APPH 4200 Physics of Fluids

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1 APPH 42 Physics of Fluids Problem Solving and Vorticity (Ch. 5) 1.!! Quick Review 2.! Vorticity 3.! Kelvin s Theorem 4.! Examples 1

2 How to solve fluid problems? (Like those in textbook) Ç"Tt=l I $T1P#( D ii;4 Lv A fj t c.t u /Z t2 tv (L (TE p Ô t /V A L L I ò C A. 6 E J/ rlf A I' io foji c. ( " '1 S _ r ç TH i S A 1) T v A.,CAL ) 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 \.' ': 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. çr;:,#3 ç Ec.tZLI lk ;1\7 C)".vEf'(tE4Â7 LOù.l/.A ç (" S Te p.. C' A. r C 1/ J L( A. t=a ce d /ê ". r A..A S / N /Tle/' i x. (J. ') Ç( i it l+ Ç'7E rp t: ç N i L '( GO;. S EI? V ia(l Ò A pi! ( C (,ol C S. i CJ A ç rk ' 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/ cgq òi,j L L L f S Y L)/l SDLU(fdl: C tar\ Y ()v ftil l t l!j /Z / A.u,7' è)., ( 2

3 Mechanical Energy Density for a Stokes Fluid 3

4 . i I I. Fluids Equations q; il cv Vi (internal) t V' ') i. ). '\ i t I.L l l: 1 ) ". a, ci "; J f J (/ \L 1 I "' 'J ; J l:: ll VJ "\ C! : ' :: \/' v ') 1 VI s: IJ '; ( v ClO "'\ i l:: a. \ l! ") q, ti () a Ii,, lù 'V i Q ';!J, VI.J r CÀ, \j f cd Ii '" t L 'J W\ t II O Q Q : lj It: CL \ Z L I 1'3 1 b I J' ') b 1, 'x 't b I :J a u I \. \. \ i re Q \t ' :) li) \. q" " J t tll ' \ IltJ (1 ì! i. J l t () l t il Qj l l'" + /' J J,, \l 1 I", l C i \l" 1'J.'" \ t Nl"ì \ i 'V 4

5 Vorticity Dynamics (Remember the importance of viscosity.) r r i \ i :J / i. J If' t "t (),J lù I :: (' Il\ (U, 'Û va l/.j, + i (V \ I :: ',f' (\ ' Q ll L I\: II b. Z l l. 'X /' \\ I l= 1: J?$ I I :J "= la F ù lrj. i I tr ' \ r 'J. 1 '.J v II 1:5, 13' 't : :i y. ) l 'j 1 l ) " f + + 'C 1"3 \. I\ 'I (" II L J. y. X b \\.u (ò C' \L 1 el \ c: X 1/ (A B) =A( B) B( A) +(B )A (A )B 5

6 Simple (2D) Fluid Rotation, : 1: i y r Q F II Q * Ç) 'i V\ ll )l J :, :i ;, tj \J L.. \I \D 'f "" " i tl J J l.. l, " \1 c\. r j,' k ) II r l( ' j w \ i rv,., 1 1 iv j, J ' i, t' ' r. l\ \l I:. I II to a \) '; (J \I fi L t v C J i ) \I S Oö ' " () \p c ' ", "../ ) ai ) " 'i " I\ \) Ql ll Q U i. V\. l,: Ll t (, :) \. i lo d? '" () Hr (' Q, r \I l t. (, VI i / t Q ri CJ \, "' v r: (j il \I \J l :J \ \f Il oj. 'X \.. I g ' ) oj \t Cl r ev CJ ' \. II C:.s ). ( C l 6

7 3D Fluid Flow CFD calculation of atmosphere (IsoVorticity contours) CFD calculation of flow in mixing tank 7

8 Today s Vorticity 8

9 , Kelvin s Theorem \t J,.r. f ( () \l l I 'ì L \: w () ':.j f LL I. '\ f L 1 ' J C\ r! VI r i '. I) l \l ' VI _ V'. '. I l X ( V) G \ i ( ( I V) \ I t ( (: : VI ly oj n t:.) (. 'C J '5 II VI l \f t v J ç 1 v ij. :i ' VI, ) F \J, Vl " " l w j J J l. \. 1 y. \J IU 1 r 1 ' J ) t,. 'ft t ( "' ( 9

10 A (General) Vector Theorem for a Moving Surface Flux \ J \ iidentity: ( L \I \. " lt: '" t U u It \ \ i IL \ \ \1 t' t cj (/ l 1 J '4 o "l f () J i., J i l '" "' a V' V' e vi" J I I \p I :: ( l C, \1, t j t t 'tt u ' N + lt ) Q c: V' Q \ I l) l f t: ta '" V\ r ',, 'i J I t( \. l l! Q t' t t. t" I ' \ rj f V "ij \J X. t \. i +. J\ J _(. Vi t: i '" 1

11 Vector Flux Identity (cont) ( \ l r ì l\t t:,, '\ r "t L c l (: ld 4\' C' ij c Vi ) " (. ""C' ( L t r!d ' L 1 + "t l (LJ l l t C' :: '. II r "S ((, \ (, \( L c l " x tt I /;: \..i " l G: l x + 1\1 ' 'l \' b "T I \i " V1 + \ l. + l\i \1 CD n i \ \ ' l" c, \ \ L li F \u t " It u:. :J LL l l\l \.\ \L C" "" g t Vi t (t ì: \J l c " L :J l. X. I ' t + '( " J\) "' t " j a \l l:j u /' I.) i; () '" 1I I + IF) \ h d tb :J Ç) \. g V), Ç) ll, \ 11

12 Two Line Vorticies c i r, iw lj J c: 9 t: N 1 l ii 11 \ C.\ \ 11 t. It ": Q 'i t. i \! "). l $ 'Ḷ. ;) Q. tf 2 o '.( i. lw Ci li vi, \. a \: \ l V (/\ 1 c: C: o l1: \J f r: t: \ II lē: fò G: 12

13 Two CounterDirected Line Vorticies L J( \! F r:\ o: : t i i W LL Q Q VI i " lj \ i 1 r r Q. lc * l J vi Q: ' t \J ; ll '5 't :J t, i=.j F U j.j t 3 (J r c: 13

14 Ring of Point Vortices 14

15 Clouds of Opposite Vortices 15

16 , Smoke Rings C\ ' c: ll I; I " \I. ( J " t) c" o; ') Q,,:. Q. v..( l J A b i I II t. "; Q ) U' tj.o LL 1 f Q i " it ") 1 l c: ( l v VI q ql.ø IL '" V' \/ Ûl v ) \ v \: ' l J 'l G c: 16

17 Toroidal Vortex Ring 17

18 Colliding Vortex Rings N f '" c6, tt_v :J :i :t 1 ' t \. J I. l :t li r \Ä :: VI I) c. $. (: t "j\t ( J =' 18

19 (Prof. Lim, Division of Fluid Mechanics, Melbourne, AU) 19

20 (Prof. Lim, Division of Fluid Mechanics, Melbourne, AU) 2

21 Surface of Rotating Bucket 21

22 How to solve fluid problems? (Like those in textbook) Ç"Tt=l I $T1P#( D ii;4 Lv A fj t c.t u /Z t2 tv (L (TE p Ô t /V A L L I ò C A. 6 E J/ rlf A I' io foji c. ( " '1 S _ r ç TH i S A 1) T v A.,CAL ) 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 \.' ': 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. çr;:,#3 ç Ec.tZLI lk ;1\7 C)".vEf'(tE4Â7 LOù.l/.A ç (" S Te p.. C' A. r C 1/ J L( A. t=a ce d /ê ". r A..A S / N /Tle/' i x. (J. ') Ç( i it l+ Ç'7E rp t: ç N i L '( GO;. S EI? V ia(l Ò A pi! ( C (,ol C S. i CJ A ç rk ' 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/ cgq òi,j L L L f S Y L)/l SDLU(fdl: C tar\ Y ()v ftil l t l!j /Z / A.u,7' è)., ( 22

23 Surface of Rotating Bucket 1. Draw a picture. 2. Dynamic or Static? 3. Coordinate system? 4. Can you apply conservation principles? 5. Use your intuition. Static corotating cylindrical (no flow in this frame) Bernoulli s 23

24 CoRotating Frame u R t u t + u u = g (1/ ) p + 2 u + u R u R = g (1/ ) p + 2 u R + R 2 2 u R with Centrifugal and Coriolis apparent forces. (Very convenient when u R =.) 24

25 CoRotating With Bucket \J l :s \Ù d LA l. 'J l:: () '\ C: ( \1, :) C: 'U \I t; \) () \ i l l U. U q. \ i 'l..j I tl\., '3 lo I :: l I C: J\) I \\ V ' " "".. '" l Force balance: ") \ I 'l \L ' ' l r: (( c: i l \) \L J \ t r' \c D (W V l \ \L 'J ù f. 1\ï i ' ' \I (V '\ t 'ù K \l \!; l') I. li + v led Q 25

26 " \ i Rigid Rotation Statics (" C,. l. li l.t lt "l \Ì rf C i a. ri CL \) J u: Q + t: 11 Ii Î U1 '1 C + C \. 'i '" \ 'f (" N Ci (VJ?: '" Ll :i Q. + + J i v V) VI ' l. II (' (\ '\. t; C' c: ': O a. r. J r\,:. t ( C' II.r 1I. 1 I Ç) "' "v U \t rj ( t () 1 \. 1\.J l: i L :: /" _. 't J l l + C ' Q "r ('e! 'r J J i. r I. \1 t' tj + '" + :: G: s: J Q. "" V" \T ", ) 'P Q. lu n i + (: O v ( t, LL ri Q.t '= T _ II ( \/ \. r. Q. (, ') 26

27 I ( j ;) l l', "" (1 ;p.s ) 1:,, " " "' "'.., ll J ' \. " /". ' "', o J Ik v " i ì: 8 t l Cl 8).L ø c:. i J ') r1 t\ G: l'.:,,\ i Q. N rj ('J Problem 5.1 Force balance: l. ( c tp, t I r: :: l" N "'J Q. trj ' lo ' tq '\ ' ' ' ". "',", ' / 't l N.( i( i: 1 r V ' \ U Q L. J t 1 v.( ṯ ' u. J tj '\ \r l i ': \) t t 7) ø (' I rj i "'l "'" "' r N \= \ ti \I f: /'./ '", " ' l ì ' 2 JL ì. :) N N 3l It! i r' ' ) o ii t ) IJ IN c. rl u N JI :: C: L '= /' \f ('. 1: Q,r \1 (\.J ' :, J + I + i il w r ; ) L (V (j c: cj Qì.) \Í (I r: ( ll ),r N 1i \h o (V r& \ f N c. r i V\ 4 'C cl II.: I: () 27

28 Summary Fluid motion can be described by vorticity dynamics. Vorticity is conserved in invicid flow. Vorticity is solenoidal, Ω =. Vorticity must either close upon itself (like a torus) or close at material boundaries (like a twirling spoon.) 28

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