APPH 4200 Physics of Fluids

Transcription

1 APPH 42 Physcs of Fluds Rotatng Flud Flow October 6, 29 1.!! Hydrostatcs of a Rotatng Water Bucket (agan) 2.! Bath Tub Vortex 3.! Ch. 5: Problem Solvng 1 Key Defntons & Concepts Ω U Cylndrcal coordnates (Appendx B) Two smple cases of crculaton: rgd rotaton and lne vortex (rrotatonal) When Ω = (rrotatonal) then flow s addtve lnear and Bernoull s Prncple apples everywhere Crculaton s Γ da Ω, and Γ s conserved n ( moves along wth) an nvscd flud. (Kelvn s Theorem) 2

2 What s the Surface of Rotatng Bucket? 3 Sold Body Rotaton (Specal Case of Rotatonal Flow) 4

3 Bernoull-lke Formula for Sold-Body Rotaton (Condton for force balance n rotatng frame) 5 Bath Tub Vortex The free surface of a tme-ndependent bathtub vortex n a rotatng cylndrcal contaner wth a dran-hole at the bottom. (a) 6 rpm, (b) 12 rpm, and (c) 18 rpm. The water n the central regon s spnnng fast and the depth of the surface dp ncreases when the rotatonal velocty of the contaner s ncreased. 6

4 VOLUME 91, NUMBER 1 P H Y S C A L R E V E W L E T T E R S week endng 5 SEPTEMBER 23 Anatomy of a BathtubVortex A. Andersen, 1,2, * T. Bohr, 1 B. Stenum, 2 J. Juul Rasmussen, 2 and B. Lautrup 3 1 The Techncal Unversty of Denmark, Department of Physcs, DK-28 Kgs. Lyngby, Denmark 2 Rsø Natonal Laboratory, Optcs and Flud Dynamcs Department, DK-4 Rosklde, Denmark 3 The Nels Bohr nsttute, Blegdamsvej 17, DK-21 Copenhagen Ø, Denmark (Receved 11 March 23; publshed 5 September 23) We present experments and theory for the bathtub vortex, whch forms when a flud drans out of a rotatng cylndrcal contaner through a small dran hole. The fast down-flow s found to be confned to a narrow and rapdly rotatng dranppe from the free surface down to the dran hole. Surroundng ths dranppe s a regon wth slow upward flow generated by the Ekman layer at the bottom of the contaner. Ths flow structure leads us to a theoretcal model smlar to one obtaned earler by Lundgren [J. Flud Mech. 155, 381 (1985)], but here ncludng surface tenson and Ekman upwellng, comparng favorably wth our measurements. At the tp of the needlelke surface depresson, we observe a bubble-formng nstablty at hgh rotaton rates (a) h [cm] r [cm] 5 (b) 4 v [cm/s] r [cm] 7 Bath Tub Vortex (Example of rrotatonal flow) 8

5 , Bath Tub Vortex (more) ' 1- '.J ll r- J r. è V.J ) v ) D ljì ') w a l - o J V J 'l - - l l(. 1 D (. 1:: J l1) 'V (l )-. 1 l),.. Y' J rl l Q) :: -p:. D ', J l (Ì t V' - ( t. ;) -. v cì tf,/ tv ( rt f t.t., t 1) L- ( ' (l L. ù 'J : 3 : ( l ra.t V' 1 b, () l- '; )1 (1 + ',, ( r c: r f Lc t ( ct :) tl.& : l (b ), (( na ) X V :s -tñ c:.t ( n l c. J,: X - ' u:, (t ' ob V 1 r.. ( r: a (ß!. c. -t L tj 1 ) lj :; lo 1: l- L q) ' Q! q; ' Q f l 'u ' 1 oj j () ) ' u.'ò Q ) 't _ 9 Bath Tub Vortex Shape 1

6 Chapter 5 Problems 5.2) Rankne vortex 5.4) Vortex mage method 5.7) Mechancal energy of crculaton 5.8) Vortcty dynamcs 11 Problem A tornado can be dealzed as a Rankne vortex wth a core of dameter 3 m. The gauge pressure at a radus of 15 m s -2 Njm2 (that s, the absolute pressure s 2 Njm2 below atmospherc). (a) Show that the crculaton around any crcut surroundng the core s 5485 m2 js. (Hnt: Apply the Bernoull equaton between nfnty and the edge of the core.) (b) Such a tornado s movng at a lnear speed of 25 mjs relatve to the ground. Fnd the tme requred for the gauge pressure to drop from -5 to -2 Njm2. Neglect compressblty effects and assume an a temperature of 25 C. (Note that the tornado causes a sudden decrease of the local atmospherc pressure. The damage to structures s often caused by the resultng excess pressure on the nsde of the walls, whch can cause a house to explode.) dnates (R, cp, x) s 12

7 '; l,. l Problem 5.2 : V ) l. f l -l : ': - Ò V r. 'R 't, - / -! / J ll -' -: j J, 1 a ) L ( ' v 1!, tv J rlt vj L : r l -t - t. V' ') ;) 4 -t (' t' )N 1: (- 1- L :: y U Q. lc- v -- ) ) '' v 1 -- N ') 't 1 :J t; t t- ' L t' l, CL (:J Q : 11; Y. l -L t. ;) () t C L.N ') q; t t tv l Lr -t c o N fì 3 ) L ' J,,. ( -- - r ' f,-j 'Ì : 14 - V 13 Problem 5.2 (cont.) (v l O t!. V r- N, lv ' ' -- (V t: M rv ( -, l. r - l. J.j l' ( 'f N.c fj C: (' t (' tv N Q. f. ', v. c- ',, J l rj N ' lj 2) rl L. r '; (', ': rv n :: 4. N ) t ) l/ ) òc - (V Q J (' ( J r: ' r; /' r- / : Q. ' 1./ f t Q (/ l: j V v t (' Q. () t- (1?- e. - f: /' '; N Q r è j l t. Ln Ç) C' lù O l Q ') ) l rj - (J ' l l :: C: cy! t= h J d 1-- C) t L :, L J t- (: H CD l-.è Ç). ÇL 14

8 Problem Consder the flow n a 9 angle, confned by the walls () = and () = 9. Consder a vortex lne passng through (x, y), and orented parallel to the z-axs. Show that the vortex path s gven by = constant. x y (Hnt: Convnce yourself that we need three mage vortces at ponts (-x, _y), (-x, y) and (x, -y). What are ther senses of rotaton? The path lnes are gven by dxjdt = U and dyjdt = v, where U and v are the velocty components at the locaton of the vortex. Show that dyjdx = vju = _y3 jx3, an ntegratop. of gves the result.) whch 15 Fo/( L /V /' Uo/2 T (CA, (:;S L( fl S t 'S V L. AV Tì í S'. 'Tl-t f J ÔA-rLt tlr ojt (2,4 ch E (N ;J-,, 't ì ). (s C-t - 'r ') A - ß,- ) J( =- C. Vl 'r L.H E/(.Ç')C ce, A-l L- c) Lo F.fow. l-û L7( jjl l. F ( S ç ' L '( 7H L 5 CJ F TH rr O( è c cjl4-7jo-u /. fj ê: -7 A-l lt /r t r1 A-'9/l v D /? -c ') l( _,. f,. E oa (, v 9 he p4 Í (s' v.( f)'l -f p l C bl 71 v r?. Tc- f S r 'A-l 5 py Problem 5.4 ( k A4,r :l, 't ) // / (l$ (j (x,-y) v o. ('2. ') l/o./f ( c' ( r2 J r td -- vhl C. A-t.L f. ljdè. FL ÙL. f/è)j/'7 (ôul'v' 77 ()A5 T D Pl c H ß J l'. 4- '( (! cj r- ( tft c) ç 1,,.A L L / t 1/ E Å. ò '2 vt 4- t- rl ÒL..c.'T' VSl-. 7F w'2 /h ),4 v- t ka- ' (r C/ ò/ (l'l F off ô So (-r Q ( 12 C () L&4Tlc) -l rev. tí f-ùv1 A-t. c: --jj D r-&/' TSc. Æt c: --rl tj plc) L- -'fue Lvf A!CD t Hl,f/).O., -l x T D t.,.t(' frla-rrz Q A A. L- ;2' L -l( s- /Lo-1 -, L P L l) ' P,/ l r tj D/l v1rll F L r ( (. l) r 7P ( (v'l:rf UO 12(1 t:,'es l f: ff 6(Y /' Ç'l LL- S A--- VL ç, 16

9 r- r' l r- ),, :t - : 3.3 () ) V c Problem 5.4 (cont.) J. t L t.r -- L - 'tnc:n f. C- r- l':. ( c. l 1 tv ( )- l - Ç:. f: t; 1-' 'J y. N. ' l' t; J,. c- cv r. J- * ' rl l r. N v N, ( 1- J- 'X t- j L /(2 r-)- /' h.)- N 1 C- ( -- 'V J x: 'f. )v. t' ll t'l) ('. t r; 'j - Y. ). -. )- Y- c: x ')( N l=- 'X C 1 lt. j + J C' f' J-- l t t, y. ) l- t- - t j t' 't )( e (Ó C' N ( tl ),j r- t- t N t:x N Q C1 t) r- -- (' íj fj F 1-, Q N t -l ( (, ), v ', 17 Problem A constant densty rrotatonal flow n a rectangular torus has a crculaton r and volumetrc flow rate Q. The nner radus s r, the outer radus s r2, and the heght s h. Compute the total knetc energy of ths flow n terms of only p, r, and Q. 18

10 Problem t- - ;.Q t' rj L -- S- tl ( c: t -- 'ì C: 'te- t :. :- * L.-C t' 3 -. N. t- V tj '. t: ' v l C-,,- ê- L 'X CJ b Q) -C -C *- -N l ( H ). - l ' 1 ' J CJ 'f. (= l-c ::.k f 19 Problem Consder a cylndrcal tank of radus R flled wth a vscous flud spnnng steadly about ts axs wth constant angular velocty Q. Assume that the flow s n a steady state. (a) Fnd fa 6). da where A s a horzontal plane surface through the flud normal to the axs of rotaton and bounded by the wall of the tank. (b) The tank then stops spnnng. Fnd agan the value of fa 6). da. 2

11 b. U, l ( j r:s ' ') L ' Problem 5.8 o -t l- c; 'v J l ct L :: 'X ) J ' tj l l -t J! t c: J () 'J. e J :: J.-. - e J' 1. r- (' j J ( r 1 J ; J G oj )- -r ( _ 'l t ' b -t' _--. C.-- b N (' j v v -t t N. ' j J -l J! (; ( ' l' ( -s V - oj.t - l- : S :Lt' 21 Summary Two specal examples of vortcty: soldbody rotaton (unform) vortcty and the lne vortex (zero vortcty away from orgn) Sample problems n rotatng flud dynamcs: Bathtub vortex Tornados, movng lne vortces Mass flow 22