APPH 4200 Physics of Fluids

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1 APPH 4200 Physics of Fluids Rotating Fluid Flow October 6, !! Hydrostatics of a Rotating Water Bucket (again) 2.! Bath Tub Vortex 3.! Ch. 5: Problem Solving 1

2 Key Definitions & Concepts Ω U Cylindrical coordinates (Appendix B) Two simple cases of circulation: rigid rotation and line vortex (irrotational) 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) 2

3 What is the Surface of Rotating Bucket? 3

4 Solid Body Rotation (Special Case of Rotational Flow) 4

5 Bernoulli-like Formula for Solid-Body Rotation (Condition for force balance in rotating frame) 5

6 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

VOLUME 91, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S week ending 5 SEPTEMBER 2003 Anatomy of a Bathtub Vortex A. Andersen, 1,2, * T. Bohr, 1 B. Stenum, 2 J. Juul Rasmussen, 2 and B.

Lyngby, Denmark 2 Risø National Laboratory, Optics and Fluid Dynamics Department, DK-4000 Roskilde, Denmark 3 The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark (Received 11

The fast down-flow is found to be confined to a narrow and rapidly rotating drainpipe from the free surface down to the drain hole.

7 VOLUME 91, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S week ending 5 SEPTEMBER 2003 Anatomy of a Bathtub Vortex A. Andersen, 1,2, * T. Bohr, 1 B. Stenum, 2 J. Juul Rasmussen, 2 and B. Lautrup 3 1 The Technical University of Denmark, Department of Physics, DK-2800 Kgs. Lyngby, Denmark 2 Risø National Laboratory, Optics and Fluid Dynamics Department, DK-4000 Roskilde, Denmark 3 The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark (Received 11 March 2003; published 5 September 2003) We present experiments and theory for the bathtub vortex, which forms when a fluid drains out of a rotating cylindrical container through a small drain hole. The fast down-flow is found to be confined to a narrow and rapidly rotating drainpipe from the free surface down to the drain hole. Surrounding this drainpipe is a region with slow upward flow generated by the Ekman layer at the bottom of the container. This flow structure leads us to a theoretical model similar to one obtained earlier by Lundgren [J. Fluid Mech. 155, 381 (1985)], but here including surface tension and Ekman upwelling, comparing favorably with our measurements. At the tip of the needlelike surface depression, we observe a bubble-forming instability at high rotation rates (a) h [cm] r [cm] (b) v [cm/s] r [cm] 7

8 Bath Tub Vortex (Example of irrotational flow) 8

9 , Bath Tub Vortex (more).t., tv 0 i :i () )0 ii ïl '" u "."'òi 0 Q ) \) 't \_ lj :; lo 1\: l- \L q) '" Q! I 0 q; '" i 'u "' 1 Q (ß\ f 0\ li 0 oj i j I c. -It,/ "" L II!. I ( rt tj 1\ f i t i t 1)i L- ( \' (l L. \ù 'J 3\ \I i: (\ l ra i.t V' 1" b, '; '",, c: l- () I + (" \ ri \0 L\ci fi :: Q) l( t \l (b Ii (( n\ c:.t \l c. -p:. ID I, 1:: JI '- IJ rl )\1 (1\ \ (" ct r :) tl\.& i: \ ob V\ )", I n\a \) X IV \:s I( \J,: X -I \' u:. 1\ \D ii J\ '\ " l (\Ì ii \l -tñ i, (ti r: \' ai 1 0 r. \\. t ( v cì tf V' - ( t. ;) -. i (. l1) \ 'V (l " )- \. i 1 l),.. Y' '0 \ 1-.J ll V.J ) vi ") ljì ') w a 0 l 0 "- \o V IJ '" r- \J r. \è 0 D - - \l 'l IJ 9

10 Bath Tub Vortex Shape 10

11 Chapter 5 Problems 5.2) Rankine vortex 5.4) Vortex image method 5.7) Mechanical energy of circulation 5.8) Vorticity dynamics 11

12 Problem A tornado can be idealized as a Rankine vortex with a core of diameter 30 m. The gauge pressure at a radius of 15 m is Njm2 (that is, the absolute pressure is 2000 Njm2 below atmospheric). (a) Show that the circulation around any circuit surrounding the core is 5485 m2 js. (Hint: Apply the Bernoull equation between infinity and the edge of the core.) (b) Such a tornado is moving at a linear speed of 25 mjs relative to the ground. Find the time required for the gauge pressure to drop from -500 to Njm2. Neglect compressibilty effects and assume an ai temperature of 25 C. (Note that the tornado causes a sudden decrease of the local atmospheric pressure. The damage to structures is often caused by the resulting excess pressure on the inside of the walls, which can cause a house to explode.) dinates (R, cp, x) is 12

13 , Problem 5.2 VI l \\ f ': VI Ii, - J -: \ i 't 0 r. i/, l\.0 l" -L i t. '- 0 \) 0 -l \: - \" Ò 'R 0/ \ -! J ll II i-' \j 1\ a L $ '- '\ vi \1 vj 0 L r i tv l\ 0 II \. V' ') l ii -t (' ii " t' i)\n 1:\ i " )"!, '- \J rlt I: -i t -t ;) 4 1- '- L v ''\ 0 $ y.- 1\ -- \) :J t; \1 II Li It' (:J 11; Y. (- :: 1\ U \ Q. v 't \t t- l, i' CL lc- \1 -- N ') Q :i ;) i \ () t \ i "C \ L I.N ') " q; t I: '- 0 t\ tv 3 l Lr -t o fì c ) L '; 0 II N '- 0 '\ J \, \, '-. \ (\ \ \ -- '- "- "" r i '" f,-j 'Ì 0 i: 14 "- V 13

14 I I l '- Problem 5.2 (cont.) (v i i N, lv "' O i t! 'i -- \. (V t: M Ii l rv \\ (I r- "V -, l. r - \l. J \.j l' ( 'f N '- N \.c.i C: Q. f \. ', '- ',, v. (' \ It (' tv '- J l" rj N '\ lj 02) rl r i '; c-i fj \L \. :: 4. N \) - \ \ (' 0, ': rv \n t II 0 ) I l/ \) I (V Q J (' ( ii ' r- r: r; /' /" " ": Q. "' 0 1./ f t Q II (/ l: j VI v t " (' I Q. () t- (1?- e. "- 0 \i /' f: '; N Q r è òc \ \J j\ \l ti 0. Ln Ç) '- C' \ lù 0 O 0 "" l \ Q ') \) i l\ rj - (J l I'" l\ \:: C: J cy! t= \ h \d 1-- t C) L i:, '-" \L J t- (: H CD l-.è Ç). ÇL 14

15 Problem Consider the flow in a 90 angle, confined by the walls () = 0 and () = 900. Consider a vortex line passing through (x, y), and oriented parallel to the z-axis. Show that the vortex path is given by = constant. x y (Hint: Convince yourself that we need three image vortices at points (-x, _y), (-x, y) and (x, -y). What are their senses of rotation? The path lines are given by dxjdt = U and dyjdt = v, where U and v are the velocity components at the location of the vortex. Show that dyjdx = vju = _y3 jx3, an integratiop. of gives the result.) which 15

16 Problem 5.4 Fo/( Li /V /' Uo/2 T (CA, (:;S I L( fl ij ÔA-rLt tlr S t I'S V L. AVI Tì íi S'. 'Tl-t f ojt (2,4 ch 0 E (N ;J-,i, 't ì ) '-"". (s C-t - 'r ') A - ß,- " \ ) J( =- 0 C. Vl 'r L.H E/I( \.Ç')C ce, ia-l L- c) Lo F.fow. l-û L7( jjl l. F ( S ç i ""I' L '( 7H LI 5" CJ F TH rr O( Iè c cjl4-7jo-u /. II fj ê: -"7 A-l i lit /r t r1 A-'9/l v D /? -c ') l( _,. f,. E oa (, v 9 I he p4 "Í (s' v 0.( f)'l -f "p l C bl I 71 v r?. Tc- fi S r '- 'A-l 5 py ( "k A4,r :l, 't ) // / (l$ (j (x,-y) v o. ('i2. ') l/o./f ( c' ( r2 J ri td -- vhl C. A-t.L f. ljdièi. FL ÙL. f/è)j/'7 (ôul'v"' 77 ()A5 T D Pl ii c H ß J l'i. 4- '( (! cj r- ( tft c) ç 1,,.A L L / t 1/'- E Å. ò '2 vt 4- t- rl ÒL..c.'T' ViSl-. 7F w'2 /h ),4 v- t " ka- ' (r C/ ò/ (l'l 0 F off ô So (-r Q ( 12 C () L&4Tlc) -l rev. tí f-ùv1 A-t. i c: --jj D r-&/' TSc. Æt c: --rl tj plc) L- -'fue Lvf "A!CD It IHl,f/) ti.,.t(' frla-rrz Q A A. L- ;2' L i.o., -l x T D -l( is- /Lo-1 -, L P L l) "' P,/ '-l ri tj D/l v1rll F L 0 r ( (". l) r 7P (i (v'l:rf UO 12(1 t:,'es II Il "" f: ff 6(Y /' 0 Ç'l LL- S A--- VL ç, 16

17 I r- I r' \ Problem 5.4 (cont.) (, ), "" v i 'I, 0 i 1-, Q N t (' íj f\j F -l \( \i l Ie Ii (Ó C' N + ( tl 10. \. ),j r- " t- - It" N t:x N Q t", y. )" l- '- t- t C1 t) r- -- \I j t' I 't r; 'j "- \. Y- t c: " i l=- Y. ) \ " -. )- x ')( N " i 00)( 0J C' f' J-- l" II. Ij + C" 1\ II 'X ii t- j L /(2 r-)- /' h.)- N "1 C- (" -- 'V IJ x: 'f I. )v. t'. ll t'l) Ii (' f:.. "' l' 1- J- \ 'X t; \J,. c- cv r" ". N J- I t; 1-' I "*\ '" 'J y. rl li r. Ç: '- v '-, II '- N N Ii Ii..r IJ ti L -- L t ii f - 'tnc:n. '-. (\ 1 ( C- r- \ l': c.\ li lt -\ r-, tv (I )- \ ),i :t -i I: 3.3 '- () i ) V\ c l\ t\ 17

18 Problem 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. 18

19 Problem 5.7 I- I " II \ t- "- ;.Q \ \ t'" tl (I \ i i c: \t II -- Ii S- rj L -- II 'ì \ \ C: 0 'te- t'- I:. 0 \ '- \:- *. t- VI 0 tj II LI.-C t' \3 -. N '. t: '" v li C-,,- ê- \ L 'X CJ b Q) -C -C *- -IN li (I ). "" \ -i l 0 '\ 1I \ H '0 \J II l-c CJ 'f II. (i= ::.k f 19

20 Problem 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. 20

21 \ b. U \, i l\ (I "" j r:s '\ ') L ' Problem 5.8 Ii c: "" J " 'J '-. 0 e () J ct 'v J ( -s VI "- oj.t - l- : S :Lt' \1 I. r- (' j J (I r 1 J ; J G " "- '- -- "- "- oj )- -r '- ( _ 'l t '0 b -t' _--. \C.-- b N ('\ j v v -t t N\ I" \. I' i j i \J -ili II J 0! l' (; \( '" J' :: J t.-. - "e L ::! -\t ij l tj \l 'X ) J 'i \\ l '- '- '- '- o II -t l- c; 21

22 Summary Two special examples of vorticity: solidbody 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

APPH 4200 Physics of Fluids

APPH 4200 Physics of Fluids 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