Lecture-9 - APPH 4200 Physics of Fluids Rotating Fluid Flow...

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Unformatted text preview: APPH 4200 Physics of Fluids Rotating Fluid Flow October 6, 2011 1.! ! Hydrostatics of a Rotating Water Bucket (again) 2.! Bath Tub Vortex 3.! Ch. 5: Problem Solving 1 Key Definitions & Concepts • Ω≡∇×U • • Cylindrical coordinates (Appendix B) • When Ω = 0 (irrotational) then flow is “additive” linear and Bernoulli’s Principle applies everywhere • Circulation is Γ ≡ ∫dA⋅Ω, and Γ is conserved in (“moves” along with) an inviscid fluid. (Kelvin’s Theorem) Two simple cases of circulation: rigid rotation and “line vortex” (irrotational) 2 What is the Surface of Rotating Bucket? 3 Solid Body Rotation (Special Case of Rotational Flow) 4 “Bernoulli-like” Formula for Solid-Body Rotation (Condition for force balance in rotating frame) 5 Bath Tub Vortex The free surface of a time-independent bathtub vortex in a rotating cylindrical container with a drain-hole at the bottom. (a) 6 rpm, (b) 12 rpm, and (c) 18 rpm. The water in the central region is spinning fast and the depth of the surface dip increases when the rotational velocity of the container is increased. 6 PHYSICA L R EVIEW LET T ERS VOLU M E 91, N U MBER 10 week end i ng 5 SE P T E M BER 20 03 A nat 91, of MBER 1 ub VOLU M Eomy N Ua Bat ht0 Vor t ex P H Y S I C A L R E V I E W L E T T E R S week end i ng 5 SE P T E M BER 20 03 A. A ndersen ,1,2,* T. Boh r,1 B. St enu m ,2 J. Juul Ra smussen ,2 a nd B. L aut r up3 1 T he Te ch n ica l Un iversit y of Den ma rk , Depa r t ment of Physics, DK - 2800 Kgs. Ly ngby, Den ma rk R isø Nat iona l L aborat or y, Optics a nd F luid Dy na m ics Depa r t ment , DK - 4 0 00 Rosk i lde , Den ma rk 3 T he Niels Boh r In st it ut e , Blegda msvej 17, DK - 2100 Cop en hagen Ø, Den ma rk ( Re ceived 11 Ma rch 20 03; publ ishe d 5 Sept ember 20 03) 2 We present exper i ment s a nd t heor y for t he ‘‘bat ht ub vor t ex,’’ wh ich for m s when a fluid d ra i n s out of a rot at i ng cyl i nd r ica l cont a i ner t h rough a sma l l d ra i n hole. T he fa st down -flow is found t o be confine d t o a na r row a nd rapid ly rot at i ng ‘‘d ra i npip e’’ f rom t he f ree sur face down t o t he d ra i n hole. Sur round i ng t h is d ra i npip e is a region w it h slow upwa rd flow generat e d by t he E k ma n layer at t he bot t om of t he cont a i ner. T h is flow st r uct u re leads us t o a t heoret ica l mo del si m i la r t o one obt a i ne d ea rl ier by Lundg ren [J. F luid Me ch. 155, 381 (1985)], but here i nclud i ng sur face t en sion a nd E k ma n upwel l i ng, compa r i ng favorably wit h ou r mea surement s. At t he t ip P HtY Snee dA L e R E V I E W L E, T T E R S of he I C lel i k sur face depression we VOLU M E 91, N U MBER 10 obser ve a bubble- for m i ng i nst abi l it y at h ig h rot at ion rat es. 11 week end i ng 5 SE P T E M BER 20 03 u m .– [13] does PACS nu mbers: 47.32n y, 47. 45.Gx not cha nge qua l it at ively when it is rot at i ng [14]. We t herefore spe culat e t hat t he cr it ica l va lue of (a) 1vor where t ng i n st velo y y) s . flow rat e) a nd va r y t he rot at ion rat e (ahe ula r abi l itcit set i n o ccu rs when t he downwa rd T he generat ion of st rongly lo ca l i ze d 0.5 t icit y is a fa sd rag be comes a rg shown n P ict u res of t he obser ve d f ree sur face profiles lare e enougih t o overcome t he upwa rd buoyci nat i ng a nd compl icat i ng i ng re d ient of a broad va r iet y of 10 sol id sur faces a i ncrea ses w d d , t nd at Fig. 1. T he sur face depression ncy force a nit h rag ahe bubbles for me d by t he capi l la r y fluid flows ra ngi ng f rom vor t ex she ddi ng at n st abi l it y lel i ke shap t a nd 18 r pm t he sur face d ip ha s a nair row, nee da long w it h e he flow. Using va lues for t he d rag (such a s padd les, sa nd r ipples, or i n sect w i ngs) over flows 9.5 on a rpm ica l ubble v l is rot at i ng ver y rapid ly (a round 104spher). Atbh igherat ah igh Reynolds number [15], we t h rough t u r bines t o la rge -sca le t or nadoes [1]. I n genera l , t hese flows a re p oorly underst o o d, si nce t he i nt er play ues of t he rot at ion rat e , t he test i mat e longer it ica l va lue of t he ver t ica l velo cit y a s ip is no t he cr stable: A i r 9 WC d ga2 =… d † n by a bet ween fa st a x ia l mot ion a nd i nt en se , lo ca l i ze d vor t icit y bubbles det ach f rom it a nd a re ragge d9o,wwhere t he is t he rad ius of t he bubble. Wit 2 a bubble i ncrea of leads t o d i fficult mat hemat ica l problem s out side t he com sur roundi ng flow a s shown i n Fig. h . W hen israd ius sed a ˆ 0:5 mm, we find WC  ÿ 8.5 25 she ddi1ng ih ich is i nuntel mea sure d ra nge of w. We do for t able rea l m of cla ssica l subject s such a s p ot ent ia l flow f u r t her, t he f re quency of bubble cm s , w ncrea ses t hi or sta nda rd bounda r y layer t heor y. F IG.82. T he t ip of t het sura i r-fil le d core extends a l l not , wayt hver,nhave mea surement s of w i m me d iat ely be he face depression is un st ablehe howe e w t h roug h t he t when do lo t s su One of t he most wel l - k nown exa mples of 0 rat e 0wst he cont a iner .6 . h ig0.8A n a i1 bubblewhahejustr face t ip for a round 22 r pm when t he t ip such flo .2 is 0.4r n 0 is d hole rot at ion of h. r t he so- ca l le d ‘‘ bat ht ub vor t ex,’’ wh ich for m s d f rom t he t ip a nd is d ragge d down t he d rben by t he stable , a nd a pre cise t est of t he cr it er ion for det ache when water a i comes un r [cm] t he t ip i n st abi l it y rema i n s t o be ca r r ie d out. d ra i ns out of a cont a i ner. T he st rong, sur round i ng floma lo ca l i ze d defor w. We t ha n k Jen s Eggers a nd Vacht a ng P ut ka rad ze for t ion of t he f ree sur face ma kes t he vor 0 x beaut i f ul ly 5t e d rawen3(a) r at t ent ion t o Ref. [12] a nd Tom Lundg ren v isible , a nd ha s made t he bat ht ub vor t exht he ow st ry p e u re is su r pr isi ngly complex. Fig u r i g ou T e fl protot uct (b) ‘‘vor t ex.’’ T h is p opula r it y is i n stashows a n otverlay of t wo flow v isua l i zat ion s for helpf ul suggestion s. We a re t ha n k f ul t o Mor t en rk cont ra st o t he made by 40 r at t ent ion wh ich t he phenomenon re ceiaddi n g fluorescent dye at t he sur face (re d) aEdnebjerg for help wit h mea surement s of t he f ree sur face ves i nt he l it erat u re. n at t he depression , a nd t o Ma rk Sch ra m C h r ist en sen , Mor t en T he few cla ssic pap ers about it eit her negle ct t he a x ia l bot30 m i n let ( g reen) , resp e ct ively. T he cont rol pa ra met ersNielsen , a nd Jacob R icht er for i nspi ration to Nørgaa rd flow [2] or con sider t he problem wit hout a f ree sur face [3]. corion tondbato ub . 1( b). T he cent ra l down -flowrougg ion ies of prepa rat or y exper i ment s. We ack nowlresp he t ht Fig t h re h a ser Si m i la rly, t extbo oks ver y seldom ment e dge i ze d , (re20 belowwithen f ree sur face dip is st rongly loca lsuppor t f rom t he Da n ish Nat ura l Science Resea rch d) dele d t h i vor t ex, a nd i f t hey do [4 , 5] t he flow is mo Counci l G . a a n t he do viscous p ot ent ia l t heor y (wit h t he i nclusion of nd ad hoc wnwa rd velo cit ies a re of t he order of 1 m sÿ1ra nt No. 2 4 -56880. core [4]) wit hout i ncor p orat i ng t he aT iaismot ion. apphe rs si m i la r t o a wea k ly con ica l d ra i npip e x h l reg ion I n t ea 10 vor t ex core t he a x ia l velo cit y ca n be htg h , a n essentcompa rable t o t hat of t he d ra i n hole. I f w i i h a rad ius ia l i ng re d ient of t he st rong ‘‘swi rl’’ [6] whh is ma kes t he flow t o t he . ‘‘esophag usub ofr tt hei nva cylixd rt he conta i ner wit h a F IG 1. T he bat ht ’’ vo ex t ich 0cor resp onds or t e n , ica l so fa sci nat i ng. Ou r a i m i n t h is L et t er is t o prov ide ba sic d ra i n hole at t he cent er of t he bot t om (not show n). T he fluid i n ‘‘mout h’’ would app ea r t o be t he su.6face reg ion aboveresent t he 0fl 1 ra 1r 2 *P underst a ndi ng of t he stat iona r y bat ht ub vor0ex: t he .4ow 0.8 e cent.2l region is rot ati ng rapid ly, a nd t hadd ress: Cor nel l Un iversit y, Dept. of T heoret ica l t th e dept h of t he d r,aan , bhe iinterde -st at ionasu]yface wip n lyends on tlhe frotat ionnd of of iehe Mech.,iner. , N Y 14853, USA. i nd t ut n t he r r flo d odep a sma l ract ionratAppl t d cont a It haca a fr [cm ree e st r uct u re , t he shap e of t he f ree sur face t such as he s t h of he e (a) was ta ken at 6 ly ( b) at ]12fr.pJm , a nd , Vat tex pF low [1 . Nat u a p endence of i mp or t a nt cha ract er ist ics he flow trat ei ze roug hPtict urd ra i n hole act uarlpm ,comes HromLug t (c)Ior .18 r(color).i n (a) Are ovnd Te chnolog y ow v isua l i zat ion s made G 3 m. F IG. 4 t h is reg ion. Nealrly a l l hof dtra i n floren’ratiusision s0v ide da bd Ktrheunp er F labad, fluid ida , 19 95). n erlay of t wo fl (color). Numer ica solut ione of he hole rad quat pro.1 cm n y t he Lundg w s e e wa s ( ieger, Mta r be r F lor T u t he cent ra l sur face depression , t he rat e of t he out -flow, by add i ng fluorescent dye at t he sur face (re d) a nd at t he bot t om7 w it h sur face t en sion ( blue) a nd wit hout sur face t enhe ( g reen)he sion [2] H. A. of t : i n a 3 ÿ1 a nd t he rot at ion rat e i n t he vorcompa reh iwitE kxpernment a l resulthehbot9 o312 Tpm.rat escont a iout -flow wereEi ns3e62 cmdgH. , Li ,, iresp e ct ively. T he hre d dye is flowi ng i n a t h i n t ex core. n h e ma i layer at dept(re10. atcm.ÿof t he T he ner. ÿI n t h is (a) t i n letn ( sreen) n P ro cee di ngs of t e Heat td t t mr (c) T he phot a nd F were ( bs 3:54 cm s 1 , a nd(a) 3:16 cm3 s 1 .Tra n sfer og raph sluid Me cha n ics In st it ut e , Sta n ford ) d) We st udy a stat iona r y bat ht ub vor t ex i n a rot at i ng v [cm/s] h [cm] DOI: 10.1103/ PhysRevL et t.91.10 4502 Bath Tub Vortex (Example of irrotational flow) 8 1- ~ , ~ ~ oe -- ) ~I 1 lJÌ a 0 .J lL .J '0 ~. 1:: l( ~Q) :: ~ ~~ ,, ~ :i~ tv 0 i ~ .t Bath Tub Vortex (more) ~q) 0 lJ (ß\~ ~!. ., ~ 3\ ~ ~ ra + ~ ~ ~ n\~ -p:. I, IJ . ~ () fI fi Ii ~ \J .t (" \ ri L\ci .. i .t ~ ~ I '" Q! I ~ ..Q \0 \l (b (( c: \l c. JI '- rl iI J\~ 1- ~ ~ l .. w ) 0 r- l) ,. \~ ~ V vi ') 0 "IJ ~ 0 f c. .. t L. \ù L- () ii .. ~~ ~ 0\ li 0 1)i ~ l ~ ~ \' II -It ,/ ti ( rt (~ l- \I b, '; '" c: (" ct ~~\ r ~ \l (\Ì " ~ I~ n\A -I \' X t i: .. )", ~ I( ,: ID '\ l ii 1\ \D , l1) '" Y' .. \J \è ") .. ~ \o V - )0 ïl Q ~ ) \L ~ .. oJ ~ 'J i: .. V' :) ~ \~ \) ~ u: 1 0 V\ v ob ~r: '" u" ."'òi ~ \) 't \_ ~ lo1\: :; ..~ ~ q; '"~ ~ ai i, \:s X tl\ ~\ l- i I.. L "" 1\ ~ tj ~ (l 1" ~ (ti -tÑ ~ IV ~ .& )\1 (1\ ~ (\ 'u ~~ "' 1i j ~ \' )- ".. r. \\. ( t cÌ ~ ~tf ~ ( t. - ... i V' - ;) ~ 'V (. ~\ (l .~ i .. .. .. ~ r. ~ D IJ 'l \l 9 Bath Tub Vortex Shape 10 Chapter 5 Problems • • • • 5.2) Rankine vortex 5.4) “Vortex image” method 5.7) Mechanical energy of circulation 5.8) Vorticity dynamics 11 Problem 5.2 Vorticity Dynamics 30 m. 2. A tornado can be idealized as a Rankine vortex with a core of diameter pressure The gauge pressure at a radius of 15 m is -2000 Njm2 (that is, the absolute any circuit is 2000 Njm2 below atmospheric). (a) Show that the circulation around between surrounding the core is 5485 m2 js. (Hint: Apply the Bernoull equation ar speed infinity and the edge of the core.) (b) Such a tornado is moving at a line sure to of 25 mjs relative to the ground. Find the time required for the gauge pres an ai drop from -500 to -2000 Njm2. Neglect compressibilty effects and assume the local temperature of 25°C. (Note that the tornado causes a sudden decrease of lting excess atmospheric pressure. The damage to structures is often caused by the resu pressure on the inside of the walls, which can cause a house to explode.) 3. The velocity field of a flow in cylindrical coordinates (R, cp, x) is UR =0 Urp = aRx Ux =0 where a is a constant. (a) Show that the vorticity components are úJR = -aR úJrp = 0 úJx = 2ax 12 G rJ -i Q.:~" 0: ~ :; ~~ ~ Problem 5.2 q; I: VI 0 't\ l~ -t 3 l~ \\ f ': ~ VI 't ,.. i/ )" !, '\J I: ~ -t ;) ~ 1- '- L t; \) ~v ~~ 0 :J ~Li It' ~~ Ii ~ -: ~ ~ rlt~ -i t ''\ II (:J Y. 11; ~ o~ fì \j \ " ~ () .0 ~ ~ - - r. J i vJ L 0 \. ii ~ \1 ~ ~ Lr~ II '\~ \, ~ '- 0 ~~ ~ -- 1\ ~ ~ ,~0 tV .. ~ c '; 0 N )L \, \ \ '- "- -l \: Ò ~-! a '- 0 II 4 ~ 0/ 1\ '\ \) ~ \t .. :i i~ t ~ ~0 l\ \" \ ll ~ ~ 1\ ~ ~ (\ .\ ~~ .. ~ -L l" t. i ~ '\) ~0 ~ 'R II L U i)\N 1:\ i r ~l\ l V' " .. (- y.~ .. ~ ~ v ~ Q l, ~\ "C J ~'-- ~ ') i-' ~ vi ~ ~ , \1 J \( tV ~ ~ i ~ -t (' Q. V "- ~ ~ ~'Ì f ~ r ~ t ~ ;) i' ~ 't lc') ii t' -- N :: \~ ~ ~ \1 \¡ .. ') ~ ~ '" i ~ "" ~ " I.N t- ~ CL ~~ ~ \i L ~ ~ ,-j .. .. ~ 0 ~ i: 14 ~ 13 ß . .r" 'J J ~ G: i V\ \, rJ V ~ ~ Problem 5.2 (cont.) (~ I Q. f '- c-I~ ri :: Q r: ": ~ ~ ?\l Ç) ~ (v ~ ~ i lv "' 'i -t:~ Ii ~ .. ~~ '.i l" ~ ~ 4. \ ~ ( " r; ~ f: ~ ~ 0 0 Q lÙ ~ I'" ~ rJ ~t \ l\ ~ J C) .. \. ',¡ v. ',, rJ~ '; 0 N ~ \I ~ Q ~ " \i ti ~O ~ \:: h L (' \ N ~ ~ \) 0 l/ iI Q. I N ~ \. M fj ~ \\ l tV (' 0 II , ~l i ~ "V (V (I r-,~ l. ~ \n ~ \. ~ ~ è \.c r ~- \l. l' 'f N C: J '\ ~ \L ~ \ òC \J \) ) ~ t- t e. r C' ~ ~ ~ l \d ~ '-" t- (: ~ t= ~ ! H v ~ (1 "- ~~ j\~ '- f /' r/" t l: ~ 0 () 0 N .. 1./ VI 02) ~rl ~ ~ rv (' ~ .. Ln ~ .., J ~ l\ cY l\ '- ~ (/ "' ~ J , ~ N ~ I ': rv lJ It '(' (V ~' II (' I Q. /' '; Q ~ t J \ ~.j ~ ~ ~O i t! ~ ~ ~ j 0. ~ \ "" \) - ') (J i~ C: 1-- ~ CD i: ~ ~ l- \L ~ ~. ÇL .è Ç) 14 úJR = -aR úJrp = 0 úJx = 2ax Problem 5.4 an Rx-plane. (b) Verify that V 06) = O. (c) Sketch the streamlines and vortex lines in Show that the vortex lines are given by x R2 = constant. 4. Consider the flow in a 90° angle, confined by the walls () = 0 and () = 900. z-axis. Consider a vortex line passing through (x, y), and oriented parallel to the Show that the vortex path is given by 11 2 2 x+ y= constant. e vortices at points (-x, _y), (Hint: Convince yourself that we need three imag tion? The path lines are given (-x, y) and (x, -y). What are their senses of rota velocity components at the by dxjdt = U and dyjdt = v, where U and v are the location of integratiop. of the vortex. Show that dyjdx = vju = _y3 jx3, an which gives the result.) ting coordinates, and prove 5. Star with the equations of"1otion in the rota Kelvin's circulation theorem D -(ra) = 0 Dt where 15 Problem 5.4 J( #s,.y po 0 AtE.. ~~ LI..~ =- 0 C. Vl~ 'r L.H E/I( \.Ç')C cE, iA-l Uo/2 T (CA, (:;S I Li /V /'~ Fo/( S t I'S V L. AVI Tì íi S' . 'Tl-t f iJ ÔA-rLt tLr ~oJT (S ç i ""I' L '( 7H ¡¡ T D Pl ii c H ß ~ J l'i. 4- '( (! cJ r- ¡( tft c) .. ç 1,, ~.A L L / (N ;J-,i, 't ì ) ~ '-~ ~:l, 't ) // ~~~ /.. (l$ .." \ ) (j~ (x,-y) (s C-~t - 'r ') IIê: - A-l i ....lit fJ "7 ~ /r l( E¿¡ oA ( , v 9 _,. f, ¡. ~"p l C ~ bl I 71 v r? . ß ,- ~t .. 1/'- E Å. ò '2 vt 4- t- rL ÒL. ~.c.'T' V~iSl-. 7F w'2 /h¿) ,4 v- t "¥kA- ¡' (r C/ ò/ (l'L ~ 0 F "". ~ 5" .... CJ F TH rr O( Iè c cJL4-7JO-u (2,4 cH 0 ~E /. A- F.fow. l-Û L7( jJL l. F L- c) Lo L( ~ fl (j off ô So (-r Q ( 12 C () L&4Tlc).. -l rEv. ~tí f-ùv1 ~ A-t. ¿i c: --jJ D r-&/' TSc. Æ..t c: rl -tJ ~ pLc) L- ti. ,. 'fue Lvf "A!CD It IHl,f/) frLA-rrz t r1 A-'9/l v D /? -c ') I he p4 "Í (s' ( "k A4,r t(' Q A A. L- ;2' L i.O., -l x T D -l( is- /Lo-1 -, ~L P L l) "' v 0 .( f)'l -f P,/.. '-l ri tJ D/l v1rlL F L 0 ~ r v o. ('i2. ') l) r 7P (i (~v'l:rf UO 12(1 t: ,'ES II Il ( ~ (". ~"" f: ff 6(Y /' 0 Ç'l ~ ~ ~LL- S Tc- fi S r¡ '- l/o./f ( c' ( r2 J ri td --.. vHl C. A-t.L f ~'A-l 5 pY . lJDIè~i. FL ÙL. f/è)J/'..7 (ôul'v"' 77 ()A..5 A--- £V~L ç , 16 ~ r- (~ i \ ~ ç; ~ 0 -J \Y i, t 0 :t -- tI ~ t" , - 'V ~. ~ :t 3 .. .. t f ~. f: lL I. "1 t 'j .. \. j.. Problem 5.4 (cont.) ~ v ,~ i ,~ ~ (' íJ f~\j ~ Ie C' + .. ~~ ~ ~ (~ N 1\ Y- 0J ii ti Y. II ~ t'~ (" )v 1- y. IJ ~~~ ..'tNc:N~ Ii r. N N '- v '- ,. r" c-r' cv". I.. 'J (' .~ r- j~ L ~ ~ /(2 ~ . Ij "¥ t- ),j I~t" C- tv .. C- ~ ii ~ Ç: li I t; \J Ii I y. t C1 'l=C" ~i~~ c: t- /' IJ .¡ t'l) ~ t; 1-' "*\ '" ~ ~ rl Ii .r ~- ~. '1~ ~ -i .3 rt:x )" t) I ~ )~ x C' ~.)x: t' ~ N l- " -- ~ ), ~ 0 ~ t F ~ ~ \. tL ~ \I r; "- ~ 't ~ r- ~ f' J-- l" I: ) c (, ~ i "" 'I N 1- ~ \ -l \( Ii N 10. tQ ~ i 00)( II l .. Q \i (Ó t\~ ~, )- \ ~ l\ ~-\~~ II I J- \ 'X "' l' h ~ " ~ lt -. N ~ ')( )- ~ 'X '- .. ~. ~ r-)- ~" + \~ N -'f ~ N J- (I c.\ ~ L ~ ~,~~~ (~ -- . (\ li ~ 'i~ ),i .. r- r- \ l': L () V\ 17 Assume that the flow is inviscid and barotropic and that the body forces are conservative. Explain the result physically. Problem 5.7 6. Consider the interaction of two vortex rings of equal strength and similar sense of rotation. Argue that they go through each other, as described near the end of Section 8. 7. A constant density irrotational flow in a rectangular torus has a circulation r and volumetrc flow rate Q. The inner radius is r¡, the outer radius is r2, and the height is h. Compute the total kinetic energy of this flow in terms of only p, r, and Q. 8. Consider a cylindrical tank of radius R filled with a viscous fluid spinning steadily about its axis with constant angular velocity Q. Assume that the flow is in a steady state. (a) Find fA 6) . dA where A is a horizontal plane surface through the fluid normal to the axis of rotation and bounded by the wall of the tank. (b) The tank then stops spinning. Find again the value of fA 6) . dA. 9. In Figure 5.11, locate point G. Literature Cited Lighthil, M. J. (1986). An Informl Introduction to Theoretical Fluid Mechanics, Oxford, England: Claren- don Press. Sommerfeld, A. (1964). Mechanics of Deformble Bodies, New York: Academic Press. (This book contains a goo discussion of the interaction of vortices.) Supplemental Reading 18 ~ ¡. t~ VI 0 tJ II '\ '0 II CJ 'f rt" ~ (i= II. ~ ~ ~ \: .. l~ \ (J Problem 5.7 ~ \ .Q ~ ~\~ (I ~\t ~ 0 ~ ~ ~I~- ~ ~~ ~I II ~ \ t- " rJ ~ -- ~ \ ~ 0 "" ). li "- ~ ~; ~ L~ 'ì H II -C -C *- C-,,- ê- \ ~ -. N t: '" ~ t' ~\ \ ~ '- \:'te- ~ ~.. S~\ ~t'" ~ Ii ~~ ~~ ~t'.-C li LI ¡~ Q) 'X b C: -~ ~ tl~ ii c: II ~ 0 I:. * .. '. v ~ .. CJ ~\ -i l~ .. 1I \ \J ~ l-C .. :: .k .. ~ f \3 L -IN (I 19 sense of rotation. Argue that they go through each other, as described near the end of Section 8. 7. A constant density irrotational flow in a rectangular torus has a circulation r and volumetrc flow rate Q. The inner radius is r¡, the outer radius is r2, and the Problem 5.8 height is h. Compute the total kinetic energy of this flow in terms of only p, r, and Q. 8. Consider a cylindrical tank of radius R filled with a viscous fluid spinning steadily about its axis with constant angular velocity Q. Assume that the flow is in a steady state. (a) Find fA 6) . dA where A is a horizontal plane surface through the fluid normal to the axis of rotation and bounded by the wall of the tank. (b) The tank then stops spinning. Find again the value of fA 6) . dA. 9. In Figure 5.11, locate point G. Literature Cited Lighthil, M. J. (1986). An Informl Introduction to Theoretical Fluid Mechanics, Oxford, England: Claren- don Press. Sommerfeld, A. (1964). Mechanics of Deformble Bodies, New York: Academic Press. (This book contains a goo discussion of the interaction of vortices.) Supplemental Reading Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press. 20 Pedlosky, J. (1987). Geophysical Fluid Dynamics, New York: Springer-Verlag. (Ths book discusses the \9 ~ l~ ~ ~ ~ i ¡ l\ ~ ~ \~ ~ (I U ~ \, ~ ~ b. ~' ~~L r:s '\ ') ~ ~ j~ "" Problem 5.8 "" J ~ " ~~ 0 ~ 'J -. '-. ~~J~ r- -s ~~( oJ .t ~ ~- VI "- j v~~-t I" ~~ ~ v ~ t N\ ~ ~\ . ~ ~ N~ ~ ('\ ~ ~ ~t~ b ~-t' .-- ~ )- ~ -r '- ~ ( ~ 'l ~ '0 ~ ~\C _~~.. _--. b l- : ~ S ~ :Lt~' \1 ~ I. (I.. ~ ~1 J ;~ J ~ G "-~ -- "- "" '- o ~ J .. .. ~ ! l iJ ~ \l tJ ~ J .. '- '- '- c; ~ "e ~ .-. ~ i ~ I' ~(; II J 0! ~ r ~~ji \J-ili ~..'" \( l' (~' ~ jJ Ii J' ~ t ~ \t .. ~ - ~ ~ '- 'i \\ l~ 'X ~ ~) L :: J :: c: ~ () e ~ ~ ~ ~ ct .. ~ 'v J ~~ II o -t l~ 21 Summary • Two special examples of vorticity: solid- body rotation (uniform) vorticity and the line vortex (zero vorticity away from origin) • Sample problems in rotating fluid dynamics: ‣ Bathtub vortex ‣ Tornados, moving line vortices ‣ Mass flow 22 ...
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