuations in Shear Flows ~ What Happens with Finite Re? J( ~ q. Rl jCJ Òo 0 ~~ ': ~ t \ -- ~ \\'1 :: ~ \ r -1-- -I \. ~ ,~ I~ J Q. If) ~ , i I i: l'\ I~\ .+- r, f\ ~ .Q f\ G r+ -I~ ). ~~ ¿~ F- ~ ~ \-tJ ~ It c: ~ .~ ~~ .J ~~ f' ~ It .. 'l ) .~ úl ~ ~ \l ~.. .- l ~ ~ ~~ t ~ .J \\iJ r ~ o LL \ '1 i \ ~ \ /_ - - - - _I ~ \ I 0- .J -l \~ ~:) 4 '- \\ ) ~ t~ ~ el \~i F ~~ t 4 . \~\~ Ii · J-l rl W . l~ 1 " I:. '" \ ~ ") - ~ VI \, ;/ (:3 i ..l o .....~, "' t ~~ C\ r ~ "": Q a .J .. ... III u_ 0' "ll ~ a ~ '" ~ '" t I v, VI ~ I. '0 ( .. ~ t ~ () VI ~ ~ v -' ~ E lJ ~ i r: .. .¿ I'" , ft =- ~ ~~ , I ~ ~ + "' ç J V\ ~~ ~~ J ~ Vl $ 1: .: ~ j.. ~ (U r;: J ~ ~ t t. \, tJ ~ (V ¿-: .. . ~ ') I=' a .. VI \C i l. Q ~ .'1 J. \ .J \l .. ~ .. )-\ J r- ,, ~ -! \. ( -~ ~ ~ ~ i- .o .. ~ - ~ 3 .. LL U.j ) V\ G ~ J .. t~ 1). '? \0. \ '" -l 1- ~~ L1 ~ lt .J t .. .. lU ~ 1; ~ ~ ~ lU ~ ~ \I It -t , 't L2 ~ V' t- .. :J ~ ~ 0 0 t- ~ ~ (- ~ :) Z) IJ r '0 Uj Q. Q. (. il :i .J Vl Vi :; () - ... t -t :c :r -3 it ~ ~.~ ). ~ :: D- ~ 0 , t- t I lU ~ ~l l. () ~ V" '- f: ~ i\6 Cë .. -r t~ \ J. (~ .. ib '- -l~ ~ i (~ ~ ¿~ .. J.~ It: II 't ~ ~ t (:: U1 ~ ( '" '\ ~ 'I J- F 1. -t J i fr ~û I. \\ cJ f" c: t ~ -L 1I f: ~ " \. ~r .. " -\~ t:s "" VI t:: -l(\ ~ l. J- ~ .~~ L\l .- ~ ~ ~ .~ l\ \ ~ ~ )~ 'i '- ~\~ ~ ~ .. J. d: a: Ët t- J ~\~ ~ (~ ~ J- .. 14 ~ Reynolds Stress May Destabilize Shear Flow lL ... E ~ Q J; ~ c: j rt - Q, ~ ~ Tu I( t)u L E/'c P e: .. -- vi l~ f Vl ~ l1 Q ø Turbulence: A Grand Challenge T u ll ßu L E f'C ! (S A-,o f /-1 C ~+ ii 01/ S' T()I?~ULE¡vcf '(EORY lS l/Eí?cc DIFFfCv'-'" - /lÒN l,(",I£A/ ~d-7L l- n. 0.0 i F¡ E S / ..17? A-c- l-G/,v~ i 77 c)"- 5 TUltß UL. (: ~clf /'õv?¿L 5' uÇi l: ~ J ~ ~ì í ; ~ ). tl .j '" -- t~ ") ÇJ ~ () lL t 1 ~ ~ ~~ ": f\ -~ \L -0 '0 "- uJ Ql \ì ~ o '- ~ r ti 't I. h . ¿ ~ ~ \. \J \c ) V' \i ~ 't j ~3 \l r: J Q lti ": ~ ~ .. 0 i 1 i r~: 3 t3 t -' .. CJ \j P '- J -- ). .. Î~ t ~ ~ lL .J :) It. () oJ t f (¡ 1- i. N y ~ 0 LL -- "- ~ lJ ~ ~ \) '" ~ ~ ) ,. r ¿,,Aç ~ ¡A ? (.: )- ~ ,J t ~ y. i .. 't V' ( ~ .ø J ~ r ~ Uj ~ ') ~ ~ v \. ~ ... .. (l (~Y. .. .J ~~ û V\ 3 ~ l': ì -0 v ~ ~ Q .. l "- .. \) ) \~ () V .. )l ~~ pr~- ~ ~ o~ ~ '- t 1- ll :: ~ ~ F ~~ V\ ~ ct A sTI? 0 Vl H ~ )' ie S ! ç71c-l-fr C'r) ,.L/r2c.-l ()". / COLLI r lo.-.J-E(, J )' ff t)C Ii: J (; E 0/ H '" S ( c j I l.~ê.- / (!J,e (' u L .477 0 -V / n. Oeu'- 'Î ~A-~cr l~ -. Q !\ (J ~~ \1 ~ FLU ( .l 0 y~~( c s I () ¡A ~ ! Ne) 2 ';i!(" r "--0 :. S ('òuj' c- é- S £/Vf.9 T' F#' l/ "" 0 ~¡f r c .A-cE ~ ri.A '"'-l ~ 1"2 ¡ÂJl--"Q ò "" / (! (-- A- ò Tl' C H D l- 00 A JcG" A-q 15 S J~ (- \t l '"' - 1- l.u u 1 '0 '- A- q R cA ,,,.. C (- t4 L L P: ~t: F / .v of CON ri ~ VU~ t) 'r /'A-~i c. r PH Y S (l- 5 - Ta. A- S' l'o,l-l 14 Ç7 7- ') (i C AL ¡,,//J ò .A ¿i-I A-~ Oò/(¡/ rEl:-.4/?ó "-.5 £' C/cJLvE ? 16 J \J ~ ~ - .. ~ ~ ~ - 15 /I A JI Y 17 18 19 20 21 22 23 24 Mathematica Notebook Lorenz_Model.nb 25 Summary • Understanding linear instabilities in fluid dynamics involves three important steps: • • Continuity and Navier-Stokes Statics Linear Dynamics Reduction using normal modes Matching boundary conditions Flow shear instabilities can be destabilized by viscosity! Nonlinearity, U⋅∇U, can drive chaotic, or turbulent, dynamics 26 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online