APPH 4200 Physics of Fluids

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1 APPH 4200 Physcs of Fluds A Few More Flud Insables (Ch. 12) Turbulence (Ch. 13) December 1, !! Vscous boundary layer and waves 2.! Sably of Parallel Flows 3.! Inroducon o Turbulence: Lorenz Model 1

2 Vscous Boundary Layer above an Oscllang Wall I. (From Landau and Lfshz, Flud Mechancs, 2nd Edon, Elsever, 2004.) ru=---==:-:-- l u.==--- =--=:--:.=:=--= - -E--c-Ln l-aC---- n.w T LSZ: Q"'.-l"" L /r-*,'/ " *",,/ 11' u_. 4-E--Lf.l-Q ó L6 I,U.. ò. --- s:_(l'_c _A_. I_n.,, C.' -"'71 ""--r -e _ I _ F-4-c)-" n_.. D" I&, AI' í)c) a" _.._.. u l zq :: _T ?---- _+/V A",. ea - '" ( /' S : _-l U c/ L'L:: _-"j r:--y 17.Y. _.. -l _._ _ l-w. M 6 -ø.l-/ r-nf.4 SI cn.,,lß4 "n_ / II r-1 c"'!â.ol_ I p _IÙ I' s_.r ": '?!.l Æ g &I w----= í IL L/I lî í'1l LS gl. 2

3 :A ßC)l-"1 'r L' Equaon for Vscous Flow Dynamcs )l - (! cllj S.T -;.: / " I l= I..?. '1 (I r:ù\ " F Av e- ''(de e.s:,-l.,. (- -\- I N G 6 U 17. U.: c:.;.l. Le "' fl e S C, L uï Ilh..s q (Y, -E) :: 17H V :: C'W=YÅ d II., ) cl1 z c ('" ) ",7+,) $ -f-. *-hl ' l Y, -)=- "0 ()S (f - '/) 12- j 3

4 Peneraon of Vscous Dsurbances - ì I I I! ' (.A 1 : ",.( l. ; I føn ï11($! í '( _ /1 S. ( c) l 4..uc.M4 é ': L'l + r l ) c.' A - I ()oi'" wa"'!;." () : 1M If l), c/i-.ati"a.$. I, l o:, '" S '1 ()!. c l" -+71 o - b.l"" G I c.e L. Acu A- ' F w ù r c) o IV "'l( r -e '( AL.S4--I-7 ø "".)',0,7,,1' c.avø (btl-,'\4,1 STIl ç V,!. (O l) A-p, -,4-lfìf 2''' -. -r Y' A. - OSc "øf'"-l FI' 61c4Cl g

5 Example: Gravy Waves H--- -.l-lf l,-u-(j----kd-eo "--. m_. u v Co. TT'. -- l'_. ----,,- -I-==-=wT.: : l_.-:.. =-=-.===-_-= _l ----Ó nn._ v-.n n......n.._n _. n n _ r I I-f---ì--I_O.--T---c-4-r- = a_!l H.*-.fl ""Cl Å LP.. ---T- fa f rø:lf I! -()c ( (b-:,--fl4 o -A;lr--+-6?-(S--c:.eì---n ==--= l_:;1 =======-:==:==_====-===n-=_._==---._= 5

6 Vscous Effecs on Parallel Shear Flow 0. \. \( - l1 "" ' v- l ) : 0. ' + \ \- v 0 l I) ) P- " - y 1 "- \). I., "- l.: r '" I. \. \. IL V\ " Ó (L "- \ '" '". l lol r- 3.j c. u. )- o I:: " I- \, V: " '" \ 1-..j '\ 1/ ' L \., ll \. "" \ 6

7 I I Orr-Sommerfeld Equaon for 2D Dynamcs n Shear Flow /' o -. I - l \I " \) x ': "' ). " o" 1j: + Ç\ ( (I ( (I ). \' r "+ ( -l\. \1 :: I' () II. b \ I,c; \U '" + b l \\ r.) -l \. 1' :.. ( 5 0 'cl. - \l' a. II I l L + \ \' \ \' ': \1. ( - - 0; =l I:G l.; \\ + + I' \ Ih (. - l.: I 'Á No I -r I L,. ) - l I "' r,: \.\.4.(1 a r' a.. V'. " ". Ip n\ -I II w p D - II Q- \ ') \' lè V' r' (' (V \' \- \\ C'. ( \1 - ( +-. "Á (': \ \'. )- o lu l ; - l (Vj) -- ''' \ Y. c II Dr). NI '" )- :: +- ). cv -1 cl + \ c 0 \ I 'ï :) ï U III ll,. (-: -l \ 'l r- p + \ c : - "- " l\ \ I' Cl rl ()., ). o \,U \U F 7

8 Soluon usng Normal Modes -- 'y., 'l "- - 'X í) 0 "- 1,- l l \ "- g - 0 ("0 1 - \ ūu. \/ \. l 1 I: \. ;) I ' 0 I. q- \L ' \: a. I, ' l: "( 'r j. 1'1 ' \ l- g - I: r- V) ': V) f Q \) '\. F- li a l. \ () 0 I' 'Y. l j 8

9 I I I I Orr-Sommerfeld Equaon Q l '\ l j. v.b \!. ') l! 4 '\ ( l lj l '" l \/ 1 Ç) (-L L r6 -. " " o ḻ Vl ' '\I) I \ -. :,.'\ \ I j \"j C' c II (':). \' \)- " \ I. ll. Ò). ll \L "" Q -. 'l ' ''l \W F \l,, l (- \). Cf C1 ' \) Ç) )( \A l 1, u \ \ 8- /' N,. -- " j- V 1 '0 11 \' c: " lx l" - '+ l Cl ' r / bl \". -f I:: c: ") çl \l ". o.3 (- (V () \ - N\ \ll Ill -- -r - ' \ " (\ (\ h 'C)\ l ( c; CL Ç'\ \I ;(." l- p v-.+ (\ \ () +- II \ \6\ r- \ ). \ \;:,. j \ -.x )- '" 9

10 Orr-Sommerfeld Equaon, Q, (Ç. Q' "\ 1- '1 "- f, l e L * 1--.j C: l ') '0., l ll Ul \,, ( "' (V I rv l""). "- (" I '1 \' f). Nh j '"I r b\ )- (' I',-,. \ V\Y.rI Il If (, Q.3 '\ r- L "0 Ic ".: )- I I: o ") ': _ r ' I \ I ( Q -. : L) lù l: C: :s o V\ 10 "- I-,a I. "' (,( - \\ \. V' - ') ll " I- 8 V' c: () I r. I f' ).. \ l l' -- /" Q \ - lj : f C' ' I. \ '" 'ù -- I- e! -.L : V\ " ) II (- ''I ç: \.; :. F V\ 4 10

11 I I I Raylegh s Inflecon Pon Creron - ") 7 : " l -j j - () r- \\ \L.Q f - b - a rl,; cl - f' rv a I. 1 V\ b). - l IV \ ) l. j. "" cv ri \. C ' - ' N U (V \ j II - \,;) (" - ) "- j- -. IU \ '; ' () : j.,) f '" rr "2 '" ; )-- -- I1 + lc '" 1-- % (U l. --, l- 0 r-.j V\ l ' \\ - (' \c \I L- -- c:..). - "" )- c: \l a 11

12 Raylegh s Invscd Creron r-. ' ṭ ç \ () " L a \ \ ri N.) \. )- l ), ' 1 -. u I \) - lu :: Ò ) :. l 0 - Q. '; \( 'I (I 9 \ l; ( ) \I (- 0 ) V"! l' )? \)- b L 3 -- \ v Ḻ \. \, \. r \ 'í \, \ \ \, \ \. \ '\ " () \\ '\ \ \ ) (; LL l \, l/ \ "" '; ( c -- v r " ).. ' 1 o " c. \l. ) ' V\ "" \ re 12

13 Wha Happens wh Fne Re? Rl j- f\ f\ 0 Òo r+ f'. F- \- 'l. I. Q ) l 4. úl -- I c: G \l a.. '" I III.- a '" u_. v, l "ll I ( '0 I. v () -' E $.: r- (U, 1: Vl f a,.o ;: 3 ) LL =-,, j \l '1 )-\. \. l..- U.j r- Q '. -! 0 - : l \. I lu \I '0 - ( Z) 0 :. --. Vl ). () - D- V :: :;.. \. " 13

14 Energecs of Flucuaons n Shear Flows l\ \.- I l. f: - - (:: )- ' -r \. \6 Cë lu Uj r lu - b II ( I=' G '). - '? 1- ( -l (V 1). r l l ': C q. (, \ -- \ r I \. \\'1 :: I : \ I \ 0-. '1 \ \ \ /_ _I I: ' U1 '\ Iṫ -L I Q. If), l'\ -I )..+- -l \ :) I el \ F W. + -: -l \0. \ '" L 1. - 'I ( '" \ \ f:- :: -l(\. \ I\ r, -l 4. \\ o., "' C\ r (:3 l 1 " I:. '" \ ") - \, ;/ "- ": Q "' ç V\ f r F 1. - û f" c: r " - Ë d: a: 1I c- "" :s -\ - ( \ 14

15 Reynolds Sress May Desablze Shear Flow r: o (- \ l () r Ç o l. "' ") :) Uj -- Q 1. ') l o ll \) '0 ": \ì '" - V\ 1- ll 1 u 1- l v :: ().j '" Ql F 1- l.u I. ì -0 \ \. Q \ \l ' j. l': ": C ) v c () - ll r 3.,. ' I. \j f ( V Q "- "- h 1 r: u 0 y. '0 ( 3!\ û l (Y. f\ 3 3 y (l "- \ \. N V' ) \c ' V' ) -' -0.ø \L LL \ ( V\ \) pr- \ ). )- ). -. -? l (.: P )l ì í ; Î 15

16 Turbulence: A Grand Challenge T u ll ßu L E f'c! ( S A- q R ca,,, C (- 4 L L P: : F /.v A-,o f /-1 C + 01/ S' of CON r VU ) 'r /'A- c. r /I A I Y,. r,,aç A PH Y S (l- 5 - Ta. A- S' l'o,l-l A sti? 0 Vl H )' e S! ç71c-l-fr C'r),.L/r2c.-l ()". / COLLI r lo.-.-e(, )' ff )C I: (; E 0/ H '" S ( c j I l.ê.- / (!,e (' u L V / n. Oeu 'Î A-cr FLU (.l 0 y( c s I () A! Ne) 2 ';!(" r "--0 :. S T()I?ULE vcf '(EORY ls l/eí?cc DIFFfCv'" - /lòn l,(",i A/ d-7l l- n. 0.0 F E S ('òuj' c- é- S /Vf.9 T' / 17? A-c- l-g/,v 77 c)"- 5 F#' l/ "" 0 f r c.a-ce r.a '"l TUlß UL. (: clf /'õv? L 5' uç 14 Ç7 7- ') ( C AL,,// ò.a -I 1"2 Âl--"Q ò "" / (! (-- A- ò Tl' C H D l- 00 A cg" A-q 15 S A- Oò/( / rel:-.4/?ó "-.5 ' C/cLvE? 16

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25 Mahemaca Noebook Lorenz_Model.nb 25

26 Summary Undersandng lnear nsables n flud dynamcs nvolves hree mporan seps: - Connuy and Naver-Sokes - Sacs - Lnear Dynamcs - Reducon usng normal modes - Machng boundary condons Flow shear nsables can be desablzed by vscosy! Nonlneary, U U, can drve chaoc, or urbulen, dynamcs 26

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