Lecture-8 - APPH 4200 Physics of Fluids Problem Solving and...

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Unformatted text preview: APPH 4200 Physics of Fluids Problem Solving and Vorticity (Ch. 5) October 4, 2011 1.! ! Quick Review 2.! Vorticity 3.! Kelvin’s Theorem 4.! Examples 1 How to solve fluid problems? How To S'òL-vl rLUi,¿ Pflc!IJLErl5? d) (A1 lJ::A-! /1-- òC:E ( AJ 0.. oI '"~-r í1 ò dic ') (Like those in textbook) Ç"Tt=l ~ I fJ t c.t u /Z t2 A D iI;4 Lv ~ tv (L (TE p Ô t- /V A- L- L I~ ~ ò C- A. 6 E 0 J/ £-rlf A I' io f-OJi c. ( "- '1 S $T1P#( ~~ _ r ç TH i S A 1) T ¡v A. ,CA-L ¿) t1 çrTI ( f If ¿i ~ '- E.. ? ~ o~ TOY /VA t- (c .) A il l' ç (. L ve.L (J -t L( .l ~( 7 A--- ò~ æ~ L lÍ ~ S' t) TH &2 \.' ': 0 l2 (¿ &...A ~C¿)) lj rf c A J) e Å."- '- 1- t~ P Yl () -1 c. c."" ~ rl t7 Âd (. c. lf r1 c).l tf ,c t F l'c ( C U t- 'r . ç-r;: ,#3 ç Ec.tZLI -lK ;10\7 ~C)".vEf'(tE4Â7 LOù.l~/.-A-~ ç (" S Te p. . C' A. -r ~ C 01/ ¡J L( A. t=A- cE 0 d /ê ". r A. .A~ S / N /Tle/'0 i x. (J. ') Ç(¡i it l+ C tAr\ Y ()v .i CJ- A- ç N¡i L '( GO;. S E-I? V iA(l Ò A- pI! ( ~ C ( ,oL C S rk 0 -' IS.- T "" ".1 #/V '7 v l ~ "'Ò '" ;£~ (v ~r E-"IS ft-' r ) p Il ( __ c ( to L If (è ~ò Â, Ç( pf.;AJ It A- i/ ~ c-GQ òi,J L L L f S Y¿L)/l SDLU(fdl--: Ç'7E rP t: ç- ftiL l t¿ l!J /Z / A.u ,7' è) ., ( 2 8 N -l l - J J :t Hints with HW #2 (i .. ~ -l ç: il ~ \~ ~ ì 1 d) ì J \J r. * ~ t~ ~ J t: li. ~ i o " t 1~ I) It . .. ~ \: .. J J II ~ 'l ~ cC 1I IJ, ~ ~ ~ ~~ ~ ! v Q I. L ./ 1 '- .. ~ )~ ~~ o1 ~v~ ~ '5 fu ~ IL 1 iJ ~ V' r. y. ~ \. Õ "u ! .. ~ I. ~ (.: ~ \¿ () ( ~Jl 'v c: o \. -. -: .. a- o ~ -0 a. sf) ') i )- ~ ~ ~\.J ~ \ \I \ t 7j-- \ \Jfrtt~ \ \ -! "- '" ''- ~ t t" -. :: es 1\ , \. " r. ~ c. ~l.- ~ ~ -- ~ \s\~ J \¡ j r. \i \ ~ \i ¡- -c ;: i r \. \ .J \. '\ \ '\ \. ~"li '\. I '\ ~ ~ '\ t,:¡ --- '( ~ \. \ ! ...1 .. 3\ ~~ ~~ \L il ~ t: it i lJ . .J I~-c j ";~ ~~ .: I4 l¡ ~~~ Y. ~ C1 ': ~ ') ~ t " - \J v a ,J /" ~ l" ~~ ~ u1 N \,. ~ ,J I. o ~ 4. ~ . -. ~ \l'J ,-, ,~ 'U ~ ~ ti: JI ~ tv::; ¿ \\ '! ~ ~ \6 '; II ~ :i -t ~ ~ .3 3 ! J /" G t \J - '- I.. )- Q ~ j- l V' .. 0 ,.. . L. Vorticity Dynamics . vA -. .J -: ~ . I~ II Q \~ -i ~ 'Û .j (U ~ 0 ~ t J ../ () ,J '.. lÙ lL "= ù F lrJ 'J 't :i ,~ )- )~ 1.. ~ - ~ 'C f- ~ ~ II ~ 'I \ c: i :J r¡r- -i \ ~ i~ -If' "t I :: (' ~ , + (V ~ ~ '-,f' ~ I ~ L b. -l-a i I tr II 13' 'j l~ + 1"3 \-. J (" ~ L X 1/ Il\ ~ ~ , l/ I :: (\ ~ 'X I\: Z '\ r1:5 y. l~ "~ ~ + I~\~ . ~~ ~ ~ \\ /' I :J y. 1~ '- X b- .. \~\ .u (ò C' -- -l l. l=- J 1: ?$ \L ~ v .. 1- ~ el ~ (Remember the importance of viscosity.) 4 ~ J ~ -0 Ç) "C ( V' ~ .J U- -. .. ~ ~ F- '0 Simple (2D) Fluid Rotation " 'i l~ ~ ~ 0 d 0 ~ Q, ~ v - ~ ~ ~ -- -. )CJ 'f ~ ~ 0 ) w ~ - -i '- () ? .. l r- ~ 1- r: ~ Il .Cl "- .. i ~ 0 ¡:.. , 0 II ~ ~ II fi S y Q~ .. ~ ~ 'i ~ J i lL )l .~ " ~ .~ J ~ t' r ~ j ,'- ,. j, ,~ \l , 1 '- 0' .. ~ rII \D ~ ~ J V\ -; \J - \I - Q :, ,.. L. ." Jl ~ ~k iv ./ \) v ) " '" ev 0 ~ ~ ~ '\I \I - c --...~ ). "0 ,: t ~, :) \. \) .. l~ ~ \I CJ .. -( ~ C Q U V\ i. .- \p ~ '- j l\ -- ~ .. ~ (J ~ () ~ g .. -' \t ~ I~ r. /3 i ri 1 l( , l. ~ 1~ .s 0 C: \... oJ - rI-' (j il ~ \f 0 oJ Q ~~ VI ~ lL I\ I:. ~ -~ ~ to .. a '; L- t CJi ~ Oö ) 0 - (- i t ~ 0 \~ \J - \I (' ~ lo Ql 'X :J \ , "' ~ -( , Hr ~ ~ ~ Ll ~ ai ) "- rv ~ ~ .. ~ Ç) 1: c\ ~ l¡ t. 3 -- ~~ \i F * .. :i tJ ~ "" \1 tl , r~ 3 ~ ~~ ~ \. II ~ 5 3D Fluid Flow CFD calculation of atmosphere (Iso-Vorticity contours) CFD calculation of flow in mixing tank 6 (9 ~ ~ ~: ).. - \J ~ X W .. ") (. -. -"" ~ it: t' ~ tL. ~ ~ t 1- V Q 1 \r I': '* 'J Kelvin’s Theorem \tJ~ ,. r. f -( ~ () \l l~ G I~ ~V) ~\~ I~ I 'ì ~ .l~ X ~ L'J r~ w () \: ': .j _ V' 10 v ~ ~~ r '\ f LL I. C\ i -. . I) \J .. ~ ~ ~ J ~ Il , (~ (( ~ ~V)~ \i t~ ~ (~ (: -: VI ~ lY oJ n t: .) ~ (. 'C J ~ '5 II ~~ VI ~ ~ fVI ! ~ L~1 :i ç .. '- \f t ~~ 0 v VI -- \J w 1 ~ "- j ~ ~( ~ "' -~ .. ( t- ~ 'fT ~ ) ,. lJ l. "- ~ F- ~ VI ,.. iJ. )Vl , ~ '0 ~ ~ \l '0 '- l .. J .. y. t ~ ~ \. ~IU 10 J ~ r1- ~ '0 ~ 7 ~ ~ 1 'I ~ ~ ~ t: , il ). 1-- J- ~ u.. '- ~ ~ u t l! ~ l¡ ~ ~.. -' Ç) J lL -t ~ J .. oi ~ ) ': '8 ~~ \. 0 A (General) Vector Theorem for a Moving Surface Flux I I' ~ .. ~ t( r- .. '" + ~ t"~ ~ -. 'i ',, I~ I l ~ ~ I~ ~~N 0 f \i J i- ~ l~ \ ~ ~ t~ lt: L- \I (~ ~\~ \ l- - J a 1¡J IL t' t u It ~ U ~ ~ .. ~ J~ rJ V ~ J ~ J ~ 0 ~ V\ lt ) Q ~ C, f l) \p\1.. t .. t- l .. ~ '4 ~ "' V' cJ (/ 0 f ~ J\ t: ~. Q ~ \ ~ I :: "iJ ~~ ~ X . i~ \. i~ ~ \ ~ ~ t'~ .. l~ \J ~ ~~ -~ ... + t ~ '" _(. Vi ~ !~ Q~ tt c: V' ,- \\ i- '" ., o "l t j ~ ~ u- J~ .. ~ 'tt ( t: tA- "- ~ ~ ~ \. '" l~ ~ ~ \1 ~ V' e vi" () ~ '- ~\ i 8 ~ t? ~ ~ ~ .. J .. Q. ~ ,j J ~ " \) ~ f' ~ v t 't ~ ~ \¡ ~ ~ ~: l- ~ lU a - ~ o (= \. ~ ~~~ ~ x tt (~\ ~ ~ ~ ~~" ~ l~ I~ /;: ~ r- ìt: ~- \. .i " ~ l\t l~ ~~ ~ l" "T \~ ~ l~ . ~ + l\i \1CD n ~ \ L lI c- ,~ ~\- i \¥ \ '- ~~ ~ ~ + ~ ~V1 " I \i .. ¡~ ~ \'~ b + ~1\1 ~ ''l ,,~ "t '\ r~ L c~ ~ G: l0x ~ l~ (: ld -4\' iJ c~ Vi - )~ (. ~ ""C' ~ C' __ "0 ~ "t \( :: '. ~ t- C' l~ (LJ .. ~~ ~ + ~(L¡ -t -r ~ !d '- ll "S II .. ~ r ~ (( , ~l~ cL- , \( ~ -- ~~ L~~ -1 Vector Flux Identity (cont) \L "" " It ~ u: ~ ~ a I~ '- L :J -, .. Ç) ll , g Ç) :J ~ ~u " ~ j "' " '( J \) ~ ~ ~ ~ ~ Vi t(t g ~ ~ t -. :J .. LL ~ \u t " ~ ~ l.. ~ c ~ \J .. .. ì: I +~ IF) \~ h d tb () '" 1I ~ i; ~ I.) /' t \. ~ - ~ -0 ~ V) -- \ ~ C" l\l \ .\ l ~ l~. ~ + t X. l:J \l ~ F 9 ~ k:' - \J ~ 1~ J 'i L -J G ~ ~ c- iw J lJ 11\.. 1- ii l-~ N c:9.. t: -i ~~, ~ ~ r.. Two Line Vorticies \ .. -£ C.\~ 11 ~ t. ": It Q 'i t - .- i - \! ") 0. l- $ 'L .. ;) Q ~ -. tf ~ .. 2 ~ o~~ ~ '0 .( i. lW ~ r: ~ fò ~.. ~ e: l- II ~0- ~ .. Ci li t: \ ~ vi ~ ,~ \. a \: ~ \ l- ~ (/\ .. ~ V 1 c: C: o l1: \J f~ G: 10 ~ ~ .. ~ 0 .j "- \J ( ~ 'G ~ I:: \J Two Counter-Directed Line Vorticies .. L ! J( \! - ~ ~ ~ ~ ~F o: ~~ :¿ t ~ i -i W LL ~ Q - ~ c: ~ -r ¿ Q ~ .. ~ 0 Q. \J VI lJ ~ i 0 "- ~ 0 't '5 Uj .J ~ ~ r l~ 0 vi lc Q: - .. ; ll i= :J t,- .J ~ ~~ ~ 3~ ~ (J r:\ \i .. ~ ~ 0 1 r * J '- ~ t F ~ ~ t ~ 11 Cà '\. \J (J E 0 ~ ~ '.. ~ Smoke Rings C\ ~ .- (¡ J~ t) ~ ~ ~ ~ ~ J A ) ~ '0 ~ LL .o c: lL 0 ~ I; I~ ~ ') ,:. 0 ~ b ~ 1 ~ "- \I ~ " c" o; Q, .( Q. l "; II Q c: 1 f- Q .. ~ 0 it .. ~ l~ i ") l~ v ~ , VI '" V' ~ ~ v -l )-- v \-- \: ~ '0 J ~ ~ 'l ~G 0 c: ~~ ~ ~ v.~ i~ I ~ ~ t. t~~ ¿ Ûl IL ~ ql q ~( U' tj " ~ 0 ~ .ø ~ \/ 12 Toroidal Vortex Ring http:/ /www.woodrow.org/teachers/esi/1999/princeton/projects/fluid_dynamics/vortex.html 13 V' lJ o Ü -i -. ~ ." c: 2 \l c; c ~ F j. ~ ~ Colliding Vortex Rings ~c6~~, Nf'" tt_~v :J :i :t ~~ ~ t "j\t $---. (: ~ 1 '~~~ .. \. t J - I. li l :t \Ä r:: VI I) ~~ c. ~ (!~J =-'- ~ 14 http:/ /serve.me.nus.edu.sg/limtt/#Video_Gallery (Prof. Lim, Division of Fluid Mechanics, Melbourne, AU) 15 http:/ /serve.me.nus.edu.sg/limtt/#Video_Gallery (Prof. Lim, Division of Fluid Mechanics, Melbourne, AU) 16 Surface of Rotating Bucket 17 How to solve fluid problems? How To S'òL-vl rLUi,¿ Pflc!IJLErl5? d) (A1 lJ::A-! /1-- òC:E ( AJ 0.. oI '"~-r í1 ò dic ') (Like those in textbook) Ç"Tt=l ~ I fJ t c.t u /Z t2 A D iI;4 Lv ~ tv (L (TE p Ô t- /V A- L- L I~ ~ ò C- A. 6 E 0 J/ £-rlf A I' io f-OJi c. ( "- '1 S $T1P#( ~~ _ r ç TH i S A 1) T ¡v A. ,CA-L ¿) t1 çrTI ( f If ¿i ~ '- E.. ? ~ o~ TOY /VA t- (c .) A il l' ç (. L ve.L (J -t L( .l ~( 7 A--- ò~ æ~ L lÍ ~ S' t) TH &2 \.' ': 0 l2 (¿ &...A ~C¿)) lj rf c A J) e Å."- '- 1- t~ P Yl () -1 c. c."" ~ rl t7 Âd (. c. lf r1 c).l tf ,c t F l'c ( C U t- 'r . ç-r;: ,#3 ç Ec.tZLI -lK ;10\7 ~C)".vEf'(tE4Â7 LOù.l~/.-A-~ ç (" S Te p. . C' A. -r ~ C 01/ ¡J L( A. t=A- cE 0 d /ê ". r A. .A~ S / N /Tle/'0 i x. (J. ') Ç(¡i it l+ C tAr\ Y ()v .i CJ- A- ç N¡i L '( GO;. S E-I? V iA(l Ò A- pI! ( ~ C ( ,oL C S rk 0 -' IS.- T "" ".1 #/V '7 v l ~ "'Ò '" ;£~ (v ~r E-"IS ft-' r ) p Il ( __ c ( to L If (è ~ò Â, Ç( pf.;AJ It A- i/ ~ c-GQ òi,J L L L f S Y¿L)/l SDLU(fdl--: Ç'7E rP t: ç- ftiL l t¿ l!J /Z / A.u ,7' è) ., ( 18 Surface of Rotating Bucket 1. Draw a picture. 2. Dynamic or Static? 3. Coordinate system? …Static …co-rotating cylindrical 4. Can you apply conservation principles? (no flow in this frame) …Bernoulli’s 5. Use your intuition. 19 Co-Rotating Frame ∂u + u · ∇u ∂t = g − (1/ρ)∇p + ν ∇2 u ∂ uR + uR · ∇uR ∂t = g − (1/ρ)∇p + ν ∇2 uR + R⊥ Ω2 − 2Ω × uR with Centrifugal and Coriolis “apparent” forces. ( Very convenient when uR = 0.) 20 \1 L- ": ~ ~ \! () .. \- u- ~ u ~ Co-Rotating With Bucket ~ v ~ 'ù ~ \!; \L '0 ~ \L \) 'l ~ ~ U. ., ~ .. ~ () C: , 0 \J 0 K t.. ù ~ .. \0 ~ \c ~ ~ l. \1 ~ I ~ '3 I- q. ~ ~ \I \L C: l :s i '0 r: \) ~ led l') + 1\ï -l ~ D \~ l (( () U I :: 'l. lo J\) l:: . "" ~ 'J ~ :) ~ ~ \Ù i ~ .J " .. '" . .. -l ") '\ ( ~ 'U d ~ I. \l 'J ~ '- r' ~ .. ~~ '- I C: \\ ~ \~ l \i l~ Q (W (V tL\~ \~ '- -l .. c: J ~ ~ li ~ 0 LA i f. V ~ '\ \I ~ ~ t I.. l t; ~ 0 V 21 ~ tJ d t ~ . à! - "\ .. ~ d: -l ~ ~ ;).. "- ~ '" C1 ~ 'ì .. J ~ ù i O: "i ,. to r: ~ ., Applying Bernoulli ~ .r -. 1- rJ t ~ .. .. ~ ( C, (" .~ \ () ~ .. ~~ ~ i \ -1 \. 1\.. -- .J ~ " l-: ~- '~r a. ri 11 -C 'i ~ +- 'f ~~ CL ~ Ii ~ (" N - '" :i ~ a.(\ l ~~ ?: '" V) ~ l.t ~ .. U1 ~ -' J lt \Ì u: Î 0 '1 \. .. VI ~ -i 0 l. ~ ~ '\ ~ ': t; ,: .. 0 0 J- 't L :: ( t Ç) _. ~ "r lu v ') '" ~ \1 .. G: ~ n ~ (: ~, ~ _¿ \/ \. ~ '= LL -t V" r ~ I. :: J -~ (, 0 Q ~~0 U - \t ~l -l .. ~ ..~ ~. lI 0 v Ll rf ~i "l -C t: (' ~ C' 1I "v J + \) Q ~ ¡. c: -C Ci (VJ + ~ ~ Q. r. ~ C' J 0. II /" + "' -C '~ 0. + ~~ r-\~ + ~ II \T Q. + ( Q.t II Q.r. I .. O-~ ~ ~ ~ i- +- J Q. ) O-~ ri s: "" tJ ~~ J ('e! t' ~ i ", 'P i- ¿ ~ T ( ~~ 22 . .. -i l. ( r1 t\ G: J ') .: ~ ~ ;p ;) --ctp ~ tI ~, r: ::~ .. .. l~o - '- -trJ Q. "'J I l" N l l' .s ~ j-~ ~ ('J i ,,\~ l Cl Q. ~ ~ l' .L ~ - ~N ~ - .. ,.. .. "" ( ) .. ~ "'- "', /-". 8 ~ 8) t- ~ ~ì: -, '- \. 1: Ik rJ __ ~ ø c: v _ -(1 , lp i - ~,,~ "-- "- "'- "'. ~. J ll J- o ~ i ~ " Problem 5.1 '\ '- '- ~ '- ". "',-", .. '0 ~ / 't l£ ~ N.( i( i: ~ -1 r- N N ~- ./ '", V ~-'~ \~U Q .. ~ ('~~I rJ "' i ~ -l .. ~ \I ~ f: 0 ~ \= \ ti r- N "'" "' L~ -~- ~ '0 tQ .- J ~~ t1 ~ ~v ~ -' .( t tJ u. J "- .. N -c- ~ IN .0 u rl JI~ :: ~ C: - ~ L '=.. /' \f (' .. .1: .. Q \1 ,r ~ (\ .J ':I , J+ ~w+- i il .. r ~ (V ~ L .. -- () ~cl -£ 'C 4- V\ i r- ~ N .. c. \f r& ~ (V 0 ~~ (j c: .. cJ Qì .) \Í I: ~- II /' ~ ~; ~ ~ ~ ~) .: ~ '- .. -l ì '0 2 JL ì. :) o ii ~ t (I ~ ( ~ ,r N )~ lL IJ ) 3l~ ~ r: .. ~ ~ 1i \h o )~ ~ It '\ \r r' ~ .. ~ '.. !i ~l -i ': ~ ~ ~ \) ~ 7) ~ t- t- ø~ 23 Summary • Fluid motion can be described by vorticity dynamics. • Vorticity is conserved in invicid flow. • Vorticity is solenoidal. Vorticity must either close upon itself (like a torus) or close at material boundaries (like a twirling spoon.) 24 ...
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