University of Southern California Dissertations and Theses

On the spin-up and spin-down of a contained fluid

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On the spin-up and spin-down of a contained fluid

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Content O N T H E S P IN -U P A N D S P IN -D O W N O F A C O N T A IN E D F L U ID by P a tr ic k D an W eid m an A D is s e rta tio n P re s e n te d to the F A C U L T Y O F T H E G R A D U A T E S C H O O L U N IV E R S IT Y O F S O U T H E R N C A L IF O R N IA In P a r t ia l F u lfillm e n t of the R e q u ire m e n ts fo r the D e g re e D O C T O R O F P H IL O S O P H Y (A ero s p a c e E n g in e e rin g ) June 1973 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA TH E GRADUATE SCHOOL U N IV E R S ITY PARK LOS ANGELES, C A LIF O R N IA 9 0 0 0 7 This dissertation, written by Patrick Dan Weidman under the direction of h.X&... Dissertation Com mittee, and approved by a ll its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R O F P H IL O S O P H Y Dean Date.....^^P^.Æ}. _ _ _ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h is m a n u s c rip t is ded icated to m y g ro o vy p aren ts M e r le and O ra 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS The author is thankful to Tony M a x w o rth y fo r pro vid in g a r e s e a rc h e n v iro n m en t unencum bered by d e ta ile d g o a ls. D iscu ssions w ith h im and F r e d e ric k B row and on the e x p e rim e n ta l aspects of this w o rk w e re alw ays h e lp fu l, e s p e c ia lly w ith re g a rd ito the in te rp re ta tio n of the s ta b ility m e a s u re m e n ts . G uidance p e rta in in g to the m a th e m a tic a l develo pm ent was provided by B . A . T ro e s c h and L . R edekopp, and p a rtic u la r thanks a re given to R . E d w ard s fo r his constant en co u rag e m e n t and often d ire c t help w ith the th e o ry . The author a ls o w ishes to exp ress his g ratitu d e to C . W in an t, T . D eg lo w , and G. K oôp, each of w hom took an in te re s t in a p a rtic u la r fa c e t of the re s e a rc h . Thanks a re due to M . F lin k fo r con structing m o s t of the e x p e rim e n ta l ap p aratu s. F in a lly , s p e c ia l acknow ledgem ent is acco rd ed A . B le e k e r fo r his p a tie n t assistan ce d uring the long hours of data a c q u is itio n . The m a n u s c rip t was typed by C . L e s lie . T h is re s e a rc h was funded by the N a tio n a l Science Foundation u nd er C o n tra c t N o. G K 19107. I l l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A B S T R A C T A n a n a ly tic a l and e x p e rim e n ta l in v e s tig a tio n is m ade of the flu id m o tio n in sid e a u n ifo rm ly a c c e le ra tin g (o r d e c e le ra tin g ) c y lin d e r. I t is m a th e m a tic a lly d e m o n s tra te d th a t an e x a c t f i t to n u m e ric a l com putations fo r the E k m a n m ass flu x w ill re s u lt in a v e lo c ity d is c o n tin u ity a t the w ave fro n t in the in v is c id equation fo r im p u ls iv e s p in -u p . L a s e r D o p p le r m e a s u re m e n ts of the a z im u th a l v e lo c ity n e a r the w ave fro n t in d ic a te th a t the d is c o n tin u ity is 1 sm oothed out in an 0 ( E ‘^) s h e a r la y e r , and aw ay fr o m the w ave fro n t th e o re tic a l and e x p e rim e n ta l re s u lts a re in acco rd . In the case of sp in -d o w n , the m e a s u re d v e lo c ity p ro file s a re in e x c e lle n t ag re e m e n t w ith th e o ry w hen th e flo w is stab le. F lo w v is u a liz a tio n studies have m ade p o s sib le the d e te rm in a tio n of s ta b ility bound a rie s fo r both the s p ir a l E km an w aves observed a t the c y lin d e r end w a lls d uring sp in -u p , and the G o r tle r v o rtic e s w h ic h ap p eared along the c y lin d ric a l w a ll d u rin g spin-dow n. A n a n a lys is of these re s u lts suggests th a t th e end w a ll in s ta b ilitie s a re c h a ra c te ris tic of Type n E k m a n w aves; the a x ia l v e lo c ity d is trib u tio n n e a r the c y lin d ric a l w a ll is found to be p a r tic u la r ly im p o rta n t in d e s c rib in g the m e chans im fo r in s ta b ility d u rin g spin-dow n. I V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS D E D IC A T IO N A C K N O W L E D G E M E N T S A B S T R A C T T A B L E O F C O N T E N T S L IS T O F T A B L E S L IS T O F F IG U R E S I IN T R O D U C T IO N I I T H E O R E T IC A L A S P E C T S 1. D e fin itio n s . 2. L ite r a tu r e S u rv e y . 3. D e riv a tio n of E q u atio n s . 4 . C u rv e F ittin g R o g e rs and L a n c e 's D a ta . 5. Im p u ls iv e S p in -U p . 6. S p in -U p at C onstant A c c e le ra tio n . 7. Im p u ls iv e S p in -D o w n . 8. S p in -D o w n a t C onstant D e c e le ra tio n . 9. B o undary L a y e rs at th e C y lin d ric a l W a ll, i n E X P E R IM E N T 1. In tro d u c tio n to L a s e r D o p p le r A nem om ettry. P age i i i i i iv V v i ii ix 1 4 4 5 9 15 19 31 39 42 47 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TA BLE OF CONTENTS (CONT'D) 2. The S c atterin g P ro c e s s . 54 3. M odes of O p e ra tio n . 55 4. The D o p p le r S h ift E q uation. 58 5. H e tero d yn e S ig n al O p tim iza tio n . 60 6. P a ra m e te rs A ffe c tin g th e S ig n al N o is e and P o w e r. 62 7. V e lo c ity M e a s u re m e n ts of a Solid. 68 8. V e lo c ity M e a s u re m e n ts in a L iq u id . 74 I V F L O W V IS U A L IZ A T IO N A N D S T A B IL IT Y 90 1. P r e lim in a r y D iscu ssio n . 90 2. F lo w V is u a liz a tio n of S p in -U p fro m R est, 91 3. E k m a n In s ta b ilitie s . 95 4. G B rtle r In s ta b ilitie s . 102 5. W ave F r o n t P o s itio n fo r S p in -U p fro m R est. I l l V D IS C U S S IO N O F R E S U L T S 113 1. T h e o ry V e rs u s E x p e rim e n t. 113 2. F lu id S ta b ility fo r S p in -U p and S pin-D ow n. 121 V I S U M M A R Y A N D C O N C L U S IO N S 127 V n A P P E N D IC E S 131 A . T H E D O P P L E R F R E Q U E N C Y S H IF T E Q U A T IO N 131 B . S IG N A L A N D N O IS E E Q U A T IO N S 138 V I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS (CONT’D) C . L E N S F O C A L L E N G T H M E A S U R E M E N T S 142 D . E R R O R A N A L Y S IS 149 E . O P T IC A L R A Y T R A C IN G C A L C U L A T IO N S 159 m n R E F E R E N C E S 166 IX T A B L E S 172 X F IG U R E S 174 V ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES T ab le IV . 1 T ab le IV . 2 T a b le IV . 3 T a b le D . 1 T ab le D . 2 T a b le D . 3 T a b le D . 4 T a b le IX . 1 O bservatio n s of tu rb u len ce in the E k m a n la y e r N o ndim ensio n al w avelengths n ea r the S ta b ility boundary A s u m m a ry of E k m a n boundary la y e r m easu rem en ts Synopsis of conditions corresp o n d in g to F ig u re X . 10 Synopsis of conditions co rresp o n d in g to F ig u re X . 11 Synopsis of conditions co rresponding to F ig u re X . 12 Synopsis of conditions co rresponding to F ig u re X . 16 E k m a n suction calcu latio n s due to R ogers and Lance [ s ] fo r v a rio u s values of s = U)/0 and C T = n/u) T ab le IX . 2 C y lin d e r dim ensions and toleran ces Page 94 99 100 156 156 156 157 173 173 V lll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF f i g u r e s : Page F ig u re I I . 1 Sketch of the com pu tations due to R o g ers and Lan ce fo r the E k m a n suction. 13 F ig u re I I . 2 C h a ra c te ris tic paths and b ou ndary valu es fo r im p u ls iv e s p in -u p . 21 F ig u re I I . 3 Sketch showing the valu es of s fo r w h ich c h a ra c te ris tic paths w il l in te rs e c t d u rin g im p u ls iv e s p in -u p . T h e m a x im u m in the c u rv e is lo cated at s^. 28 F ig u re I I . 4 C h a ra c te ris tic paths and b o u n d ary valu es fo r sp in -u p at constant a c c e le ra tio n . 32 F ig u re I I I . 1 S ch em atic of L D V in th e d e te c to r optics m o d e. 56 F ig u re I I I . 2 S e lf-fo c u s in g L D V set up in (a) the re fe re n c e b e a m m ode o r (b) the d iffe r e n tia l D o p p le r m o d e. 57 F ig u re I I I . 3 Sketch of the s e lf-fo c u s in g (c ro s s -b e a m ) la s e r optics used fo r m e a s u rin g the a z i m u th a l v e lo c ity o f a ro ta tin g In c ite w h e e l. 69 F ig u re I I I . 4 C onstant a c c e le ra tio n c o n tro l u n it. 77 F ig u re A . 1 A s ta tio n a ry sou rce o f ra d ia tio n as seen by a m oving p a r tic le . 132 F ig u re A . 2 A m oving source of ra d ia tio n as seen by a s ta tio n a ry o b s e rv e r. 134 F ig u re C . 1 T op v ie w of o p tic a l s e t-u p fo r lens fo c a l m e a s u re m e n ts . 142 F ig u re C . 2 C o o rd in ate sys te m fo r th e fo c a l p o sitio n m e a s u re m e n ts . 143 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES (Cont'd) P age F ig u re C . 3 F o c a l depth v a ria tio n a cro s s the lens d ia m e te r as d e te rm in e d by R e s e c tio n M eth od I . 146 F ig u re C .4 F o c a l point v a ria tio n s in the plane as d e te rm in e d by R e s e c tio n M ethod I . 147 F ig u re C . 5 F o c a l depth v a ria tio n s acro s s th e lens d ia m e te r as d e te rm in e d by R e sectio n M eth od I I . 148 F ig u re D . 1 T h e functio ns (a) ng), (b) A ( 9 .,n ), and (c) B (9 ., ng) fo r use in th e e r r o r a n a ly s is of the ro ta tin g disc e x p e rim e n t. 158 F ig u re D . 2 The 1 /e p ro b e v o lu m e . 152 F ig u re D . 3 L o c atio n o f p ro b e vo lu m e fo r m e a s u re m e n ts a t r / a = 0. 975 . 154 F ig u re E . 1 R ay tra c in g d ia g ra m fo r a s p h e ric a l r e f r a c t ing s u rfa c e . 160 F ig u re E . 2 C onstants used in th e ray tra c in g calcu latio n s fo r the c ir c u la r c y lin d e r. 162 F ig u re E . 3 F o c a l p oin t p o s itio n in sid e th e c y lin d e r as d e te rm in e d by ra y tra c in g th e o ry . 165 F ig u re X . 1 V e lo c ity p ro file s fo r sp in -u p at constant a c c e le ra tio n as com puted fro m equations (11.60) and (n . 61 ). 175 F ig u re X . 2 A n g u lar v e lo c ity of the in te r io r flu id fo r spin-dow n a t constant d e c e le ra tio n as com puted fr o m equations (II. 90) and (II. 9 1 ). 176 F ig u re X . 3 O scillo sc o p e and s p e c tru m a n a ly z e r d is p la y s of the D o p p le r s ig n a l (Vp a 60 K H z ). 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES (Cont'd) F ig u re X , 12 S p in-up fr o m r e s t fo r = 2 . 34 X 10 188 F ig u re X . 13 S pin-dow n to r e s t fo r E ^ = 2 5 .4 X 10 189 F ig u re X . 14 Spin-dow n to re s t fo r E ^ = 1 9 .9 X 10 190 F ig u re X . 15 Spin-dow n to r e s t fo r E = 1 3 . 8 X 1 0 ^ . 191 P age F ig u re X . 4 V e lo c ity m e a s u re m e n ts of the ro ta tin g In c ite w h e e l. 178 F ig u re X . 5 S ch em atic d ia g ra m of the ap p a ra tu s fo r flu id v e lo c ity m e a s u re m e n ts in s id e the ro ta tin g c y lin d e r in clu d in g th e L a s e r O p tic s , the M o to r D r iv e S y ste m , the R . P . M . In d ic a to r S ystem , and the D o p p le r S ig n a l E le c tro n ic s . T h e lis t of com ponent p a rts is given in the fo llo w in g tw o p ag es. 179 F ig u re X . 6 S ignal p ro c e s s in g e le c tro n ic s fo r the R .P . M . In d ic a to r S y ste m . 182 F ig u re X . 7 S e lf-fo c u s in g la s e r optics fo r th e flu id v e lo c ity m e a s u re m e n ts . 183 F ig u re X . 8 P lo t of the D o p p le r fre q u e n c y Vj^ as a functio n o f th e lens p o s itio n L fo r solid body ro ta tio n of the flu id . 184 F ig u re X . 9 S p in -u p fro m re s t fo r E ^ = 27. 10 X 10 185 F ig u re X . 10 S p in-up fro m re s t fo r E ^ = 1 1 .7 5 X 10 186 -6 F ig u re X . 11 S p in -u p fro m r e s t fo r E = 8 .7 4 X 10 . 187 F ig u re X . 16 S pin-dow n to r e s t fo r E ^ = 9. 92 X 10 192 F ig u re X . 17 V e lo c ity p ro file s co rres p o n d in g to F ig u re X . 15. 193 x i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES (Cont'd) Page F ig u re X , 18 V e lo c ity p ro file s co rresp o n d in g to F ig u re X . 16. 194 F ig u re X . 19 A c c e le ra tio n fro m one an g u la r v e lo c ity to ano ther at s m a ll R ossby n u m b e r fo r E = 1 6 .2 X 1 0 -6 . 195 a F ig u re X . 20. D e c e le ra tio n fro m one an g u la r v e lo c ity to ano ther at s m a ll R ossby n u m b er fo r E ^ = 1 5 .7 5 X 1 6 -6 . 196 F ig u re X . 21 Photographs o f the E km an in s ta b ilitie s fo r spin-up fro m re s t a t Eq^ = 7. 64 X 10~6 . T h e re a r e th re e -s e c o n d in te r v ^ s betw een fra m e s ; fra m e 2 co rresp o n d s to t 6: 0 and fra m e 31 correspon d s to t at 1. 0 . 197 F ig u re X . 22 P rin ts of A e d e ta ile d E k m a n flo w at (a) t a 0. 18, (b) t 0. 43, (c) t ~ 0. 79, and (d) t ~ 1 .0 4 . 198 F ig u re X . 23 O bservatio n s of E km an s ta b ility and tra n s itio n to tu rb u le n c e . 199 F ig u re X . 24 T im e fo r the f ir s t app earance of G o rtle r v o rtic e s as d e te rm in e d fro m the flow v is u a liz a tio n o b s e rv a tio n s . 200 F ig u re X . 25 P lo t of the c r itic a l G b rtle r s ta b ility p a ra m e te r as a functio n of the E k m a n n um ber fo r the flo w v is u a liz a tio n o b servatio n s given in F ig u re X . 24. 201 F ig u re X . 26 T im e develo pm ent of th e m o m en tu m thickness and G B rtle r s ta b ility p a ra m e te r corresponding to the v e lo c ity p ro file s of F ig u re s X . 17 and X . 18. 202 F ig u re X . 27 O bservatio n s of the p o s itio n of the w ave fro n t d u rin g sp in -u p fo r E^^ = 1 9 .6 7 X 10"6. 203 x ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES (Cont'd) P age F ig u re X . 28 O b servatio n s o f the w ave fro n t d u rin g sp in - up fo r E = 11.18X 10"® . 203 0. F ig u re X . 29 O b servatio n s of the p o s itio n of th e w ave fro n t d u rin g sp in -u p fo r E ^ = 7. 64 X 10"^. 204 F ig u re X . 30 O b servatio n s of the p o s itio n of the w ave fro n t d u rin g sp in -u p fo r E ^ = 2 .12 X 10"^. 204 F ig u re X . 31 T h e in te r io r flu id ro ta tio n a l v e lo c ity tUj as a function of the w a ll ro ta tio n a l v e lo c ity Q at the f ir s t ap p earan ce of th e G o rtle r c e lls . 205 F ig u re X . 32 T h e c r itic a l v e lo c ity ra tio ( ^ j/^ ) c r it the f ir s t app earance of the G o r tle r c e lls as a function o f the E k m a n n u m b e r. 206 F ig u re X . 33 T h e re s id u a l sp in -u p tim e (tg -t^ ) as a function o f the no n d im en sio n al tim e re q u ire d fo r the c y lin d e r to a tta in its fin a l angular v e lo c ity . 207 F ig u re X . 34 C o rre la tio n s of the sp in -u p tim e fo r la rg e E k m a n n u m b e r. 208 F ig u re X . 35 C o rre la tio n s of the sp in -u p tim e fo r s m a ll E k m a n n u m b e r. 209 X lll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h ap ter I IN T R O D U C T IO N The w o rk in this tre a tis e is concerned w ith the unsteady d yn am ics of a flu id contained in a ro ta tin g c y lin d e r. T h e o re tic a l analyses can be found in the lite r a tu r e fo r both the lin e a r [ l ] and n o n lin e ar [ 2 , 3,4, 5] tre a tm e n t of the spinQup p ro b le m . A p p a re n tly , h o w e ve r, th ere is a paucity of corresponding e x p e rim e n ta l d ata, even fo r the sim p le g eo m e try of a c y lin d e r. One notable exception is the u n published w o rk of In g e rs o ll and V e n e zia n [ 6 *j. The d iffic u lty a ris e s fro m the fa c t th at conventional con tact probes s ig n ific a n tly d is tu rb the flu id m otion; in fa c t, probes in s e rte d in flow s w ith la rg e s w ir l and low a x ia l v e lo c ity w ill sense distu rb an ces due to th e ir own w ake w hich d isru p ts the flow one w ishes to m e a s u re . The noncontact la s e r D o p p ler v e lo c im e te r (L D V ) probe was of p a rtic u la r in te re s t as a tool fo r geo physical e x p e rim e n ta l p ro je c ts being developed in the A e ro sp ac e E n g in e e rin g D e p a rtm e n t under the d ire c tio n of D r . Tony M a x w o rth y . B e fo re applying the L D V to a la rg e scale e x p e rim e n t, we decided to s e t up a s m a ll-s c a le flu id dynam ics system in o rd e r to obtain p ra c tic a l firs t-h a n d in fo rm a tio n on its o p e ra tio n . Consequently* an enclosed c y lin d ric a l c o n ta in e r con structed of c le a r p la s tic w a lls was fille d w ith w a te r and ro ta te d Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by a v a ria b le speed m o to r. W ith th is s im p le ap p aratu s w e w e re in a p o s itio n to re a c h tw o o b je c tiv e s . T h e f ir s t goal w as to get the L D V in w o rk in g o rd e r and m a k e m e a s u re m e n ts o f s im p le s o lid body r o ta tio n , c o v e rin g the v e lo c ity ran g es one w ould expect to encounter in a la rg e scale e x p e rim e n t; th e second and m o re in te re s tin g o b je c tiv e w as to obtain m e a s u re m e n ts o f th e tra n s ie n t v e lo c itie s d u rin g sp in -u p \ since the L D V seem ed p e rfe c tly suited fo r th is p ro b le m . T h e body of th e th esis is s e p arated into fiv e m a jo r ch a p te rs ; th e o ry , e x p e rim e n t, flo w v is u a liz a tio n , d iscu s sio n , and conclusion. In C h a p te r I I a s u rve y is m ad e of the e x is tin g th e o rie s w ith s p e c ia l a tten tio n placed on the w o rk of W e d e m e y e r [ s ] w h ic h , though a p p ro x i m a te in n a tu re , p ro v id e s a fra m e w o rk w h ic h takes in to account the fu ll n o n lin e a rity of the sp in -u p p ro b le m . A n im p o rta n t lim ita tio n of W ed e m e y e r*s th e o ry is pointed out, and his ideas a re extended to the p ro b le m of spin-dow n fr o m so lid body ro ta tio n . C h a p te r I I I d eals w ith the e x p e rim e n ta l en d eavo r, in clu d in g a d iscu ssio n on the th e o ry of o p e ra tio n o f la s e r D o p p le r m e a s u re m e n t s y s te m s . Som e p r e lim i n a ry v e lo c ity m e a s u re m e n ts of a ro ta tin g d isc and a flu id in solid body ro ta tio n a re p re s e n te d . F in a lly , w e p ro v id e som e unsteady flu id flo w m e a s u re m e n ts fo r sp in -u p and spin-dow n in the ro ta tin g c y lin d e r. F lo w v is u a liz a tio n studies of th e tra n s ie n t flo w ^ p re s e n te d in C h ap ter IV , point out the existen ce of both E k m a n [ 4 ] and G o rtle r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 [ 4 , 3 l ] in s ta b ilitie s . T h e s ta b ility b o u n d ary fo r th e observed E k m a n w aves on the c y lin d e r end p la te s is d e te rm in e d fo r a la rg e range of n eg ative R ossby n u m b e rs . A n a n a ly s is of the G o r tle r v o rtic e s observed d u rin g spin-do w n points out the p o s sib le im p o rta n c e of a Ludw ieg [ 4 6 ] in s ta b ility . C h a p te r V p re s e n ts a d iscu ssio n of the re s u lts ; the e x p e rim e n ta l d ata obtained fro m the L D V a re co m p ared to the th e o ry developed in C h ap ter I I . E m p h asis is p laced on the r e lia b ility o f the th e o ry a n d /o r the a c c u ra c y o f th e m e a s u re m e n ts , p a r tic u la rly w hen th e re is a d is a g re e m e n t b etw een the tw o . T h e concluding re m a rk s a r e p rese n te d in C h a p te r V I. In o rd e r to aid the re a d e r, w e have often in clu d ed fig u re s and illu s tra tio n s w ith in the te x t. F ig u re s p e rta in in g to the e x p e rim e n ta l and th e o re tic a l re s u lts , h o w e v e r, a re given in C h ap ter X in o rd e r to not crow d and obscure the m a in d ev elo p m en t. T h e appendices, b ib lio g ra p h y , and ta b le s a r e p rese n te d in C h a p te rs V I I , V I I I, and IX , re s p e c tiv e ly . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h ap ter I I T H E O R E T IC A L A S P E C T S 1. D e fin itio n s In te rm s of the ja rg o n c u rre n tly em ployed , w e a r e concerned h e re w ith the flu id d y n a m ic a l p ro b le m o f sp in -u p . In the re c e n t l i t e r a tu re "s p in -u p " has been d efin ed ra th e r lo o sely as the p ro ce ss by w h ich a state of rig id ro ta tio n of a contained flu id is estab lish ed , e ith e r by an im p u ls iv e in c re a s e o r d e c re a s e in the an g u la r v e lo c ity of the c o n ta in e r. T h is d e fin itio n is , in one sense, r e s tr ic tiv e b e cause one is only d ea lin g w ith im p u ls iv e v e lo c ity changes of the boundary; in ano ther sense it is too g e n e ra l because it does not d istin g u ish betw een the in h e re n tly d iffe re n t n a tu re (e. g. , s ta b ility ) of the tra n s ie n t m otions obtained by an in c re a s e as opposed to a d e c re a s e of th e an g u lar v e lo c ity of the solid b o u n d a rie s . F o r the purposes of th is te x t, th e r e fo r e , w e s h a ll use the t e r m "sp in -d o w n " as w e ll as sp in -u p . S p in -u p w ill be defined as the p ro cess by w h ich a contained flu id ach ieves a fin a l state of solid body ro ta tio n , e ith e r fr o m a s m a lle r in itia l state of solid body ro ta tio n o r fro m re s t. L ik e w is e , spin-dow n w ill r e fe r to the d e c e le ra tio n of the flu id by d e c re a s in g the system fr o m an in itia l state of u n ifo rm an g u la r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v e lo c ity e ith e r to a fin a l state of solid body ro tatio n o r to re s t. Note th at the above d e fin itio n s of sp in -u p and spin-dow n do not p lace any re s tric tio n s on the m a n n e r in w h ic h the solid bou ndaries a tta in th e ir fin a l state. Thus one can ta lk about im p u ls iv e sp in -u p , sp in -u p at constant a c c e le ra tio n , im p u ls iv e spin-dow n, etc. 2. L ite ra tu r e Survey T h e f ir s t m a jo r w o rk on the sp in -u p p ro b le m w as p rese n te d by G reenspan and H o w ard [ l ] , h e re a fte r denoted G & H . T h e y co n sid er ed s m a ll changes about a state of s o lid body ro ta tio n of a contained flu id ; if the in itia l an g u la r v e lo c ity w as Q then the fin a l state was (l+ e)Q , w h e re e is a s m a ll p a ra m e te r. T h is affo rd e d a lin e a riz a tio n of the equations of m o tio n w h ic h th ey proceeded to solve by the L ap lac e tra n s fo rm m ethod. It w as shown th a t the sp in -u p tim e of 2 i the lin e a riz e d p ro b le m is given by (L w h e re L is th e c h a r a c te ris tic d im en sio n along the axis of ro ta tio n and V is the k in e m a tic v is c o s ity of the flu id . T h e ir a n a ly s is p ro v id e s the fo llo w in g p h y s ic a l d e s c rip tio n of 2 spin-up fo r s m a ll E k m a n n u m b er E = v /O L . C o n sid er a c y lin d e r of flu id w h ic h is given a s m a ll im p u ls iv e in c re a s e in its an g u la r v e lo c ity . The im m e d ia te re s u lt is a shear la y e r w h ic h fo rm s on the top and botto m d iscs and th icken s by viscous d iffu s io n to fo r m q u a s i steady E k m a n la y e rs . A ls o th e re e x is t in e r tia l o s c illa tio n s w ith a fre q u en c y 2 0 , but th ey a r e of v e ry s m a ll am p litu d e and have v ir tu a lly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 no e ffe c t on the sp in -u p p ro c e s s . The E k m a n la y e rs on th e d iscs cen trifu g e flu id r a d ia lly o u tw ard thus inducing a s m a ll v e r tic a l suc tio n into the boundary la y e rs fro m the geo strop hic in te r io r . C o n s e r vatio n of m ass n ec essita te s an eq u ally s m a ll r a d ia l in flu x in the flu id in te rio r to balance the outw ard r a d ia l tra n s p o rt in th e E k m a n bound- 1 a ry la y e rs . T h e re s u ltin g secondary m o tio n is o f o rd e r E^ s m a lle r than the an g u la r v e lo c ity . Since the in te r io r flo w is e s s e n tia lly in v is ç id , the a n g u la r m o m e n tu m of a rin g of flu id m u s t be co n served. C onsequently, the a n g u la r v e lo c ity of the in te r io r flu id is in c re a s e d as it m oves ra d ia lly in w a rd to re p la c e flu id lo s t to the E k m a n bound a ry la y e rs . A lte rn a te ly , the flu id v o rtic ity is in c re a s e d b y the stretc h in g of v a r te x fila m e n ts u n til the steady state v e lo c ity (l+e)fl is reached at w h ich tim e the E k m a n boundary la y e rs d ecay. T h e d im en sio n less tim e scale is of o rd e r E how ever the in e r tia l -1 o s c illa tio n s p e rs is t u n til a la te r tim e of o rd e r E . B y th is tim e the E k m a n boundary la y e rs have d iffu se d in to the in te rio r so th a t the flo w is no lo n g er in v is c id ; th e re m a in in g in fe rtia l o s c illa tio n s a re then d estro yed by viscou s e ffe c ts . W ith tim e n o n d im en s io n alized by the ro ta tio n a l v e lo c ity Q, the tra n s ie n t response to a s m a ll im p u ls iv e change in the c y lin d e r ro ta tio n ra te can be s u m m a riz e d as fo llo w s : T h e E k m a n bou ndary la y e rs and s m a ll am p litu d e m o d al o s c illa tio n s fo r m in a tim e (}(1); the sp in -u p pro cess pro ceed s on a tim e scale 0 (E ^); fin a lly the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. viscou s decay of the in e r tia l o s c illa tio n s by th e d iffu sed b ou ndary la y e rs continues to a tim e 0 (E ^). T h e re e x is t, of c o u rs e , tra n s ie n t b ou ndary la y e rs on the c y lin d ric a l side w a lls , but th e ir p re s e n c e has no im p o rta n t e ffe c t on th e m a in fe a tu re s of th e sp in -u p phenom enon. T h e lin e a riz e d a n a ly s is of G & H w as extended b y G reen sp an and W ein b au m [ z ] to include n o n lin e a r e ffe c ts by an expansion in pow ers of s, the R ossby n u m b e r. T h e con clu sion of th e au th o rs of th is p ap er (h e re a fte r denoted G & W ) w as th a t the n o n lin e a r in te ra c tio n s >are3 ra th e r w e a k w hen co m p a re d w ith th e b a s ic visco u s p ro ce sses in the R o ssby n u m b er range < e < %. T h e ir re s u lts fo r a s p h e ric a l con ta in e r show only a s m a ll d iffe re n c e fro m th e lin e a r th e o ry , except in the re g io n n e a r th e equ ato r w h e re the flu id spins up som ew hat fa s te r fo r la rg e e . T h e p r im a r y re s u lt is th en th a t th e n o n lin e a r in te r actio n s fo r m o d e ra te R o ssby n u m b e r does not a lte r th e tim e scale fo r sp in -u p as d e te rm in e d by th e lin e a riz e d th e o ry . It should be noted th a t the a n a ly s is of G & W is not u n ifo rm ly v a lid n e a r a v e r tic a l bou ndary since th e ir p e rtu rb a tio n expansion is based on a n o n d im en - J L s io n al boundary la y e r th ic k n e s s of o rd e r and thus does not a d m it 1 /3 - the p o s s ib ility of unsteady S te w a rt son la y e rs o f o rd e r E and E * w h ich w e re discussed in th e lin e a riz e d tre a tm e n t of G & H . A n e n tire ly d iffe re n t a n a ly s is w as g iven b y W e d e m e y e r [ 3 ] who co n sid ered the fu lly n o n lin e a r p ro b le m of th e sp in -u p fro m re s t of a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 flu id in a c y lin d ric a l c o n ta in e r. W ith an a p p ro x im a te a n a ly s is o f th e E k m a n boundary la y e rs he w as ab le to fo rm u la te a single n o n lin e a r p a r tia l d iffe re n tia l equation fo r the a z im u th a l v e lo c ity . W ith an a d d itio n al assum ption, e s tim a te d to in c u r an e r r o r on th e o rd e r of 15 p e rc e n t, a closed fo r m so lution in te rm s of exp o n en tial functions could be obtained. Altôiough a p p ro x im a te in n a tu re , W e d e m e y e r's a n a ly s is q u a lita tiv e ly points out the existen ce of a "w ave fro n t" w h ic h o rig in a te s a t the c y lin d ric a l boundary and p ropagates r a d ia lly in w a rd . T h e fro n t sep a ra te s q u iescen t flu id a t s m a lle r r a d ii fro m flu id at la r g e r r a d ii w h ic h has a c q u ire d an g u la r m o m en tu m . G reenspan [ 4 ] , in his book on ro ta tin g flu id s , has extended W e d e m e y e r's w o rk to in clu d e s p in -u p fro m fin ite an g u la r v e lo c ity fo r a r b itr a r y a x ia lly -s y m m e tr ic c o n ta in e rs . In ad d itio n , an e x c e l le n t e x p e rim e n ta l v e rific a tio n of the w ave fro n t phenom enon is given by the photograph in F ig u re 1 .4 of his book. P erh ap s the b est p h y s ic a l in te rp re ta tio n o f th e re s u lts th at can b e obtained fro m the W e d e m e y e r ap p ro ach is g iven by V e n e z ia n [s ]. In a fo llo w in g p ap er V e n e z ia n [ 7] co n sid ered in d e ta il th e flo w in the neighborhood of the w ave fro n t. T h ese w o rk s w ill be discussed in th e th e o re tic a l p re s e n ta tio n of th e th e s is . In the fo llo w in g sections w e s h a ll use the ideas&of W e d e m e y er to develop the equations fo r sp in-dow n as w e ll as s p in -u p , w ith a c r itic a l lo o k a t the assum ptions and th e consequent v a lid ity of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solution s. W e s h a ll then p ro ceed to solve th e equations w ith a«vfew a p p ro x im a tio n s as p o s s ib le , using the co m p u ter w hen n e c e s s a ry . In th is w a y one can perhaps d e te rm in e i f W e d e m e y e r's r e la tiv e ly s tra ig h tfo rw a rd app ro ach can p ro v id e m o s t of the d e ta ils of th e flo w , o r if one m u s t re s o rt to a p e rtu rb a tio n technique s im ila r to th a t of G & W . A lthough the tre a tm e n t is a p p lic a b le to g e n e ra l a x ia lly - s y m m e tric c o n ta in e rs , w e s h a ll co n cern o u rs e lv e s w ith only the rig h t c ir c u la r c y lin d e r. In a d d itio n , s p e c ia l a tte n tio n w ill be focused on solutions in w h ich the c y lin d e r ach ieves its new an g u la r v e lo c ity b y a constant a c c e le ra tio n o r d e c e le ra tio n , since thèse cases a re perhaps the m o st p h y s ic a lly re le v a n t. A n im p u ls iv e change can then be thought of as the lim itin g case of a la rg e an g u la r a c c e le ra tio n . 3. D e riv a tio n of Equations W e s h a ll use an in e r tia l re fe re n c e sys te m w ith the o rig in lo cated a t the c e n te r of the c y lin d e r. T h e n a tu ra l co o rd in ates a re c y lin d ric a l r ( r , 0, z) and the re s p e c tiv e v e lo c ity com ponents w ill be denoted by u(u, V , w ). T h e v e r tic a l z -a x is is alig n ed along th e g e o m e tric a l axis of the c y lin d e r, w h ic h its e lf is co in c id e n t w ith the ro ta tio n v e c to r 0 . T h e fla t lid s of the c y lin d e r a re lo cated a t z = ± h /2 and the c y lin d r i c a l b ou ndary is at r = a . W e r e s tr ic t o u rs e lv e s to the case E « 1 2 w h e re E = v /flh is the a p p ro p ria te ly d efin ed E k m a n n u m b e r. Since _! the E k m a n la y e r thickness 6 is equal to (v /0 )^ , w e conclude th a t 6 /h « 1 . Thus the h o riz o n ta l b ou ndary la y e rs a r e confined to a th in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 reg io n n e a r the top and b o tto m lid s . W e sh all now fo rm u la te the equations governin g th e flo w in the in te r io r reg io n , i . e . , the re g io n aw ay fro m th e E k m a n la y e rs on the h o riz o n ta l bou ndaries and th e shear la y e rs on the v e r tic a l c y lin d r i- c a l.w a ll. The N a v ie r-S to k e s equations fo r an in c o m p re s s ib le flo w w ith a x ia l s y m m e try a re given by ôu 3u 3u v^ Ô or oz r or \ 2 / ' r ' ^ + (U ,2 ) Ov 9v + + + (11,3) Ot Or Oz Oz w h e re = and the con tinuity equation is Since the secondary v e lo c itie s u ,w w e re shown in G & H to be of o rd e r s m a lle r than the a z im u th a l v e lo c ity v , the fo reg o in g equa tions can be g re a tly s im p lifie d . T h e re s u ltin g equations, c o rre c t to o rd e r E (cf. [ é ] , pp. 1 6 2 -1 6 3 ), a re then I ? = T ’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 I î + “ ( l 7 + r ) ôp â f = ® • ( n .7 ) Equations (II. 4 ) - (II. 7) w ith a p p ro p ria te bou ndary .conditions a re s u ffic ie n t to d e s c rib e the O '(l) in te r io r m o tio n . F r o m equation (II. 7) w e see th a t the p re s s u re is independent of the v e r tic a l co o rd in ate z and Mence p = p ( r ,t ) o n ly. T h en (II. 5) gives v = v ( r , t ) and equation ( II. 6) reduces to W e now need an a d d itio n a l re la tio n s h ip betw een u and v to co m p lete the p ro b le m fo rm u la tio n . A n a p p ro x im a te re la tio n s h ip fo r spin -u p fr o m re s t w as given by W e d e m e y e r; th is a p p ro x im a tio n w ill now be g e n e ra liz e d to include the case of spin-dow n. F r o m the re s u lts of G & H w e know th a t in response to a change in w a ll speed, the E km a n la y e rs grow in a tim e 0 ^ and thus the boundary la y e rs fo rm "in s ta n tly " on the sp in -u p tim e scale o r o rd e r 1 /E ^ n . T h e re fo re , except fo r the f ir s t few re v o lu tio n s , the E km an suction w ^ (r) a t the edge of the boundary la y e rs can be used as the boundary condition im posed on the in te r io r flu id . W e co n sid er a m ass balance a cro s s a c y lin d ric a l sheet of height h a t a ra d iu s r fr o m the v e r tic a l a x is . W e s h a ll r e fe r to th e ra d ia l v e lo c ity in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 b ou n d ary la y e r as u , and th at in th e in te r io r flu id as s im p ly u. T h e equation fo r the m ass b alan ce a t each tim e t can be w ritte n + 2TTrPu(r,z,t)(h-20) = 0 (H .9) ^2TTrP J U g (r, z ,t)d z ^ w h e re the f ir s t te r m in p a ren th e ses re p re s e n ts the flu x in one E k m a n la y e r and hence m u st be m u ltip lie d b y a fa c to r of tw o to b alan ce the re tu rn flu x . Since w e a r e con cerned w ith only E « 1, w e can n eg lec t 26 w ith re s p e c t to h and equ ation (II. 9) can then be w ritte n 6 u ( r ,t ) = tL g (r, z ,t)d z . (11.10) T h a t u is independent of z can also be d e te rm in e d fro m equation (II. 8). R o g ers and Lan ce [s ] have n u m e ric a lly com puted the n o n lin e a r b ou ndary la y e r flo w produced by an unbounded flu id in so lid body ro ta tio n u o above a fla t d is c of ro ta tio n a l v e lo c ity 0 . T h e flu id and w a ll w e re co n sid ered to be of in fin ite r a d ia l exten t so th a t the edge effe cts produced by a fin ite d isc w o u ld not have to be co n s id e re d . V e lo c ity p ro file s w e re obtained fo r e ig h t valu es of s = /Q in the ran g e 0 ^ s ^ 1 and fo r eig h t v alu es o f C J = O/tJu in the range 0 ^ 1. Thus th ey have co n sid ered th e e n tire range of re la tiv e v e lo c itie s fro m the K a rm a n flo w (s = 0 ), fo r w h ic h the d isc is ro ta tin g b elo w a s ta tio n a ry flu id , to the B odew adt flo w {o = 0) w h ic h is the re v e rs e case. T h e ir re s u lts fo r th e E k m a n suction a t the o u ter edge of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 boundary la y e r can be w ritte n in the fo r m x i n 2 f ( s ) | / o ^ s ^ n w j r ) = -v2< > for < ? Iw^g(cJ)/ f 0 ^cr ^ 1/ w h e re the functions f(s ) and g (a) a re obtained by d raw in g a sm ooth c u rv e th ro u g h the n u m e ric a l points given by R & L . T h e n u m e ric a l com putations a re p rese n te d in T a b le IX . 1 and a sketch of the cu rves is p rese n te d in the fig u re b e lo w . W edem ey e r * s ap p r o x im a tio n 0. 8 0 .4 f(s) 0.4 -0 .4 - 0. 8 - 1.2 F ig u re I I . 1. Sketch of the co m p u tatio n s due to Rogers and Lance fo r th e E k -m a n su ctio n . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 F ro m , the equation o f co n tin u ity ( II. 4) i t is easy to show th a t the E k m a n flu x is re la te d to th e E k m a n suction as fo llo w s J u ^ ( r , z , t ) d z = - Y w j r ) . (11.12) Equations (II. 10), ( II. 11), and (II. 12) then p ro v id e the d e s ire d r e la tio n betw een u and v 4 . 1 7 1 . (u)2g(a)) u ( r ,t ) = - — , (11.13) 'g(c): and the equation of m o tio n (II. 8) becom es % a \ \ / ^ X T Tr\ 0 . (II. 14) f y o r r / 'g(C) Solution o f the above n o n lin e a r p a r tia l d iffe re n tia l equation w ith the p h y s ic a lly a p p ro p ria te b ou ndary conditions p ro vid es the 0^(1) in te rio r flo w . W e note th a t in th is a p p ro x im a tio n the bou ndary con ditio n a t the edge of the E k m a n la y e rs shows up as a fo rc in g fu n ctio n fo r the 1. convection te r m . F o r s p in -u p th e e x p re s s io n O^f(s) is used; th is is the equation W e d em ey e r d e riv e d . In the case of spin-dow n the a p p ro p ria te te r m is U)^g(o). T h e fo rm of s and a depends on how the w a ll v e lo c itie s a r e a lte re d . F o r im p u ls iv e changes s and a w ill be functions o f r and v alo n e , but in the event th a t the w a lls a re a c c e le r ated o r d e c e le ra te d at a fin ite ra te th ey w il l a ls o be a functio n of tim e . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 L e t us focus our atte n tio n on the assum ptions im p lic it in the d e riv a tio n of equation (II. 14). W e have taken th e m ass flu x fro m the steady n o n lin e a r solution of R & L and used it to lo c a lly d e te rm in e the flo w in the unsteady p ro b le m . T h e fo rm a tio n o f th e E k m a n la y e rs is v ir tu a lly instantaneous fo r th e values of 0 n e c e s s a ry to s a tis fy the condition E « 1 fo r com m on flu id s and c o n ta in e rs of s u ffic ie n t s iz e . T h e re fo re , the flo w in th is sense is q u a s i-s te a d y in a c c o rd w ith the assu m p tio n . The m o re serio u s asp ect of the a p p ro x im a tio n lie s in the lo c a l n a tu re of its a p p lic a tio n . W e have assum ed th a t the E k m a n flu x depends only on the lo c a l d iffe re n c e betw een the flu id and w a ll v e lo c itie s ; indeed it m ay also depend on h ig h er o rd e r d e riv a tiv e s in regions w h e re the flu id v e lo c ity changes ra p id ly w ith ra d iu s . F o r now, h o w eve r, w e w ill s im p ly p ro ceed to th e solution and then use th e re s u lts to d e te rm in e i f th e o rig in a l assum ptions a re both p h y s ic a l ly and m a th e m a tic a lly co n sisten t. 4 . C u rv e F ittin g R o g ers and L a n c e 's D a ta F o r the case of spin up fro m re s t W e d e m e y e r has fu rth e r s im p lifie d equation (II. 14) by using a s tra ig h t lin e a p p ro x im a tio n fo r the E k m a n flu x . H e con sid ered the lin e a r f it f(s ) = k ( l - s ) , (H . 15) w h e re k = 0. 885 and thus s a tis fie s the end points of the n u m e ric a l com putations of R & L fo r th e case 0 ^ s ^ 1 , as shown in F ig u re I I . 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 T h is w as the b as is fo r d e te rm in in g th a t the e r r o r in his a n a ly s is w as on th e o rd e r of 15 p e rc e n t. G reen sp an [ 4 ] notes th a t o th er authors have em ployed the lin e a r a p p ro x im a tio n , but w ith d iffe re n t v alu es of k . The choice k = 1 corresp o n d s to the E k m a n flu x fo r lin e a riz e d m otion as re p o rte d by Jacobs [9] . A n ex p re s s io n fo r th e c o rre c tio n fro m the lin e a riz e d th e o ry w as d e riv e d b y R & L . T h e ir re s u lt, p resen ted h e re in te rm s of the R ossby n u m b e r, 2 3 4 f(e) = - e + 0 . 3 e - 0. 0875e + 0(e ) (11.16) is v a lid fo r |s | « 1. T he e xp re ssio n fo r g(e) is then g(e) = ^ 1 f(e ) (11.17) (l+ e )2 w h e re e = s -1 = ( l - a ) / a . Th e m a in th ru s t of th is e ffo rt, h o w e v e r, is to look fo r a c c u ra te ap p ro xim atio n s to the cu rve s f(s ) fo r s p in -u p and g(CT) fo r spin-dow n in o rd e r to see i f e x p e rim e n ts w ould b e a r out W e d e m e y e r's b asic assum ptions. A n a ly tic a l fits w e re sought w h ic h could perhaps lend th em selves to u s e fu l closed fo rm solutions to the d iffe re n tia l equa tio n . T h is w as not p o ssib le fo r th e function f(s ), but a v e ry a c c u ra te le a s t-s q u a re s f it to the eig h t data points of R & L is given by the p o lyn o m ial 7 f(s ) = ^ a ^ s * ( II. 18a) n=0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w h e re the co e ffic ie n ts a a r e d e te rm in e d to be n 17 1 ^0 = 0 .8 8 4 4 6 0 0 ^4 = -3 8 .4 5 3 7 9 8 5 ^1 = 1 .0 7 2 5 8 0 3 ^ = 3 8 .2 3 6 6 1 5 1 ^2 = -9 .4 4 2 6 3 6 7 ^6 = -2 1 .0 6 5 8 8 9 3 . ^3 = 23. 8819285 ^ = 4 .8 8 6 7 3 9 3 ( II. 18b) T h e fu n ctio n (II. IS ) has a standard d e v ia tio n of 10 ^ and is e v e r y w h e re sm ooth and continuous. F o r the case of spin -d o w n , a m o re fo rtu n a te c irc u m s ta n c e o cc u rs. T h e functio n g(a) can be c lo s e ly a p p ro x im a te d b y an equa tio n o f the fo rm g(G) = -C^ o ^ a ^ l ( II. 19) w h ic h c le a r ly s a tis fie s th e n o -p u m p in g co n d itio n a t a = I . A n o th er im p o rta n t con ditio n to be m e t is g(0) = -1 .3 6 9 6 1 w h ic h co rresp o nds to the E k m a n suction fo r Bodew adt flo w . T h e th ird condition is obtained by e x a c tly passin g the c u rv e th ro u g h one of the re m a in in g six points com puted by R & L . U sing the co n ditio n g (0. 8) = -0.2 1272 gives the b es t o v e ra ll f it , and so lution fo r th e tw o unknown constants T h e au th o r is g ra te fu l both to T e r r y D eglo w and D r . B . A . T ro e s c h who independently a r r iv e d a t th is re s u lt. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 y ie ld s = 3. 80828 and = 2 .7 8 0 5 6 . T h e re s u ltin g c u rv e f it is e v e ry w h e re w ith in 1 .0 6 p e rc e n t of the com putations of R & L . E q u atio n ( II. 19) can a ls o be w ritte n g(s) = l ^ s ^ » (11.20) w h e re K = 1. 36961 and b = 0. 35964. T h is is th e fo rm of the c u rve f it w h ich w ill be c a lle d upon in la te r sections since it is convenient fo r d ire c t su b stitution in to th e equation of m o tio n . It is in te re s tin g to note th a t C a r r ie r [lo], in his exten sio n o f the O seen m ethod app lied to s w irlin g flo w b o u n d ary la y e rs , has also a r r iv e d at an a p p ro x im a tio n fo r the E k m a n suction g(cr) w h ic h has about the sam e a c c u ra c y as equation (II. 20). H is fo rm u la , h o w eve r, is c o n s id e ra b ly m o re c o m p lic a te d than the e x p re s s io n g iven above, and consequently w as not em p lo yed in th is a n a ly s is . Th e above a p p ro xim a tio n s fo r f(s ) and g(a) c o v e r th e e n tire ran g e of the independent v a ria b le s , w h ic h is a n e c e s s a ry co n d itio n i f w e co n sid er sp in -u p fro m re s t o r spin-dow n to re s t. H o w e v e r, in th e case o f sp in -u p o r sp in -d o w n fro m one a n g u la r v e lo c ity to another, a lin e a r a p p ro x im a tio n about s = a = 1 m ay be s u ffic ie n t. T h e only r e s tr ic tio n is th at a t no tim e m a y the in te rio r flo w lo c a lly exceed th e range o f s or a fo r w h ic h th e a p p ro x im a tio n is v a lid . T h is in clu d es flo w s of s m a ll to m o d e ra te R ossby n u m b e r, depending on the a c c u ra c y d e s ire d . F o r the case of im p u ls iv e s p in -u p o r s p in - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 dow n, the la rg e s t p e rm is s ib le r a tio o f th e m a x im u m (an g u lar) v e lo c ity to the m in im u m v e lo c ity is d ir e c tly d e te rm in e d by th e v a lid ran g e of scor cr. F o r spinPup o r s p in -d o w n at constant a c c e le ra tio n , h o w e v e r, the r a tio is d e c id e d ly la r g e r i f the a c c e le ra tio n s a re s m a ll. T h is re s u lts fr o m the fa c t th a t the lo c a l flu id v e lo c ity does not lag f a r behind the lo c a l w a ll v e lo c ity , and th e Rossby n u m b er re m a in s e v e ry w h e re s m a ll. L in e a r a p p ro x im a tio n s w h ic h w ill be u s e fu l fo r the cases cited above a re given by f(s ) = k '( l - s ) , 0 .8 ^ s ^ 1, k ' = 1 .0 2 0 (11.21) g(a) = - K '( l - a ) , 0 .8 ^ a ^ 1 , K ' = 1 .0 3 0 (11.22) and the e r r o r s in c u rre d can be d e te rm in e d fr o m the Rossby n u m b er expansions ( II. 16) and (II. 17). It is found th a t in the ranges in d icated f(s ) is e v e ry w h e re c o rre c t to w ith in 2 p e rc e n t w h ile the m a x im u m e r r o r fo r g(a) is about 3 p e rc e n t. 5. Im p u ls iv e S p in -U p W e s h a ll c o n s id e r f ir s t the case of im p u ls iv e sp in -u p fro m an in it ia l state o f solid body ro ta tio n Q, to a fin a l state Q^. In th is case s = and equation ( II. 14) becom es Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 T h e d im e n s io n a l v a ria b le s ( v * ; r * , t * ) a re m ade u n itie s s w ith the c h a ra c te ris tic values [aO^;a, One then obtains the n o n d im en- sio n al d iffe re n tia l equation 5v a t w ith the corresponding b ou ndary conditions v ( r , 0) = rY 0 ^ r ^ 1 v ( l , t ) = 1 t s 0 (n .2 5 ) 2 w h e re = v / Q ^ is the E k m a n n u m b e r and Y = It tu rn s out to be m o re convenient to w r ite the equation in te rm s of th e non- d im e n s io n a l c irc u la tio n F = r v , and w ith th is change of v a ria b le equation (II. 24) can be w ritte n ar at ■ T h e equation is now in standard fo rm to be solved by the m ethod of 2 c h a ra c te ris tic s . The solution states th a t the c irc u la tio n re m a in s constant ^ = 0 (11.27) 2 T e c h n ic a lly rv is the a n g u la r m o m en tu m p e r u n it m a s s , but w e s h a ll s im p ly r e fe r to it as the c irc u la tio n throughout th is te x t. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 along c h a ra c te ris tic paths defined by the d iffe re n tia l equation f t - -4 ( II. 28) w h e re T is the constant valu e of the c irc u la tio n . Since f(s ) ^ 0 o e v e ry w h e re , w e conclude th a t th e c h a ra c te ris tic s a re a ll le ft-ru n n in g . T h e boundary valu es in th e r - t p lan e a re dep icted in the sketch below . t 1 2 F ig u re I I . 2. C h a ra c te ris tic paths and boundary valu es fo r im p u ls iv e s p in -u p . T h e c h a ra c te ris tic s shown a r e th ese q u a lita tiv e ly co rresp o n d in g to the lin e a r a p p ro x im a tio n fo r f(s ). T h e a rro w in th e expansion fan in d icates the d ire c tio n of in c re a s in g c irc u la tio n w h ic h takes on the value Y at the d ivid in g c h a ra c te ris tic r^ (t) and th e valu e 1. 0 at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 the c y lin d ric a l boundary. In g e n e ra l equation (11.28) fo r the c h a r a c te ris tic s in Regions I and I I would have to be n u m e ric a lly in te g rated . H o w e v e r, we s h a ll f ir s t co n sid er the s im p lifie d case in w hich the E k m a n m ass flu x is ap p ro x im a te d by a s tra ig h t lin e . 5. 1 In te rio r Solution fo r f(s ) = k ( I- s ) . W ith the aid of (II. 15) equation (11.26) can be w ritte n in the fo rm (II. 29) w h ere = k E ^ . T his is the equation f ir s t solved by W e d e m e y e r fo r sp in -u p fro m re s t and w ill th e re fo re be r e fe r r e d to as W e d e m e y e r's equation. The g en eral so lu tio n fo r a r b itr a r y in itia l a n g u la r v e lo c ity (cf. [ 5 ] ,p g . 221) is given below . R egion I: 2 T ( r ,t ) = R egion II: T ( r ,t ) = 1 e x p (-2 p ^ t) r - e x p (-2 p ^ t) 1 - exp(-àPj^t) r ^ r ^ (t) (11.30) r ^ r ^ (t) (11.31) The d ivid in g c h a ra c te ris tic is d e te rm in e d fro m the so lu tio n of (11.28) w ith the in itia l condition = y = L t t = 0. One re a d ily obtains the fo llo w in g re s u lt: r^ (t) = [ y + ( l-Y )e x p (-2 p ^ t)]’ ^^. ( II. 32) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 The solutions show th at although P (and hence v) is continuous a cro s s 9 r ôv the w ave fro n t r (t), the s p a tia l d e riv a tiv e i r - (and a ls o is d is - o o r or continuous. The s tre n g th of the d isc o n tin u ity , given by the e x p re s sion ( e x p ( Z p ^ t ) - l ) ' o is in itia lly in fin ite and approaches z e ro as the fin a l state of ro ta tio n is reached. The r a d ia l v e lo c ity is e a s ily com puted fro m equation (II, 13), w hich in this case becom es u (r, t) = (11.34) w h ere u (r , t) has been a p p ro p ria te ly n o n d im en s io n alized . Since u is p ro p o rtio n a l to P, we conclude that w h ile both the u and v v e lo c itie s a re continuous a c ro s s the w ave fro n t, th e ir d e riv a tiv e s a re discontinuous and so d isco n tin u ities in s h e a r a re propagated along 3 the w ave fro n t. The value of the r a d ia l v e lo c ity a t the w ave fro n t 3 F o r the purposes o f this w o rk d isco n tin u ities in v e lo c ity w ill n a tu r a lly be te rm e d " v e lo c ity d is c o n tin u itie s " w h ile d isc o n tin u itie s in ju s t the v e lo c ity d e riv a tiv e s w ill be r e fe r r e d to as "s h e a r discontinuities'J A sh e ar la y e r , of co u rs e , is a g e n e ra l te rm r e fe r r in g to any la y e r n e c e s s a ry to sm ooth out a v e lo c ity o r a s h e ar d is c o n tin u ity . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 u (r ^ .t) = (11.35) is ju s t eq u al to r^ (t) so that the w ave fro n t is a con tact s u rfa c e . T his p ro vid es a convenient p h y s ic a l d is tin c tio n betw een the two so lution re g io n s ; R egion I contains flu id w hich has alw ays re m a in e d in the in te r io r ahead of the w ave fro n t, w h ile a ll the flu id in R egion I I has passed through an E k m a n boundary la y e r . The v o rte x lin e s in Region I a re pushed ahead of the fro n t and s tre tc h e d as flu x is lo s t to the boundary la y e rs - - thus the s p in -u p m ech a n is m h e re is one of v o rte x stre tc h in g as in the lin e a r solution of G & H . Behind the w ave fro n t, on the o th e r hand, the sp in -u p process is achieved by the flu id a c q u irin g an g u la r m om entum fro m the so lid bou ndaries. In the lim it Y -+ 0, equation (II. 30) shows that the a h im u th a l v e lo c ity re m a in s z e ro fo re v e r in R egion I, but the volum e of th is re g io n disap p ears as t . T h e re can be no v o rte x stre tc h in g in this re g io n because the flu id v o r tic ity is in itia lly z e ro . Thu s, in the s p e c ia l case of sp in -u p fro m r e s t, a ll the flu id m u st pass through an E k m a n la y e r . The v e r tic a l v e lo c ity can be obtained fro m the equation of continu ity w hich has the non dim en sio n al fo rm . (11.36) ReprocJucecJ with permission of the copyright owner. Further reproctuction prohibitect without permission. 25 P e rfo rm in g the in te g ra tio n on z , using the s y m m e try boundary co n di tion w (r , o ,t) = 0, one obtains w (r , z , t ) = " I v ) (11.37) w h ere equation (II. 34) has been used to e lim in a te u. W e observe that w is discontinuous a t the w ave fro n t; in fa c t it can be shown that w changes sign acro ss the dividing c h a ra c te ris tic such th at the E k m a n boundary la y e r is con verg en t in R egion I and d iv e rg e n t in R egion I I. 5 .2 V e r tic a l B oundary and F r e e Shear L a y e rs . An in v e s tig a tio n of the propagating fre e s h e ar la y e r in the neigh borhood of r (t) correspo n d in g to W e d e m e y e r's solution fo r sp in -up o fro m r e s t was given by V e n e zia n [? ]. He re ta in e d the (3(E) viscous s tre s s te rm s in the ang u lar m om entum equation (II. 2), and assum ed that the secondary v e lo c itie s u and w w e re s t ill d e te rm in e d by equa tions (11.34) and (II. 3 7 ), re s p e c tiv e ly . The non d im en sio n al equation w ritte n in te rm s of the c irc u la tio n becom es 2 E ^ a t w h ere Æ, = a /h is the c y lin d e r asp ect ra tio and we have again em ployed the lin e a r a p p ro xim atio n fo r the E k m a n flu x . A change of coordinates was made such th at the w ave fro n t could be lo cated a t a constant value of one o f the co o rd in ates . F o r E q « ÆI the d iffe re n tia l equation fo r r is reduced to B u rg e r's equation, w hich gives a tra n s itio n la y e r of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 6 ; i thickness ©’(E ^ ). W ith this so lu tio n a ll the v e lo c itie s v a ry continuous - 1 /3 ly fro m R egion I to R egion I I and, consequently, an unsteady 0 (E ) S tew arts o n la y e r is not re q u ire d . W e s h a ll now con sid er the v e r tic a l boundary a t r = 1. F o r a ll tim e s a fte r the in itia l im p u ls e , so lution (II. 31) fo r the in te r io r p ro b le m shows th at the a z im u th a l v e lo c ity n ea r the c y lin d ric a l boundary does not lag f a r behind the w a ll v e lo c ity . V e r y n ea r the w a ll, then, üu/fi^ cr 1 and the lin e a r a p p ro x im a tio n given by k = 1 is indeed v a lid . B y sub stituting the solution fo r F into equation (II. 38) one finds that the b racketed te rm on the rig h t hand side is id e n tic a lly z e ro , and hence the in te rio r solution s a tis fie s the fu ll viscous equation. T h is is not s u rp ris in g since the in te r io r solution a lre a d y s a tis fie s the n o -s lip con ditio n v ( l , t ) = 1. The r a d ia l v e lo c ity is also continuous and s a tis fie s u ( l , t ) = 0, but the v e r tic a l v e lo c ity w is discontinuous a t the w a ll. 1 Thus a n O (E '*) v e r tic a l boundar la y e r does not e x is t n e a r the w a ll, a l l / 3 though an 0-(E ) la y e r is expected in o rd e r to s a tis fy the condition w ( l, t) = 0. 1, In the lin e a r p ro b le m of G & H the double la y e rs of o rd e r E+ and 1 /3 E re m a in attached to the w a ll. In the n o n lin e a r case V e n e zia n has shown th at the E + la y e r is propagated aw ay fro m the w a ll along the d ivid in g c h a ra c te ris tic r^ (t). W ith this a n a lys is one concludes that the n o n lin e a rity sep arates the double s tru c tu re of the v e r tic a l bounda a r y la y e r . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 5 .3 A C lo s e r Look a t the W ave F ro n t. The p re s e n ta tio n up to now has been e s s e n tia lly a re v ie w o f the published w o rk re le v a n t to W e d e m e y e r's fo rm u la tio n o f the s p in -u p p ro b le m , a ll of w hich assum es a lin e a r f it to f(s ). I t is exp ected that a m o re a c c u ra te a p p ro x im a tio n w ould p ro vid e a b e tte r so lu tio n fo r v ( r , t) w h ich could then be c o rra b o ra te d w ith e x p e rim e n ta l d ata. In o rd e r to obtain the "e x a c t" so lu tio n , the path o f the c h a ra c te ris tic s in equation (11.28) was n u m e ric a lly in te g ra te d w ith the b e s t-fit fo r the E k m a n flu x as given by equation (II. 18). These c a lc u la tio n s gave the unexpected re s u lt th at the c h a ra c te ris tic s alw a ys in te rs e c te d in the re g io n of the w ave fro n t fo r sp in -u p fro m r e s t. It was e v e n tu a lly decided th at this m u st s te m fro m the non m o noto nicity of the fu n ctio n f(s ). T h a t this co n jec tu re is in fa c t tru e can be seen fro m the fo llo w ing a n a ly s is . We r e w rite equation (11.28) fo r the c h a ra c te ris tic paths in the fo rm -P 'd t = (11.39) w h ere P' = E q . The in te g ra tio n of this equation fo r a r b itr a r y (w e ll- behaved) f(s ) s a tis fy in g the in itia l con ditio n r = 1 a t t = 0 gives X P«t = - J ---------------------------------------------------(11.40) 1 x f(F ^ /x ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 W ith the change of v a ria b le s x § = the c h a ra c te ris tic cu rve s s a tis fy the re la tio n . F / / H (r ,t,r ) = P 't - o J j, O d§ 5%5) = 0, (11.41) and the envelope of in te rs e c tin g c h a ra c te ris tic s , i f it e x is ts , is given by (see, fo r e x a m p le , [ l l ] , pp. 1 7 2 -1 7 3 ) ^ ( r , t ,F ^ ) = 0 . o ( II. 42) D iffe re n tia tio n of (11.41) w ith the aid of L e ib n itz 's ru le then gives f(F ) = f(F / r ) o o (II. 43) as the condition fo r the in te rs e c tio n of c h a ra c te ris tic paths. Since 2 0 ^ r ^ 1 we m ust have F / r 2 : r , w h ich is the situ a tio n depicted o o below . f(s ) F ig u re I I . 3. Sketch showing the valu es of s. fo r w hich c h a ra c te ris tic paths w ill in te rs e c t d u rin g im p u ls iv e s p in -u p . The m a x im u m in the c u rve is lo cated a t S ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 Thus we conclude th a t the c h a ra c te ris tic s m u s t c ro ss i f f(s ) is not a m onotonie functio n . N o te th a t nothing has been said about the in itia l con ditio ns. In the g e n e ra l case of a r b it r a r y in itia l an g u la r v e lo c ity , only th at p o rtio n o f the c u rve fo r w hich y ^ s ^ 1 is neces - s a ry to d e s c rib e the sp in -u p p ro c e s s . If the p o s itio n of the m a x im u m is lo c a te d a t s = then only flow s fo r w hich Y ^ w ill condition (11.43) be s a tis fie d . (F ro m the le a s t squ ares f it , s ^ = 0 .0 7 5 0 0 5 and f ( S j) = 0 .9 2 0 7 3 4 .) T h is includes the p a rtic u la r case of s p in -u p fro m r e s t in w hich case the c h a ra c te ris tic s w ill alw ays in te rs e c t. S a tis - fe-ction of equation (II. 43) m eans th at the in te r io r so lu tio n is c h a r a c te riz e d by a d isco n tin u ity in the 0 (1 ) a z im u th a l v e lo c ity . The fo reg o in g a n a ly s is has d e m o n s tra te d th a t an y a n a ly s is based on W e d e m e y e r's in te rio r so lution cannot pro vid e a ll the d e ta ils of the m oving sh ear la y e r fo r c e rta in values of s:. An a n a ly s is s im ila r to V e n e z ia n 's fo r sp in -u p fro m r e s t m ig h t be u se fu l if the position and s tre n g th of the d isco n tin u ity w e re known a p r io r i, but this is no lo n g e r the case. If one w e re to pro ceed w ith this type of ap p ro ach, the c ru x of the p ro b le m would lie in the d e te rm in a tio n of the p o sition o f, and the ju m p conditions a c ro s s the v e lo c ity d is c o n tin u ity . In the a p p ro x im a te so lution r^ (t) was u n iq u ely d e te rm in e d by k , Y, and E ^ , The "e x a c t" so lution fo r Y ^ is c h a ra c te riz e d by only a sh ear d is c o n tin u ity , and the p o sitio n of the w ave fro n t can alw ays be found by n u m e ric a l in te g ra tio n . In fa c t, fo r s lig h tly la r g e r Y» say Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Y > 0. 2 , the W ed em ey er and V e n e zia n analyses a re expected to p ro vid e an acc u rate d es crp tio n o f the whole sp in -u p p ro ce ss. T h e re fo re , o n ly flow s fo r w hich Y < w ill a new in v e s tig a tio n of the w ave fro n t re g io n be n e c e s s a ry . W e m ig h t now re -e x a m in e the o rig in a l assum ptions in lig h t of the p recedin g, re s u lts , s p e c ific a lly the lo c a l a p p ro x im a tio n fo r the m ass flu x in the E k m a n la y e rs . The existen ce of an a z im u th a l v e lo c ity d isco n tin u ity is in co n s isten t w ith the o rig in a l assu m p tio n th at lo c a lly the in te r io r v e lo c ity is a p p ro x im a te d by so lid body ro ta tio n . H o w e v e r, an an a lys is s im ila r to th a t given by V e n e zia n would s u re ly re m o v e the d is c o n tin u ity in a la y e r of o rd e r E * , and the flo w would ap p ear lo c a lly sm ooth when view ed fro m the E k m a n la y e r . Thus we m ust con cern o u rse lve s w ith the m eaning of lo c a l. The E k m a n flu x given by equations (II. 11) and ([I. 12) depends only on the "lo c a l" flu id v e lo c ity re la tiv e to the m oving p late and not on its h ig h er o rd e r d e r iv a tiv e s . If ah E k m a n la y e r of thickness (^(Ea) lie s beneath a v e r tic a l — Ô 9 s h e a r la y e r of thickness 6 , then fo r Ô » one has <— and ■ ' s s 9 r 9 z hence the lo c a l h o riz o n ta l v e lo c ity g rad ien ts a re c o m p a ra tiv e ly s m a ll F o r this scalin g the lo c a l a p p ro x im a tio n should be v a lid . On the o th e r hand, if 6^ ~ 0 (E 2 ) then the E k m a n flu x m ay W ell depend on the h ig h e r o rd e r d e riv a tiv e s so that the flo w w ill be no n lo cal. The ideas p resen ted above a re known to be ap p licab le to lin e a r iz e d ro ta tin g flo w s . F o r e x a m p le , the E k m a n c o m p a ta b ility condition Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ...........................................................31; — 1 /3 d e riv e d by Jacobs [ 9 ] is su c c e s s fu lly used beneath 0(E'* ) and 0'(E ) S tew artso n la y e rs . A p e rtu rb a tio n an a ly s is of the equations solved by R & L m ig h t be a fr u itfu l w ay of pro vin g this c o n je c tu re fo r the n o n lin e a r case. 6. S p in -U p a t C onstant A c c e le ra tio n We s h a ll now co n sid er the sp in -u p process fo r w h ich the c y lin d ri c a l w a lls a re u n ifo rm ly a c c e le ra te d fro m one a n g u la r v e lo c ity to an o th e r. In this case s = iu/(0;-KXt) d u rin g the a c c e le ra tin g phase and the d im en sio n al fo r m of equation (II. 14) is given by ÔV* — — - — r * at* h w h e re is the in itia l an g u la r v e lo c ity and a is the constant ra te of i _ J L a c c e le ra tio n . W e n o n d im e n s io n a lize the v a ria b le s w ith [a ^ a ;a ,a and obtain 0 ,11.45, w h e re t. = Ù J J K and now the E k m a n n u m b e r is defined as E = 11 c x 2 v /v S h . T h is equation can be som ew hat s im p lifie d i f w e m a k e the _! tra n s fo rm a tio n t = (t+t^)^ and ag ain use F as th e dependent v a ria b le . One then obtains | E = 0 (11.46) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and th e boundary conditions a re r(r,t?) = r^t. r(l,T) = T 32 0 ^ r ^ 1 Â T ^ t f 1 (11.47) E q uation (11.46) is now in can o n ical c h a ra c te ris tic fo r m and, as the the im p u ls iv e case, states th at the c irc u la tio n m ust be constant r = r. ( II. 49) along c h a ra c te ris tic paths defined by dT dT = -2 E (11.49) The c h a ra c te ris tic s a re a ll le ft-ru n n in g and the fo llo w in g d ia g ra m shows ty p ic a l paths in the r - T p la n e fo r the case of lin e a r f(s ). I I I IV 1 n 1.0 o 'L IM IT .2 F ig u re 114. C h a ra c te ris tic paths and boundary valu es fo r sp in -u p a t constant a c c e le ra tio n . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 i w h e re = 2 k E ^ . W e s h a ll f ir s t co n s id e r the so lution in R eg io n I. _! Since th e in itia l c irc u la tio n along the b o u n d ary t = t? is p ro - 2 p o rtio n a l to r (c f. F ig . I I . 4 ), w e seek a solution of the fo r m r ( r ,T ) = r^G (T ) , (11.51) and su b stitu tio n o f th is in to equ ation (II. 50) gives the f ir s t o rd e r, n o n lin e a r d iffe r e n tia l equation fo r G (T ) a o + 2P ^(G -T )G = 0 (11.52) w h ic h m u st be solved w ith th e in it ia l condition 1. G (t^) = t . . (11.53) E quation (II. 52) is a B e rn o u lli equation and can be s im p lifie d w ith the substitutions i = (2Pg)^/^G , X = (2 p g )^ /\ (11.54) g ivin g a lin e a r , f ir s t o rd e r d iffe r e n tia l equation fo r u (x ), n a m e ly ^ + X u = 1 . ( II. 5 5 ) T h e solution of th is equation is e a s ily found, but u n fo rtu n ately m u st be le ft in te rm s of an in te g ra l. A fte r going b ac k th ro u g h tra n s fo rm a tio n s ( II. 54) and s a tis fy in g th e in itia l co n d itio n ( II. 5 3 ), one obtains the fo llo w in g e x p re s s io n fo r the c irc u la tio n in Rëjgion I 3 2 t . e x p (C T ) r r(r,T) = ^ (11.56) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 w h e re C = the in te g ra l is d efined by r 3 I(x ) = J exp(§ )dÇ . (11.57) Thus w e see th a t as in the case of im p u ls iv e s p in -u p , th e flu id in 4 R eg io n I has an a z im u th a l v e lo c ity p ro file p ro p o rtio n a l to r . In R egion I I w e m u s t d e te rm in e the c h a ra c te ris tic paths fo r ( II. 50) fro m the d iffe r e n tia l equation I f = ^ (^o - =^T^) (11.58) w h e re is th e v alu e of th e c irc u la tio n along the c h a ra c te ris tic . W e 2 can sep arate v a ria b le s b y su b stitu tin g u = r . E q u atio n (11.58) then becom es w h ic h is id e n tic a l in fo r m to equation ( II. 55) and can be re a d ily solved along w ith the bou ndary co n ditio n r = T ^ > u = l , T = T (11.60) o i 1 T h e solution is then r^(T) = e x p ( - C T ^ ) { e x p ( C T ^ ) + 2 P _ T ^ C ' ^ ^ ^ [ l ( C ^ ^ ^ T ) . I ( C ^ ^ \ ) ] } ^ (11.61) 4 W e note th a t th is does not re p re s e n t s o lid body ro ta tio n sin ce, b e cause of the secondary m o tio n , the flu id p a rtic le s a re being con tin u a lly re d is trib u te d in the m e rid io n a l p lan e. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ................................................................................................... 36 w h e re the constant C and the in te g ra l I(x ) have been previously- d e fin e d . T h e d ivid in g c h a ra c te ris tic is obtained by setting T = t f in the above equation. Once the in te g ra l I(x ) is eva lu a te d , w e can d e te rm in e the c irc u la tio n a t each p o s itio n and tim e d u rin g the phase of constant a c c e le ra tio n . In p a r tic u la r, one can com pute the conditions along the lin e T = t^ w h ic h w ill p ro v id e the in itia l valu es n e c e s s a ry to c a r r y th e solution into R egions I I I and IV . W e s h a ll now co n sid er th is p o s t-a c c e le ra tio n phase of the m o tio n . E q u atio n ( II. 51) gives ^ e fo r m of the solution along the bound- 1 a r y T = t^ w h ic h m a rk s th e beginning of R egio n I I I . W e see th a t F ( r ,t ^ ) = r^ G (t^ ) = r^ g ^ (11.62) w h ic h co rresp o n d s to th e tim e t = t . - t . , say t = t . in o rd e r to be f 1 o co n sisten t w e m u st now n o n d im e n s io n a lize the im p u ls iv e sp in -u p equation (11.23) w ith the sam e c h a ra c te ris tic q u a n tities used in th is i _ ji a n a ly s is ,n a m e ly [oL^a;a, a ^]. W e a lso r e a liz e th at th e w a lls w ill alw ays be a t the an g u la r speed 0^ so th a t s = Cu/0^. T h e equation of m o tio n then becom es ? ( — 4 -------) E = ° ± 1 . w h e re now = k E ^ /t^ , and the bou ndary con ditio n is a p p ro p ria te ly w ritte n as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 r ( r .‘o> = and can be d e te rm in e d fro m equations (II. 56) and (II. 6 2 ). A g ain 2 one obtains a solution in the fo r m F ( r ,t ) = r g(t) w h e re g(t) m u st s a tis fy % + 2p^g(g_t^) = 0 (II. 64) and the boundary conditio n g(t ) = g . T h is equation can be solved o o by sep aratio n of v a ria b le s and the e x p re s s io n fo r the c irc u la tio n in R egio n I I I is s im p ly T ( r ,t ) = y - ------------------------------------------ (11.65) 1 - ( — ^ ------)e x p {-2 P g t^ (t-t )] F in a lly w e s h a ll ou tlin e the solution in Region TV. A g ain equation (11.63) is a p p lic a b le and thus th e equation fo r the c h a ra c te r is tic paths is given by 2 w h e re T ^ is the va lu e of the c irc u la tio n along each path. The in itia l 1 conditions along T = t^ c a lc u la te d fro m ( II. 61) can be w ritte n in the fu n ctio n al fo rm T = T (§); r = Ç, t » t . (11.67) ■ I i. o A fte r re a rra n g in g (II. 66) to re a d Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 ^ ( r ^ ) = -2P^(t^r^ - T j) (11.68) w e see th at the equation is s e p e ra b le and the solution s a tis fy in g (11.67) is re a d ily found to be 1 r^ (t) = ^ + i f ( t - t ^ ) | . (11.69) 2 The d iv id in g c h a ra c te ris tic r^ (t) c a r r ie s the c irc u la tio n = t^ ^ and its lim itin g r a d ia l exten t ( c f . , F ig . I I . 4) is obtained by le ttin g t Thus once th e steady state m o tio n is re a c h e d , a ll th e flu id at r a d ii la r g e r than (Ü./Q ^)^ w ill have passed th ro u g h an E k m a n boundary la y e r . A s in the case of im p u ls iv e sp in -u p , a shear d is c o n tin u ity is propagated along the d iv id in g c h a ra c te ris tic and the a z im u th à l v e lo c ity s a tis fie s its bou ndary con ditio n a t the c y lin d ric a l w a ll. T h e secondary v e lo c itie s can be re a d ily c a lc u la te d , but w ill not be d e a lt w ith h e re . It suffices to say th a t th e m oving w ave fro n t m u st be c h a ra c te riz e d by an la y e r to sm ooth out the sh e ar d is c o n tin u ity , 1 /3 w h ile an E la y e r re m a in s attach ed to the c y lin d ric a l w a ll in o rd e r to s a tis fy the bou ndary con ditio n fo r the v e r tic a l v e lo c ity . A sam ple c a lc u la tio n of the v e lo c ity p ro file s d u rin g the a c c e le r a tio n phase of sp in -u p fro m re s t as d e te rm in e d b y equations ( II. 60) ( - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 ! and (II. 61) is given in F ig u re X . 1 in th e b ac k o f the th e s is . W e note th a t the solution is single - v a lu e d in tim e and space because of th e lin e a r a p p ro x im a tio n to f(s ). T h e n u m e ric a l solutions obtained w ith th e fu ll e x p re s s io n fo r f(s ) w ill be p re s e n te d in a la te r section fo r c o m p a riso n w ith the e x p e rim e n ta l re s u lts . 7. Im p u ls iv e S p in-D o w n W e now co n sid er th e flu id m o tio n w ith in th e c y lin d e r w hen the w a lls a r e im p u ls iv e ly d e c e le ra te d fr o m an in itia l an g u la r speed Q. to som e fin a l speed T h e flu id a ls o is assum ed to be in itia lly ro ta tin g a t the sam e a n g u la r v e lo c ity as th e c y lin d e r. In th is case w e assu m e th at the flu id a n g u la r v e lo c ity is e v e ry w h e re g re a te r than o r equal to the w a ll an g u la r v e lo c ity fo r the in te r io r m o tio n , and so w e choose the E k m a n pum ping g iven by g(c) as d ep icted in F ig . I I . 1. F o r the im p u ls iv e d e c e le ra tio n G = and in s e rtin g the a p p ro x im a tio n (II. 20) in to ( II. 14) one obtains I f + K ^ ( v * r * f + ( B r + S ) = 0 < “ • fo r th e equation o f m otio n fo r th e in te r io r flo w . W e scale the v a ria b le s w ith the c h a ra c te ris tic v a lu e s [Q .a ;a ,0 . and a r r iv e at the nondim en sio n al equation ôv at (v/r + ti b ) ( B + r ) = “ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 — 2 w h e re 1 - 1 = 0 / 0 P = K E ^ , and E-, = v / h 0, is th e E k m a n n u m b e r. I 1 4 U 1 T h e co rresp o n d in g bou ndary con ditio ns a re (H .7 3 ) v ( r , 0) = r O ^r ^l v ( l , t ) = l-L t > 0 A g a in the equation s im p lifie s som ew hat i f the c irc u la tio n is used as the dependent v a r ia b le . W ritin g T = r v w e im m e d ia te ly obtain 2 l / r - M r \ \r + tiir^ / ôr = 0 (II. 74) and since the flu id is in itia lly in s o lid body ro ta tio n , w e seek a solution of the fo r m r(r,t) = r^F^(t) . (11.75) T h e s p a tia l dependence is then re m o v e d and w e a r e le ft w ith the d iffe re n tia l equation fo r th e te m p o ra l function (II. 76) w h ic h m u s t s a tis fy th e in itia l co n d itio n F (0 ) = 1. T h is equ ation is sep arab le and can be re a d ily in te g ra te d w ith th e a id of p a r tia l ■ fra c tio n s . T h e im p lic it solution fo r F (t) (n . 77) T + f iV \F -t-i' s a tis fie s th e in it ia l co n d itio n , but th e bou ndary con ditio n a t th e w a ll, F = M.^, is not s a tis fie d . I f w e take th e lim it of ( II. 77) as p.-* 0 w e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 obtain the e x p lic it solution F(t) = 1/(1 +P^t) (n.78) c o rresp o n d in g to spin-dow n to re s t. In th is case the c irc u la tio n d is trib u tio n is s im p ly r(r,t) = r^/(l+P^t)^. (11.79) H e re the c irc u la tio n d is c o n tin u ity a t th e w a ll takes on the valu e one a t t = 0 and d im in is h e s to ze ro as t Since th is corresponds to an (}( 1 ) v e lo c ity d is c o n tin u ity , one w ould expect an 0 ’(E'*) v e r tic a l shear la y e r attach ed to th e c y lin d ric a l bou ndary. W e o b serve th at the solution fo r a lin e a r a p p ro x im a tio n to g(a) is included in the e xp re s s io n (II. 7 7 ). One need only set b equal to z e ro and change the constant K to th e a p p ro p ria te v a lu e K ' given by equation ( II. 2 2 ). T h e one advantage of th is s im p lific a tio n , v a lid fo r r e la tiv e ly s m a ll im p u ls iv e changes of th e w a ll v e lo c ity , is th at an ex p lic ity e x p re ssion fo r the c irc u la tio n can be obtained. In th is a p p ro x im a tio n the solutio n is given by i 1 i n 1 + IJ 2 + (l-ia 2 )e x p (-2 p ^ )i2 ^ ) 1 i 1 _1 +[i^ - (l-H 2 )e x p (-2 p li2t)_ ( II. 80) 4" T h e secondary v e lo c itie s can be c a lc u la te d fro m equation ( II. 13) along w ith the equation of c o n tin u ity . T h e re s u lts given below ( 4 ^ ) \F + bU/ u ( r ,t ) = P ^ F ^ — r (11.81) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 ( 4 ^ ) Vf + bu / w ( r , t ) = -2 8 F | : y — j z (11.82) Vf +bii a re p resen ted in te rm s of th e fu n ctio n F (t) since it is g e n e ra lly an 2 im p lic it functio n of tim e . N ow F so (II. 82) shows th a t the EkKian m an la y e rs a r e alw ays d iv e rg e n t and th e re fo re a ll th e re tu rn flu x m u st pass th ro u g h a v e r tic a l sh e ar la y e r at the c y lin d ric a l bou ndary. W e als o note th at the secondary v e lo c itie s do not s a tis fy th e n o -s lip condition a t the w a ll. A d iscu s sio n of the v e r tic a l b ou ndary la y e r w ill be d e fe rre d to a la te r sectio n . 8. S p in -D o w n a t C o nstant D e c e le ra tio n . W hen the c y lin d ric a l c o n ta in e r is d e c e le ra te d a t a constant ra te fro m an in it ia l state of solid body ro ta tio n 0 ., w e have c r = (n.-at)/üü and the d im e n s io n a l equ ation of m o tio n becom es 1 w h e re a is again the a c c e le ra tio n ra te . W e now sca le the v a ria b le s w ith [a ^ a ;a ,a in o rd e r to obtain the d im e n s io n le s s equation w h ich m u s t sa tis fy the boundary and in itia l conditions v ( r , 0) = t . r 0 ^ r ^ 1 v ( l , t ) = t , - t t ^ O . (II. 85) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 “ 2 In the p rece d in g exp ressio n s = v /h /v /5 T , and t. = A fte r m akin g the obvious tra n s fo rm a tio n s r = r v ; T = t^ -t ( n . 86) equation (II. 84) becom es and the co rresp o n d in g boundary conditions a re r ( r ,t ^ ) = r^ t. 0 ^ r ^ 1 r(l,T) = T 0 . A g ain a solution can be found in th e fo r m r ( r , T ) = r V ( T ) and upon substitution in to (II. 87) w e obtain w h ic h m u st s a tis fy the in itia l condition 1 ( II. 88) (n. 89) ( ^9 ~ ' 1 = 0 (11.90) ^ (t.) = t f . (II. 91) U n fo rtu n ately an a n a ly tic sblution could not be found fo r the fu n ctio n J^(T), but n u m e ric a l in te g ra tio n is s tra ig h tfo rw a rd . In F ig u re X . 2 w e p re s e n t som e com puted solutions fo r equation (II. 90), showing the e ffe c t of the d e c e le ra tio n ra te . T im e is nondim en s io n a l- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 iz e d by w h ic h is the tim e it takes fo r the c y lin d e r w a lls to com e to r e s t, i . e . , t^ = 0 . /a . F o r r e la tiv e ly slow d e c e le ra tio n ra te s g(CJ) is w e ll a p p ro x im a te d by equation (11.22) and a closed fo r m so lu tio n can be obtained. W e co n sid er th is case in the fo llo w in g sectio n . 8. 1 In te r io r S olution fo r g(a) = - K '( l - a ) T h e a n a ly s is of the sp in -d o w n p ro b le m fo r d e c e le ra tio n s in w hich the lin e a r expressiony^or g(CT) is s u ffic ie n t fo llo w s e x a c tly as above. In equation (II. 90) w e set b = 0 and o b tain th e R ic a tti-ty p e equation ^ - P ^ ( ^ - T) = 0 (11.92) w h e re now the a p p ro p ria te c o e ffic ie n t is = K ’E^^. T h is equation can be re a d ily solved w ith the sub stitutions y = ; X = P^^^T (11.93) fo llo w ed by the tra n s fo rm a tio n y = - u '/ u (11.94) w h ic h y ie ld the lin e a r , second o rd e r d iffe r e n tia l equation ,2 ^ - X U = 0 . (H .9 5 ) dx T h is is reco g n ize d as A ir y 's equ ation and has the g e n e ra l solution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 i u(x) = C j^A^(x) + C^B^Cx) w h ere A^(x) and B^(x) a re the tabulated (cf. [ l 2 ] , pp. 4 7 5 -8 ) A ir y functions of the f ir s t and second kind. T ra n s fo rm in g back to the o rig in a l v a ria b le s , one finds th at the in te rio r so lution w hich s a tis fie s the in itia l condition (II. 91) is given by ,2 /3 r ( r , t ) = 2 /3 _ 1 A :(x ~^ ' t) + k B.'(x - r j t) 1 o 6 o 1 o 6 (II. 96) in w hich k = - o b !(s ) + x ^ B .(x ) I o o 1 o J ’'o = In these exp ressio n s the p rim e denotes d iffe re n tia tio n w ith re s p e c t to the a rg u m e n t. The above solutions co rresp o n d to the d e c e le ra tin g phase of the spin -d o w n m o tio n ; they a re a p p lic a b le up to the tim e the c y lin d e r reac h es the fin a l steady state speed. In the p o s t-d e c e le ra tio n phase the solution is given by the im p u ls iv e sp in -d o w n equations. If the a n g u la r v e lo c ity of the flu id a t the end of the d e c e le ra tio n phase is 0 ^ , then the so lu tio n in the p o s t-d e c e le ra tio n phase is given by equation (II. 77) w ith 0^ e v e ry w h e re sub stituted by n^. W hen the d e c e le ra tio n begins w ith a high a n g u la r v e lo c ity and if the d e c e le ra tio n ra te is not too la rg e , equation (II. 96) w ill pro vid e an a c c u ra te so lu tio n fo r a la rg e o v e ra ll change in the w a ll a n g u la r v e lo c ity . If the w a ll speed approaches z e ro , h o w e v e r, the lo c a l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 : valu e of a w ill also tend to ze ro and the so lution w ill no lo n g er be a p p lic a b le . In this event one m u s t r e s o r t to a n u m e ric a l so lu tio n of equ ation (II. 90). F o r com pleteness we include the exp ressions fo r the secondary v e lo c itie s in the in te r io r flu id u ( r . T ) = - y -~ - I r (11.97) I f e ] f f c ] - ■ w (r , T) = ^ 1 z . (11.98) F r o m these exp ressio n s it is a s im p le e x e rc is e to show th at the 2 s tre a m lin e s in a m e rid io n a l plane a re of the fo rm r z = const. I t is perhaps of som e in te re s t to c o n tra s t the spin-dow n m ech an is m w ith th at of sp in -u p . W e see fro m the im p u ls iv e spin-dow n equation (II. 74) th at the c h a ra c te ris tic paths a re given by d r a - à ( '^o = p r ‘ 4 r + b n r^ I. o and since the rig h t hand side is alw ays n eg ative, the c h a ra c te ris tic s a re a ll rig h t-ru n n in g . T h is holds tru e irre s p e c tiv e of the m anner in w h ich the d e c e le ra tio n is acc o m p lish e d , and shows th a t the d ivid ing c h a ra c te ris tic propagates tow ards in c re a s in g r a d ii, i . e . , outside the c y lin d e r boundary. T h is , of co u rs e, has no p h ysical sig n ific a n c e, exc ep t to point out that a ll the flu id in the c y lin d e r re m a in s w ith in R egio n I and hence the s p in -d o w n m e c h a n is m is e n tire ly that of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 : v o rte x -s tre tc h in g , o r ra th e r v o rte x -s h rin k in g . T h e v o rte x lin e s a re convected to w ard s in c re a s in g r a d ii and shrun k as flu id p a rtic le s I flo w into the in te r io r fro m the E k m a n la y e rs . W e note th a t the E * la y e r w h ic h propagates along the d iv id in g c h a ra c te ris tic cannot le a v e th e flu id co n tain e r and hence m u s t re m a in attach ed to the c y lin d ric a l w a ll. T h e s tru c tu re of th is la y e r w ill be th e topic of discu ssio n in the fo llo w in g section. 9. B o undary L a y e rs a t th e C y lin d ric a l W a ll. A lthough a ll the v e lo c ity com ponents fo r sp in-dow n a r e not continuous at the c y lin d ric a l b ou ndary, th e only (}( 1 ) d isco n tin u ity is in the a z im u th a l v e lo c ity . W e expect a sh ear la y e r of thickness O'i'E*) and, since th is is m an y tim e s la r g e r than the E k m a n la y e r th ic k n e s s , w e assum e th a t th e lo c a l a p p ro x im a tio n fo r th e E k m an m ass flu x is v a lid . T h e equation fo r the a z im u th a l v e lo c ity is then obtained by re ta in in g the viscou s s tre s s te rm s in the an g u la r m o m en tu m equation (II. 2 ). W e sh a ll be concerned w ith only the case of spin-dow n a t constant d e c e le ra tio n since the a n a ly s is fo r im p u ls iv e spin-dow n is v irtu a lly id e n tic a l. T h e equation of m o tio n in the v e r tic a l boundary la y e r is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 w h e re the constants P , E , and JR. have been a lre a d y d efin ed . T h e 5 O C botm dary conditions w h ic h m u st be s a tis fie d a re F (r , T) --------* 3^{'T)r^ @ in te rio r (II. 100) r ( l , T ) = T t. ^ 0 w h e re (T) s a tisfies equation ( II. 90 ). W e a r e now dealin g w ith a n o n lin e a r, s e c o n d -o rd e r p a r tia l d iffe r e n tia l equation and, although it cannot be solved in closed fo r m , som e s im p lific a tio n can be m ad e. W e begin by re w ritin g the equation and boundary conditions in te rm s of the an g u la r v e lo c ity (U = r / r ^ w h ic h then yield s and U )(r,T) ---------^ J(T) = J ^ ( t ) @ i n t e r i o r ( II. 102) W(1, T ) = T t^ ^ TS 0 . T h e de pendent r v a ria b le is then scaled in a m an n e r a p p ro p ria te to a boundary la y e r a n a ly s is , n a m e ly ' .L e ric rL u = ^ L e t us now co n sid er ju s t the a c c e le ra tio n te r m in equation ( II. 101). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 In te rm s of the new v a ria b le w e have 1 ^ = (1 + cp»t + cpilf») (II. 104) w h e re the p rim e denotes d iffe re n tia tio n w ith re s p e c t to T . T h e g e n e ra l fo rm u la fo r ' in te rm s of th e o rig in a l v a ria b le s is 105) (J-T) and w e note th at th is e x p re s s io n is id e n tic a lly z e ro both âlt the in te rio r w h e re U ) = J and at the w a ll w h e re (U = t . One m ig h t assum e then th a t is a w e a k ly v a ry in g fu n ctio n of tim e and hence t ~ 0 throughout the bou ndary la y e r . T h is ass u m p tio n , though d e fin ite ly not m a th e m a tic a lly rig o ro u s , can a lw a y s be v e r ifie d once th e solu tio n is obtained. I t is eq u iva len t to saying th a t the a n g u la r a c c e le r atio n âU)/ÔT depends only on the lo c a l d iffe re n c e b etw een th e flu id and w a ll a n g u la r v e lo c itie s . W ith th is s im p lific a tio n equation (II. 101) can be red u ced to an o rd in a ry d iffe r e n tia l equ ation w ith the tim e T as a p a ra m e te r. T h e equ ation of m o tio n then becom es f ç JR~ \ d r ' w h e re T has been re p la c e d b y T ^ t® e x e m p lify th e fa c t th a t the d iffe re n tia l equation is o rd in a ry . Since the equation is also ô f second o rd e r w e need an a d d itio n a l boundary co n d itio n , obtained by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 m atch in g d e riv a tiv e s a t th e o u te r edge o f the bou ndary la y e r . A t each tim e w e then m u s t s a tis fy the fo llo w in g conditions U )(r ,T j) » J (T j) ^ ^ ( r , T ^ ) -----» 0 @ in te r io r (11.107) UJ(1,T^) = T j t^ i T h e fo rm u la tio n of th e p ro b le m as it stands is p a r tic u la rly w e ll suited fo r n u m e ric a l so lu tio n b y the shooting technique. Id e a lly , one begins the in te g ra tio n a t som e ra d iu s outside the boundary la y e r , using the in te r io r conditions in d ic a te d in (II. 107), and continues the solution up to the w a ll w h e re th e valu e o f U ) is eva lu a ted . I f t^(l, T^) / a new ra d iu s is selec te d to b eg in th e in te g ra tio n and the solution is ite ra te d u n til the bou ndary co n d itio n a t the w a ll is s a tis fie d . In p ra c tic e , h o w e v e r, one m u s t p e rtu rb the in te rio r conditions s lig h tly . T h is becom es re a d ily a p p a re n t w hen Ive observe fro m equation (II. 106) th a t the second o rd e r d e riv a tiv e depends in itia lly on only th e in te r io r solution plus the a d d itio n a l t e r m ^ . O utside th e boundary la y e r th e in te r io r solution is id e n tic a lly z e ro and in ad d itio n th e in itia l slope is set equal to z e ro . T h e in itia l c u rv a tu re , th e re fo re , is com puted to be z e ro and, i f th e ro u n d -o ff e r r o r s in the com putations a r e a c c u m u la tiv e ly s iria ll, th e in te g ra te d solution w il l give s im p ly U)(r, T^) = J (T j^ ) through th e e n tire bou ndary la y e r . Since J(T^) > T^, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 a n eg ative p e rtu rb a tio n w as m ade on the in itia l v a lu e of w, i. e . , U ) = J (T j) - e. V e lo c ity p ro file s c o rres p o n d in g to e x p e rim e n ta l E k m an n u m b ers w e re com puted and w ill be p res e n te d la te r along w ith the la b o ra to ry m e a s u re m e n ts . F o r m o st of these com putations a p e rtu rb a tio n of e = 10 ^ w as s u ffic ie n t to s ta rt th e in te g ra tio n alogg the c o rre c t path. T h e n u m e ric a l solution shows th a t the a z im u th a l v e lo c ity d is co n tin u ity is sm oothed out in a bou ndary la y e r w h ic h ra p id ly grow s J L to a th ickn ess of o rd e r E '’’. T h e ra d ia l v e lo c ity is lik e w is e con tinu ous and s a tis fie s the boundary co n d itio n a t r = 1, although the a x ia l v e lo c ity re m a in s discontinuous at the w a ll. One w ould expect that the bou ndary con ditio n on w could be s a tis fie d by inclu d in g ad d itio n al 1/3 te rm s in the m o m en tu m equations, undoubtedly by using an E scaling fo r th e boundary la y e r th ic k n e s s . A n a n a ly s is of th is d e ta il of the flo w fie ld w as not pursued, but one should b e a r in m in d that it is in th is th in la y e r ad jacen t to the c y lin d ric a l w a ll th a t m ost of the flu id re tu rn s to the E k m a n la y e rs on the top and bottom d is c s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p te r I I I E X P E R IM E N T 1. In tro d u ctio n to L a s e r D o p p le r A n e m o m e try Thé D o p p ler freq u en c y s h ift of a signal is a fa m ilia r o c c u r- | I rence to persons who have sensed the change in pitch of aud ible I freq u en cies fro m the w h is tle of a passing tra in . The am ount of i freq u en cy s h ift is p ro p o rtio n a l to the m agnitude and re la tiv e v e lo c ity betw een the source and the o b s e rv e r. A s tro n o m e rs use the "re d sh ift" D o p p ler effe c t to d e te rm in e the re c e s s io n v e lo c ity of j d istan t s ta rs ; in this case the source em its its own ra d ia tio n . M o re I i re c e n tly CW (continuous w ave) la s e rs have been used to m ake la bora - 1 ! to ry m easu rem en ts of re la tiv e v e lo c itie s by s c a tte rin g h ig h ly co - | h e re n t m on o ch ro m atic lig h t fro m m oving o b jects. The D o p p le r freq u en cy is obtained by hetero d yn in g (o p tic a lly m ix in g ) the illu m in atin g ra d ia tio n w ith the D o p p le r shifted ra d ia tio n on the face of a photo d e te c to r. Y eh and C um m ins [1 3 ] w e re the f ir s t to use the D o p p ler s h ift of ra d ia tio n s c a tte re d fro m s m a ll p o ly s ty re n e spheres suspended in a m oving flu id to d e te rm in e v e lo c ity a t a p o in t. M an y other in v e s tig a to rs have since used the L a s e r D o p p le r V e lo c im e te r (L D V ) to obtain v e lo c ity m eas u rem en ts in liq u id s and gases fo r a v a rie ty of flo w conditions, e . g . . F o re m a n , et a l. [ 1 4 ] , G oldstein & K rie d [151 , and B ien & F e n n e r [1 6 ] to nam e a few . 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 I The LJDV has som e d is tin c t advantages o ve r m a te r ia l probes : i I such as h o t-w ire a n e m o m e te rs , ra k e s , p ito t tubes, etc . The basic | sho rtcom ing of these con ventio nal d evices is th at they n o ticeab ly j d is tu rb the flu id and th e re fo re c o rre c tio n fa c to rs m u st be introduced | to re -e s ta b lis h the unpertu rb ed con ditio ns. In a d d itio n , none of the | conventional probes m e a s u re v e lo c ity d ir e c tly . M o re o v e r, w ith the I exception of the h o t-w ir e , m ost probes have poor s p a tia l re s o lu tio n | j and a re sluggish in so far as freq u en cy response is con cerned. In ; c o n tra s t, the n o n in tru sive L D V v ir tu a lly e lim in a te s the p ertu b atio n j i effects; the frequency response is ty p ic a lly in the m e g a H e rtz range; j i m easu rem en ts can be m ade in inaccessib le a re a s such as c le a r a ir | I turbulence detection; v e lo c itie s have been m eas u red fro m as lo w as I i -4 3 10 c m /s e c to speeds of 10 m /s e c [1 7 ] and fin a lly , w ith the aid of - 6 d iffra c tio n lim ite d len ses , probe volum es on the o rd e r of 10 cc a re re a d ily obtained. The L D V does re q u ire s c a tte rin g p a rtic le s suspended in a s e m i-tra n s p a re n t m ed iu m fo r flu id d yn am ic m e a s u re m ents; m ost flu id s of in te re s t, fo rtu n a te ly , possess this la tte r q u a lity . It is easy to see, then, w hy la s e r D o p p le r a n e m o m e try is becom ing an in c re a s in g ly im p o rta n t to o l fo r e x p e rim e n ta l flu id d y n a m ic is ts . A survey of the m o st re c e n t developm ents in the fie ld of la s e r D o p p ler a n e m o m e try is given in a re p o rt [1 8 ] on E U R O M E C H 36. M uch of the la te s t e ffo rts have been d ire c te d to w ard s im p ro v e m en t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 of the sig n al processing e le c tro n ic s w ith em phasis on tu rb u le n t flo w i m e a s u re m e n ts . O f p a rtic u la r in te re s t w as the p re s e n ta tio n of pow er s p e c tra data fo r fu lly developed pipe flo w . U n til the p re s e n t th is type of m e a s u re m e n t could only be obtained w ith h o t-w ire s . I 2. The S c atterin g P ro c e s s I In o rd e r to obtain a D o p p le r s ig n a l, co h eren t m o n o c h ro m atic | lig h t m u st be s c a tte re d fro m a m oving o b ject. In the case of a solid (say a m oving piece of m a c h in e ry ) the su rface can u s u a lly be tre a te d so th a t it has high re fle c tiv e p ro p e rtie s and hence one can incorpo^ ra te low pow er la s e rs . In som e instances m ir r o r s can be m ounted | on the m oving object to in s u re v ir tu a lly no loss in the s c a tte re d | beam p o w er. j ! F o r flu id d y n a m ic a l a p p lic a tio n s , h o w e v e r, the d e te c tio n of ! I s c a tte re d lig h t becom es a m a jo r p ro b le m . One m ig h t in itia lly hope j I th at th e re would be enough s c a tte re d in te n s ity fro m the m o lecu les to i p ro vid e an observable h etero d yn e s ig n a l. F o r a gas, w h e re the index o f re fra c tio n is n e a rly u n ity and the m o le c u le s a re m uch s m a lle r than the illu m in a tin g w avelen g th of the la s e r beam , R a y le ig h s c a t te rin g o ccu rs. In ty p ic a l la b o ra to ry e x p e rim e n ts the ra tio of the — 13 s c a tte re d to the in cid en t lig h t in te n s itie s is of the o rd e r 10 D e te c tio n o f R a yle ig h s c a tte rin g has been m ade possib le w ith the aid | ' I o f h ig h pow ered pulsed ru b y la s e rs [ 1 9 Î , but the r e la tiv e ly low pow ers asso ciated w ith CW la s e rs m ake th is m e a s u re m e n t im p o ssib le. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 ! On. the o th er hand, if the d im en sio n of the s c a tte rin g p a rtic le s is on the o rd e r of the illu m in a tin g w avelen g th (o r la r g e r ), a m uch s tro n g e r s c a tte rin g c a lle d M ie s c a tte rin g w ill p re v a il. F o r this reas o n p re s en t day re s e a rc h in la s e r D o p p le r a n e m o m e try re q u ire s the ad d itio n o f co n tam in an t p a rtic le s . T h e y a re u s u a lly of m ic ro n s ize so that they w ill e a s ily tra c e the flu id m otion (cf. re fe re n c e [ 2 0 ] ) , and of low enough num ber d en sity so as to not s e rio u s ly a ffe c t the flu id m o tio n . O rd in a ry tap w a te r often contains enough n a tu ra l im p u ritie s , but a ir g e n e ra lly needs to be seeded w ith sm oke o r som e o th er con ta m in a n t. 3. M odes of O p eratio n T h e re a re a num ber of o p eratin g c o n fig u ratio n s fo r the L D V . One o f the o rig in a l m ethods, so m etim e s r e fe r r e d to as the d e te c to r optics m eth o d ,is sketched in F ig u re I I I . 1. T h e illu m in a tin g optics s im p ly p ro vid e the focused ra d ia tio n a t a point P in the flu id . The h e te ro dyning is done by the d e te c to r o p tics. R a d ia tio n passing through lendes L 2 and L 3 , lik e w is e focused a t P , is alig n e d on the face of a p ho todetecto r by m ir r o r s M l , M 2 , and a beam s p litte r . The a p e r- a tu re in su res that the ob served s ig n a l o rig in a te s fro m a r e la tiv e ly s m a ll probe v o lu m e . This technique has the disadvantage of re q u ire ing high a c c u ra c y in the position ing of the o p tic a l elem en ts since a la rg e .re d u c tio n in sig n al c u rre n t is o b s erved fo r s lig h t m is a lig n m e n ts . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 D e te c to r O ptic s M 2 A p e rtu re - M l e te c to r L3 B e a m S p litte r L 2 Illu m in a tin g O ptic s < |> L 1 C W L a s e r F ig u re I I I . 1. S chem atic of L D V in the d e te c to r optics m ode. Shipp, e t a l. [2 1 ] and a t the sam e tim e G o ld s te in and K rie d [ 15] im p ro v e d on the m ethod by in tro d u cin g the c ro s s -b e a m s y s te m . In this m ode the re fe re n c e and s c a tte re d w avefro n ts a re alig n e d by s im p ly causing two beam s to focus a t a com m on p o in t. S c a tte re d ra d ia tio n fro m both beam s consists of m a in ly s p h e ric a l w aves c e n t e re d about the com m on fo c a l point; thus the hetero d yn in g is effe cte d a t the probe vo lu m e. A ra th e r ingenious v a ria tio n due to B ray to n [ 2 2 ] is the s e lf-fo c u s in g illu m in a tin g o p tic s . The basic e le m e n t in tiiis s e t-u p , sketched in F ig u re I I I . 2, is the p a ra lle l s u rfa c e fla t. The f la t sp lits the incom in g ra d ia tio n into two p a r a lle l beam s w hich a re a u to m a tic a lly focused a t a com m on point by the s im p le lens L l . Since the heterodynin g takes place a t the fo c a l point, the detection optics can th e o re tic a lly be alig n e d along an y rad iu s em an atin g fro m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 . DetectbPi O p tic s n itim in a tin g O ptics CW L a s e r 9 /2 L 2 D e te c to r L l P a r a lle l F la t F ig u re I I I . 2. Se If-fo c u s ing L D V set up in (a) the re fe re n c e beam m ode o r (b) the d iffe re n tia l D o p p ler m ode. that point. Two o p tim u m positions a re illu s tra te d in the sketch. The re fe re n c e beam m ode (a) is ty p ic a lly used when the co n cen tratio n of s c a tte rin g p a rtic le s is f a ir ly high; h e re the e^ beam is v ir tu a lly u n s c a tte re d and the ra d ia tio n fro m the e^ b eam is s c a tte re d through an angle 6 . In the d iffe re n tia l D o p p ler mode (b) both beam s a re s c a tte re d through an angle 8 /2 ; this co n fig u ratio n is g e n e ra lly used when the s c a tte re d lig h t in te n s itie s a re lo w . The d iffe re n tia l D o p p ler m ode is also known as the "frin g e m ethod" because the beam s form , a frin g e p a tte rn in the volum e of in te rs e c tio n . R e c e n tly [2 3 ] this frin g e p a tte rn has been used to m m e a s u re the v e lo c ity of in d iv id u a l p a rtic le s in a flo w m ed iu m . If the flo w is w e a k ly seeded so that the fo c a l volum e contains only one p a rtic le a t a tim e , the tim e of flig h t betw een frin g e s can be m e a s u red . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 I Know ing the distances betw een frin g e s , one can e a s ily d e te rm in e the speed of the p a rtic le . The o p tic a l arra n g e m e n ts in F ig u re s I I I . 1 and i n . 2 a re s e t up fo r fo rw a rd s c a tte rin g . Iln som e cases, fo r exam ple in the àbsence of a second window in a w ind tunnel, a b ac kw a rd s c a tte rin g sys te m m ay be d e s ira b le . In this case the d e te c to r optics and illu m in a tin g optics a re lo cated on the sam e side of the te s t c h a m b e r. The backw ard s c a tte rin g m o d e, h o w ever, u s u a lly re q u ire s f a ir ly p o w erfu l la s e rs (ty p ic a lly 1 -5 w atts) even fo r flow s w hich a re h e a v ily seeded. The state of the a r t in la s e r D o p p ler a n e m o m e try is p ro g res sin g so fa s t that the modes of op eratio n discussed above m a y soon be antk- quated. One should co n sid er the techniques presen ted h e re as p a rt of the evo lution of a ra p id ly changing fie ld . 4. The D o p p le r S h ift E q u atio n The equation re la tin g the D o p p ler fre q u e n c y s h ift to the v e lo c ity v of the s c a tte rin g p a rtic le s is d e riv e d in Appendix A . The n o n re la tiv is tic re s u lt is given by n(e - e ) • V Vd = — -------------------------- (in. I) O w h e re n is the absolute index of re fra c tio n of the m ed iu m surroun ding the s c a tte rin g p a rtic le s , is the vacuum w avelength of the illu m in atin g ra d ia tio n , and e^ and e^ a re the re s e p c tiv e u n it ve c to rs ReprocJucecJ with permission of the copyright owner. Further reproctuction prohibitect without permission. ...................... 59 p e rp e n d ic u la r to the in cid en t and s c a tte re d w ave fro n ts in the flu id m ed iu m . We now co n sid er the s p e c ific o p e ra tio n a l m odes p re v io u s ly discussed. In the d e te c to r optics a rra n g e m e n t (F ig u re I I I . 1) the w ave fro n ts along e^ can be alig n ed w ith the illu m in a tin g ra d ia tio n w ith lens 1,3. The s c a tte re d ra d ia tio n is then picked up by lens 1 2 along the d ire c tio n e^. Thus (e^ - e^) = -s in 0 i + (cos 0 - l ) j V = - V I and consequently ( III. 2) o so we see th at the D o p p le r sh ifted fre q u e n c y depends on the s e t-u p (6) of the d e te c to r optics s y s te m . If one w ished to m ake a v e lo c ity tra v e rs e acro s s the flo w fie ld , a ll the lenses w ould have to be m oved sim u lta n e o u s ly . A ls o , if d ir e c t v e lo c ity m e as u rem en ts a re d e s ire d , one w ould need an a c c u ra te d é te rm in a tio n of the included angle 8. In the s e lf-fo c u s in g c ro s s -b e a m sys te m (F ig u re I I I . 2) the d ir e c tions e^ and e^ a r e uniqu ely d e te rm in e d by the illu m in a tin g optics. F o r this a rra n g e m e n t we have Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (e - e ) = 2 s in (3 /2 )j s o _ V = vj and equation ( III. 1) gives I ^ . 2- V (9/2) , . (III. 3^ Ï In a la te r sectio n w e s h a ll show how the angle 0 can be m ea s u re d | I w ith good a c c u ra c y . W e a ls o note in th is system th a t v e lo c ity t r a - j v e rs e s can be m ade w ith r e la tiv e ease by s im p ly tra n s la tin g the j focusing lens L l . | 5. H etero d yn e S ignal O p tim iz a tio n | A p h o to m u ltip lie r tube (P M T ) is g e n e ra lly used to m e a s u re the D o p p le r s h ift fre q u en c y . Used as a d e te c to r, the P M T is pow ered a t the lo w e r lim it of its d e te c tio n a b ility in o rd e r to reduce the d a rk c u rre n t (th e rm io n ic c u rre n t e m itte d fro m the s e n s itize d coating in the absence of in cid en t lig h t). W hen used as a m ix e r , on the o ther hand, the P M T is m o st e ffic ie n t w hen op erated n e a r the peak output le v e l. In th is case the shot noise (noise accom panied by the signal due to s ta tis tic a l flu ctu a tio n s of the pho toem ission c u rre n t) is m uch g re a te r than the d a rk c u rre n t. C onsequently, cooling of the m u lti p lie r tube, g e n e ra lly of co n s id e ra b le b en efit fo r the red u ctio n of d a rk c u rre n t, is not n e c e s s a ry w hen the P M T is used as a m ix e r. T h e re a re a m u ltitu d e of fa c to rs w hich d ir e c tly a ffe c t the out - put (anode) c u rre n t of the p h o to m u ltip lie r. These include the re la tiv e pow er le v e ls of the in cid e n t and s c a tte re d beam s, the h eterodyning Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 e ffic ie n c y , the s ize of the probe v o lu m e, the co n cen tratio n of s c a t te rin g p a rtic le s , the quantum e ffic ie n c y of the P M T , etc. W hat we s h a ll be concerned w ith h e re is , given a m a x im u m pow er P d e - m tected by the photocathode, how can one o p tim ize the output sig n al? The equation fo r the anode c u rre n t, d e riv e d in A p pendix B , is given by ^ ^ cos t + 6 ) ] (U I. 4) w h ere P and P a re the pow er le v e ls of the re fe re n c e and s c a ttered o s beam s in cid en t on the photocathode, r) is the hetero d yn in g e ffic ie n c y , (Dg is the c ir c u la r fre q u en c y of the D o p p le r s h ift, G is the o v e ra ll gain of the P M T , and 6 is the constant phase sh ift betw een the two b eam s. Now suppose the P M T has a m a x im u m safe c u rre n t le v e l (I ) , then the m a x im u m in cid en t pow er is lim ite d by a m (I ) ■ '“ -51 The m a x im u m c u rre n t in ( III. 4) occurs w hen T| = cos '((Wjjt.+ Ô ) = 1, so th at P in ( III. 5) is re la te d to the re fe re n c e and s c a tte re d ra d ia tio n m pow er by the equation p = p + p + 2 V P P ■ . (n i. 6) m o s o s W e w ish to m a x im iz e the m agnitude of the D o p p le r s ig n al 1 =20 y P P ' cos (u)^ + 6) ( III. 7) I s a o s D i I subject to the re s tric tio n ( III. 0). It is a s im p le e x e rc is e to show Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 th at 2 G v P P is a m a x im u m when a o s P = P = -4 - P (IH. 8) o s 4 m in w hich case the am p litu d e of the D o p p le r sig n al is ( I^ ) ^ /2 . O f even g re a te r im p o rtan ce in the detectio n of the D o p p ler sig n al is the anode s ig n a l-to -n o is e pow er r a tio , denoted (S /N ) . In A ppendix B it is shown th at ( 4 ) / ^ ( ^ ) w h ere is the lum inous s e n s itiv ity of the cathode, e the unit e le c tro n ch a rg e , A ll) the e ffe c tiv e noise bandw idth, and K is the m u ltip lie r noise fa c to r. A g a in it is a sim p le m a tte r to show th at equation ( III. 8)1 gives the re la tiv e beam pow er con ditio n w hich m a x im iz e s the s ig n a l- j I to -n o is e pow er r a tio a t the output of the p h o to m u ltip lie r. I I F r o m the above co n sid eratio n s we conclude th at both the s ig n al j pow er and the s ig n a l-to -n o is e ra tio a re m a x im iz e d by m atch in g the | 1 in te n s itie s of the re fe re n c e and s c a tte re d beam s. O ften, h o w e v e r, I P^ is less than P ^ . In th is case the sig n al can g e n e ra lly be o p tim ized by o p eratin g the P M T a t the peak a llo w a b le voltage and in creasin g the re fe re n c e pow er P^ u n til the m a x im u m a llo w a b le c u rre n t is reached. 6. P a ra m e te rs A ffe c tin g the S ignal N oise and P o w er The fo reg o in g an a ly s is was p re d ic a te d on a system of p e rfe c tly heterodyned beam s o p tim a lly alig n ed on the face of the pho todetector. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 W e now b r ie fly co n sid er the p a ra m e te rs w hich a ffe c t the D o p p le r s ig n al p r io r to the d etec tio n u n it. M uch of the in fo rm a tio n is taken ! I fro m re fe re n c e s [17] and [2 4 ] . A d d itio n a l re fe re n c e s w ill be cited j I w hen n e c e s s a ry . I 6. 1 In s tru m e n t B roadening The p h o to m u ltip lie r detects not a single D o p p le r fre q u en c y but | ra th e r a band of fre q u en c ie s a p p ro x im a te ly in a G aussian d is trib u tio n about a c e n te r fre q u e n c y . It is d e s ira b le to m in im iz e this spec- | 1 t r a l broadening of the re c o rd e d sig n al fo r tra c k in g and m e a s u re m e n t | purposes. In s tru m e n ta tio n broadening, as the nam e im p lie s , re fe rs j to broadening caused by the o p tic a l d etectio n and e le c tro n ic p ro c e s s - | i ing units. S p ectru m a n a ly z e rs , fo r exa m p le, have a fin ite band- | w id th w hich co n trib u tes to the s p e c tra l broadening. M is a lig n m e n t of the beam s along the a x is of the P M T as w e ll as the m u tu a l m is a lig n - \ m en t of the re fe re n c e and s c a tte re d w aves both co n trib u te to signal pow er loss and s p e c tra l broad ening. This la tte r e ffe c t is of m a jo r con cern w hen using the L D V in the d e te c to r optics m ode. In s tru m e n ta l broadening is also a function of both the s c a t te rin g angle 0 and the solid angle AQ detected by the p h o to m u ltip lie r. A sim p le an a lys is shows th at the freq u en cy sp read AVq Is re la te d to the detection g e o m e try by / 4 An V TT y (H I. 10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 4 1 C le a rly one w ould lik e to have s m a ll s c a tte rin g angles and s m a ll | a p e ra tu re s in o rd e r to m in im iz e the s p e c tra l bandw idth. \ 6 . 2 D iffu s io n B roadening D iffu s io n broadening is the te r m ap p lied to s p e c tra l broadening | I due to the ran d o m B ro w n ian m o tio n of the s c a tte rin g p a rtic le s in a | flu id . One can use k in e tic th e o ry to show th a t p a rtic le s of m ic ro n | ! size have m ean speeds on the o rd e r of 1 m m /s e c . It is not only th is 1 instantaneous m o tio n w hich co n trib u tes to the d iffu s io n broad ening, j I but also the ran d o m w a lk of the p a rtic le s . Random w a lk th e o ry | (see [2 5 ] , pp. 4 1 -4 9 ) states th at the m e a n squ are d istance a p a rtic le j I d rifts is d ir e c tly p ro p o rtio n a l to tim e ; since a p a rtic le has a fin ite j i resid en c e tim e in the probe vo lu m e, it w ill d iffu se w h ile it is co n - | vected along w ith the m ean flu id v e lo c ity . T h is e ffe c t becom es e s p e c ia lly im p o rta n t a t low speeds w h e re the re s id e n c e tim e s a re la rg e , and a c tu a lly d e te rm in e s the lo w e r lim it on m e a s u re a b le v e lo c itie s . E d w a rd s , et a l. [2 6 ] have shown th a t re s o lu tio n of the D o p p le r sig n al becom es im p ra c tic a l fo r v e lo c itie s V < ^ sin (6 /2 ) ( III. 11) o w h e re (6 /2 ) corresponds to the s e lf-fo c u s in g optics m ode. A ll the j sym bols a re fa m ilia r except D w h ich is the d iffu s io n c o e ffic ie n t of j the p a rtic le s . F o r ty p ic a l v a lu e s , e .g . , 6 = 1 0 °, = 6328 ^ , n= 1, j and D = 10 ^ c m ^ /s e c , one finds the lo w e r g T a c tic a l lim k t v = 2 x 10 ^ | I — cm/aÆ-C^--------------------------------------------------------------------------------------------- 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 6. 3 V e lo c ity G ra d ie n ts In a c tu a l flu id m ea s u re m e n ts the detected s ig n a l has c o n tr i butions fro m the s c a tte re d lig h t ra d ia te d by each m oving p a rtic le in the fo c a l vo lu m e . If a v e lo c ity g ra d ie n t exists w ith in the probe v o lu m e , the pho todetecto r w ill re c o rd a c o m p a ra b le range of D o p p ler fre q u e n c ie s . Thus in a d d itio n to the obvious d e s ire to obtain good s p a tia l re s o lu tio n , the e x p e rim e n ta lis t is encouraged to use s m a ll fo c a l volum es in o rd e r to m in im iz e the a s s o c ia te d s p e c tra l b ro a d e n in g . M o re o v e r, E d w a rd s , e t aj^ [2 6 ] have pointed out th a t in la rg e v e lo c ity grad ien ts the G a u s s ia n -lik e d is trib u tio n of the D o p p le r s ig n a l becom es a s y m é trie , and the fre q u e n c y s h ift co rres p o n d in g to the v e lo c ity a t the c e n te r of the probe vo lu m e is no lo n g e r concident w ith the peak in the s p e c tra l d is trib u tio n . T h is e ffe c t was o b served in th e ir la b o ra to ry m e a s u re m e n ts , but the v e lo c ity g ra d ie n ts w e re q u ite la rg e and the d is p la c e m e n t of the d e s ire d fre q u e n c y fro m the peak was r e la tiv e ly s m a ll. A ls o , the d istan ce o v e r w hich such steep g ra d ie n ts g e n e ra lly occu r is so s m a ll th a t one is m o re concerned w ith the s p a tia l re s o lu tio n p ro b le m due to the fin ite s ize of the probe v o lu m e . T u rb u le n t v e lo c ity flu c tu a tio n s w ith in the probe vo lu m e w ill a lso give ris e to s p e c tra l bro ad en in g , but of cou rse one w ould lik e to m e a s u re this tru e D o p p le r s ig n a l. F o r this reas o n it is o f p r im a ry im p o rta n c e to keep the in s tru m e n ta lib ro a d e n in g a t a m in im u m in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ — 66" tu rb u len ce m e a s u re m e n ts in o rd e r to p ic k out the s p e c tra l d is t r i bution w hich a c tu a lly re s u lts fro m the t e a l v e lo c ity flu c tu a tio n s . 6..4. S c a tte rin g A n gle P e rh ap s the single m o st im p o rta n t e ffe c t on the sig n al pow er is the angle a t w hich the s c a tte re d ra d ia tio n is d etected . A s one m ig h t exp ect, M ie s c a tte rin g is s tro n g ly peaked in the fo rw a rd d ire c tio n . T h is corresponds to s m a ll 6 in F ig u re s I I I . 1 and I I I . 2. Thug one is fo rc e d to m ake a co m p ro m is e in s e le c tin g the s c a tte rin g ang le since both s p e c tra l broadening and p robe volum e in c re a s e w ith d e c re a sin g 0 . (F o r a d iscu s sio n of the probe v o lu m e , see A p pendix D , p a ra g raph 2 . 1 . ) 6. 5^ P ath len g th Coherence The id e a l la s e r fo r use in L D V system s w ould be one fo r w hich a ll the pow er is in a sin g le a x ia l m ode (T M ^ ^ ) , h o w ever m an y a x ia l m odes of fre q u e n c y spacing Av = c / 2 L (c is the speed of lig h t and L is the la s e r c a v ity length) e x is t in CW la s e rs . F o re m e n [2 7 ] has shown th at a m is m a tc h in the pathlengths of the h eterodyned beam s can re s u lt in an in te rfe re n c e o f these a x ia l m odes of e x c ita tio n , causing a re d u c tio n in sig n a l p o w e r. The g re a te r the pathlength d iffe re n c e s , the m o re s ev ere is the pow er re d u c tio n . A ls o , fo r a g iven m is m a tc h , h ig h e r pow er la s e rs w ill e x p erien ce a g re a te r p e r centage pow er lo ss. T h is is because the la s e r c a v itie s a re lo n g er and hence the a x ia l m odes a re m o re c lo s e ly spaced and can m o re Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 j re a d ily in te rfe r e . j 1 6. 6 L a s e r N oise | A n o th er p a ra m e te r w hich can a ffe c t the D o p p le r sig n al is the | lo w -fre q u e n c y noise in h e re n t in a ll la s e rs . Logan [2 8 ] has shown i that the m agnitude of this noise m eas u red by a p h o to m u ltip lie r is I in v e rs e ly p ro p o rtio n a l to the fre q u en c y scanned by a sp ectru m a n a ly z e r . D e te c tio n of the D o p p le r sig n al a t lo w e r fre q u en c ies b e- | com es m o re d iffic u lt, th e re fo re , not because the sig n al pow er is red u ced , but ra th e r because the s ig n a l-to -n o is e ra tio is d im in is h ed . 6. 7 H ertodynin g E ffic ie n c y The heterodynin g e ffic ie n c y T ) (or coherence loss fa c to r) is affected by the g e o m e tric a lig n m e n t of the two beam s, the source coherence of the illu m in a tin g beam , the tra n s m is s io n path co h erence. lo sses , and the unequal pathlengths p re v io u s ly discussed. The geo m e tric a lig n m e n t p ro b le m is v ir tu a lly e lim in a te d by the self-fo c u s in g optics system . E ven if the fo c a l volum es of the two in te rs e c tin g beam s do not e x a c tly co in cid e, th e re w ill be no loss in alig n m e n t of the ra d ia tio n em anating fro m the p a r tia lly overlap p in g vo lu m es, although th e re w ill be a s ig n ific a n t red u c tio n in sig n al p o w er. Source coherence can o nly be im p ro ved by obtaining a b e tte r q u a lity la s e r. T ra n s m is s io n path coherence losses a re caused by d en sity flu c tu a tions along the la s e r beam path w hich a lte r the re fra c tiv e index of the m e d iu m . U n d er n o rm a l conditions in la b o ra to ry e x p erim en ts Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 this e ffe c t is co n sid ered to be n e g lig ib le . 7, V e lo c ity M easu re m e n ts of a Solid B ecause of the num erous fa c to rs in fluencing the heterodyned s ig n a l, it was decided to fo llo w the advice o f Logan and f ir s t t r y to o bserve the D o p p ler sig n a l produced by s c a tte rin g lig h t fro m a r o ta ting fro s te d lu c ite w h eel. The fro s te d s u rface p ro vid es e x c e lle n t s c a tte rin g cen ters fo r the in c id e n t la s e r beam and hence strong s c a tte re d ra d ia tio n . A ls o , since the g e o m e try is s im p le , i t is not d iffic u lt to m ake d ire c t v e lo c ity m e a s u re m e n ts , th e re b y gaining a d d itio n a l e x p erien ce in the use of la s e r D o p p ler a n e m o m e try . 7. 1 E x p e rim e n ta l S e t-U p The s e lf-fo c u s in g o p tic a l a rra n g e m e n t fo r the ro ta tin g w h eel e x p e rim e n t is sketched in F ig u re I I I . 3. A c ir c u la r d isc was m a c h ined fro m a stock piece of % - inch lu c ite plate selected fo r its f la t n ess. The d isc was sandblasted on one side w ith fin e g ra in sand a t low a ir p re s s u re , thus p ro vid in g a u n ifo rm ly h azy s u rfa c e . The disc w as then m ounted on the d riv e s h a ft of an e le c tric m o to r w hich in tu rn w as m ounted on a tw o -d im e n s io n a l tra v e rs in g m ech a n is m . One axis of the tra v e rs e provided r a d ia l position ing of the fo c a l point w h ile the m u tu a lly p e rp e n d ic u la r axis of the second tra v e rs e pro vid ed fo r m o vem en t o f the disc in and out of the fo c a l plane. In this m an n er fin e fo c a l ad ju stm en ts could be e a s ily m ade a t each ra d ia l position w ithout a lte rin g e ith e r the illu m in a tin g o r detecting o p tics. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 L u c ite W h eel L a s e r Lens P h o to d etecto r P a r a lle l F la t F ig u re I I I . 3. Sketch of the s e lf-fo c u s in g (c ro s s -b e a m ) la s e r optics used fo r m e a s u rin g the a z im u th a l v e lo c ity of a ro ta tin g lu c ite w h eel. The illu m in a tin g ra d ia tio n was p ro vied by and E le c tro -O p tic s M o d e l 125 H e -N e CW la s e r. The sig n al was detected by an R C A 8645 I j (S -20 response) p h o to m u ltip lie r w hich w as pow ered by a K rie th le y I I 245 pow er supply. The sig n al w as fed d ire c tly to a T e k tro n ix M o d el 564 oscilloscope equipped w ith a s p e c tru m a n a ly z e r p lu g -in . In itia l e x p e rim e n ts w e re c a rrie d out in a d arkened ro o m in o rd e r to e lim in ate any spurious lig h t fro m h ittin g the face o f the P M T , The la s e r w as focused on the fro s te d side of the ro ta tin g w h e e l and the pow er to the P M T was g ra d u a lly in c re a s e d u n til a D o p p le r sig n al w as ob served a t about 1300 v o lts . T h is is w e ll below the o p tim u m o p erating pow er w hich is a p p ro x im a te ly 1500 volts fo r the H e -N e w avelength. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 W ith a fu rth e r in c re a s e to about 1350 vo lts the P M T could e a s ily d e te c t the D o p p le r s ig n al even w hen m oved 1 0 -1 5 ° out of lin e of the re fe re n c e b e a m . T h is m eans th at the h etero d yn ed s ig n a l con sisted of s c a tte re d lig h t fro m both b ea m s , d e m o n s tra tin g th a t the s y s te m had m o re than enough pow er and d etectio n a b ility . The s ig n a l-to - noise ra tio ranged fr o m about 15 a t 100 K H z to o n ly 2 a t 400 K H z. Som e exam p les of the o b s erved signals a re p rese n te d in F ig u re X . 3. A n a e e u ra te d e te rm in a tio n of the D o p p le r fre q u e n c y w as m ade by alig n in g an o s c illa to r s ig n a l w ith the peak in the s p e c tru m s to re d on the d isp lay s c re e n of the o scillo sco p e as in d ic a te d in F ig u re X . 3d. The o s c illa to r fre q u e n c y was re c o rd e d by a d ig ita l c o u n te r. 7. 2 V e lo c ity M easu re m e n ts D ir e c t m e a s u re m e n ts of the w h eel speed w e re m ade in the fo llo w ing m a n n e r. The tra v e rs in g m e c h a n is m was c a re fu lly a lig n e d as p re v io u s ly d e s c rib e d so th a t ju s t the a z im u th a l v e lo c ity com ponent w ould be m e a s u re d . The la s e r beam s w e re focused on the c e n te r of the w h eel and the v e r n ie r read in g was re c o rd e d . The m o to r was then s ta rte d and a llo w ed to ru n fo r o n e -h a lf h o u r so a l l m oving p arts could : w a rm up. The a n g u la r speed of ro ta tio n was obtained by a v e ra g in g th re e s e p a ra te tim in g s o f 100 w h eel re v o lu tio n s w ith a stop w atch. The d isc was then tra v e rs e d to d iffe re n t r a d ii and the co rres p o n d in g v e r n ie r positions and D o p p le r fre q u e n c ie s w e re noted. A t the end of the ru n the a v e ra g e speed of the w h eel was again m e a s u re d to in s u re Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 th at the ro ta tio n ra te was con stant. W h eel speeds m e a s u re d by the stop w atch w e re c a lc u la te d fro m the re la tio n v = iQ w here O was the a v e ra g e an g u la r v e lo c ity of the s ix tim in g s . No m e a s u re d ro ta tio n ra te d eviated m o re than 1% fro m a n o th e r, and since the r e la tiv e e r r o r in the rad iu s m e a s u re m e n ts was le s s than 0. 004 the e r r o r in the v e lo c ity m eas u re m e n ts should s a tis fy dv V tim e d = The v e lo c ity d e te rm in e d fro m the D o p p le r s h ift can be c a lc u la te d fr o m equation ( III. 3). The constants n and a re known to c o n s id e r able a c c u ra c y so the only re m a in in g m e a s u re m e n t to m ake is the ang le 0. Know ing the fo c a l len g th f of the le n s , one could s im p ly m e a s u re the distance betw een the p a r a lle l beam s and c a lc u la te the angle a t w hich the beam s c ro s s . T h e b eam s e p a ra tio n , h o w e v e r, cannot be a c c u ra te ly d e te rm in e d w ith s im p le m e a s u rin g d e v ic e s . If the p a r a lle l su rfa c e f la t is w e ll m ad e, w hich indeed i t m u s t be in o rd e r to a s s u re th at the fo c a l points of the two beam s w ill be c o in - 5 c id en t, the angle 0 can be c a lc u la te d fro m the equation 5 T he p a r a lle l fla t used in these e x p e rim e n ts had su rface s f la t to w ith in 1 /2 0 and p a r a lle l to w ith in 0. 01 arcs eco n d s . M e a s u re m ents w ith a m ach in ists s c a le showed no p e rc e p tib le d e v ia tio n betw een the beam s a c ro s s a 2 0 -fo o t ro o m . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 I _i/sia8A| — taa^G/Z) = cos 6^ tan J sin I— -— I I ( III. 13) obtained fro m the g e o m e try of the illu m in a tin g optics in F ig u re I I I . 3. I t m ig h t a t f ir s t a p p e a r that this m ethod w ill y ie ld a less a c c u ra te d e te rm in a tio n than d ir e c t m e a s u re m e n t since i t has in tro d u ced th ree a d d itio n a l fa c to rs , n a m e ly , & the thickness of the p a r a lle l fla t, n^ its r e fra c tiv e index, and 8^ the angle of incidence of the la s e r b eam on the fla t. B ut these constants can be d e te rm in e d w ith co n sid erab le p re c is io n . F o r e x a m p le , the re fra c tiv e index is re p o rte d to be 1 .5 1 7 4 ± 0. 0005 and the fla t thickness can e a s ily be m eas u red to w ith in one p a rt in a thousand. The angle 8^ can be a c c u ra te ly d e te r m in ed in m an y w ays; one m ethod is to m ea s u re the v e r tic a l distance betw een in c id e n t and re fle c te d beam s a t la rg e distances fro m the fla t and c a lc u la te 8. fro m trig o n o m e tric re la tio n s . The angle 9 was calc u la te d in the m anner d es c rib e d above and the v e lo c ity -fre q u e n c y re la tio n was found to be V = , (H I. 14) -5 w h ere = 4. 29 X 10 c m /s e c /H z . V e lo c itie s c a lc u la te d fro m this equation w ere plotted a g a in s t the tim ed m e a s u re m e n ts , but, although th ere was lit tle s c a tte r about a s tra ig h t lin e , tthe slope of the lin e was a lm o s t 3% in e r r o r . T h is m eans th at the constant of p ro p o rtio n a lity ^yas in c o rre c t. Since a ll the constants except Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 the fo c a l len g th in equation ( III. 13) w e re a c c u ra te ly known, it was presu m ed th&t the n o m in al fo c a l.le n g th re p o rte d by the m a n u fa c tu re r (f = 115 m m ) was a t fa u lt. Subsequently a p ro g ra m was in itia te d to d e te rm in e the fo c a l len g th of the lens in c o rp o ra te d in this e x p e rim e n t and o th er lenses fo r fu tu re use. M easu rem e n ts w e re m ade a t the J e t P ro p u ls io n L a b o ra to ry in P asadena w here the au th o r had access to a p re c is io n o p tic a l bench, tra v e llin g m ic ro s c o p e , and asso ciated equ ipm en t. The fo c a l Impositions of in d iv id u a l p a ra lle l ra y s passing through v a r i ous d ia m e tric a l positions acro s s the lenses w e re d e te rm in e d by the m ethod of re s e c tio n . A n exp lan ation of this p ro ce d u re and a p re s e n ta tio n of som e m easu rem en ts a re given in A ppendix C . F o r the p u r poses h e re we s im p ly re p o rt th a t the fo ch l length of the lens used in this e x p e rim e n t turned out to be f = 118. 4 ± 0. 45 m m . The new constant of p ro p o rtio n a lity in ( III. 14) was then c a lc u la te d to be K jj = 4. 41 X 10 ^ c m /s e c /H z , w hich accounted fo r the d iscrep a n cy in slope. The re s u lts a re p resen ted in F ig u re X . 4. An e r r o r an alysis was und ertaken to d e te rm in e the a c c u ra c y of the absolute v e lo c itie s m e a s u re d by the L D V . T h e a n a ly s is , p r e sented in Appendix D , shows th at the expected re la tiv e e r r o r should be dv V D o p p ler = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 : F r o m ( III, 12) and ( III, 15) we see that h o riz o n ta l and v e r tic a l e r r o r b ars fo r F ig u re X , 4 should both be about 1% of the m ea s u re d values and hence the d eviatio n fro m the 4 5 ° slope is expected tofcbe a p p ro x i m a te ly v (1)^ + (0 .0 0 9 6 )^ o r 1.39% . Indeed the m a x im u m d e v ia tio n was found to be 1 .3 4 %, although u s u a lly the e r r o r was less than 1. 0%. This e x e rc is e in m ea s u rin g the D o p p le r s ig n a l fro m a m oving so lid o b ject provided u sefu l fa m ilia riz a tio n w ith the L D V fo r la te r e x p e rim e n ts in a flu id m e d iu m , e s p e c ia lly in the re c o g n itio n of the D o p p le r sig n a l. A ls o , the an a ly s is in A ppendix D shows that the m a jo r co n trib u tio n (çiO, 8%) to the e r r o r in the m e a s u re d v e lo c ity was due to the fre q u en c y m e a s u re m e n t. T h is in fa c t w ould be the o nly e r r o r in c u rre d if the L D V w e re c a lib ra te d w ith a w e ll-k n o w n v e lo c ity , ra th e r than used to m ake d ir e c t v e lo c ity m eas u rem en ts as in this e x p e rim e n t. 8. V e lo c ity M easu rem e n ts in a L iq u id In this section we d es c rib e in d e ta il the m e th a n ic a l, e le c tro n ic , and o p tic a l s e t-u p fo r the m eas u rem en ts of flu id v e lo c ity in a ro tatin g c y lin d e r. F o llo w in g that is a p re s e n ta tio n of the in itia l m e a s u re m ents fo r steady flo w w ith the flu id in so lid body ro ta tio n . F in a lly we p re s e n t the unsteady flo w m eas u rem en ts obtained fo r s p in -u p and spin-dow n. 8. 1 C y lin d e r C o n stru ctio n The c y lin d ric a l w a ll was m ade out of c le a r lu c ite c y lin d ric a l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 I sto c k a p p ro x im a te ly 12 inches in le n g th , 6 inches in d ia m e te r, w ith a ; w a ll thickness of 5 /8 inches. The p la s tic was not c o n sid ered sm ooth enough fo r o p tical m e as u re m e n ts due to irr e g u la r itie s on the in n e r and o u te r s u rfa c e s , a p p a re n tly s e t up d u rin g the cooling stage of the castin g p ro ce ss. T h e re fo re , the c y lin d e r waLlls w e re c a re fu lly m ach in ed c o n cen tric to w ith in ± 0 . 001 in . and then po lish ed sm ooth. P o lis h in g s e rv e d the dualfpurpose of re m o v in g the w a ll roughness due to the m achining cuts of the la th e tool and re s to rin g the w a ll tra n s p a re n c y fo r the la s e r b eam s. C ir c u la r glass plates w e re m ounted a t each end o f the c y lin d e r ag a in s t su rfa c e s m achined fla t the p a r a lle l. Th e plates w e re h eld in place by p la s tic a n n u la r fra m e s scre w e d s e c u re ly into the c y lin d e r end w a lls . The c y lin d e r was m ade w a te r tig h t w ith the a id of ru b b e r g as kets. Tw o thin rin g b ea rin g s w e re fitte d snugly o ve r the ou ter d ia m e te r and butted up ag a in s t a sho ulder m ach in ed about two inches fr o m each end of the c y lin d e r. An a lu m in u m fra m e was co n stru c ted to a c c e p t the o u te r ra c e s o f the b ea rin g s and hold them r ig id ly in p la c e . F in a l a lig n m e n t, a cc o m p lish e d w ith the a id of s h im stock, p ro v id e d fo r a sm ooth and fr e e ly tu rn in g c y lin d e r. As shown in F ig u re X . 5, the c y lin d e r was b e lt d riv e n by a v a ria b le speed m o to r. The d riv e was e ffe cte d a t one end in o rd e r to keep the m o to r fr o m in te rfe rin g w ith the o p tic s , but this design had the disadvantage of putting a to rq u e on the c y lin d e r w hich could induce a s lig h t wobble in its ro ta tio n . F o r this reas o n b ea rin g s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 w hose r o lle r s had a v e r y s m a ll d ia m e tric a l c le a ra n c e (0 ,0 0 1 - 0. 002 in . ) had to be s p e c ia lly o rd e re d . F in a l dim ensions and t o le r ances fo r the c y lin d e r a re p ro vid ed in T a b le IX . 2. O f p r im a ry im p o rta n c e fo r the sp in -u p m e a s u re m e n ts is the v a ria tio n of the c y lin d e r h eig h t h betw een the glass p la te s . In the th e o re tic a l tre a tm e n t p rese n te d in C h ap ter I I we have im p lic itly assum ed th at the end w a lls a re p a r a lle l So that th e re w ill be no s tre tc h in g of the v o rte x lin e s due to g e o m e try . A c co rd in g to P e d lo sky and G re en sp a n [ 2 9 ] , the end w a lls m ay be co n sid ered p a r a lle l as long as th e ir slops s a tis fy the re la tio n 0 < = 6/h (m. 16) W w h ere Ô = z /v /O is the E k m a n th ic kn ess . The m e a s u re d d eviatio ns in T a b le IX . 2 can be used to c a lc u la te the a v e ra g e w a ll slopes a s s u m ing the v a ria tio n s o cc u r g ra d u a lly o ver the e n tire d ia m e te r, w hich indeed was found to be the c as e. Using the a c c u m u la tiv e d e v ia tio n (±0. 00095 in . ) of the c y lin d e r end s u rfa c e and glass p late fla tn e s s , one finds a m a x im u m ave ra g e w a ll slope of 9^ = 3 .2 X 10 ^ ra d ia n s . Now the h ig h es t a n g u la r v e lo c ity encountered in the s p in -u p m e a s u re m ents w as about 110 r a d /s e c . T h is corresponds to an E k m a n th ic k ness 6 ~ 0. 0032 in. fo r w a te r a t ro o m te m p e ra tu re . One re a d ily find s that (6 /h ) . ~ 2. 8 X 10 and so the c r ite r io n given by m in ( III. 16) is s a tis fie d . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 8. 2 M o to r D r iv e S ystem | 1 A H P v a ria b le speed m o to r w as used to d riv e the c y lin d e r. ! - 1 I Its r . p . m . w as re g u la te d by a constant a c c e le ra tio n c o n tro l u n it, | I designed to a u to m a tic a lly p ro vid e u n ifo rm a c c e le ra tio n o r d e c e le ra - ; i I tio n of the c y lin d e r. A s illu s tra te d in F ig u re I I I . 4, th e usu al s in g le tu rn p o tie n tio m e te r fo r th e v a ria b le speed c o n tro l m o n ito rin g the d riv e m o to r w as re p la c e d by a 1 0 -tu rn p o te n tio m e te r, w hich its e lf w as m e c h a n ic a lly d riv e n a t constant speed by a s m a ll 1/70 H P Bodine m o to r. W ith th is a rra n g e m e n t c u rre n t w as supplied a t a constant ra te to the d riv e m o to r, w hich in th eo ry w ould u n ifo rm ly a c c e le ra te the c y lin d e r. A v a ria b le speed c o n tro l u n it fo r the 1 /7 0 H P m o to r w as used to s e le c t the d e s ire d ra te of a c c e le ra tio n . N ot shown in the fig u re a re the clu tch and lim it sw itches w hich h elped in p ro vid in g re p e a ta b le ru n s . 120 V 120 V V a ria b le Speed C o n tro l 1 /7 0 H P M o to r J X A 10 - tu rn P o t V a ria b le Speed C o n tro l C u rre n t to D riv e m o to r F ig u re I II . 4. C onstant a c c e le ra tio n c o n tro l u n it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 The c o n tro l u n it, u n fo rtu n a te ly , was by no m eans p e rfe c t. F o r one th in g , a c c e le ra tio n fro m re s t n e v e r s ta rte d u n ifo rm ly but alw ays w ith a s m a ll in itia l ju m p in ro ta tio n speed w hen the to rq u e of the d riv e m o to r o ve rc am e the s ta tic fric tio n of the s y s te m . A t high ra te s of a c c e le ra tio n the ju m p in an g u la r speed was not discernible^ but it becam e in c re a s in g ly obvious a t the lo w e r ra te s . S im ila rly , d e c e le ra tio n to r e s t alw ays ended w ith a s m a ll sudden stop. This p ro b le m was kept a t a m in im u m by fre q u e n t lu b ric a tio n of the c y lin d e r support bearin g s and the b ea rin g s fo r t h e R . P . M . In d ic a to r, a lso pow ered by the c y lin d e r d riv e m o to r. The second and m o re serio u s p ro b le m has to do w ith the u n i fo rm ity of the a c c e le ra tio n s . W ith n o -lo a d the a c c e le ra tio n and d e c e le ra tio n ra te s w e re reas o n a b ly constant o ve r the e n tire speed range of the d riv e m o to r, but s ig n ific a n t d ev iatio n s (5-10% ) so m e tim e s occured under load. The a c c e le ra tio n ra te s , h o w e v e r, could be m ade reas o n ab ly constant (1-3% ) w ith one o r two speed a d ju s t m ents of the 1/70 H P m o to r d u rin g the a c c e le ra tio n p e rio d . The d e c e le ra tio n ra te s obtained in th is m an n e r w e re som ew hat (2 - 2%) b e tte r. 8. 3 R . P . M . In d ic a to r System The ang ular speed o f the c y lin d e r was m o n ito re d e le c tro n ic a lly j by the R . P . M . In d ic a to r S ystem . A s shown in F ig u re X . 5, a s m a ll j I shaft , fre e to ro ta te in its b ea rin g p illo w b lo cks, w as g eared up | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 j I (9:1) to the H P d riv e m o to r. E ig h t m agnets w e re p re s s -fitte d ; I I into holes d r ille d a t equal angles around a c ir c le s c rib e d on the | face of a p la s tic hub. W ith the hub m ounted on the h igh speed shaft, I the m agnets w e re c a re fu lly positioned so th ey w ould sw eep close by i a fix e d m agnetic tape head . The output s ig n a l fro m th e tape head | w as e le c tro n ic a lly conditioned to p ro vid e a constant am p litu d e one I v o lt square w ave corresp o n d in g to the pulsed fre q u e n c y of the mag;- n ets. A c irc u it d ia g ra m of the sig n a l conditioning u n it is p rese n te d | in F ig u re X . 6. The square w ave sig n al w as then fed into a fre q u en cy m e te r whose analog output signal could be plotted v ers u s tim e on an j i X - Y re c o rd e r. | The g e a rin g up of the ro ta tio n a l speed of the m agnets helped to | p ro vid e good fre q u en c y re s o lu tio n fo r even the fa s te s t s p in -u p and | j spin-dow n ra te s . The p r im a r y re a s o n fo r th is e ffo rt, h o w e v e r, was I to o verco m e a p ro b le m in low fre q u e n c y d etectio n . The H e w le tt- j i P a c k a rd fre q u en c y m e te r g en erate s and e le c tric s ig n a l (p ro p o rtio n a l I in am p litu d e to the input fre q u en c y) a t h alf'th e: re c o rd e d fre q u e n c y . | The High . freq u en cy o s c illa tio n s a r e dam ped out by the m e c h a n ic a l in e r tia of the m e te r needle (o r by the m e c h a n ic a l in e r tia of the r e c o rd in g pen w hen the analog output o f the fre q u en c y m e te r is fed into an X - Y r e c o rd e r), but fre q u e n c ie s below about 40H z w ill e x h ib it s ig n ific a n t o s c illa tio n s . The ra p id ly ro ta tin g m agnets pro vid ed high Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 ! enough fre q u e n c ie s to o v e rc o m e th is p ro b le m . ! i 8. 4 L a s e r O ptics System A z im u th a l v e lo c itie s w e re m e a s u re d w ith the L D V set up in the I s e lf-fo c u s in g , re fe re n c e beam m ode as sketched in F ig u re X . 7. I I The com ponents c o m p ris in g th e illu m in a tin g op tics w e re m ounted on | i 1 a rig id a lu m in u m channel w h ich could be e a s ily elevated o r pivoted ! fo r a lig n m e n t p u rp o ses. The p a r a lle l fla t was m ounted in a s m a ll I I a lu m in u m fra m e w h ich could be ro ta te d to give a v a rie ty of b eam j se p a ra tio n d is tan ces . The focusing lens ( f 2 5 0 m m ) was m ounted | on a p re c is io n tra v e rs e d riv e n by a 1 /7 0 H P v a ria b le speed Bodine m o to r. The fo c a l axis w as c a re fu lly alig n ed w ith the in cid en t beam s so th a t the fo c a l point re m a in e d a t the sam e re la tiv e h eig h t th ro u g h out the exten t of its tra v e r s e . The lens p o sitio n w as m o n ito re d by a V e e d e r Root c o u n te r, a ls o d riv e n by the m o to r was shown in F ig u re X . 5. W ith this s e t-u p the r e la tiv e lens p o s itio n could be ra p id ly d e te rm in e d to w ith in ± 0 .0 0 2 in . N e xt the a lu m in u m fra m e was c a re fu lly o rie n te d so th a t the focused beam s w ould s y m m e tric a lly cut equ al ang les (6 /2 ) w ith the h o riz o n ta l d ia m e te r o f the c y lin d e r. F i n a lly , a defocusing lens was placed in fro n t of th e R C A 8645 m u lti p lie r , both of w hich w e re alig n ed w ith the re fe re n c e beam . The lens served to spread the illu m in a tio n o v e r the face of the photocathode and th e re b y p re v e n t breakdow n of the m u lti-a lk a li se n s itize d coating. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 W ith this a rra n g e m e n t, ra p id tra v e rs e s a c ro s s the c y lin d e r ra d iu s could be m ade w ith o u t re p o s itio n in g the p h o to m u ltip lie r tube. T h is was possible because the re fe re n c e b eam m ade o n ly a s m a ll v e r tic a l e x c u rs io n throughout the e n tire tra v e rs e fro m c y lin d e r w à ll to c e n te r due to the s m a ll c ro ss in g angle (8 jy 4. 0 d e g re e s ) of the la s e r b eam s. Thus the re fe re n c e b eam was alw ays picked up by the defocusing lens and illu m in a te d on the photocathode. One a d d itio n al im p ro v e m e n t of the la s e r optics s y s te m was m ade to enhance the pow er of the re fe re n c e b ea m . A th in f ilm of s ilv e r was vacuum deposited on the o u te r face of the p a r a lle l fla t w h e re the la s e r b eam m akes its second in te rn a l re fle c tio n (cf. F ig u re X . 7). T h is in c re a s e d the pow er of the re fe re n c e b eam by about 98%. 8. 5 D o p p le r S ignal E le c tro n ic s The e le c tro n ic units used to re c o rd the D o p p le r s ig n a l a re shown in F ig u re X . 5. The c u rre n t fro m the p h o to m u ltip lie r was m o n ito re d by a d. c. a m m e te r to in s u re th at the m a x im u m safe v alu e (0. 5 m a ) was not exceeded. The s ig n a l was then fe d through a fie ld e ffe c t tra n s is to r (F E T ) in co rd e r to p ro vid e b e tte r im pedance m atch in g betw een the anode of the P M T and the re c o rd in g e le c tro n ic s . N e x t the s ig n a l was filte r e d , a m p lifie d , and o b s erved on both an o scillo sco p e and a s p e c tru m a n a ly z e r. The s p e c tru m a n a ly z e r was c a lib ra te d w ith a s ig n a l g e n e ra to r whose fre q u e n c y was a c c u ra te ly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. _ ^ I re c o rd e d by a d ig ita l co u n te r. In the steady flo w m e a s u re m e n ts the c a lib ra tio n could be m ade a t the tim e of the m e a s u re m e n t as in } j the ro ta tin g d isc e x p e rim e n t (cf. F ig u re X . 3d). In the unsteady j m e a s u re m e n ts the c a lib ra tio n w as c a r r ie d out a fte r each ru n . j 8. A Steady F lu id M o tio n M e a s u re m e n ts The in itia l m eas u re m e n ts w e re m ade w ith the flu id in solid I I body ro ta tio n . Tw o drops of a liq u id so lu tio n of tin y (0. 5 m ic ro n | d ia m e te r ) p o ly s ty re n e b a lls w e re added to the d is tille d w a te r d u rin g | 1 the fillin g p ro c e s s . C a re was ta k e n to r id the c y lin d e r of a ll a ir bubbles fo r even a s lig h t d e n s ity inhom ogen iety was observed to ; 1 m a rk e d ly a ffe c t the sp in -u p m o tio n . A bubble tra p p ed in the h o r i- I z o n ta lly m ounted c y lin d e r was ob served to fo llo w an ir r e g u la r t r a - } je c to ry since its bouyant fo rc e w as alw ays d ire c te d v e r tic a lly up w ard s w h ile the p re s s u re fo rc e on the bubble due to the c ir c u la r i I m o tio n of the flu id was alw ays d ire c te d r a d ia lly in w a rd s . T h is e r r a t ic bubble m o tio n w as accom panied by w aves set up throughout the a x ia l length of the c y lin d e r. W ith the two m a jo r im p ro v e m e n ts o ver the ro ta tin g disc L D V s ys te m , n a m e ly the s ilv e r d ep o sitio n on the p a r a lle l fla t fo r enhanced, beam pow er and the ad d itio n of the F E T , the D o p p le r sig n al could be r e a d ily d etected . T h e re was s ig n ific a n t atte n u atio n of the s ig n a l- to -n o is e r a tio , h o w e v e r, due to the ro ta tin g w a lls ; w hen the cylinder; w as stopped suddenly so th a t the flu id w as s t ill ro ta tin g , the S /N I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ' 83 w ould n o ticeab ly in c re a s e . T h is was a ttrib u te d to the s c a tte rin g of i I the la s e r beam s fro m im p e rfe c tio n s on the in n e r and o u ter su rfaces | of the c y lin d e r; although the p le x i-g la s s w as sm ooth to the touch, it | ! w as by no m eans o p tic a lly sm ooth. These im p e rfe c tio n s a c tu a lly I pro ved quite useful in lo catin g the in n e r w a ll w hich served as a re fe re n c e fo r the fo c a l p o s itio n m e a s u re m e n ts . The sig n al was | I observed to a lm o s t trip le in am p litu d e as the fo c a l point passed i through the in te rfa c e . W ith the p o s itio n of m a x im u m s ig n al in te n s ity | taken to be the lo catio n of the in n er w a ll, its p o sitio n could be d e te r - ! I m in ed to w ith in about ± 0 . 010 in. i Some observation s a r e re p o rte d h e re since they w e re p re s e n t I throughout the course of e x p e rim e n ts . These com m ents con cern | the s p e ctru m a n a ly z e r sig n al a t high and low fre q u e n c ie s . M e a s u re - | m ents w e re m ade o ve r a w ide freq u en cy range fro m less than 7 K H z ! to a lm o s t 1 M H z . In the in te rm e d ia te range (4 0 -5 0 0 K H z) the D o p p ler sig n al was e a s ily detected w ith s ig n a l-to -n o is e ra tio s on the o rd e r of 5. Above 500 K H z the S /N d im in is h e d to about 1. 5 a t 900 K H z . T his set the p ra c tic a l upper lim it fo r the freq u en cy m e a s u re m e n ts . B elow 40 K H z th e re w e re re g u la r la rg e a m p litu d e peaks in the sp e ctru m w hich in crease d w ith d ec re a s in g fre q u e n c y . This was p ro b ab ly the 1 /v la s e r noise re p o rte d by Logan and discussed in p a ra g ra p h 6. 6. W ith the c y lin d e r w a lls at r e s t and the flu id in m o tio n (as in the j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 I i fin a l stages of sp in -d o w n ), the D o p p le r s ig n al could be e a s ily d e te c - | j ted down to 7 K H z . D u rin g s p in -u p , h o w e v e r, d etectio n of signals I below 40 K H z w as re n d e re d d iffic u lt by the co m b in atio n of both the low freq u en cy la s e r noise and the loss in S /N due to the m oving | w a lls . I In o rd e r to d e te rm in e the fo c a l p o sitio n re la tiv e to the in n er j w a ll, one m u st tra c e the paths of the la s e r beam s as they r e fr a c t through the a ir -lu c ite and lu c ite -w a te r in te rfa c e s . This r a y tra c in g p ro ce d u re is exp lained in d e ta il in A p pendix E ; also included is the c a lib ra tio n cu rv e (F ig u re IE. 3 ) re la tin g the fo c a l p o s itio n x to the lens m ovem ent L . The D o p p le r freq u en cy m e a s u re m e n ts fo r w a te r in solid body ro ta tio n a re plotted versus L in F ig u re X . 8. A t f ir s t it seem ed s u rp ris in g th a t the re s u ltin g c u rve was lin e a r ; h o w e v e r, in A ppendix E w e show th a t fo r so lid body ro ta tio n d - V r , n n " [e + @(e )] (m.i7) ^o I I and so fo r s m a ll 6 the slope is indeed constant. (T h e n o n lin e a rity i ap p ears in the c o rre c tio n te rm . ) U sing the e x p e rim e n ta l valu es ! Q = 93. 0 r a d /s e c , n = 1. 000 fo r a ir , \ = 6. 328 x iO ^cm , 6 =0. 0672 a o ra d ia n s , one p re d ic ts fro m equation (III. 17) dVj, c , = - 9.95 X 10 (s e c -c m ) (in. 18) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 and the av e ra g e slope found in F ig u re X . 8 AV 5 I =-9. 89 X 10 (s e c -c m ) (H I. 19) i AL is seen to be in close a g re e m e n t. The D o p p le r fre q u e n c y m e a s u re m e n ts can be used as a check on the r a y tra c in g c a lc u la tio n s . The cu rve in F ig u re X . 8 in d icates th at the lens tra v e lle d a d istan ce L = 3. 112 in . w h ile its fo c a l p oint ^ ! tra v e rs e d fro m the inside w a ll to the c y lin d e r a x is . T h is co m p ares I ! fa v o ra b ly w ith the com puted d istan ce L = 3; 119 in . re p o rte d in Appen-j- d ix E . I I 8. 6 U nsteady F lu id M o tio n M e a s u re m e n ts ! In the unsteady m o tio n asso ciated w ith sp in -u p and sp in-dow n one needs to re c o rd the D o p p le r fre q u e n c y as a functio n of tim e . A n | e le c tro n ic freq u en cy tra c k in g d ev ice em ploying a p h a se-lo c ked loop would p ro vid e an id e a l system since the sig n a l could be plotted versus! j tim e on an X - Y re c o rd e r. Some tra c k in g d evices a re a v a ila b le in thej fo rm of e a s y -to -u s e in te g ra te d c irc u its , but th ey have som e d ra w backs. F ir s t of a ll the sig n al m u st be v e ry c lean w ith a high s ig n a l- to -n o is e ra tio , u nlike the broad G aussian signals observed in the e x p e rim e n t. Secondly, the tra c k in g ranges a re g e n e ra lly about ±70% of the c e n te r freq u en cy; in the s p in -u p m eas u re m e n ts one would lik e to have a freq u en cy range of about 100:1 . E v en m o re sop histicated freq u en cy tra c k e rs w ould have d iffic u lty fo llo w in g the D o p p le r s ig n al Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 I ! at low fre q u e n c ie s in th is e x p e rim e n t, because of the s tro n g ly peaked' I : low fre q u en c y la s e r n o is e . | Due to the above c o n s id e ra tio n s , the idea of e le c tro n ic a lly 1 tra c k in g -th e sig n al was abandoned. The fre q u e n c ie s w e re m a n u a lly j tra c k e d in the fo llo w in g m a n n e r. The c y lin d e r r . p . m . w as re c o rd e d ! v ers u s tim e on the X-Y -pW tter. W hen the D o p p le r sig n a l crossed a r e f - ; eren c e lin e on the s p e ctru m a n a ly z e r, a b lip was m ade on the X - Y | re c o rd e r by m o m e n ta rily d ep re ssin g the " z e ro check" button and a t I I the sam e tim e the fre q u en c y settin g of the s p e c tru m a n a ly z e r was j i noted. Then the decade scale was changed to catch up w ith the j D o p p le r s ig n a l and the p ro cess was re p e a te d . A t the end of each ru n j i the re fe re n c e lin e a t each decade setting was c a lib ra te d w ith an oscil4 la to r and the fre q u en c y was re c o rd e d by a d ig ita l co u n te r. T his p ro ce d u re w o rked v e ry w e ll fo r slow sp in -u p and sp in- ; down ra te s w hen s u ffic ie n t data could be had in a single ru n . F o r thei m o re ra p id a c c e le ra tio n ra te s runs had to be rep ea te d in o rd e r to | obtain a w e ll-d e fin e d c u rv e . M o s t of the unsteady m eas u rem en ts j i w e re m ade a t th re e r a d ii, n a m e ly r / a = 0 .3 3 4 , 0 .5 1 7 , and 0 .8 0 7 . The re s u lts a re plotted ag a in s t the n o n d im en sio n al tim e t = t /t ^ . w h e re t = o ^i ■ ^f (in. 20) a and a is the a v e ra g e a c c e le ra tio n ra te as d e te rm in e d fro m the c y lin d e r r . p . m . cu rv e on the X - Y re c o rd e r. T y p ic a l w a ll v e lo c itie s a re Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 included in the fig u re s fo r each o f the e x p e rim e n ta l cases d es c rib e d j below . ' i F ig u re s X . 9 - X . 12 a re a p re s e n ta tio n of the re s u lts fo r s p in - ■ up fro m r e s t co v erin g a range of E k m a n num bers fro m E —2 ,3 4 X 1 0 ^ a _6 to 27,0 X 10 . The solid lin e s co rresp o n d to the th e o re tic a l ! I re s u lts fo r the in te rio r solution; fo r t < t the solutions a re c a lc u - o la te d fro m equations (11.48, 11.49) and fo r t,> t the solutions a re o com puted fro m equations (11.27, 11.28). The bro ken lin e s in d icate the d o u b le-valu ed n ess of the in te rio r so lu tio n n e a r the w ave fro n t. j I ! M e a s u re m e n ts of sp in-dow n to re s t fro m an in itia l state of | I solid body ro ta tio n a re p resen ted in F ig u re s X . 13 - X . 16. O nly r e la j I tiv e ly slow d e c e le ra tio n ra te s w e re in vestig ated since the flo w was ! observed to go unstable ra th e r q u ic k ly a t the h ig h e r ra te s . The in - I t e r io r solution indicated by the so lid lin e s a re com puted fro m eq u a- ■ tio n (II. 90) fo r t < t and equation (II. 77) fo r t > t . The data in ! o o j F ig u re s X . 15 and X . 16 a re c ro s s -p lo tte d vers u s the rad iu s in Fig-: I u re s X . 17 and X . 18 to e x h ib it the boundary la y e r c h a ra c te r n e a r the | I i c y lin d ric a l w a ll. The solid lin e s re p re s e n t the co rresp o n d in g n u m e r ic a l solutions fo r the v e lo c ity p ro file s n e a r the w a ll given by equation (11.106) w ith boundary conditions (11.107). F ig u re X . 19 shows som e e x p e rim e n ta l re s u lts fo r a re la tiv e ly slow sp in -u p fro m one solid body ro ta tio n ra te (Q^ = 39. 0 ra d /s e c ) to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 another = 104 .0 r a d /s e c ). T h e o re tic a l re s u lts a re given by- equations (II. 56) and (II. 60, I I. 61) fo r t < t ; fo r t > t the solutions o o I a re com puted fro m equations (11.65) and (11.67, 11.69). The lin e a r | I a p p ro x im a tio n fo r the E k m a n flu x given by equation (II. 21) was used j I in the c a lc u la tio n s . I ! F ig u re X . 20 is an exam ple of a re la tiv e ly slow d e c e le ra tio n fro m Q = 104. 4 ra d /s e c to Q = 39. 8 ra d /s e c . T h e o re tic a l re s u lts j 1 f I fo r the in te rio r solution a re com puted fro m equation (II. 96) fo r t < t I ° I and equation (II. 80) fo r t> t^ . In these calcu latio n s equation (II. 22) | was used to a p p ro x im a te the E k m a n m ass flu x . | I F r o m the e r r o r an a ly s is in A ppendix D we see that the re la tiv e ! : - - - - - - - - - - - - - - i e r r o r s in the fre q u en c y, tim e , and p o sition m e as u rem en ts a re ty p i- i I c a lly ± 0 .0 1 5 , ± 0 .0 0 5 , and /iO .O IO , re s p e c tiv e ly . M o re s c a tte r is I p re s e n t in F ig u re s X . 10 - X . 12 and X . 16 because of the s lig h t non- I re p e a ta b ility of the m u ltip le runs n e c e s s a ry to define the v e lo c ity p ro file s . In som e cases w h e re a p ro file could be d efined in a single ru n , one can see s m a ll consistent excu rsio n s of the e x p e rim e n ta l data fro m the th e o re tic a l in te rio r solution in reg io n s (away fro m the v e r tic a l shear la y e rs ) w h ere they should a g re e . This is because the th e o re tic a l curves a re calcu lated fro m the a v e ra g e e x p e rim e n ta l a c c e le ra tio n ra te s , when in fa c t the d riv e m o to r v e lo c ity cu rve was only lin e a r to w ith in & - 3% . The e x p e rim e n ta l points w e re alw ays observed to re fle c t the tru e w a ll v e lo c ity . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 W e should keep in m in d the fa c t th at a ll v e lo c itie s m e a s u re d in the e x p e rim e n t re p re s e n t an a v e rag e both in tim e and space. In the m a n n e r the data was taken, s m a ll am p litu d e fre q u e n c y exc u rsio n s of the m easu red v e lo c ity less than about 1 /5 c y c le /s e c p ro b ab ly could not be d is c e rn e d , and hence the re c o rd e d s ig n al re p re s e n ts an a v e r age o v e r as m any as 3 c y lin d e r revo lu tio n s a t the h ig h es t ro ta tio n ra te s . A t low v e lo c itie s , on the o th er hand, the D o p p le r s ig n a l could be re s o lv e d o ve r a ju s t as s m a ll fra c tio n of a re v o lu tio n , and exc lu d ing the exception discussed b elow , no v e lo c ity flu ctu atio n s w e re obserÿ& d. Since the low v e lo c ity m eas u re m e n ts exh ib ited a z im u th a l s y m m e try , we assum e the flo w was also s y m m e tric a t the h ig h e r ro ta tio n ra te s . The one notable exception was during the fin a l stages (t > 1 .0 ) of spin-dow n when s ig n ific a n t v e lo c ity flu ctu atio n s (A v /v ~ 0. 15) w e re ob served . Thç flow v is u a liz a tio n stu d ies, w hich a re the su b jec t of discussio n in the fo llo w in g c h a p te r, c o n clu siv ely d em o n strated that this unsteadiness in the flo w was due to the b reakin g up of G B rtle r c e lls . These v e lo c ity m e a s u re m e n ts , th e re fo re , re p re s e n t an a v e r age of the observed low fre q u en c y flu c tu a tio n s . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h ap ter IV F L O W V IS U A L IZ A T IO N A N D S T A B IL IT Y 1. P r e lim in a r y D iscu ssio n P re v io u s ly s e v e ra l types of in s ta b ilitie s have been ob served in ro ta tin g flu id m o tio n s. C e rta in ly the b est known a re the T a y lo r [SO ] v e rtic e s o b served in the gap betw een c o n cen tric c y lin d e rs ro ta tin g a t d iffe re n t a n g u la r v e lo c itie s . A s im ila r so->called G o r tle r [ S l ] in s ta b ility can occur in the boundary la y e r of a fb iid s tre a m as it flow s o v e r a concave w a ll. O th e r typés of in s ta b ilitie s w hich o ccu r in ro ta tin g flo w s a re those asso ciated w ith the Z k m a n boundary la y e r . Such in s ta b ilitie s have been o b served and studied by m any in v e s tig a to rs . Some f ir s t w orks w e re due to S m ith [ 3 2 ] , G re g o ry , S tu a rt, and W a lk e r [ 3 3 ] , and F a lle r [ 3 4 ] . M o re re c e n t e x p e rim e n ta l studies have been c a rrie d out by T a tro and M o llo -C h ris te n s e n [3 5 ] and C a ld w e ll and V a n A tta [ 3 6 ] . T h e o re tic a l com putations havenbeen published by F a lle r and K a y lo r [3 7 ] and L illy [ 3 8 ] . The fo r m of E k m a n boundary la y e r in s ta b ilitie S 3 a re g e n e ra lly co n sid ered to be of two types. The Type I w aves a re asso ciates w ith an in fle c tio n a l in s ta b ility in the v e lo c ity p ro file n o rm a l to the wave 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 I bands; these w aves a re g e n e ra lly o b s erved to be s ta tio n a ry w ith re s p e c t to the boundary o r have r e la tiv e ly low phase v e lo c itie s . The Type I I in s ta b ility , on the o th e r hand, re s u lts fro m an in te ra c tio n of s h e a r and C o rio lis fo rc e s ; the phase speeds of these w aves a re f a ir ly ra p id and can , th e o re tic a lly , be m o re than one -h a lf the geos tro p h ic v e lo c ity . Some of the above m en tio n ed in s ta b ilitie s w e re o b served during s p in -u p and spin-dow n in the p re s e n t in v e s tig a tio n . S m a ll, fla t a lu m in u m fla k e s (alu m in u m p a in t p ig m en t) suspended in the flu id s e rv e d w e ll fo r flow v is u a liz a tio n . E k m a n w aves could be seen a t the c y lin d e r end w a lls du rin g s p in -u p fo r s u ffic ie n tly s m a ll E k m a n nu m b ers E ^ . G e r tle r v o rtic ie s w e re alw ays o b s erved to fo rm a t the c y lin d ric a l w a lls a t som e tim e d u rin g sp in -d o w n fo r a ll the d e c e le r a tio n ra te s in v e s tig a te d . In the fo llo w in g sections we s h a ll discuss som e of the o b servatio n s and t r y to access th e ir e ffe c t on the in t e r io r m o tio n . 2. F lo w V is u a liz a tio n of S p in -U p fro m R e s t. O b servatio n s of the in te r io r flu id mofâon and end w a ll in s t a b il itie s w ere m ade using the a lu m in u m p a rtic le flo w v is u a liz a tio n technique. B e fo re each ru n the flu id in s id e the c y lin d e r was m ix ed w e ll in o rd e r to obtain a u n ifo rm suspension of alu m in u m fla k e s . Then, once the m otion had ceased , the c y lin d e r was a c c e le ra te d Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 ; fro m r e s t to som e fin a l so lid body ro ta tio n r a te . The tra n s ie n t flow p attern s on the c y lin d e r end w a lls could alw ays be v is u a liz e d since the E k m a n suction continuously e n tra in e d the a lu m in u m p a rtic le s fro m the quiescent in te r io r (R egion I) and tra n s p o rte d them r a d ia lly o u t w a rd along the end plates and then back into the flu id in te r io r behind the w ave fro n t (R egion II) . As the flu id began to spin up in R egio n I I, the a lu m in u m p a rtic le s c e n trifu g e d outw ards and e v e n tu a lly a ll the pigm ent was deposited on the c y lin d ric a l bou ndary, le a v in g a c le a r in te r io r flu id . The flo w v is u a liz a tio n o b servatio n s a ffo rd the fo llo w in g g e n e ra l d e s c rip tio n of the flow fie ld d u rin g sp in -u p fro m re s t. The flu id was alw ays observed to be tu rb u le n t a t the c o rn e r of the c y lin d e r d u ring the in itia l stages of s p in -u p . A je t of tu rb u le n t flu id was eje c te d fro m the E k m a n boundary la y e r up along the c y lin d ric a l b ou ndary. 2 A t the low a c c e le ra tio n ra te s ( O - 0. 2 ra d /s e c ) the tu rb u le n t je t w ould q u ic k ly die out and the ensuing m o tio n was o b served to be e n tire ly la m in a r ; no E k m a n in s ta b ilitie s w e re o b served a t the end 2 w a lls . A t m o d erate a c c e le ra tio n ra te s (a ~ 1 .0 r a d /s e c ) the tu rb u le n t jets eje c te d a t the c o rn e rs fro m each end p late w ould m e e t a t the c e n te r, thus fo rm in g an annulus of tu rb u le n t flu id enclosing the la m in a r in te r io r . The tu rb u le n t colum n w ould th ic ken s lig h tly , but then q u ic k ly disap p ear a t the c e n te r and rece d e b ac k to the end w alls le a v in g an e n tire ly la m in a r flo w d uring the r e s t of the sp in -u p Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 p ro c e ss. Ekm an. in s ta b ilitie s w e re observed on the end w a lls a t la rg e r a d ii during a s h o rt p o rtio n of the s p in -u p tim e . 2 A t h ig h e r a c c e le ra tio n ra te s ( a ~ 2. 0 r a d /s e c ) the tu rb u len t colum n fo rm e d along the c y lin d ric a l boundary w ould th icken c o n s id e r a b ly and the flu id a t the o u te rm o s t r a d ii w ould re la m in a riz e as it was spinning up; thus we have the p ic tu re of a tu rb u le n t a n n u la r colum n of ro ta tin g flu id bounded on both sides by la m in a r flo w . The tu rb u le n t colum n w ould e v e n tu a lly disapp ear (alw ays f ir s t a t the c e n te r and then re c e d e back to the end w a lls ) n ea r the end of the a c c e le ra tio n phase. E k m a n w aves w e re p rese n t d uring as m uch as h a lf the sp in -u p tim e a t these a c c e le ra tio n ra te s . In ad d itio n , a c o m p licated w ave p a ttern so m etim e s ap p eared in the reg io n betw een the re g u la r la m in a r waves and the tu rb u le n t flu id w hich its e lf did not e x h ib it any w ave'rlike c h a ra c te r. P re s u m a b ly this p a tte rn was the re s u lt of an in te ra c tio n of both Type I and Type I I E k m a n w aves as discussed by F a llè r and K a y lo r [ 3 9 ] . A t the h ig h es t a c c e le ra tio n ra te obtainable ( a ~ 20 2 r a d /s e c ), the tu rb u le n t annulus consum ed a s ig n ific a n t p o rtio n of the flu id volum e ; the E k m a n in s ta b ilitie s w hich ap p eared a t the c y lin d e r end w a lls p e rs is te d fo r a co n sid erab le tim e a fte r the c y lin d e r had a tta in e d its fin a l an g u lar v e lo c ity . A s e rie s of photographs of the tra n s ie n t flo w pattern s on the c y lin d e r end w a lls w e re obtained w ith a h ig h -sp ee d fla s h u n it ■ ' o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 I syn ch ro n ized w ith a N iko n c a m e ra . The photographic sequences w e re taken fo r a c c e le ra tio n ra te s a p p ro x im a te ly co rresp o n d in g to the ; E k m a n num bers in F ig u re s X . 9 - X . 12 fo r w hich the v e lo c ity p ro file s a re known. One can e s tim a te the m a x im u m a n n u la r w id th (A r ) and d u ra tio n (At) of the tu rb u le n t E k m a n boundary la y e r. W e note th at this does not n e c e s s a rily co rres p o n d to the tu rb u le n t annulus in the flu id in te rio r ; indeed the E k m a n boundary la y e r m a y be tu rb u le n t w h ile the in te r io r flow is la m in a r o r even s ta tio n a ry . In T db le IV . 1 the a n n u la r w id th is n o n d im en sio n alized by the ra d iu s , a , and the d u ra tio n tim e is nondim ens io n a liz e d by the sp in -u p tim e , t^, w hich is taken to be the tim e a t w hich a ll the flu id had obtained about 99% of its fin a l an g u la r v e lo c ity . These m e as u rem en ts should be co n sid ered m o re q u a lita tiv e than q u a n tita tiv e . The fo u rth colum n in the table M a x im u m W id th D u ra tio n of P ro d u c t of of T u rb u le n t T u rb u len c e (A r/a ) tim es E k m a n N u m b e r E k m a n la y e r (A r/a) (A t/tg ) (A t/tg ) 1 9 .6 7 X 10"^ 0. 15 0 .0 4 0. 006 11. 18 X 10"^ 0 .2 0 0 .0 5 0. 010 7 .6 4 X 10"^ 0 .3 0 0 .2 0 0. 060 -6 2. 12 X 10 0. 80 0 .3 5 0. 280 Table IV . 1. O b servatio n s of turbulence in the E k m a n la y e r. gives a crude in d ic atio n of the o v e ra ll m a x im u m percentage of tu r b u len t pumping during the sp in -u p p ro cess. We conclude that the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 : flo w w as p re d o m in a n tly la m in a r fo r the th re e la rg e s t E k m a n n u m b e rs . F o r = 2. 12 X 10 ^ , h o w eve r, the am ount of tu rb u le n t pum ping m a y be s ig n ific a n t. B ecause o f the m o re e ffic ie n t m ix in g associated w ith tu rb u le n t flo w , one w ould expect the sp in -u p tim e to be som ew hat s h o rte r than th a t p re d ic te d by la m in a r th e o ry . 3. E k m a n In s ta b ilitie s A sam ple set o f photographs fo r sp in -u p fro m r e s t at a m o d e ra te 2 a c c e le ra tio n ra te (a £ s : 1 .5 ra d /s e c ) is p re s e n te d in F ig u re X . 21. T h e c a m e ra w as a u to m a tic a lly trig g e re d at th re e -s e c o n d in te rv a ls by a r e la y d riv e n by a s ig n a l g e n e ra to r. In th e f ir s t fra m e the c y lin d e r is a t re s t, and the second fra m e w as taken ju s t as the c y lin d e r had begun to a c c e le ra te . T h e c y lin d e r has reac h ed the steady st state a n g u la r v e lo c ity = 1 0 9 ra d /s e c ) by the 31 fra m e . W e see th at the E km an la y e r exh ib its la rg e scale w aves d u rin g the in it ia l stages o f the a c c e le ra tio n , but the d istu rb an ce soon d is a p p e a rs . T h is ir r e g u la r w avy s tru c tu re in the E k m a n la y e r w as observed even w hen the c y lin d e r a c c e le ra te d f a ir ly sm oothly fr o m re s t, and thus it cannot be a ttrib u te d e n tire ly to the s m a ll im p u ls iv e ju m p in the ro ta tio n speed th at often o c c u rre d a t the beginning of th e a c c e le ra tio n s fro m re s t. One can also c le a r ly see the c o re o f q u iesc en t flu id c o rre s p o n d ing to R egio n I; the c o re d ia m e te r is seen to d im in is h ra p id ly a t f ir s t and then m o re slo w ly aâ the speed of the w ave fro n t d e c re a s e s . The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 ; th q u ie s c e n t flu id is s t ill v is ib le in the 24 fra m e w h ich corresponds to ; t = 0 . 83. The w hite rin g w hich ap p ears in fra m e s 1 0 -1 3 and 3 5 -3 6 a ls o ap p eared in som e of the o th e r pho tographic sequences taken a t d iffe re n t a c c e le ra tio n ra te s . The reas o n fo r its ap p earan ce is not fu lly und ersto od, but i t does not re p re s e n t the advancing w ave fro n t, n o r does it alw ays co rres p o n d to a re g io n of tu rb u le n t flo w . E n la rg e d p rin ts of fra m e s 7, 13, 2 3 , and 32 a re p resen ted in F ig u re X . 22. The ir r e g u la r w avy d istu rb an ce in the E k m a n la y e r is c le a r ly v is ib le in fra m e 7. Inlithè 13^^ fra m e the flo w is e v e ry w h e re la m in a r and som e E k m a n w aves have begun to a p p e a r n e a r the c y lin d ric a l w a ll. In fra m e 23 the in s ta b ility is c le a r ly v is ib le ; close in sp ectio n of this photograph re v e a ls a c o m p lic a te d p a tte rn o f b i fu rc a tin g s p ira ls . In fra m e 3 2 , h o w e v e r, the b ifu rc a tio n s a re v ir t u a lly absent and one o b serves a tig h tly wound s p ir a l o f a lm o s t constant w avelen g th . The s ta b ility boundaicy/for the o b served E k m a n w aves was d e te r m in e d in the fo llo w in g m a n n e r. C lo se s c ru tin y of F ig u re X . 22 w ill re v e a l th ree c irc le s on the glass end w a lls of the c y lin d e r. These c o rre s p o n d to the th ree r a d ii w h e re the v e lo c ity p ro file s w e re taken in F ig u re s X . 9 - X . 12. F r o m an an a ly s is of the photographs one could d e te rm in e if a p a rtic u la r c ir c le co rres p o n d ed to an E k m a n flo w th at was la m in a r ly s ta b le , la m in a r ly u n s ta b le , o r tu rb u le n t. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 I F o r e x a m p le , in. F ig u re X . 22d the flo w was noted to be la m in a rly stable a t the in n e r c ir c le ( r /a ) = 0. 334) and la m in a r ly unstab le a t the two o u te r c irc le s ( r / a = 0. 517 and 0. 807 ). If a c irc u m fe re n c e a t one o f the th re e r a d ii ap p eared to be o n ly p a r tia lly u n s tab le, o r if lit s e p a ra te d a re g io n of la m in a r flo w fro m the E k m a n in s ta b ilitie s , then i t was co n sid ered to be " m a rg in a lly s ta b le . " The m e a s u re d v e lo c ity p ro file s w e re then used to d e te rm in e the lo c a l flu id and w a ll v e lo c it ies fo r the p a rtic u la r tim es the photographs w e re taken. The c o r r e s ponding R ossby (Ro) and Reynolds (R e) n um bers w e re c a lc u la te d using the fo llo w in g d e fin itio n s , w a ll and w h e re = v -rQ is the lo c a l geos tro p h ic v e lo c ity outside the E k m a n boundary la y e r . The s ta b ility observatio n s a re p rese n te d in F ig u re X . 23. O f cou rse one w ould lik e to d e te rm in e w h eth er we a re o b servin g the a lm o s t s ta tio n a ry Type I w aves o r the ra p id ly m oving Type I I w aves. U n fo rtu n a te ly the photographs do not give this in fo rm a tio n since the c a m e ra fre e z e s the m o tio n . M ea s u re m e n ts of the w avelengths (band spacing n o rm a l to the w ave fro n ts ) w e re m ade w h en ev er the w aves Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 a p p e a re d a t one of the th re e s c rib e d c ir c le s . The w aves seem ed to f a ll into two groups as in d icated in F ig u re X . 23. R esults fro m 69 m e a s u re m e n ts in d ic ated th at the w aves fo r Ro > -0 . 23 had an a ve rag e w avelen g th of \ / 6 = 20 . 4 w ith a stan d a rd d eviatio n o f a = 2 . 4. These bands (in d ica ted by the sym bols O ,# ) w e re a ll o rie n te d to the rig h t of the geos tro p h ic flo w a t angles v a ry in g fro m 1 -7 d e g re e s . In Tab le IV . 2 w e have lis te d som e of the m e a s u re d w avelengths along the s ta b ility boundary. W e see th at th ere is a p p a re n tly no tre n d w ith R o ssby o r R eynolds n u m b e r. H o w e v e r, the m e as u rem en ts did in d i cate th at the w avelengths in c re a s e som ew hat fro m the s ta b ility boundary ( \ / 6 ~ 19) to the boundary fo r the onset of turbulence ( X / 6 ~ 2 3 ) . The few re m a in in g m e as u rem en ts (in d icated by the s y m b o ls ^ ,^ ) g e n e ra lly d ec re ased in w avelength fro m about \ / 6 = 18 n e a r Re = - 0 .2 3 to about X /6 = 10 a t Re ~ - 0 .3 5 . The in s ta b ilitie s w hich ap p eared in the neighborhood of -0 . 20 > Ro > =0. 25 often ap p eared as a b ifu rc a tin g p a tte rn , and so only an av e ra g e w avelength could be d e te rm in e d in these cases. These w aves w e re alw ays o rie n te d to the r ig h t of the geos tro p h ic flo w , w ith one exception . O b servatio n s n e a r R e = -0 . 30 in d ic a te d w aves o rie n te d both to the rig h t and to the le ft of the geos tro p h ic flo w . These w aves, h o w e v e r, w e re in close p ro x im ity of the c y lin d ric a l w a ll (cf. F ig u re X . 22b), and this p e r p e n d ic u la r boundary m ay w e ll a ffe c t the b e h a v io r of th eiin s ta b ilitie s . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 Ro R e (X /6 ) 0 .0 5 5 77 18. 7 0 .0 5 9 82 17. 1 0. 063 90 2 0 .2 0. 107 198 2 2 .0 0. 113 101 19.0 0. 120 110 18. 8 0. 141 124 19.2 0. 165 145 2 0 .4 0. 188 166 19. 7 0 .2 2 6 199 2 0 .0 T ab le IV . 2 . N o n d im en sio a a l w avelengths n e a r the s ta b ility boundary. Included fo r co m p ariso n in F ig u re X . 23 a re the re s u lts of som e m eas u rem en ts by S m ith and G re g o ry , S tu a rt, and W a lk e r a t Ro = - 0 .5 w hich corresponds to the case of a spinning disc in the p re s ence of a q uiescent flu id . The la tte r in v e s tig a to rs m ade a s e rie s of o bservatio n s and the re s u lts given h ere re p re s e n t the a ve rag e of th e ir m easu rem en ts fo r both the Type I c r itic a l R eynolds nu m b er and the Reynolds n um ber corresp o n d in g to the onset of tu rb u len ce. The lin e a r re g re s s io n cu rves fo r the c r itic a l Reynolds n um ber fo r Types I and I I waves (m e asu red only a t p o sitive R ossby n u m b ers) due to T a tro and M o llo -C h ris te n s e n a re also included. In this sam e R ossby n um ber range we have plotted (a p p ro x im a te ly ) the c r itic a l R eynolds num ber m easu rem en ts fo r Type I I w aves due to C a ld w e ll and V a n A tta . T h is cu rve re p re s e n ts a le a s t squares f i t to th e ir data and includes the c o rre c tio n fo r fin ite am p litu d e effe cts due to F a lle r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 & K a y lo r [37^. O nly one Type I and T yp e I I s ta b ility m e a s u re m e n t' in the in te rm e d ia te negative R ossby n u m b er range could be found in the lite r a tu r e , and these o b servatio n s a re due to F a lle r and K a y lo r [ 39 ] . T h e ir m easu rem en ts of the re s p e c tiv e c r itic a l Reynolds n u m b ers a re also in d icated in F ig u re X . 23. A s u m m a ry of the co m p a riso n data is p resen ted in the fo llo w in g t a b le . The w ave angle is denoted by c p and is p o sitive when the w ave lie s to the le ft of the geos tro p h ic flo w . The nondim e ns io n a l w avelength fo r the Type I w aves o b served by G re g o ry , S tu a rt and W a lk e r was calcu lated by F a lle r [ 3 4 ] . In v e s tig a to r W ave T yp e pa Ro (Re)^ X/6 c p G re g o ry , e t a l. [3 3 ] I - 0 .5 0 436 2 1 .5 14° F a lle r & K a y lo r [3 9 ] I -0 .2 1 6 210 15. p o sitive F a lle r & K a y lo r [3 9 ] I I -0 .2 1 6 234 38. negative T a tro , e t a l. [3 5 ] I = 124. 5 + 7. 32Ro 11. 8 1 4 .6 ° T a tro , e t a l. [3 5 ] n (R e)c = 5 6 .3 + 116. 8R0 2 5 -3 3 0 to 8° C a ld w e ll, e t a l. [3 6 I I 0 .0 56. 7 - - T ab le IV . 3. A S u m m ary of E k m a n bou ndary la y e r m e a s u re m e n ts . W e r e a liz e of course th at the p re s e n t observation s w e re m ade u n d er unsteady flo w conditio ns. W ith the single exception o f the observatio n s m ade by F a lle r and K a y lo r, the c o m p a riso n data w e re obtained fo r steady flow con ditio ns. H o w e v e r, it can be a rg u e d that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 ; the u nsteady m e as u rem e n ts taken in this e x p e rim e n t should also be v a lid fo r steady flo w . It seem s reas o n ab le to assu m e that the E km an in s ta b ilitie s grow in a tim e s ca le c o m p a ra b le to the E k m a n la y e r fo rm a tio n tim e (cf. F a lle r and K a y lo r [3 9 ])» w h ich is n e g lig ib ly s m a ll a t the h ig h ro ta tio n ra te s w h ere the o b s ervatio n s w e re m ade. T h is a rg u m e n t is supported by the fa c t th at the m e a s u re m e n ts define a f a ir ly co n sisten t s ta b ility boundary even though the data was o b tain ed fro m a w ide range of a c c e le ra tio n ra te s . The s c a tte r in the data is due p a r tly to the photographic m e a s u re m e n t technique and p a rtly to the fa c t th at the E k m a n num bers in the photographs did not e x a c tly c o rres p o n d to the av e ra g e E k m a n num bers in the v e lo c ity p ro file s . Because of these q u a lific a tio n s , the data in F ig u re X . 23 is co n sid ered to be o nly 5 -1 0 % a c c u ra te . A c o m p a riso n of the p re s e n t m eas u re m e n ts w ith those in J a b le IV . 3 suggests that we a re ob servin g Type I I E k m a n w aves, a t le a s t in the R ossby n u m ber range - 0 . 23 < Re < -0 . 07. I t is d iffic u lt to t e ll w h ic h type of w aves a re p re s e n t fo r Ro < - 0 .2 3 . F u r th e r discussions of these o b servatio n s w ill be given in C h ap ter V . The E k m a n in s ta b ilitie s w e re not re a d ily o b served d u rin g sp in -d o w n because, as s h a ll be exp lain ed in the fo llo w in g s ec tio n , the a lu m in u m p a rtic le s re m a in e d attach ed to the c y lin d ric a l w a lls d u rin g the in itia l stages of the d e c e le ra tio n . W hen the v e rtic a l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 b ou ndary la y .e rà ' d i& in o tig o u n s tab le, the p a rtic le s w e re pum ped into the E k m a n la y e rs , but by this tim e the d istu rb an ce caused by the G o r tle r c e lls was often stro n g enough to m ask out v is u a liz a tion of the re g u la r E k m a n w aves. F o r som e d e c e le ra tio n ra te s , h o w e v e r, the in s ta b ilitie s w e re c le a r ly v is ib le ; w aves o b served d u rin g sp in -d o w n , of co u rs e , co rres p o n d to p o s itiv e R o ssby n u m b e rs . A fin a l co m m en t is m a d e lh e re w ith re s p e c t to the E k m a n in s ta b ilitie s . E ven though the r a d ia l ex te n t and d u ra tio n o f the tu rb u le n t E k m a n la y e r was s m a ll fo r the th re e la rg e s t E k m a n n um bers in T ab le IV . 1, the sam e was not tru e fo r the la m in a r in s ta b ilitie s . T h e y did occur o ver la rg e reg io n s (A r/a > 0. 6) ând fo r long tim e s (At/1^ > 0. 5) and yet the v e lo c ity p ro file s a re w e ll p re d ic te d by the th e o ry in regions aw ay fro m the m oving w ave fro n t. Thus one concludes that the la m in a r pum ping p re d ic te d by R o g ers and Lance is not s ig n ific a n tly a lte re d by la m in a r in s ta b ilitie s in the E k m a n bound a r y la y e r . 4. G o r tle r In s ta b ilitie s The a lu m in u m p a rtic le v is u a liz a tio n technique p re v io u s ly d is cussed was also used to o b serve the p resen ce of G o r tle r v o rtic e s d u rin g sp in-dow n. In itia lly the a lu m in u m p a rtic le s w e re u n ifo rm ly c e n trifu g e d ag a in s t the c y lin d ric a l w a ll. A b rig h t lig h t was placed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 I I n e a r the end w alls to illu m in a te the in te r io r . T h e n , as the c y lin d e r ! ! was spun down, a lte rn a tin g lig h t and d a rk c irc u m fe re n tia l s tre a k s in d ic atin g the ero s io n p a tte rn of the G o r tle r v o rtic e s w ould ap p ear acro s s the length of the c y lin d e r. The v o rtic e s fo rm e d d is c re te c irc le s n e a r the m id -p la n e z = 0 and they s p ira le d (so m etim es in a b ifu rc a tin g p a tte rn , and o th er tim es as d is c re te h e lix e s ) g e n tly o u t w ard s tow ards the end w a lls a t z = ± h /2 . The a v e ra g e angle of the h e lic a l m o tio n in c re a s e d a t a f a ir ly u n ifo rm ra te fro m m id -p la n e to end w a ll. T his can be re a d ily exp lained if we assum e th a t the 1 /3 v o rtic e s a re em bedded in the E la y e r ad jace n t to the c y lin d ric a l w a ll. In this la y e r the a x ià l v e lo c ity is d ire c te d aw ay fro m the c e n te r and to w ard the end w alls since it m u st re tu rn the m ass flu x to the E k m a n boundary la y e rs . If the c e n te r o f the v o rtic e s lie a t r^ = a - e , then a t an y given tim e we have (cf. equations (II. 89) and (II. 98)) w (r^ , z ;t) = k ^ z v (r ^ , z ;t) = k^ (IV . 3) and the su p erp o sitio n of these two v e lo c ity fie ld s w ould give ris e to the h e lic a l m otion d e s c rib e d above. These e s tim a te s a re onlyapproxr- im a te since they a re based on the E 4 la y e r s o lu tio n s, but they a re p ro b a b ly q u a lita tiv e ly c o rre c t. F o r e x a m p le , in the lin e a riz e d 1 /3 v e r tic à iO ’(E ) boundary la y e r so lution of G reen sp an and H o w ard [ l ] , i t is shown that aw ay fro m the h o riz o n ta l end plates w is an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ................... 104 1 ! odd,: inc re a s in g ^function of z. 2 V e r y ra p id d e c ele ra tio n s (a ~ -1 0 ra d /s e c ) fro m the high in itia l i ro ta tio n ra te (Q. ~ 109 ra d /s e c ) a lm o s t in stan tan eo u sly re s u lte d in an unstable flow n e a r the w a ll. The v o rtic e s ap p eared so ra p id ly that one could not d e te rm in e if th ere was a p re fe re n tia l a x ia l p o sitio n fo r the f ir s t in s ta b ility ; indeed the flo w seem ed to go unstable e v e ry w h e re a t once. A t slow d e c e le ra tio n ra te s , on the o th er hand, the flo w was o bserved to go unstable f ir s t n e a r the end w a lls and then p ro g res s ra p id ly tow ards the c e n te r. T h is is not in accordance w ith the n u m e r ic a l com putations of B r ile y and W a lls [4 0 ] fo r im p u ls iv e spin-dow n to re s t. (T h e ir re s u lts show th at the c e llu la r m o tio n f ir s t ap p ears a t the c y lin d e r m id -p la n e .) F o r the slo w est d e c e le ra tio n ra te s (C t ~ 2 - 0 .2 ra d /s e c ) the flow was ob served to be stable d uring m o re than h a lf of the d e c e le ra tio n phase, but the flu id alw ays u ltim a te ly w ent u n stab le. The v is u a liz a tio n technique was used to d e te rm in e an upp er bound on the tim e of the f ir s t in s ta b ility as a fu n ctio n of the E k m a n n u m b e r. In s ta b ilitie s observed in th is m an n er give ran upper bound (a la te tim e ) fo r two reas o n s. F ir s t of a ll it takes a fin ite tim e fo r the a m p lific a tio n ra te s of a given w avelength to becom e la rg e enough to scour aw ay the a lu m in u m p a rtic le s on the c y lin d ric a l w a ll. M o re im p o rta n tly , the la rg e c e n trifu g a l fo rc e s holding the heavy a lu m in u m fla k e s ag a in s t the w a ll w ill s u b s ta n tia lly r e s is t the e ro s iv e actio n of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 ; the in s ta b ilitie s . O n ly a r e la tiv e ly d ilu te suspension of the alu m in u m ; p a rtic le s was used in hopes of a m e lio ra tin g these e ffe c ts . Just enough p a rtic le s w e re used to lig h tly coat the c y lin d e r w a ll a t the in itia l ro ta tio n ra te of 0^ =1109 r a d /s e c . The in s ta b ility tim es a re p resen ted in F ig u re X . 24. The re s u lts suggest th at i f the c y lin d e r w e re im p u ls iv e ly spun down to re s t fro m this high in itia l ro ta tio n r a te , the flow w ould go unstable a lm o s t im m e d ia te ly . No observa»». -6 tions a re given fo r E k m a n num bers la r g e r than 13; 5 X 10 because of the d iffic u lty in c o n s is te n tly d e te rm in in g when the flo w w ent un sta b le ; this p ro b le m was m o s t lik e ly due to the r e la tiv e ly w eak a m p lific a tio n ra te s a t these low d e c e le ra tio n s . . T h is also suggests th at the " fir s t tim e " o b servatio n s m a y be s ig n ific a n tly la te a t the h ig h e r E k m a n num bers in F ig u re X . 24. In o rd e r to co m p are the p re s e n t ob servatio n s w ith those of o th er in v e s tig a to rs , the m o m en tu m thickness and G o r tle r c rite rio n w e re c a lc u la te d fo r each of the in s ta b ility tim es given in F ig u re X . 24. The m om entum thickness fo r the spin-dow n boundary la y e r was c a l culated fro m the equation (9 is the m o m en tu m th ic kn ess ) 2 2 ^ 2 9a (w -O) = J (U)-0)(U) -U))r d r ( I V . 4) 0 w here is the an g u la r v e lo c ity a t the w a ll given by the in te rio r solu tio n . The v e lo c ity p ro file U)(r) was d e te rm in e d fro m equation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 i ( II. 106), The c r itic a l G b r tle r s ta b ility p a ra m e te r was then c a lc u la te d a c c o rd in g to the d e fin itio n f p a0(w -0 ) The s ta b ility re s u lts a re p resen ted in F ig u re X . 25. W e can co m p a re these data w ith those of T illm a n n |4 l] and M a x w o rth y [4 2 ] f o r im p u ls iv e spin-dow n to re s t. T illm a n n 's m e a s u re m e n ts w e re m ade in a c o n c e n tric a lly ro ta tin g c y lin d e r a p p a ra tu s . The in n e r and o u te r w a lls ro ta te d a t the sam e ra te u n til the flu id betw een them was spun up. Then the o u te r c y lin d e r was suddenlÿ'/stopped and m o tio n p ic tu re s w e re taken of the ensuing la m in a r in s ta b ilitie s and the sub sequent tra n s itio n to tu rb u len ce. T illm a n n used the fo llo w in g equa tion d e riv e d by G rohne I = ( 4) = '^ G r o h n e ( 4 ) d V - to c a lc u la te the m om en tu m thickness a t the o u ter boundary r = a . H is re s u lts in d ic ated an a lm o s t con stant v a lu e of theC Q brtler p a ra m e te r fo r the onset of the c e llu la r in s ta b ility o v e r a range o f e ffe c tiv e c u rv a tu re (6 /a ) , v iz . . G . = 6. 15 ; 3 X 10"^ < - < 7 X lO "^ . (IV . 7) c r it ~ a In M a x w o rth y 's e x p e rim e n t a c y lin d e r of flu id w ith â fre e su rface was spun up to so lid body ro ta tio n on a tu rn ta b le and then suddenly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 stopped. The m om entum thickness and G B rtle r s ta b ility p a ra m e te r w e re c a lc u la te d assum ing the side w a ll boundary la y e r to be of the R a y le ig h type, so that i " " ° R a y l e i g h ( ^ ) • The G B rtle r c r ite r io n was c a lc u la te d fro m the equation (C . G (IV . 9) crit vi w here the valu e of k = 4. 5 was d e te rm in e d fro m the e x p e rim e n t. The re p o rte d valu e G . = 2 5 .8 is in d ic a te d in F ig u re X . 25 along w ith c r it T illm a n n 's m ean valu e given by equation (IV . 7). T hese data a re a p p ro p ria te ly lo cated on the axis = 0 w hich corresponds to im p u l s ive spin-dow n. The ra th e r la rg e valu e of G ^^.^ re p o rte d by M a x w o rth y is e v i d en tly due to his assum p tio n of a R a y le ig h type boundary la y e r . We note fro m equations (IV . 6) and (IV . 8) that Q G r ohne _ _ q. 414 (IV . 10) R a y le ig h and so the c r itic a l s ta b ility p a ra m e te r g iven by M a x w o rth y is r e - 3/2 duced by a fa c to r (0 .4 1 4 ) when based on G ro h n e's th e o re tic a l a n a ly s is . The re s u ltin g valu e G ^^.^ = 6. 86 is seen to co rres p o n d w e ll w ith T illm a n n 's d ata. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ............ ' 108... i The e ffe c tiv e c u rv a tu re fo r M a x w o rth y 's e x p e rim e n ta l data can als o be c a lc u la te d acc o rd in g to the new va lu e o f G . . F r o m eq u a- c r it tion (3) in his pap er we have: I - ■ p v .li,: a n The e x p e rim e n ta l valu e of given by (IV . 9) was found to be constant only fo r T < 0 < 2 (ra d /s e c ), even though data w as taken up to n = 5 r a d /s e c . The d isc re p a n c y a t ffie h ig h e r ro ta tio n ra te s was a ttrib u te d to the fa c t that the c y lin d e r could not be suddenly brought to r e s t due to the in e rtia of the tu rn ta b le a t the h ig h e r ro ta tio n ra te s . 2 ^ In s e rtin g the e x p e rim e n ta l valu es a = 5. 08 c m and v = 0. 01 cm /sec into equation (IV . 11) fo r the v a lid range of ro ta tio n ra te s , one obtains G . , = 6. 86 ; 12 X 10"^ < i < 19 X 10"^ (IV . 12) c r it a ' w hich, when co m p ared w ith T illm a n n 's re s u lts , shows th at the c r i t i c a l s ta b ility p a ra m e te r fo r the app earance of G b r tle r c e lls is v ir t u - -3 a lly in s e n s itiv e to the e ffe c tiv e c u rv a tu re up to 8 /a ~ 20 X 10 Though im p u ls iv e sp in -d o w n was not possib le w ith the ap p aratu s used in the p re s e n t e x p e rim e n t, it is tem pting to e x tra p o la te the m e a s u re m e n ts to = 0. L a c k of data points in this re g io n m akes P r iv a te co m m u n icatio n . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ...... ÏQ 9 the e x tra p o la tio n d iffic u lt, but one m ig h t exp ect an in te rc e p t in the neighborhood of = 10, in f a ir a g re e m e n t w ith T illm a n n 's and M a x w o rth y ’s m e a s u re m e n ts . 4. 1 C o m p ariso n W ith the V e lo c ity P ro file s The tim e of the f ir s t app earance of the G B rtle r v o rtic e s a c c o rd ing to F ig u re X . 24 is in d ic ated in F ig u re s X . 17 and X , 18 fo r the two sets of m eas u red v e lo c ity p ro file s . The tim e a t w hich the flo w f ir s t becom es c e n trifu g a lly unstable acco rd in g to the R a y le ig h c rite rio n was also ca lc u la te d . The R a y le ig h c rite r io n [ 4 3 ] states th at in v is c id c ir c u la r flow s a re stable only if the sq u are of the c irc u la tio n in c re a s es w ith ra d iu s . In itia lly the flo w is in s o lid body ro ta tio n and hence is e v e ry w h e re stab le, but as the sp in-dow n com m ences the flu id n e a r the w a ll is sh eared into an unstable v e lo c ity p ro file . The c a l cu lated R a y le ig h in s ta b ility tim e s should p ro vid e a lo w e r bound fo r the f ir s t appearance of the G S rtle r c e lls since v is c o s ity has â s ta b iliz in g in flu en c e. Indeed we see fro m the v e lo c ity p ro file s in F ig u re s X . 17 and X J 18 that the e x p e rim e n ta l points a re in a cc o rd w ith th eo ry fo r tim es s m a lle r than the in v is c id in s ta b ility tim e . The e x p e rim e n ta l v e lo c ity p ro file s p ro b a b ly p ro vid e the best in d ic a tio n of in s ta b ility . In F ig u re X . 17 the v e lo c ity p ro file s suggest th at the flo w was a lre a d y unstable a t T = 0. 45 fo r E ^ = 13. 8 X 10 and in F ig u re X . 18 it app ears th at in s ta b ility has o c c u rre d n e a r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 t = 0. 15 fo r E = 9. 92 X 10 The flo w -v is u a liz a tio n in s ta b ility a tim es a re som ew hat la te in both cases. As p re v io u s ly discussed, this is m ost lik e ly due to the re la tiv e ly w eak a m p lific a tio n ra te s a t these low d e c e le ra tio n ra te s . The th e o re tic a l tim e develo pm ent of the m om entum thickness 0 and G b 'rtle r s ta b ility p a ra m e te r G„ was c a lc u la te d fo r the two E k m a n 0 num bers w here the v e lo c ity p ro file s have been m e a s u re d . The r e sults given in F ig u re X . 26 show th at the boundary la y e r a c q u ire s its g re a te s t m om entum d efe c t n e a r t = 0. 15 in both c a s e s ,b u t the s ta b ility p a ra m e te r continues to grow u n til a m uch la te r tim e . T his m eans that, acco rd in g to the s ta b ility curves fo r flo w along a h ig h ly concave w a ll given by A . M , O. S m ith [ 4 4 ] , som e w avelengths w ill be sub jected to in c re a s in g ly g re a te r a m p lific a tio n ra te s up to about r = 0. 50. A fin a l com m ent is m ade h e re w ith re s p e c t to the side w a ll in s ta b ilitie s . The G o r tle r v o rtic e s m u s t s u re ly a lte r the E k m a n pum ping ra te s n e a r the c y lin d ric a l w a ll because the v e lo c ity p ro file s a t la rg e r a d ii a re d e c re a s e d below the th e o re tic a l v a lu e . One can see , h o w eve r, th at the v e lo c ity d is trib u tio n a t s m a lle r r a d ii is s t ill w e ll p red icted by the th e o ry . F o r in stan ce, the v e lo c itie s fo r r a d ii less than r / a = 0. 5 in F ig u re s X . 17 and X . 18 s t ill fo llo w the th e o re tic a l p re d ic tio n s , and ob servatio n s in d icate th at the d istu r;- - bance ■which m oved ra p id ly in w a rd a fte r tj= 0 . 8 was due t& th e b re a k in g Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill; up of the G S rtle r c e lls . T h is im p lie s th a t the E k m a n guctidngs âiii: fa c t d e te rm in e d b y the lo c a l gd CKStnophic v e lo c ity g ra d ie n t a t th e a v a il. 5. W ave F ro n t P o s itio n fo r S p in -U p fro m R e st T h e flo w v is u a liz a tio n photographs in F ig u re X . 21 give an in d ic a tio n of the p o s itio n o f the w ave fro n t fo r sp in -u p fr o m r e s t. M e a s u re m ents o f th e a v e rag e rad iu s of th e u n d istu rb ed flu id w e re ta k e n fro m the photographic sequences and a re p lo tted as a fu n ctio n o f tim e in F ig u re s X . 27 - X . 30. T h e f u ll th e o ry p rese n te d in C h a p te r I I does not p re d ic t the w ave fro n t p o s itio n fo r sp in -u p fr o m r e s t OT fo r s p in - up fo r s = U)/n < Sj^. H o w e v e r, it is perhaps w o rth w h ile to c o m p a re the p re s e n t data w ith the w ave fro n t p o s itio n c a lc u la te d fro m the lin e a riz e d th e o ry using the m a x im u m suction f = f^ as in d ic a te d in F ig u re I I . 3. I f the v e lo c ity d is c o n tin u ity a t the w ave fro n t is w e a k , then th is m a x im u m valu e fo r th e E k m a n suction should give the a p p ro x im a te fa s te s t p ro p a g atio n speed o f the w ave fro n t. F r o m equation ( II. 61) the w ave fro n t p o s itio n fo r sp in -u p fro m re s t is given by r^ (T ) = e x p (-C T ^ ) (IV . 13) and in accordance w ith the above d iscu ssio n w e ta k e the constant C 4 i to be — f ^ E ^ , so th a t in te rm s of the n o n d im en sio n al tim e t w e have r^(t) = exp | - - | fjE ^ t^ ^ ^ | . (IV . 14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 T h is equation is only v a lid fo r t < t . F o r t > t one can d e te rm in e o o the continued w ave fro n t p o s itio n fro m equation ( II. 32) w ith the a p p ro p ria te n o n d im e n s io n a liza tio n . T h e w ave fro n t p o sitio n s fo r the flo w v is u a liz a tio n data w e re c a lc u la te d fro m equ ation (IV . 14) above and a re p rese n te d in F ig u re s X . 27 - X . 30. S till an o th er d e fin itio n of th e w ave fro n t can be m ade fro m the v e lo c ity p ro file s . In lin e w ith the above d iscu s sio n , w e s h a ll d efine the w ave fro n t as the tim e a t w h ic h the m e a s u re d v e lo c ity (fo r a given ra d iu s ) equals the v e lo c ity c o rres p o n d in g to the "kn ee " at the w ave fro n t in the th e o re tic a l solutions p re s e n te d in F ig u re s X . 9 - X . 12, i . e . , the tim e at w h ic h d v /d t is in fin ite acco rd in g to the in te r io r solution . T h ese data a re p lo tted in F ig u re s X . 28 - X . 30 w h e re the E k m a n n u m b ers fo r the v e lo c ity m e a s u re m e n ts a re c o m p a ra b le to the E k m a n n u m b ers fo r the flo w rv is u a liza tio n .d a ta -. T h e E k m a n n u m b er in F ig u re X . 9 w as not s u ffic ie n tly close to th a t in F ig u re X . 27 fo r a c o m p a riso n to be m a d e . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter V D IS C U S S IO N O F R E S U L T S 1. T h e o ry v e rs u s E x p e rim e n t In C h a p te r I I w e have p rese n te d som e solutions fo r the tim e - dependent m o tio n o f a hom ogeneous flu id enclosed in a u n ifo rm ly a c c e le ra tin g o r d e c e le ra tin g c y lin d e r. In C h a p te r I I I w e have p r e sented a v a rie ty of e x p e rim e n ta lly d e te rm in e d v e lo c ity p ro file s co rresp o n d in g to the th e o ry , and in A ppendix D it w as shown th at these m e a s u re m e n ts can g e n e ra lly be expected to have an a c c u racy of b e tte r than 3%. And in C h a p te r IV w e have p rese n te d som e v is u a l iz a tio n o b servatio n s of th e m o tio n w ith in the c y lin d e r to d e te rm in e w hen th e conditions on w h ic h the th e o ry is based w e re m e t, i. e . , w hen the side w a ll bou ndary la y e rs w e re stable and w hen the E km an la y e r pum ping w as la m in a r. In the fo llo w in g sections w e s h a ll c o m p a re th e th e o re tic a l and e x p e rim e n ta l v e lo c ity d ata in the lig h t of the s ta b ility flo w o b s e rv a tio n s . 1. 1 S p in -U p fro m R e s t T h e re s u lts given in T a b le IV . 1 in d ic a te th a t the pum ping in the E k m a n bou ndary la y e r is e s s e n tia lly one of la m in a r flo w fo r the E k m a n n u m b ers in F ig u re s X . 9 - X , 11. Thus it is not s u rp ris in g 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .............................................................. 114 '1 th a t the e x p e rim e n ta l v e lo c ity p ro file s a re in good a g re e m e n t w ith the th e o re tic a l re s u lts aw ay fro m the w ave fro n t. T h e m e a s u re m en ts in the reg io n of the w ave fro n t (c f. e s p e c ia lly F ig u re X . 11) suggest th at the in v is c id v e lo c ity discontinuity p re d ic te d by the in te r io r solution is indeed sm oothed out in a b ro ad m oving viscous la y e r s im ila r to th at p re d ic te d b y V e n e zia n [ ? ] . F u rth e r evidence fo r th is la y e r is given by the n u m e ric a l solutions of G o lle r and Ranov [4 5 ] in th e ir calcu la tio n s of s p in -u p fro m re s t of a fre e su rface flu id contained in a v e r tic a l c y lin d e r. In th is w o rk th e fu ll viscou s a z im u th a l m om en tu m equation is n u m e ric a lly solved w ith the "exact" cu rve f it to R o g ers and L a n c e s ' E k m a n suction (but w ith som e fre e su rface a p p ro x im a tio n s ), and th e ir v e lo c ity p ro file s e x h ib it the c h a ra c te ris tic E ^ la y e r a t th e w ave fro n t. Although a d ire c t co m p ariso n w ith the p re s e n t e x p e rim e n t cannot be m ad e, the q u a lita tiv e correspondence is expected to be v a lid . In the case of a lm o s t im p u ls iv e s p in -u p fro m re s t in F ig u re X . 12, the flo w observation s have in d ic ated th a t th e re m ay be a f a ir am ount of tu rb u le n t pum ping in the E k m a n boundary la y e rs . T h e r e fo re , one cannot a ttrib u te the la rg e d is c re p a n c y n e a r the w ave fro n t e n tire ly to a sm oothing out of the r e la tiv e ly la rg e v e lo c ity dis = co n tin u ity in a m oving la m in a r sh ear la y e r . T h e exp lan atio n fo r the d is c re p a n c y betw een th e o ry and e x p e rim e n t n e a r th e w ave fro n t in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 th is case is p ro b a b ly due to a com bination o f the fo llo w in g ; i) T h e w ave fro n t tra v e ls fa s te r than la m in a r th e o ry w ould p re d ic t due to the increasedgpum ping in the tu rb u le n t E k m a n la y e rs . ii) E v en i f the flo w w e re e n tire ly la m in a r, the w ave fro n t m ay propagate c o n s id e ra b ly fa s te r than the in te r io r solution p resen ted h e re p re d ic ts , because of the strong v e lo c ity d isco n tin u ity in d ic a te d in F ig u re X . 12. T h is la tte r co n jectio n is supported by the w ave fro n t o b s e rv ations p rese n te d in F ig u re s X . 27 - X . 30. A ll in d ic atio n s a re th at the in itia l d istu rb an ce propagates ahead of the d iv id in g c h a ra c te ris tic given by the s im p lifie d (i. e . , in c o m p le te ) in v is c id solu tio n . A c tu a l ly , the observatio n s in th e E k m a n boundary la y e r m a y not be in d ic a tiv e of the lead in g d istu rb a n c e in th e flu id in te r io r a t the lo w e r E k m a n n u m b e rs , since the boundary la y e r flo w m ay be tu rb u le n t o r e x h ib it E k m a n w aves beneath the quiescent in te rio r flu id . A t the slow est a c c e le ra tio n ra te in F ig u re X . 27, h o w eve r, the E k m a n la y e rs w e re observed to be e n tire ly la m in a r in the re g io n of the w ave fro n t (except fo r the ir r e g u la r w avy boundary la y e r m o tio n o bserved at s m a ll tim e s ) and so th e in itia l d is tu rb a n c e observed in th is case m o re than lik e ly in d ic ate s the tru e lead in g edge of the p ropagating fre e shear la y e r. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 No e ffo rt w as m ade to solve the fu ll viscous equation n e a r the w ave fro n t, but th e re a re at le a s t tw o approaches one could ta k e . O bviously the fu ll viscous p a r tia l d iffe re n tia l equation fo r th e a z im u th a l v e lo c ity could be n u m e ric a lly in te g ra te d . U n fo rtu n a te ly , th is p ro c e d u re is expensive since it takes a ra th e r long tim e (fro m fiv e to s ix hours on an IB M 3 6 0 -6 7 c o m p u ter) to obtain a reas o n ab le 7 set of v e lo c ity p ro file s . A n a lte rn a te p ro c e d u re w ould be to use a s h o c k -fittin g technique s im ila r to th a t used in c o m p re s s ib le gas dyn am ics in o rd e r to d e te rm in e the stren g th and speed of the p ro p a gating v e lo c ity d is c o n tin u ity . O f co u rse one w ould f ir s t have to d e te rm in e the c o rre c t ju m p conditions b y co n sid erin g the c o n s e rv atio n equations in a c o n tro l vo lu m e enclosing th e d is c o n tin u ity . A fte r the stren g th of the shock and its p o s itio n w e re known, one could perhaps p e r fo r m a th e o re tic a l an a ly s is s im ila r to V e n e zia n *s in o rd e r to obtain the a z im u th a l v e lo c ity d is trib u tio n th ro u g h the shear la y e r . Though the n u m e ric a l ap p ro ach is undoubtedly the e a s ie s t, the shock an a ly s is w ould perhaps p ro vid e m o re in s ig h t in to th e b eh avio r of th e w ave fro n t. 7 T h is e s tim a te w as co m m u n icated to the author by D r . R . G. H u ssey who has p e rfo rm e d som e in te g ra tio n s in the E k m a n nu m b er range 10 < E q < 10“'^ fo r im p u ls iv e sp in -u p fro m re s t. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 : 1 .2 T h e S p in -U p T im e A n in te re s tin g fe a tu re of the e x p e rim e n ta l re s u lts fo r sp in -up fro m re s t is th a t even though the d e ta ils of the m o vin g w ave fro n t a r e not exp lain ed by th e s im p lifie d in te r io r solutions p re s e n te d in C h ap ter I I, the sp in -u p tim e s a r e v e ry a c c u ra te ly p re d ic te d fo r the th re e la rg e s t E k m a n n u m b e rs in v e s tig a te d . E v en in th e case of a l m o s t im p u ls iv e sp in -u p (cf. F ig u re X . 12), the tim e a t w h ic h the flo w a t each ra d iu s reac h es 99% of th e fin a l a n g u la r v e lo c ity is in good a g re e m e n t w ith th e o ry . F r o m an en g in e e rin g point of v ie w , one w ould lik e to know how the sp in -u p tim e v a rie s w ith the non d im en sio n al p a ra m e te rs of the p ro b le m . O th e r than th e d im e n s io n le s s tim e , th e re a re b a s ic a lly only tw o p a ra m e te rs w h ic h a p p ear in th e in te r io r equations of m o tio n fo r sp in -u p fr o m re s t, and th ese a re E ^ = v and E ^ = v /Q h ^ . T h e aspect r a tio Æ. = a /h only a ffe c ts the v e r tic a l b o u n d ary la y e rs and does not in flu en ce th e sp in -u p tim e as long as A R ~ (}(1). A n u m e ric a l e x p e rim e n t w as u n d ertak en to d e te rm in e the e ffe c t of ju s t th e a a c c e le ra tio n r a te (throu gh the p a ra m e te r E ^ ) on th e sp in-up - tim e , tg , defined as abo ve, i . e . , the tim e at w h ic h the flo w lo c a lly a tta in s 99% o f the fin a l an g u la r v e lo c ity . W ith th is d e fin itio n , the sp in -u p tim e is a ls o a functio n o f the ra d iu s . A s u m m a ry of the re s u lts o f th is study is given in F ig u re X . 33, 1. w h e re the re s id u a l s p in -u p -tim e , d efined as O ^EQ(tg-t^), is p lo tted Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 as a functio n o f the (nond im en sio n al) tim e i t takes fo r th e c y lin d e r w a lls to re a c h th e ir fin a l an g u la r v e lo c ity , i . e . , t ^ N o te th at in th is g raph t^ = 0 co rresp o n d s to im p u ls iv e sp in -u p fro m re s t (t^ = tq), and th a t as t^ t^ t^ . T h e in te re s tin g fe a tu re of th ese calcu latio n s is th a t the r a d ia l dependence begins to d ro p out as t^ gets la rg e . T h is is because the w h o le of th e in te rio r flu id is v ir t u a lly in so lid body ro ta tio n b y the tim e the c y lin d e r reach es its fin a l an g u la r v e lo c ity . T h e e ffe c t of the p ro p ag atin g w ave fro n t is lo s t fo r these long sp in -u p tim e s . In F ig u re X . 34 w e have n o n d im e n s io n a lize d the sp in -u p tim e w ith the a c c e le ra tio n ra te and p lo tte d it as a functio n of E ^ . F o r > 15 X 10 ^ w e see th a t the s p in -u p tim e is a c c u ra te ly p re d ic te d fo r the range o f r a d ii in d ic a te d b y the equation ^f 1 1 f = — + - p 2170 e x p (-1 2 1 5 E ^ ) . (V . 1) s a ^ a I f one accepts the m o re g e n e ra l d e fin itio n o f sp in -u p as the tim e fo r w h ich th e flu id a t r / a = 0. 5 reac h es 99% 0^, then equation (V . 1) is ap p licab le fo r E ^ > 3 X 10 F o r s m a ll E k m a n n u m b e rs co rres p o n d in g to the m o re ra p id a c c e le ra tio n s the data can b est be p re s e n te d in te rm s o f th e new % re s id u a l sp in -u p tim e d efin ed as Q ^ ^ (t^ - t^ ), in w h ic h case t^ -♦t^ as E^ 0. T h e se re s u lts a re p lo tted in F ig u re X . 35 as a function 2 — 6 of E ^ . F o r E ^ < 5 X 1 0 the sp in -u p tim e a t the th re e ra d ii can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 w e ll a p p ro xim ated b y the equation t_ = t^ (r ) + s f (V .2 ) w h e re k ( 0 . 334) = 3 .0 0 X 10^° k (0 . 517) = 2 .1 5 X 10^0 k ( 0 . 807) = 1 .5 9 X 10^° and t^ (0 . 334) = 9 5 .8 8 sec t ^ ( 0 .5 l7 ) = 8 0 .7 7 sec tj^(0. 807) = 5 8 .3 4 sec . H e re obviously the p rese n c e of the w ave fro n t has a strong in flu en ce on the lo c a l sp in -u p tim e . 1. 3 S p in-D o w n to R e s t T h e th e o re tic a l p ro file s fo r spin-dow n to re s t a re w e ll supported by the e x p e rim e n ta l m e a s u re m e n ts . The in te r io r so lu tio n given in F ig u re s X . 13 - X . 16 is seen to be in e x c e lle n t a g re e m e n t w ith the e x p e rim e n ta l data fo r a ll r a d ii not affected by the side w a ll boundary la y e rs o r G b rtle r in s ta b ilitie s . M o re o v e r, the q u a s i-s te a d y assum ption m ade to obtain the ap p ro x im a te equation (II. 106) d e s c rib in g the O -Ç E "*^ ) side w a ll boundary la y e r is a p p a re n tly v in d icated by these a g re e m e n t betw een the e x p e rim e n ta l and th e o re tic a l p ro file s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 in F ig u re s X . 17 and X . 18 b e fo re th e G S rtle r v o rtic e s d is ru p t the flo w . A g ain , th e re a re perhaps tw o w ays of solving the fu ll p a ra b o lic viscous equation fo r the boundary la y e r flo w . B esid es p e rfo rm in g the tim e consum ing n u m e ric a l in te g ra tio n s , i t app ears th a t one can app ly tw o -tim in g techniques to the p ro b le m since th e re exists both a "fa s t" spin-dow n tim e and a "s lo w " viscous d iffu sio n tim e . T h is la tte r m ethod of solution m a y p ro v e to be an in trig u in g w a y of a s c e rta in in g the conditions of v a lid ity fo r th e q u a s i-s te a d y assu m p tio n invoked in the p re s e n t a n a ly s is . N a tu ra lly the solution w ould only be v a lid fo r tim e s p r io r to the m a n ife s ta tio n of the G S rtle r in s ta b ility . 1 .4 S m a ll R ossby N u m b e r F lo w s I t is q u ite re w a rd in g to see th e good a g re e m e n t betw een the closed fo r m solutions to the a p p ro x im a te th e o re tic a l equations and the e x p e rim e n ta l p ro file s in F ig u re s X . 19 and X . 20. E v e n though the o v e ra ll v e lo c ity changes a re la rg e , the flo w s a re tr u ly " lin e a r " , inasm uch as the absolute valu e of the lo c a l Rossby n u m b e r is alw ays less than 0. 2. In the case of sp in -u p , the th e o ry te lls us that the w ave fro n t is c h a ra c te riz e d by a w eak shear d is c o n tin u ity w h ic h passes by the r a d ia l p o s itio n r / a = 0. 807 a t the tim e t = 0 .4 7 4 as in d icated in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 F ig u re X . 19. W e see th at in p ra c tic e the s h e a r d is c o n tin u ity is im p e rc e p tib le a t this ra d iu s w h ere it has a lre a d y tra v e le d h a lf of its to tal distance fro m the w a ll to its fin a l p o sition a t (r /a) ^ = 0. 6125. o I jI j NÆ IT F o r sp in -d o w n , of c o u rs e , the s h e a r la y e r re m a in s attach ed to the in n er w a ll, A s h o rt e x tra p o la tio n of the u pp er bound tim e fo r in s ta b ility in F ig u re X . 24 in d icates th a t G o r tle r v o rtic e s m ay have ap p eared n ear the end of the run; if this was the case, the effe cts of the in s ta b ility a re e iv d e n tly not fe lt fo r r a d ii s m a lle r than r / a = 0. 807 as shown in F ig u re X . 20. 2. F lu id S ta b ility fo r S p in -U p and S p in -D o w n P erh ap s the m o st in trig u in g p a rt of the e x p e rim e n ta l w o rk re p o rte d h e re a re the tra n s ie n t in s ta b ilitie s o b served d u rin g sp in -u p and spin-dow n. A discu ssio n of this a s p e c t of the flu id d y n a m ic a l p ro b le m is given in the parag raphs below . 2. 1 E k m a n In s ta b ilitie s The v o r tic a l m otion s o b served in the E k m a n boundary la y e r a re e v id e n tly c h a ra c te ris tic of the Type I I w aves re p o rte d by o th er in v e s tig a to rs . The n eg ative w ave o rie n ta tio n of the s p ir a l bands in the range - 1 ° to - 7 ° , fo r e x a m p le , a r e v e ry clo se to the angles ( 0 ° t o - 8 ° ) o bserved by T a tro and M o llo -C h ris tens en, though the ave rag e w avelength X / 6 = 2 0 .4 re p o rte d h e re (negative RcJ is som ew hat s m a lle r than th e ir (p o sitive Ro) o b s e rv a tio n s . The c r itic a l w a v e length of \ / 6 = 24 fo r Type I I waves p re d ic te d by L illy 's ze ro Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R ossby n u m b er calcu latio n s is also s lig h tly la r g e r , although I the s ta b ility boundary defin ed by th e shaded re g io n in F ig u re X . 23 m ig h t be expected to have a z e ro Rossby n u m b er in te rc e p t in the neighborhood o f R e = 60 w h ic h is ty p ic a l fo r T yp e I I w a v e s . P e rh ap s the stro n g es t a rg u m e n t in fa v o r of th e T y p e I I w aves, at le a s t fo r s m a ll R ossby n u m b e rs , is given by th e n u m e ric a l c a l cu latio n s of th e a m p lific a tio n ra te s re p o rte d by L illy and F a lle r &io K a y lo r. T h e ir re s u lts show th a t the m a x im u m gro w th ra te s fo r the T yp e I I w aves a r e c o n s id e ra b ly s tro n g e r than those co rresp o n d in g to the T y p e I w aves a t lo w R eynolds n u m b e rs . Since th e ir p e rtu rb a tio n equations a r e based on the lin e a riz e d E k m a n s p ir a l (Ro -» 0), th e ir re s u lts m u st be eq u a lly v a lid fo r s m a ll n eg ative as w e ll as s m a ll p o s itiv e R ossby n u m b e rs . Though a ll eviden ce suggests th a t the in s ta b ility w e o b serve fo r Ro > - 0 . 23 a r e T y p e I I w a v e s , it is less obvious w h at is happening at s m a lle r R ossby n u m b e rs . F a lle r and K a y lo r o b serve an onset of T yp e I I w aves w h e re w e note th e onset of tu rb u le n c e (cf. F ig u re X . 23), and th e ir m e a s u re m e n t of the c r itic a l Reynolds n u m b er fo r the T yp e I w aves is close to o ur T yp e I I s ta b ility bou ndary. It is not c le a r w hy w e do not o b serve both w ave system s as do F a lle r and K a y lo r. It m ay be th a t our v is u a liz a tio n technique fa v o rs the o b s ervatio n of a p a r tic u la r m ode. Since the photographs w e re taken fro m the b o ttom of the bou n d ary la y e r , one m ig h t be convinced th a t the w ave bands w h ic h lie c lo ses t to the so lid b o u n d ary w ould be th e m o s t re a d ily Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .......................................................................... " ...... ..................... 12 J v is ib le . B ut the lin e a riz e d c a lc u la tio n s of L illy as w e ll as the fin ite j am p litu d e studies of F a lle r and K a y lo r suggest th a t a s e le c tiv ity based on height alone should not be too stro n g , since the c e n te rs of the v o rtic e s a ll g e n e ra lly lie about tw o E k m a n thicknesses above the solid bou ndary. T h e lib e r ty w as ta k e n to m a tc h th e p re s e n t m e a s u re m e n ts fo r the onset of tu rb u le n c e in F ig u re X . 23 w ith those re p o rte d by S m ith and G re g o ry , et a l. at Ro = - 0 . 5. In the re g io n of s m a ll p o s itiv e R ossby n u m b er (Ro = 0. 15), C a ld w e ll and V a n A tta note a tra n s itio n to tu rb u le n c e at R e = 148, although T a tr o and M o llo - C h r i sten s en re p o rt a valu e above R e = 200. A n e x tra p o la tio n of th e p re s e n t data c o m p a re s m o re fa v o ra b ly w ith C a ld w e ll and V a n A tta 's o b s e rv a tio n s , but the tra n s itio n R eynolds n u m b e r m a y w e ll depend on th e te s t fa c ility and o th e r fa c to rs w h ic h have not been e x p e rim e n ta lly c o n tro lle d . 2. 2 C b r tle r In s ta b ilitie s T h e m e c h a n is m fo r in s ta b ility a t th e c y lin d ric a l w a ll d u rin g spin -d o w n m ig h t be in te rp re te d as one of a lm o s t in v is c id in s ta b ility due to the la rg e in it ia l ro ta tio n ra te and th e g e n e ra lly high d e c e le ra tio n ra te C L . T o see th is w e have p lo tted the ro ta tio n a l v e lo c ity of the in te r io r flu id as a fu n ctio n o f th e w a ll an g u la r v e lo c ity (at the o b served tim e of in s ta b ility ) in F ig u re X . 31. A lso in cluded a re the tw o points co rres p o n d in g to th e in s ta b ility tim e s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124! in fe rr e d fro m the v e lo c ity p ro file s in F ig u re s X . 17 and X . 18 w h ich w e re discussed in C h ap te r IV , p a ra g ra p h 4 , 1. Included fo r c o m p a r- ; iso n is the c u rv e = Q w h ic h , of c o u rs e , is e v e ry w h e re stab le. T h e m e a s u re m e n ts in th is fig u re c lo s e ly re s e m b le the a sy m p to tic (high R eynolds n u m b e r) b e h a v io r o f the s ta b ility calcu latio n s due to T a y lo r [s o ] fo r th e case of v o rtic e s produced in a j n a rro w gap betw een c o n c e n tric a lly ro ta tin g c y lin d e rs . An analogy betw een these flow s can be m ade i f the in n e r:w a ll in T a y lo r 's p ro b le m is re p la c e d by the in te r io r flu id in the p re s e n t case, and then the gap w id th is d e te r m in ed by th e s ize of the G o rtle r c e lls . T h e above analogy suggests the p o s sib le existen ce of a lim itin g slope (dtu^/dO) fo r in s ta b ility a t high ro ta tio n ra te s (o r high Reynolds n u m b ers based on the in te r io r flu id v e lo c ity and the c e ll d ia m e te r). F ig u re X . 32 shows the v a ria tio n of th is p a ra m e te r w ith the E k m a n n u m b e r. W e note th a t the s ta b ility o b servatio n s fro m the v e lo c ity p ro file d ata give re s u lts w h ic h a re s ig n ific a n tly d iffe re n t fro m 'th o s e o f the flo w v is u a liz a tio n o b s e rv a tio n s . A t the lo w e r E k m a n n u m b ers th e flo w v is u a liz a tio n data is p ro b a b ly f a ir ly good, since the la rg e g ro w th ra te s associated w ith these r e la tiv e ly intense in s ta b ilitie s w ould give ris e to a m o re a c c u ra te tim e m e a s u re m e n t fo r the d e te rm in a tio n of the in s ta b ility . On th e o th e r hand, the v e lo c ity p ro file s obviously give a b e tte r in d ic a tio n of th e in s ta b ility tim e at the h ig h e r E k m a n n u m b e rs . T a k in g th ese co n sid eratio n s into account Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ' ................................ - ...... .- 1251 one can e x trap o late the d ata to = 0 w h ich co rresp o n d s to the la rg e s t valu e of fo r in s ta b ility in th e p re s e n t m e a s u re m e n ts . T h e v alu e (du) /dO) , = 1 .0 6 , assum ing the ro ta tio n a l v e lo c itie s a r e 1 c r it s u ffic ie n tly la rg e , gives an e s tim a te to the proposed asym p to tic lim it . T h e d e ta ile d d e s c rip tio n of the in s ta b ility m e c h a n is m is c e rta in ly m o re co m p licated than s im p lifie d a n a lys is in tim a te d above. A point in fa c t is the d is c re p a n c y betw een the a x ia l p o s itio n of the f ir s t v is ib le c e lls in the p re s e n t e x p e rim e n t and the a x ia l p o s itio n p r e d icted by B r ile y and W a lls as noted in C h ap ter IV , p a ra g ra p h 4 . W hat w e a re p ro b a b ly o b s ervin g is a Ludw ieg type in s ta b ility . Lud w ieg [4 6 ] has e x p e rim e n ta lly d em o n strated th at a s p ira l in s ta b il ity (s im p ly a T a y lo r in s ta b ility w ith a superim posed a x ia l flo w ) can occur in the gap betw een co n c e n tric ro ta tin g c y lin d e rs w hen the in n e r c y lin d e r m oves along its ro ta tio n a l a x is . H is th e o re tic a l re s u lts in d ic ate the p o s s ib ility of cro s s in g a s ta b ility boundary in spin*;down b e fo re the usu al R a y le ig h c rite r io n is s a tis fie d , as long as the flo w has an a p p re c ia b le r a d ia l g ra d ie n t o f the a x ia l v e lo c ity (d w /d r). Though L u d w ieg 's s ta b ility an a ly s is is d ir e c tly a p p licab le only to the flo w betw een co n c e n tric ro ta tin g c y lin d e rs , one m ig h t expect an analogous exp lan atio n fo r th e s p ira l in s ta b ilitie s observed in spin-dow n. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 A d e te rm in a tio n o f w h e re the flo w w ould f ir s t go unstab le based ■ on L u d w ieg 's s ta b ility an a ly s is w ould depend on the d e ta ils of the 1 /3 a x ia l v e lo c ity d is trib u tio n in the E la y e r , but fro m a lin e a riz e d an a lys is one vo u ld expect the la rg e s t v a lu e of d w /d r to occur n e a r the c y lin d e r end p la te s ; th is being th e case, one w ould have q u a lita tiv e ag re e m e n t w ith the spin-dow n o b servatio n s a t la rg e E k m a n n u m b e rs in the p re s e n t e x p e rim e n t. It is not c le a r , th en , w hy the calcu latio n s of B r ile y and W a lls show th a t the in s ta b ility f ir s t occurs a t the m id -p la n e of th e c y lin d e r. P e rh ap s th e n o n lin e a r s tru c tu re of 1 /3 the unsteady E la y e r is co n s id e ra b ly d iffe re n t fro m the lin e a riz e d case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p te r V I S U M M A R Y A N D C O N C L U S IO N S The lin e a riz e d (Ro -* 0) a n a ly s is fo r im p u ls iv e s p in -u p in a c y lin d e r given by G reen sp a n and H o w ard shows that fo r « 1 the tra n s ie n t flo w consists of fo u r d is tin c t re g io n s : the 0(E^ ) E k m a n la y e rs on the top and bottom lid s W hich tra n s p o rt the m ass flu x e ith e r r a d ia lly o u tw ard (sp in -u p ) o r r a d ia lly in w a rd (sp in -d o w n ); the in te r i m io r flu id w hich con stitutes the m a in body of the flo w ; the 0(E'^) sh ear la y e r along the c y lin d ric a l b ou ndary w hich serves to b rin g the a z i m u th al and ra d ia l v e lo c itie s to th e ir c o rre c t valu e a t the w a ll; and the 1 /3 “ <3(E ) la y e r w hich lie s beneath the <3-(E'*) la y e r and serves to tra n s p o rt the m ass e ith e r to (sp in -d o w n ) o r fro m (spin-up) the E k m a n la y e rs . In the n o n lin e a r p ro b le m , the com bined re s u lts of W e d e m e y er and V e n e zia n have shown th at the flow can be d e s c rib e d e x a c tly as j , above excep t th a tiin sp in -u p the O (E^) sh ear la y e r sep a ra te s fro m the w a ll and propagates as a s h e a r d is c o n tin u ity into the flu id in te rio r . W hen the flo w is in itia lly in s o lid body ro ta tio n , the flu id ahead of the w ave fro n t then spins up by v o rte x s tre tc h in g , w h ile the flu id behind the w ave fro n t spins up b y exchanging an g ü la r m o m en tu m w ith 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 the so lid b o u n d aries. In the s p e c ia l case o f sp in -u p fro m re s t, the flu id ahead of the w ave fro n t does not ro ta te , but its vo lu m e d im in ishes to z e ro . In spin-dow n the wave fro n t trie s to propagate r a d ia lly o u tw ard , but since it is confined to the c y lin d e r it re m a in s attached to the w a ll. In this case the in te r io r flu id is spun down e n tire ly by the co n tractio n of v o rte x fila m e n ts . The w o rk presented h e re p ro vid es a n a tu ra l extension o f the p receeding analyses to include the cases of sp in -u p and spin-dow n fo r w hich it takes a fin ite tim e fo r the w a lls to ach ieve th e ir fin a l a n g u la r v e lo c ity . A g ain , the physics of the n o n lin e a r flo w (when it is stab le) can be in te rp re te d e x a c tly as d e s c rib e d abo ve, w ith but one exception. T h is exception p e rta in s to s p in -u p (e ith e r im p u ls iv e o r a c c e le ra tin g ), in w hich case we have shown th at fo r s = uu/0 < 0. 075, the propagating w ave fro n t is c h a ra c te riz e d by a v e lo c ity d is c o n tin u ity , the position and s tre n g th of w hich a re not d e te rm in e d by o ur s im p lifie d a n a ly s is . In the single case of s p in -u p fro m re s t the v e lo c ity d is c o n tin u ity propagates the e n tire distance fro m the c ir c u la r w a ll to the c e n te r of the c y lin d e r. The a z im u th a l v e lo c ity 1 in the (E^) la y e r fo r sp in -d o w n can be d e s c rib e d by a single a p p ro x i m ate equation (analogous to the viscous equation d e riv e d by Wede% e 1 /3 m e y e r fori s p in -u p ), but the d e ta ils of the 0-(E ) la y e r fo r both s p in - up and spin-dow n can o n ly be obtained by includin g a d d itio n a l te rm s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 fro m the o th er m om en tu m equations. The above th e o re tic a l p red ictio n s a re io r n e i out by the e x p e ri m e n ta l v e lo c ity p ro file s p resen ted in the th e s is . In p a rtic u la r we fin d that the v e lo c ity d isco n tin u ity in the neighborhood of the wave fro n t is sm oothed out in a sh e ar la y e r q u a lita tiv e ly s im ila r to that p re d ic te d by V e n e zia n (shown h e re to be a good a p p ro x im a tio n fo r U)/Q > 0. 075). F u r th e r evidence fo r this propagating fre e sh e ar la y e r is given by the n u m e ric a l re s u lts of G o lle r and Ranov fo r s p in -u p fro m re s t. The flo w v is u a liz a tio n studies have shown th at the tra n s ie n t m otion in the boundary la y e rs can often be unstable both durin g s p in - up and sp in -d o w n . M eas u re m e n ts of the w avelengths and o rien tatio n s of the s p ir a l Ekm an. bands ob served durin g s p in -u p a re e v id e n tly c h a ra c te ris tic of the Type I I w aves noted by o th e r in v e s tig a to rs . F u r th e r c o rro rb o ra tio n fo r this conclusion is given by the lin e a r s ta b ility calcu latio n s of IL illy and F a lle r & K a y lo r w hich show that the a m p lific a tio n ra te s of Type I I waves a t s m a ll Reynolds num bers a re m uch g re a te r than those fo r the Type I w aves. T h e ir re s u lts should be a p p licab le fo r s m a ll neg ative Ro w h ere we o b serve the in s ta b ility , as w e ll as fo r s m a ll positive Ro w h ere m o st of the previo u s e x p e r i m e n ta l observatio n s have been m ad e. G b r tle r in s ta b ilitie s o cc u r in the side w a ll boundary la y e r during sp in -d o w n . An e x tra p o la tio n of the p re s e n t m ea s u re m e n ts of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 c r itic a l Gb’r t le r s ta b ility p a ra m e te r = 0 gives an a p p ro x i m ate value of 10, w hich is in f a ir a g re e m e n t w ith the m o re p re c is e m easu rem en ts due to T illm a n (G _ = 6. 15) and the new valu e given c r it * h e re (G _ = 6. 86), w hich re p re s e n ts a c o rre c tio n o f the o b s e rv e n t ations re p o rte d by M axw o rth y . Though one can e a s ily reco g n ize the analogy betw een the co n cen tric c y lin d e r in s ta b ility (T a y lo r c e lls ) and the spin-dow n in s ta b ility in the p re s e n t e x p e rim e n t (G b rtle r c e lls ), the flow can p ro b a b ly be b es t d es c rib e d as a Ludw ieg type in s ta b ility . H o w ev er, in s o fa r as the above analogy e x is ts , one would exp ect an in v is c id lim itin g valu e of (diD^/dQ) a t in s ta b ility ; the e x p e rim e n ta l re s u lts in the p rese n t e x p e rim e n t, assum ing the R eynolds num bers a re la rg e enough, give a f ir s t e s tim a te of ( d u ) ^ / 06 fo r the asym p to tic valu e. F in a lly , perhaps the m ost valu ab le resulü to com e out of this in v e s tig a tio n p ertain s to the v a lid ity of using R o g ers and L an ce's com putations fo r the E k m a n la y e r tra n s p o rt. T h e re is no in d ic a tio n fro m the p rese n t m easu rem en ts th at the lo c a l a p p ro x im a tio n m ade by W é d e m e y e r w ill give in c o rre c t re s u lts . P re s u m a b ly , an a p p li cation of R o gers and L an c e 's re s u lts to o ther m o re co m p licated ro tatin g flow p roblem s w ill prove eq u ally su ccessfu l. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter VII A P P E N D IX A T H E D O P P L E R F R E Q U E N C Y S H IF T E Q U A T IO N 1. The L o re n tz T ra n s fo rm a tio n As pointed out by G o ld s te in and K rie d [ 1 5 ] the D o p p le r e ffe c t fo r s c a tte re d w aves is som ew hat d iffe re n t than fo r e m itte d w aves. In the fo llo w in g two sections we s h a ll d e riv e the r e la tiv is tic D o p p le r fre q u en c y fo r both cases, i. e . , a s ta tio n a ry and m oving sou rce of ra d ia tio n . In the fin a l sectio n we s h a ll com bine these re s u lts to obtain the D o ppler s h ift fo r use in la s e r D o p p le r a n e m o m e try in a la b o ra to ry . The b asic tool is , of co u rs e , the L o re n tz tra n s fo rm a tio n . F o u r ve c to rs (x, ct) and (k, v /c ) a re both tra n s fo rm e d by the sam e m a trix fo r re la tiv e m otion betw een two fra m e s of re fe re n c e . F o r e x a m p le , if we co n sid er (k, v /c ) to be known in one coo rd inate s y s te m , then the fo u r v e c to r ( k ', v '/ c ) is given by (cf. Jackson [ 4 ? ] , pg. 356) k' y k' \ v . / I / Y P 0 ±Py 0 1 0 0 0 0 1 0 ±Py 0 0 Y k y I'J (A . 1) 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 w h e re c = V' Ik - V 1 - V (A . 2) and V is the re la tiv e v e lo c ity along the x -a x is betw een the two m oving fra m e s . E q u atio n (A . 1) does not re p re s e n t the m o s t g e n e ra l L o re n tz tra n s fo rm a tio n since k , k ' and v m u s t a ll lie in the sam e p la n e , but th is w ill be s u ffic ie n t fo r this a n a ly s is . The signs (± ) in the tra n s fo rm a tio n a re chosen to be n eg ative i f the re fe re n c e fra m e s a re m oving a p a rt and p o s itiv e if they a re app ro aching each o th e r. 2. A S ta tio n a ry Source of R a d ia tio n C o n sid er a source of ra d ia tio n v lo c a te d a t 0 and p e rc e iv e d by a ra p id ly m oving p a rtic le a t P as shown in the sketch below . The s ta tio n a ry re fe re n c e fra m e is (x, y , z) and the m oving fra m e is denoted ^ ^ I’ ^ l ’ ^1^* w ave v e c to r k is o rie n te d a t an angle 6 w ith re s p e c t to the s ta tio n a ry x -a x is such that k = jk |e ^ . W e w is h to obtain the y 0 F ig u r e A . 1. A s ta tio n a ry so u rce of ra d ia tio n as seen by a m oving p a r tic le . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 a p p a re n t fre q u en c y seen by the m oving p a rtic le , but r e fe r r e d to the v e c to r e^ in the s ta tio n a ry re fe re n c e fra m e . F r o m the la s t equ a tion in the L o re n tz tra n s fo rm a tio n given by (A . 1) we have (using the n eg ative sign fo r p) V _ E = - p y k + Y - (A . 3) c X c and since = |k | cos 6 we o b tain , w ith the a id of equation (A . 2), V ( l - ^ ) • (A . 4) Since 6 is r e fe r r e d to the s ta tio n a ry fra m e we r e a d ily o b tain the d e s ire d r e s u lt, v i z . , V / V . e \ V = • ( A - 5) 3. A M o ving Source of R a d ia tio n Now le t us co n sid er a m oving p a rtic le w hich gives off ra d ia tio n a t a ra te in the re fe re n c e fra m e (Xj^, yj^, za nd we c a lc u la te the a p p a re n t fre q u e n c y seen by a s ta tio n a ry o b s e rv e r a t S in the fra m e As in d ic a te d in F ig u re A . 2 the w ave v e c to r k ^ ^ is o rie n te d a t angle 0 w ith re s p e c t to the m oving x ^ -a x is . W e ag ain ap p ly the L o re n tz tra n s fo rm a tio n to obtain (now using the p o s itiv e sign since the re fe re n c e fra m e s a re app ro aching each o th e r) V V — = +Pyk + Y — (A . 6) c x^ c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 y X , X. 0 F ig u r e A . 2. A rrioving sou rce of ra d ia tio n as seen by a s ta tio n a ry o b s e rv e r. and since k = k i cos 8 1 we obtain V C O S 9 1 V / V cos 9 \ (A . 7) T h is , h o w e v e r,is not the d e s ire d re s u lt since we w ish to r e fe r e v e r y thing to la b o ra to ry (s ta tio n a ry ) co o rd in a te s . T h is can be a c c o m p lis h ed by using the f ir s t equation in (A . 1) w h ic h read s k = Yk + PY ^2 ^1 and since k = — cos 0. one obtains X . c 1 1 V - Y — cos 6 = — P cos 9 - - Y P c i v 6 P (A. 8) (A . 9) w h ic h re la te s 6^in the m oving fra m e to Gg in the s ta tio n a ry fra m e , In s e rtin g (A .9 ) into (A .7 ) we fin d a fte r som e s im p lific a tio n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V s Y ( l - ^ cos 02^ 135 (A . 10) and, consequently, V s P (A . 11) T h is is the d e s ire d re s u lt since e^ is r e fe r r e d to the la b o ra to ry re fe re n c e fra m e . 4. The D o p p ler S h ift In la s e r D o p p ler a n e m o m e try we need to u tiliz e both of the r e sults p re v io u s ly d e riv e d . In p ra c tic e we have a la s e r source v w hich s c a tte rs ra d ia tio n off a m oving p a rtic le a t P and is picked up by a photodetector a t S. If we assu m e that the p a rtic le s c a tte rs ra d ia tio n a t the sam e ra te a t w hich i t p e rc e iv e s it in the m oving re fe re n c e fra m e , we can use equations (A . 5) and (A . 11 ) to obtain A Æ V V : : (A . 12) W e note th a t the re la tiv is tic fa c to r Y disapp ears since the la s e r source and photodetector a re not in re la tiv e m otion . Now the D o p p ler s h ift is defined as Vd = ^^s ■ ^ (A . 13) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 and so w e obtain V ,— — . — • (e - e )V Vjj = 7 -------------------------------------------------------------- (A . 14 ) w hich is the re la tiv is tic D o p p le r freq u en cy sh ift one w ould m e a s u re in the la b o ra to ry . F o r n o n re la tiv is tic p a rtib le v e lo c itie s | v /c | « 1 w e obtain V - '' d " c ■ (A- 15) and since X = c /v w e have V *(e -e ) ( 11- > (■ > N o te th at nothing has been said about th e m e d iu m surroun ding the p a rtic le ; w e have ta c itly assum ed th at th e m ed iu m w as u n ifo rm and th at (e ^ -e ) and X w e re m e a s u re d in th a t m e d iu m . In m any cases of in te re s t the la s e r and photodetector a re in another m e d iu m (usually a ir ); th is can be broug ht to m in d b y using the vacuum w a v e length X^ = X /n fo r the m o n o c h ro m atic sou rce, w h e re n is the absolute index o f re fra c tio n of the m e d iu m surroun ding the s c a tte r ing p a rtic le s . E q uation (A . 16) then becom es n v (e -e ) Vd = -------r - ^ (A . 17) o but i t m ust be re m e m b e re d th a t (e -e ) - is s t ill m ea s u re d in the s o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 s c a tte rin g m ed iu m . E q u atio n (A . 17) is th e n o n re la tiv is tic fo rm u la quoted as equation ( III. 1) in the te x t. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX B S IG N A L A N D N O IS E E Q U A T IO N S I . H e tero d yn e S ignal E q u atio n The anode c u rre n t of a p h o to m u ltip lie r tube is p ro p o rtio n a l to the pow er in cid en t on its fa c e . W e can le a rn how to m a x im iz e th is pow er by s im p ly d e te rm in in g the co n trib u tio n s due to the in cid en t a n d s c a t t e r e d e l e c t r i c f i e l d s . W e d e n o te E O sin U ) t a s th e e l e c t r i c o Q f i e l d o f th e r e f e r e n c e r a d i a t i o n w it h c i r c u l a r f r e q u e n c y (u)= 2 tt v )- The s c a tte re d ra d ia tio n alig h ed w ith the re fe re n c e ra d ia tio n is given | by E Fsin (u u t + 6) » w h e re = U ) - ( W is the D o p p le r sh ift and Ô is | s s s o ! a constant phase s h ift. If the to ta l gain (a m p s /w a tt) of the p h o to m u l- I tip lie r is denoted G ^, then the anode c u rre n t is | = ( B . l ) w h ere E^ is the h etero d y n ed w a v e fo rm . Now since ' = [ E ^ é im (oü^t + 6) + E^isiiiüu^t (B . 2) one re a d ily find s th a t the c u rre n t has c o n trib u tio n s fro m fre q u e n c ies 2(Ug, 2(1)^, (f«g + 0)^) . (u)g - uu^). (B . 3) 14 The f ir s t th re e com ponents a re o p tic a l fre q u e n c ie s of o rd e r 10 H z and cannot be seen by ty p ic a l p h o to m u ltip lie rs w hich have cu to ff 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 g fre q u e n c ie s on the o rd e r of 10 H z . C o nsequ ently, the anode c u rre n t w ill contain only tw o te rm s , a d. c. te r m and an a . c. te r m w hose fre q u en c y of o s c illa tio n is th e fo u rth com ponent in (B . 3). T h e anode c u rre n t can be w ritte n .2 _2 I = G a a F e + E 1 [ - 2 - ^ + co s («j,t+ 6)J IB . 4) fro m w h ic h w e conclude th a t the p h o to m u ltip lie r, w hen used as a high fre q u en c y m ix e r , acts as a fre q u en c y c o n v e rte r w h ic h a m p lifie s ju s t th e d e s ire d (D o p p le r) fre q u e n c y . W ritte n in te rm s of th e lum inous p o w er in c id e n t on the cathode (P^ = fo r the i— b e a m ), equation (B .4 ) becom es I = G [P f P cos(üU^t+ô)] (B .5 ) a a o :.s o s D ■ ' w h e re the fa c to r T ] (0 ^ T ) ^ 1), c a lle d th e h eterodynin g e ffic ie n c y , is included to account fo r th e coh erence loss of th e heterodyned w aves. 2. P h o to m u ltip lie r N o ise W e now p ro ce ed to c a lc u la te the s ig n a l-to -n o ia e .pow er ra tio fo r the anode c u rre n t assu m in g the h etero d yn in g p ro ce ss is 100% e ffic ie n t. A t the cathode th e c u rre n t is given by = G ^[P ^ + Pg + 2 , v ^ ^ cos(Uj^t+6)] (B . 6) fro m w h ic h w e id e n tify Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 I = 2G y P P c o s ( U ü t + ô ) (B .7 ) s c o s D as the D o p p le r signal and is the lum inous s e n s itiv ity of the cathode. T h e rm s shot noise (o r n o is e -in -s ig n a l) c u rre n t can be c a lc u la te d fro m the equation (see, fo r e x a m p le , [ 1 ? ], pg. 2 .5 -2 0 ) = 2eAu)I (B .8 ) n c w h e re e is the u n it e le c tro n ic ch a rg e; Au) is the e ffe c tiv e n o ise bandw idth, and is the cathode c u rre n t. T he s ig n a l-to -n o is e p ow er r a tio a t the cathode is g iven by the ra tio of the a v e ra g e d m e a n square c u rre n ts , i . e . , .2 , < V < i^ > n w h ic h can e a s ily be calc u la te d fro m equations (B . 6), ( B .7 ), and (B .8 ) giving G P P (B.9) eAui) (P + P ) * o s The s ig n a l-to -n o is e ra tio a t the anode is given b y ([ 1 7 ], p . 2 .5 -2 3 ) % = K ( 4 w h e re K = ( a - l ) / a and a is the a v e ra g e gain p e r state of th e m u lti p lie r chain; K is c a lle d th e m u ltip lie r noise fa c to r. T h e s ig n a l-to - noise a t the anode is then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 ( « 1 " ( p ^ ° p “ ) <®- and does not depend on the o v e ra ll gain G^, i. e . , the n u m b e r of dynodes. T h is is exp lain ed by the fa c t th at w h ile som e shot noise is gen erated a t each dynode in the m u ltip lie r ch a in , the b ig g est c o n trib u tio n com es fro m the cathode and the f ir s t dynode since th is noise undergoes the g re a te s t a m p lific a tio n . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX C L E N S F O C A L L E N G T H M E A S U R E M E N T S F o c a l positions of p a r a lle l lig h t beam s in c id e n t on a lens can be c a lc u la te d by the m ethod of re s e c tio n . The m e a s u rin g techniques and the re s e c tio n equations used to d e te rm in e the fo c a l p oint v a ria tio n a c ro s s a lens d ia m e te r a re o u tlin ed in the fo llo w in g p a ra g ra p h s . The lens in question is m ounted on an o p tic a l bench in fro n t of a tra v e llin g m ic ro s c o p e as shown in the sketch b elo w . It is lo c a te d f a ir l y close to a g rid w h ich is illu m in a te d fro m behind by p a r a lle l lig h t fro m a c o llim a to r lens w hich has a fo c a l len g th f^ . The g rid , w hich m ay be s im p ly a glass s la te w ith fin e d a rk lin e s etched on its I N o m in a l [Im ag e I P la n e T ra v e lin g M ic ro s c o p e G rid Lens 0 T h e o d o lite F ig u re C . 1. Top view of o p tic a l s e t-u p fo r lens fo c a l m e a s u re m e n ts . 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 su rfa c e in a squ ared p a tte rn , is p laced e x a c tly the distan ce fro m the c o llim a to r le n s . The te s t lens is then a lig n e d w ith the o p tic a l axis of the bench so th at its h o riz o n ta l d ia m e te r lie s in the sam e plane as the tra v e llin g m icro sco p e and a h o riz o n ta l g rid lin e . The shadow of a g rid lin e w ill f a ll n ea r the n o m in a l im ag e plane a t the position P(x^) as in d ic ated in F ig u re C . 1. The tra v e lin g m ic r o scope is then focused on P and the re la tiv e p o sitio n x^ is m e a s u re d . The th eo d o lite, when placed up close to the g rid (w ith the tea t lens rem o ved ) and focused a t in fin ity , m eas u res the re la tiv e angles p. = of lig h t as if it o rig in a te d fro m the r e a r nodal point of the c o llim a tin g le n s . H aving x^ and fo r a ll g rid lin e s , one can d e te r m in e the fo c a l points a cro s s the len s d ia m e te r fro m the equations X N x^ tan - Xg tan tan - tan . = ( X ^ - X ^ ) t a n a ^ (C . 1) (C .2 ) w h eretth e co o rd in ate s y s te m is g iven by X F ig u re C .2 . C o o rd in ate sy s te m fo r the fo c a l position m e a s u re m e n ts, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 and the angles in (C . 1) and (C . 2) a re c a lc u la te d fro m the re la tio n s = a. + p. ■ (C .3 ) 1+1 I I "^2 ■ "‘ S ° 'l (x 2 ~ X i)c o t + (x^^-x^)cot Pg (C .4 ) w h ere is the in itia l an g le . T h e re is a c e rta in a rb itra rin e s s in the se le c tio n of P(Xj^) and P(Xg) fo r use in equations (C . 1) and (C .2 ), b e cause one could s e le c t s e p a ra te as w è ll as ad jace n t g rid lin e s . If one is in te re s te d in the lo c a l v a ria tio n of the fo c a l p o sition a c ro s s the lens d ia m e te r, a d ja c e n t g rid lin e s would be p re fe rra b le . L o c a l v a ria tio n s a re of p a rtic u la r in te re s t in p re c is io n photographic w o rk ( e .g ., the s u rv e illa n c e of the m oon fro m an o rb itin g s a te llite ) w h ere one would lik e to in s u re th at a ll rays fa llin g on a photographic p late a re eq u ally in focus. This s y s te m of data red u ctio n w ill be c a lle d R esectio n M ethod I. F ig u re s C . 3 and C . 4 show re s u lts fo r the a c h ro m a t lens used in the ro ta tin g disc e x p e rim e n t (c f. C h ap ter I I I , p a ra g ra p h 7. 2) and a p lan o -co n v ex lens w ith n o m in al fo c a l lengths of 115 m m and 90 m m , re s p e c tiv e ly . F ig u re C. 3 shows the v a ria tio n of the fo c al depth a cro s s the lens d ia m e te r and F ig u re C .4 gives the s p rea d in the h o riz o n ta l plane. The av e ra g e fo c a l len g th as d e te rm in e d fro m this m ethod o f re s e c tio n gives a value 10% lo n g e r than n o m in al fo r the p lan o -co n v ex le n s , and about 2% lo n g e r fo r the a c h ro m a t lens . In la s e r D o p p le r a n e m o m e try one is m o re in te re s te d in the fo c a l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 po sitio n of two p a r a lle l rays w ith an in itia l s ep aratio n d istan cé d. The re la tiv e positions selected fo r use in the resectixam equa tions would then depend on d. T h is p ro ced u re fo r d e te rm in in g the fo c a l points we denote as R e sectio n M ethod I I. In the ro ta tin g disc e x p e rim e n t we find d ^ 1 7 .0 m m . The re s u lts of this fo c a l length d e te rm in a tio n fo r the two lenses a re given in F ig u re C. 5. Though the fo c a l lengths a re a p p ro x im a te ly the sam e as those given by R e sectio n M ethod I, the d eviatio n acro s s the lens d ia m e te r is con s id e ra b ly s m a lle r. It is this fo c a l length fo r the a c h ro m a t le n s , f = 1 1 8 .4 m m , w hich is used to d e te rm in e the beam in te rs e c tio n ave angle 9 fo r the ro ta tin g disc e x p e rim e n t. We note that the m ethod of re s e c tio n does not ^ive in fo rm a tio n about the skewing of rays out of the h o riz o n ta l plane of the lens sw ept through by the tra v e lin g m ic ro s c o p e . One im p lic itly assum es th ere a re no la rg e a z im u th a l v a ria tio n s in slope on both faces of the le n s . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD ■ D O Q. C g Q. ■ D CD C/) o' 3 O 8 ■ D 3. 3" CD CD ■ D O Q. C a O 3 ■ D O CD Q. ■ D CD C/) C/) RESECTION METHOD 1 o I PLANO CONVEX 90 o o o o o o 95 97 99 101 28 P ACHROMAT 1 1 5 2 0 o 16 K 0 = 117.4 MM AVE 0 0 =LENS . 1 1 2 1 1 4 LENGTH 116 ■(MM) lis 120 122 124 FOCAL F ig u re C .3 . F o c a l depth v a ria tio n across the lens d ia m e te r as d e te rm in e d by R esection M ethod I. 147 i n Ü J z < _j Q. < O e w m z ÜJ . l g _ o % CD o c r X o < o o (lAIIAI)/^ 30NV1SIQ n V 3 lld 3 A = '^X c v j O Q O X I— L Ü a C O ÜJ X C O d o § T T C V I d 8 X w 1 8 O z < _l C l — ---------- 0 1 I a < M C M G O C D = i M " lU = U z < = « Q < O o u. > ? cn N cn m C D 'U 0 •B 1 § •D u W « K T D D Ü « T3 m a 0 Ü 1 m D •ê (d h (d > -w a o a. Id u o h U « g) •W Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. : : o CD ■ o S Û . c g Û . T3 CD g O = J 8 T3 C 5 - CD C 3. 3" CD c B ■ D 3 Q . C a O 3 " O 3 CD Q . ■ D CD 00 C/) RESECTION METHOD 2 PLANO CONVEX 90 0 6 1 J 95 97 99 1 0 1 Y^=LENS 28 20 16 12 z o H (/) O Û. < u q: 8 uj < 4 if) g o' I o p ? o ACHROMAT 1 1 5 I FOCAL 1 1 2 1 1 4 LENGTH lie 118 ^(MM) 120 122 124 F ig u re C .5 . F o c a l depth v a ria tio n s acro ss the lens d ia m e te r as d eterm in ed by R esectio n M ethod I I, 00 A P P E N D IX D E R R O R A N A L Y S IS 1. R o tating D is c E x p e rim e n t In the ro ta tin g d isc e x p e rim e n t the v e lo c ity is calcu lated fro m the exp re s s io n (D. 1) V = 2 sin (0 /2 ) w hich is equation (III. 3) w ith n set eq u al to one. The angle 0 is given by equation ( III. 13) in the te x t ~ T ( “t ) - ? ^®i , n ) = cos 0 g so th a t u ltim a te ly V = v i X t n , 0 ., j&, f, Vrx). o g 1 J J (D. 3) The p a ra m e te rs fo r th is e x p e rim e n t take on the fo llo w in g values: \ = 6, 328 ±0. 0000 (a n g stro m s) ; AX /X o o 0.0000 n = 1. 5174 ±0. 0005; g 0. = 38. 95 ± 0 .2 5 (d eg rees ) ; j 6 = 0. 9480 ± 0. 0005 (inches); f = 1 1 8 .3 6 iO. 45 (m illim e te rs ) V = 100 - 400 (K H e rtz ) ; o An /n = ± 0 .0 0 0 3 g g A6J0^ = ± 0 .0 0 6 4 Aje/j0 = ± 0 .0 0 0 5 A f/f = ± 0 .0 0 3 8 A V jj/V|u = ± 0 .0 0 8 3 (D .4 ) The illu m in a tin g w avelen g th fo r the H e -N e la s e r is assum ed to be _______________________ 149 ________________________________ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 known to w ith in fo u r d e c im a l place a c c u ra c y . The r e fra c tiv e index fo r the p a r a lle l fla t was re p o rte d by the m a n u fa c tu re r, w h ile 8^ and | j0 w e re obtained by d ire c t m e a s u re m e n t. The fo c a l length and its j e r r o r e s tim a te re s u lts fro m the a n a ly s is given in A p pendix C. F r e quency m e a s u re m e n t e r r o r w as d e te rm in e d by o b s ervin g how a c c u r a te ly a known fre q u en c y fro m a sig n al g e n e ra to r could be alig n ed w ith the D o p p le r sp e ctru m d isp layed on the sto rag e s c re e n of the s p e ctru m a n a ly z e r. (See F ig . X .3 d ). . The con tributions of these e r r o r s to the v e lo c ity m e a s u re m e n t can be d e te rm in e d fro m equations (D . 1) and D .2 ) . F ro m equation (D . 1) we have d ire c tly dv and using both (D. 1) and (D .2 ) one e a s ily finds ( t “ ) ^ Using the d e fin itio n fo r ç (0^, n^) in equation (D. 2) gives the two re m a in in g e r r o r con trib u tio n s ( f ^ ) e , = ( I ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (v), = - 2 0 — — COS — B (8 151 ; (D. 10)1 w here 1 1 2 cos 0. r c î E ^ - - \ ' 1' g* (dng/Og) ■ C (e ..n ^ ) ( D . l l ) (D . 12) and C ( 9 i.“ g) = V ^ (D . 13) The functions A (9 .,n ), B (0 .,n ), and § (0 .,n ) a r e p lo tted as a i ê 1 ê ^ ë fu n ctio n of 0. in F ig u re D . 1 fo r the n o m in al valu e of n given in (D. 4). 1 ë The m a x im u m in 1 (0 ., n ) a t 0, = 4 9 .5 ° corresponds to the m a x im u m 1 ë ^ possible s e p a ra tio n distance betw een the two p a r a lle l b ea m s . Furfc- th e r in c re a s e in beam s e p a ra tio n can only be obtained by using a new p a r a lle l fla t w ith a la r g e r w id th ) betw een faces . W e a re now in a positio n to ca lc u la te the expected d e v ia tio n in the m eas u red v e lo c ity . A t 0. = 38. 9 5 ° one finds fro m F ig u re D . 1 th a t A (0 .,n ) = 0. 46 and B (0 .,n ) = - 1 .2 3 . The angle 0 is com puted 1 ë ^ ë to be 8 .2 3 ° . U sing the re la tiv e e r r o r s in ( D .4 ), equations (D . 5) - (D . 10) give (d v /v )^ = ± 0 .0 0 0 0 o (d v /v ) = ± 0 .0 0 8 3 (dv/v)g = ± 0 .0 0 3 8 (dv/v)^ = ± 0 .0 0 0 5 (d v /v )n n ë = ± 0 .0 0 0 4 (D . 14) (d v /v )e = ± 0 .0 0 2 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 i and the to ta l re la tiv e e r r o r , c a lc u la te d as the ro o t m e a n squ are of the in d iv id u a l com ponents, becom es I— I = 0 .0 0 9 6 (D . 15) ^ 'D o p p le r We note th at the g re a te s t co n trib u tio n to the to ta l e r r o r in this e x p e rim e n t is due to the D o p p ler fre q u e n c y m e a s u re m e n t, 2. U nsteady V e lo c ity M easu re m e n ts In the tim e dependent v e lo c ity m eas u rem en ts w ith in the ro ta tin g c y lin d e r one is in te re s te d in the a c c u ra c y of the p o s itio n , fre q u e n c y , and tim e d ata. An an alys is of the expected e r r o r s is g iven in the fo llo w in g p a ra g ra p h s . 2. 1 P ro b e V o lu m e and P o s itio n A ccording to [ 2 4 ] , we define the probe volum e as the e llip s o id on w hich a s c a tte r c e n te r w ill g en erate a D o p p le r c u rre n t of 1 /e the m a x im u m am p litu d e a t the cen ter o f the v o lu m e . This vo lu m e is ju s t 2 "bounded" by the 1 /e in te n s ity contours of the re fe re n c e and s c a tte re d b e a m s . The p ro b e volu m e fo r eq u al d ia m e te r beam s is sketched in the fig u re below " 2 ^ l / e In te n s ity C o nto ur ’2 f o ___ . 2 s in (9 /2 ) Jz b O c o s (9 /2 ) 1 i : J L 1 \/Z h o j F ig u re D . 2 . l / e p ro b e v o lu m e . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 i 2 The fo c a l ra d iu s fo r the l / e in te n s ity contours is g iven by ! b = (2 / t t) F \ (D . 16) O o w h e re F is the lens F -n u m b e r defined as F = ^ . (D . 17) 2 H e re f is the len s fo c a l len g th and 2b is the l / e in te n s ity d ia m e te r of the input la s e r b eam . In the p re s e n t e x p e rim e n t F ~ 100 w h ic h gives b^ = 0. 04 m m . Now the angle a t w hich the beam s in te rs e c t inside the flu id m e d iu m v a rie s a c ro s s the c y lin d e r d ia m e te r. T he r a y t r a c ing c alcu latio n s (cf. A ppendix E ) show th at hhe beam in te rs e c tio n angle 0 v a rie s lin e a r ly fro m 0. 0528 ra d a t r = a to 0. 0672 ra d . a t the c y lin d r ic a l axis r = 0. The la r g e s t fo c a l volum e occu rs a t the c y lin d r ic a l w a ll. F o r this case the c h a ra c te ris tic lengths a re ^ = -6 2. 12 m m , A ^ Z ~ m m and the fo c a l vo lu m e is 3. 60 x 10 cc . The ra th e r la rg e sp rea d (6 of the fo c a l volum e in the r a d ia l d ire c tio n caused som e d iffic u lty in obtainin g the v e lo c ity m e a s u re m ents n e a r the w a ll d u rin g s p in -d o w n to r e s t. M e a s u re m e n ts w e re obtained a t r / a = 0. 975 and the p o sition of the fo c a l vo lu m e re la tiv e to the w a ll is d ep icted in the sketch on the fo llo w in g page. D u rin g the in itia l stages of spin-dow n to re s t, two d is tin c t fre q u e n c ie s , one co rres p o n d in g to the in n e r w a ll v e lo c ity and the o th e r co rres p o n d in g to the flu id v e lo c ity a t the c e n te r of the probe v o lu m e , could be d etected . E v e n though the l / e contour d id not in te rs e c t the c y lin d e r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 0 . 84 m m C l / e p o w er a m p litu d e contour _L W a te r 12 m m I t 2 . 12 m m ^ I 0. 057 m m F ig u re D . 3. L o c atio n of probe vo lu m e fo r m eas u re m e n ts a t r / a = 0 .9 7 5 . w a ll, the intense s c a tte rin g a t the w a te r-In c ite in te rfa c e gave a s ig n a l whose stren g th was c o m p a ra b le to the flu id v e lo c ity s ig n a l. D u r - | i ing spin -d o w n these th re e fre q u e n c ie s c ro ss ed each o th e r and one hadj to be c a re fu l to re c o rd the c o rre c t one. T h is d iffic u lty was n o t | p re s e n t in the m eas u rem e n ts a t s m a lle r r a d ii. As m entio ned in the te x t, the re la tiv e p o s itio n of the fo c a l point ; ! could be d e te rm in e d to w ith in ± 0 . 002 in . w h ile the re fe re n c e point a t | I the in n e r w a ll was lo cated to w ith in ± 0 . 010 in . T h is gives a m a x i m u m re la tiv e e r r o r (calc u la te d a t r = r . = 0 .3 3 0 ) in the p o s itio n i m e a s u re m e n ts o f a p p ro x im a te ly ± 0 . 010. 1 I I 2. 2 F re q u e n c y and T im e M e a s u re m e n ts In the sp in -u p and sp in-dow n e x p e rim e n ts the tim e and fre q u en c y had to be m e a s u re d sim u lta n e o u s ly . B ecause the fre q u e n c y m e a s u re m e n t was m ade und er unsteady flo w co n d itio n s, the a c c u ra c y in the m e a s u re m e n t was som ew hat less than th at obtained und er stead y flow conditions (re la tiv e e r r o r ~ ± 0 . 00 8 ). The exp ected re la tiv e e r r o r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 i I w as e s tim a te d to be ± 0 .0 1 5 . The a c c u ra c y in the tim e m e a s u re m en ts I is b a s ic a lly lim ite d to the a c c u ra c y w ith w hich the tim e could be re a d i I fro m the X - Y re c o rd e r p lo t. The sweep ra te was alw ays adjusted so 1 th at the f u ll scale corresp o n d ed a p p ro x im a te ly to the s p in -u p tim e . I i In this m a n n e r the re la tiv e e r r o r based on the sp in -u p tim e was held I to ±0. 005. T his includes e r r o r s in the tim e d elay betw een noting the | fre q u en c y on the s p e c tru m a n a ly z e r and d ep ressin g the z e ro check button on the X - Y re c o rd e r. F o r the single case of the high a c c e le r atio n ra te corresponding to F ig u re X . 12 the re la tiv e e r r o r was m o re | lik e ±0. 02. | 2. 3 R e p e a ta b ility | j In the cases w h e re a single ru n was s u ffic ie n t to define the e x p e rim e n ta l cu rve a t a given ra d iu s , the s c a tte r is seen to be s m a ll, i in keeping w ith the e r r o r an a lys is discussed above. S lig h t co n sis ten t | d eviations betw een th e o ry and e x p e rim e n t can be detected in reg io ns | aw ay fro m the m oving w ave fro n t in s p in -u p , and aw ay fro m the side - | w a ll boundary la y e rs in sp in -d o w n . T h is is because the th e o re tic a l com putations w e re based on a constant a c c e le ra tio n ra te when in fa c t the a c c e le ra tio n curves w e re only lin e a r to w ith in j - 3%. The e x p e rim e n ta l points w e re alw ays seen to r e fle c t the tru e w a ll a c c e le r a tio n s. M any o f the cu rve s defined by the e x p e rim e n ta l points w e re obtained fro m s e v e ra l rep ea te d ru n s . Though the m e a s u re m e n t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 e r r o r s a re s t ill s m a ll, the data has m o re sp read because of the in a b ility to e x a c tly rep ro d u ce a given E k m a n n u m b e r. In a ll cases, the averag e E k m a n n u m b e r was c a lc u la te d fo r each rad iu s and the th e o re tic a l cu rve was com puted fo r that a v e ra g e . The fo llo w in g tables lis t the ave rag e E k m a n num bers used to m ake the th e o re tic a l calculations, fo r the m u ltip le ru n c u rv e s . A lso included is the a p p ro x im a te m a x im u m d ev iatio n about the a v e ra g e . r / a A v e ra g e E ^ % D e v ia tio n N o. of Runs 0.330 11.92 X 10^^ ± 1 .5 k 2 0 .514 11. 74 X 10"^ ± 0 .5 Z 2 . 0.804 11.59 X 10"^ ± 1 .5 2 2 T ab le D . 1. S îynopsis of conditions co rresp o n d in g to F ig . X . 10. r / a A v e ra g e E ^ % D e v ia tio n N o . of Runs 0.334 8. 70 X 10“^ ±2. 0 3 3 0 .5 1 7 8. 71 X 10"^ ±2. 5 6 0. 807 8. 75 X 10"^ ± 3 .0 5 T ab le D .2 . Synopsis of conditions co rresp o n d in g to F ig . X . 11. r / q Average E ^ % D eviation No. of Runs 0.334 2 .3 5 5 X 10”^ ± 2 .0 7 0 .517 2 .2 6 0 X 10"^ ±4. 0 7 0. 807 2 .3 6 0 X lo"^ ± 3 .0 9 T ab le D . 3. Synopsis of conditions c o rres p o n d in g to F ig . X . 12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 r / a A v e ra g e % D e v ia tio n N o . of Runs 0. 330 10 .2 9 X 1 0 -^ ±1. 0 3 0 .5 1 4 1 0 .3 3 X 1 0 -‘ ± 1 .5 4 0. 626 9 .5 9 X 1 0 -^ ± 1 .0 3 0. 727 9 .5 6 X 10-^ ±0. 7 3 0. 804 10. 18 X 10-^ ± 2 .5 9 0. 850 9. 75 X 1 0 -^ ±1. 0 3 0. 900 9. 70 X 10-^ ±0. 5 3 0. 925 9. 82 X 10-^ ±1. 0 3 0 .9 5 0 9 .6 2 X 10-^ ±4 . 0 3 0 .9 7 5 1 0 .4 8 X 10-^ ±1. 5 3 T a b le D . 4. Synopsis of conditions co rres p o n d in g to F ig . X . 16. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 •o Q CM CD ' CE - in c ô " CD CD CD CD C D “ C M CD u m m jQ C M CM CM (A — Z m œ < C O C E CD in CD % ^ o o y o 3 = (Z/g)uo& 4 00 CO C V J ro 0 Q Q • r 4 t— 4 1 h O V (U n 3 k O 00 m 73 d n J b o < ® -2 % Q j ^ % I I II a l 0 ) M & k Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX E O P T IC A L R A Y T R A C IN G C A L C U L A T IO N S 1. F o c a l P o s itio n C a lib ra tio n fo r the C y lin d e r A unit m o vem en t o f the lens outside the c y lin d e r does not c o r r e s pond to an equal m o vem e n t of th e fo c a l p o in t w ith in the c y lin d e r. In fa c t the re la tio n of these d isp lacem en ts is not one o f s im p le p ro p o rtio n a lity , but m ust be c a lc u la te d fro m ra y tra c in g techniques. In the fo llo w in g p arag ra p h s w e s h a ll ou tlin e the ra y tra c in g p ro ce d u re used in the d e te rm in a tio n of the fo c a l p o sitio n s w ith in the c y lin d e r. C o n sid er f ir s t a ra y in cid en t on a re fra c tin g s p h e ric a l su rface at P w hich in te rc e p ts the fo c a l a x is a t Q ^as shown in the d ia g ra m on the fo llo w in g page. T h e c e n te r of the re fra c tin g su rfa c e is a t C and the pole is lo cated a distan ce r^ aw ay a t A . W e assu m e the r e fr a c t- tiv e indices a r e n^ and n as in d ic ated and th a t th e in c id e n t ra y a t P is d e s c rib e d by L ^ and cp^. T h e p ro b le m is to c a lc u la te the new angle c p taken by the re fra c te d ra y and its in te rc e p tio n d istan ce x . F r o m tria n g le P C B w e have (L - r ) sin I = ---------------- sin c p (E . 1) o r^ o 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 F ig u re E . 1. R ay .^racing d ia g ra m fo r a S p h e ric a l r e fra c tin g S u rface. and fr o m S n e ll's law of re fra c tio n n T o . _ sin I = sin I n o (E .2 ) A ls o fro m the fig u re c p = c p + I - I o o and, fin a lly , fro m tria n g le PC Q sin I (E .3 ) X = r , + r . (E .4 ) s in c p 1 1 Successive a p p lic a tio n of the above equations giverthe q u an tities c p and X w h ic h sp ecify the re fra c te d ra y P Q . T h e se a re known as the re fra c tio n equations (cf pg. 191). N ow suppose th a t, as in the case of a th in w a lle d c y lin d e r, the re fra c te d ra y in te rc e p ts a second re fra c tio n su rface b e fo re it cro sses the fo c a l a x is . I f th e pole of the new su rfa c e is located a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 d istan ce d fro m A , then the in cid en t valu es fo r th is new su rface a re given by = X - d (ID. 5) c p ' = c p (E . 6) o w h e re is now r e fe r r e d to the po le of the second s u rfa c e . E q ua tion s (E . 5) and (E . 6) a re c a lle d the tra n s fe r eq u atio n s. T h e in c i dent values a re then substituted in to the re fra c tio n equations (E . 1 )- (E . 4) in o rd e r to solve fo r the new re fra c te d ra y path. The above equations w e re used to d e te rm in e th e fo c a l p osition X a t the flu id in te r io r as a functio n of the re la tiv e lens d isp lacem en t L outside the c y lin d e r. T h e c alcu latio n s w e re c a rrie d out fo r the conditions in d ic ated in F ig u re E . 2. The angle cp^ w as d e te rm in e d fro m equation ( III. 13) a fte r having m e a s u re d a ll the n e c e s s a ry constants. (Note th at c p = 9 /2 ). The valu es n , n , and n in d i- o a g w cated a re those a p p ro p ria te to a ir . In c ite , and w a te r, re s p e c tiv e ly . T h e d istan ce betw een poles in the tra n s fe r equation (E . 5) is , of co u rs e, given b y d = r ^ - r ^ . The com puted re s u lts , p resen ted in F ig u re E . 3, show th a t the lens is d isp laced a d istan ce L = 3. 119 in . w h ile the fo c a l point x in the flu id m oves fro m c y lin d e r w a ll to c y lin d e r c e n te r. Since the angle cp^ w as not a lte re d d u rin g the co u rse of the e x p e rim e n t. F ig u re E . 3 p ro vid es th e c a lib ra tio n c u rv e fo r a ll m e a s u re m e n ts . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w r = 3. 459 in . r_ = 2 . 975 in . c p = 0. 336 r ad. o n = 1. 4 9 0 _ J k F ig u re £ . 2 . C onstants used in th e ra y "tracin g c a lc u la tio n s fo r the (C irc u la r (c y lin d e r. 2. S m a ll A ngle C o n sid eratio n s T h e re fra c tio n equations show th at c p (o r 9 /2 ) v a rie s w ith x and th e re fo re the c a lib ra tio n fa c to r in the v e lo c ity -fre q u e n c y re la tio n K D V 2n V = I T (E .7 ) is now a function of the ra d iu s . F o r solid body ro ta tio n , th e re fo re , the D o p p le r freq u en cy is not a lin e a r funfction of x . In itia l m e a s u re m ents of h o w eve r, did p lo t lin e a r ly w ith L , a re s u lt not a t a ll in tu itiv e ly expected. In the fo llo w in g an a lys is w e show th a t is indeed a lin e a r function of L , but only fo r s m a ll in cid en t angles cp^. F o r s im p lic ity w e co n sid er th e single re fra c tin g su rface in F ig u re E . 1. W e assum e th a t a ll angles a re s m a ll so sin a ~ a and cos a ~ 1. Then equation (E . 3) becom es (w ith re p la c e d by L ) (E .8 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 and hence d c p d L c p / n \ (E.9) E q u atio n ( E .4 ) s im p lifie s to re a d = = ?! + 1^^ IT (E- ^0 ) and hence - c p n c p n , N ow fo r solid body ro ta tio n v = O r = 0 (r ^ -x ) and fro m (E .7 ) one obtains Vd = c p n (ri-x ) (E . 12) o and consequently S u bstituting (E . 8), (E . 9), (E . 10), and (E . 11) in to (E . 13) one finds the s im p le re la tio n dv 2n n _ [ * o + * ( 9 ) i , (E .1 4 ) and so w ith v œ-x the c a lib ra tio n fa c to r K ^ (x ) changes in such a w ay th a t Vj^ “ L . O f cou rse w e have c o n sid ered only one r e fra c tiv e s u rfa c e , but a ra y tra c in g a n a ly s is th ro u g h a second m e d iu m gives Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 the sam e re s u lt as long as the r e fra c tiv e in d ices of a ll th re e m ed iu m s (in th is case a ir . In c ite , and w a te r) a re o f the sam e o rd e r of m ag n itu d e. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 3 CD " O 3 Q . C g a " O CD C/) C O o' 3 o 8 " O ci- 3 " i 3 CD 3. 3 " CD 3 " O o Q . C g o 3 " O o CD Q . ■ D CD C /) C /) 3 . 2 GEOMETRICAL CENTER AT X = Ro = 2. 97 5 i n . 2.8 3 2 . 4 X=R b 2 . 0 'EXP. RAY TRACING THEORY ------- X = L FOCAL POINT POSITION CALIBRATION L. 0 .8 0 . 4 INNER WALL 2 . 0 2 . 4 2 . 8 3 . 2 3 . 6 0 . 4 0 . 8 1 . 2 1 .6 0 LENSE POSITION /-L (IN ) F ig u re E . 3 . F o c a l poiht position inside the c y lin d e r as d e te rm in e d by ra y tra c in g th eo ry. O' O l C h a p te r V I I I R E F E R E N C E S 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 R E F E R E N C E S 1. G reenspan, H . P . and H o w a rd , L . N . (1963); "O n a T im e D ependent M o tio n of a R o tatin g F ln id , " J. F lu id M e c h . , V o l. 17, pp. 3 8 5 -4 0 4 . 2. G reenspan, H . P . and W ein b au m , S. (1965): "O n N o n -lin e a r S p in -U p o f a R o tatin g F lu id , " J. M a th , and P h y s . , V o l. 44, pp. 6 6 -8 5 . 3. W e d e m e y e r, E . H . (1964): " T h e U n steady F lo w W ith in a S p in ning C y lin d e r, " J^_JQuid_Mech^ , V o l. 20, pp. 3 8 3 -3 9 9 . 4 . G reenspan, H . P . (1969): T h e T h e o ry of R o tatin g F lu id s , C a m b rid g e U n iv e rs ity P re s s , G re a t B r ita in . 5. V e n e z ia n , G . (1969): "S p in -U p of a C ontained F lu id ," T o p ic s in O cean E n g in e e rin g . V o l. 1, pp. 2 1 2 -2 2 3 . 6. In g e rs o ll, A . P . and V e n e z ia n , G. (1968): " N o n -lin e a r S p in - Up of a C ontained F lu id , " (unpublished c o m m u n ic a tio n ). 7. V e n e z ia n , G. (1970): " N o n -L in e a r S p in -U p ," T o p ics in O cean E n g in e e rin g , V o l. 2 , pp. 8 7 -9 6 . 8. R o g e rs , M . H . and L an ce, G . N . (I9 6 0 ): "T h e R o ta tio n a lly S y m m e tric F lo w of a V isco u s F lu id in the P re s e n c e o f an In fin ite R o tatin g D is k ," J. F lu id M e c h . , V o l. 7, pp. 6 1 7 -6 3 1 . 9. Jacobs, S. J. (1964): " T h e T a y io r C o lu m n P r o b le m ," J. F lu id M e c h ., V o l. 20, pp. 5 8 1 -5 9 1 . 10. C a r r ie r , G. E . (1971): " S w irlin g F lo w B o u n d ary L a y e rs , " J. F lu id M e c h .. V o l. 4 9 , P a r t 1, pp. 1 3 3 -1 4 4 . 11. C o u ran t, R . (1936): D iffe r e n tia l and In te g ra l C a lc u lu s . V o l. 2, In te rs c ie n c e P u b lis h e rs , In c ., N e w Y o rk . T ra n s la te d b y E . J . M cShane. 12. A b ra m o w itz , M . and Stegun, A . (1965): Handbook of M a th e m a tic a l F u n c tio n s . D o v e r P u b lic a tio n s , In c ., N .Y . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - - - - ^1681 13. Y eh, Y . and C u m m in s , H . Z . (1964); " L o c a liz e d F lu id F lo w M e a s u re m e n ts w ith an H e -N e L a s e r S p e c tro m e te r, " A p p lied P h y s ic s L e tte r s , V o l. 4 , N o . 10, pp. 1 7 6 -1 7 7 . 14. F o re m a n , J. W . , G e o rg e , E . W ., and L e w is R . D . (1965): "M e a s u re m e n ts of L o c a liz e d F lo w V e lo c itie s in G ases w ith L a s e r D o p p le r F lo w m e te r ," A p p lie d P h ysics L e tte r s , V o l. 34, N o . 4, pp. 8 1 3 -8 1 8 . 15. G o ld s te in , R . J . and K r ie d , D . K . (1967): "M e a s u re m e n t o f L a m in a r F lo w D e v e lo p m e n t in a Square D u c t U sing a L a s e r D o p p le r F lo w m e te r, " J. A p p l. M e c h . , V o l. 34, N o . 4 , pp. 8 1 3 - 818. 16. B ie n , F . and P e n n e r, S. S. (1970 ): " V e lo c ity P ro file s in Steady and U nsteady R o tatin g F lo w s fo r a F in ite C y lin d ric a l G e o m e try , " P h ysics o f F lu id s , V o l. 13, N o . 7, pp. 1 6 6 5 -1 6 7 1 . 17. R o lfe , E . , S ilk , J. K . , B o oth , S ., M e is te r , K . , and Young, R . M . (1968): " L a s e r D o p p le r V e lo c ity In s tru m e n t, " N A S A C R -1 1 9 9 , W ashington, D . C . 18. D u rs t, R . , M e llin g , A . , and W h ite la w , J. H . (1972): " L a s e r A n e m o m e try : A R e p o rt on E U R O M E C H 36, " J. F lu id M e c h ., V o l. 56, P a r t 4 , pp. 1 4 3 -1 6 0 . 19. Seaton, S. L . (1967): "G as D e n s ity M e a s u re m e n ts U sin g L ig h t S c a tte rin g , " N A S A T N D -3 8 9 5 , W ashington, D . C . 2 0. M a rb le , F . E . (1962): " D y n a m ic s of a Gas C o ntaining S m a ll S olid P a r tic le s , " C o m b u stio n and P ro p u ls io n . F ifth A G A R D C o llo q u iu m . E d ite d b y: H a g e rty , R . P . , L u tz , O . , J a u m e tte, A . L . , and P e n n e r, S. S . , P e rg a m o n P re s s , d is trib u te d b y the M a c m illa n C om pany, N . Y . 2 1 . Shipp, J . I . , G o e th e rt, W . H . , and S n yd er, W . T . (1967): " E x p e rim e n ta l M e a s u re m e n ts of E n tra n c e R eg io n V e lo c ity P ro file s U sing a L a s e r V e lo c im e te r . " P a p ervp res en ted a t the M e e tin g o f the A m e ric a n P h y s ic a l S o cie ty, D iv is io n o f F lu id i: D y n a m ic s , B e th le h e m , P e n n . 2 2 . B ra y to n , D . B . (1969): " A S im p le L a s e r , D o p p le r S h ift, V e lo c i m e te r w ith S e lf - A lig n in g O p tic s , " P re s e n te d a t th e E le c tr o - O p tic a l S ystem s D e sig n C o n fe re n c e , Sept. 1 6 -1 8 , N ew Y o rk C o lis e u m , N . Y . A ls o , A E D C -T R -7 0 -4 5 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ........................................ ■ ; . T69.. 23. B la k e , K . A . and Jes p erso n , K . I. (1972): " T h e N E L L a s e r V e lo c im e te r, " N E L R e p o rt N o . 510. 2 4 . L e n n e rt, A . E . , et a l. (1971); " L a s e r M e tro lo g y , " P re s e n te d a t the C o n feren ce on L a s e r Tech n o lo g y in A e ro d y n a m ic M e a s u re m e n ts , June 1 4 -1 8 , von K a rm a n In s titu te , R h o d e -S a in t- G enese, B e lg iu m . N o tes pub lished in V K I L e c tu re S e rie s 39, V o l. 1. 25. F ey n m an , R . P . , L eig h to n , R . B . , and Sands, M . (1966): T h e F ey n m an L e c tu re s on P h y s ic s . V o l. 1, Addis o n -W es le y P u b lish in g C o ., In c ., R ead in g , M a s s . 2 6 . E d w a rd s , R . V . , Angus, J. C . , F re n c h , M . J . , and D unning, J. W . J r . (1971): "S p e c tra l A n a ly s is o f th e S ig n al fro m the L a s e r D o p p le r F lo w m e te r; T im e -Ind ependent S y s te m s ," J. A p p l. P h ys. , V o l. 4 2 , N o . 2, pp. 8 3 7 -8 5 0 . 2 7. F o re m a n , J . W . J r . (1967): " O p tic a l P a th -L e n g th D iffe re n c e E ffe c ts in P h o to m ixin g w ith M u ltim o d e Gas L a s e r R a d ia tio n , " A p p lied O p tics, V o l. 6, N o . 5, pp. 8 2 1 -8 2 6 . 2 8. Logan, S. E . (1970): " A M o d est L a s e r D o p p le r In s tru m e n t fo r L a b o ra to ry M o tio n M e a s u re m e n ts . " P re s e n te d at the A IA A 20th A nnual R egio n V I Student C o n fe re n c e , A p r il 30 - M a y 1, C a lifo rn ia State P o ly te c h n ic C o lle g e , San L u is O bispo, C a lif. 29. P e d lo s k y , J . and G reen sp an , H . P . (1967): " A S im p le L a b o r a to ry M o d e l fo r the O ceanic C ir c u la tio n ," J. F lu id M e c h ., V o l. 27, pp. 2 9 1 -3 0 4 . 30. T a y lo r , G. I . (1923): " S ta b ility o f a V isco u s L iq u id C ontained B etw een Tw o R o tatin g C y lin d e rs , " P h il. T ra n s . R o y. S o c ., (London), V o l. 223a, pp. 2 8 9 -3 4 3 . 31. G d r tle r , H . (1940): " Ü b e r eine d re id im e n s io n a le In s ta b ilita t la m in a re r G renza.chichtena.n konkaven W anden, " N a c h r. G e s. W is s . . G ottingen, M a th -P h y s . K l . 2, pp. 1 -2 6 . 32. S m ith , N . H . (1947): " E x p lo ra to ry In v e s tig a tio n s of L a m in a r B o u n d ary L a y e r O s c illa tio n s on a R o tatin g D is k ," N A C A T N N o . 1227, W ashington, D . C . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 3 3 . G re g o ry , N ., S tu a rt, J. T . , and W a lk e r, W . S. (1955); "O n the S ta b ility of T h re e -D im e n s io n a l B o u n d ary L a y e rs w ith A p p li c atio n to the F lo w D u e to a R o tatin g D is k , " P h il.T r a n s . R o y . Soc. (London), V o l. 24 8 A , pp. 1 5 5 -1 9 9 . 34. F a lle r , A . J . (1963): " A n E x p e rim e n ta l Study of th e In s ta b ility o f the L a m in a r E k m a n B o u n d ary L a y e r, " J. F lu id M e c h ., V o l. 15, P a r t 4 , pp. 5 6 0 -5 7 6 . 35. T a tr o , P . R . and M o llo -C h ris te n s e n , E . L . (1967): " E x p e r i m ents on E k m a n B o u n d ary L a y e r In s ta b ilitie s , " J. F lu id M e c h . , V o l. 28, P a r t 3, pp. 5 3 1 -5 4 3 . 36. C a ld w e ll, D . R . and V a n A tta , C . W . (1970): " C h a ra c te ris tic s o f E k m a n B o u n d ary L a y e r In s ta b ilitie s , " J. F lu id M e c h ., V o l. 44, P a r t 1, pp. 7 9 -9 5 . 37. F a lle r , A . J. and K a y lo r , R . E . (1966): " A N u m e ric a l Study of the In s ta b ility of the L a m in a r E k m a n B o u n d ary L a y e r, " J. A tm o s p h e ric S cien ce, V o l. 23, pp. 4 6 6 -4 8 0 . 38. L illy , D . K . (1966): "O n the In s ta b ility of E k m a n B o undary F lo w , " J. A tm o s p h e ric Science, V o l. 23, pp. 4 8 1 -4 9 4 . 39. F a lle r , A . J. and K a y lo r, R . E . (1965): "In v e s tig a tio n s of S ta b ility and T ra n s itio n in R o tatin g B o u n d ary L a y e rs , " D y n am ic s o f F lu id s and P la s m a s . E d ite d by: S. I . P a id , A c ad em ic P re s s , N . Y . , pp. 3 0 9 -3 2 9 . 4 0 . B r ile y , W . R . and W a lls , H . A . (1970): " A N u m e ric a l Study of T im e -D e p e n d e n t R o tatin g F lo w in a C y lin d ric a l C o n ta in e r at L o w and M o d e ra te R eynolds N u m b e rs , " P ro c eed in g s of the Second In te rn a tio n a l C o n fe re n ce on N u m e ric a l M ethods in F lu id D y n a m ic s . E d ite d by: M a u ric e H o lt, S p rin g e r-V e rla g , H e id e lb e rg , G e rm a n y , pp. 3 7 7 -3 8 4 . 4 1 . T illm a im , W . (1967); "D e v e lo p m e n t of T u rb u le n c e D u rin g the B u ild -U p o f a B o u n d ary L a y e r a t a C oncave W a ll, " P h y s ic s of F lu id s Supplem ent: B o u n d ary L a y e rs and T u rb u le n c e , pp. s l 0 8 - s l l l . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 I 4 2 . M a x w o rth y , T . (1971): " A S im p le O b s e rv a tio n a l Technique fo r the In v e s tig a tio n of B o u n d a ry -L a y e r S ta b ility and T u rb u le n c e , " T u rb u le n c e M e a s u re m e n ts in L iq u id s . P ro c eed in g s of S ym posium on T u rb u le n c e M e a s u re m e n ts in L iq u id s , Sept. 1969, E d ited by: P a te rs o n , G . K . and Zakin^ J. L . , P u b lish ed by the D e p a rtm e n t of C h e m ic a l E n g in e e rin g , U n iv e rs ity of M is s o u ri - R o lla , pp. 3 2 -3 7 . 4 3 . R a y le ig h , L o rd (1917): "O n the D y n a m ic s of R e vo lvin g F lu id s ," P ro c . R oy. Soc. (London), V o l. 93A , pp. 1 4 8 -1 5 4 . 4 4 . S m ith , A . M . O . (1955): "O n the G ro w th of T a y lo r -G o r tle r V o rtic e s Along H ig h ly Concave W a lls , " Q u a rt. A p p l. M a th ., V o l. 13, pp. 2 3 3 -2 6 2 . 4 5 . G o lle r, H . and R anov, T . (1968): "U n stead y R o tating F lo w in a C y lin d e r w ith a F r e e S u rfa c e , " A S M E T ra n s a c tio n s , J o u rn a l of B a s ic E n g in e e rin g , V o l. 90, S e rie s D , pp. 4 4 5 -4 5 4 . 4 6 . Lud w ieg, H . (1964): " E x p e rim e n te lle N achprüfung d e r S ta b ilita ts th e o rie n fxir re ib u n g s fre ie Strom ugen m it s c h ra u b e n lin ie n fo rm ig e n S tro m lin ie n , " Z . F lu g w is s . 12, H e ft 8, pp. 3 0 4 -3 0 9 . 4 7 . Jackson, J. D . (1963): C la s s ic a l E le c tro d y n a m ic s , John W ile y and Sons, In c ., N ew Y o rk . 4 8 . B o rn , M a x and W o lf, E m il (1970): P rin c ip le s of O p tic s , F o u rth E d itio n , P e rg a m o n P re s s , G re a t B r ita in . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h ap ter DC T A B L E S 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 s f(S) a -g (c ) 0. 0 0. 88446 0 .0 1 .3 6 9 6 1 0. 1 0 .9 1 7 6 9 0. 1 1. 19516 0 .2 0 .8 6 1 7 5 0 .2 1 .0 3 0 8 0 0. 4 0 .6 5 9 9 6 0 .4 0 .7 2 6 0 8 0. 6 0 .4 3 0 7 7 0. 6 0 .4 5 3 7 8 0. 8 0 .2 0 8 0 1 0 .8 0. 21272 0 .9 0 .1 0 2 0 1 0 .9 0 .1 0 3 0 9 1 .0 0 .0 100 0. 0 T ab le IX . 1. E k m a n suction calc u la tio n s due to R o g ers and Lance [ s ] fo r v a rio u s valu es of s = uVOand an= 0 /w . D im e n s io n (inches) D e v ia tio n (inches ;)a ) A c c u ra c y (inches) O utside d ia m e te r 6 .9 1 8 ± 0 . 001 ± 0 .0 0 0 2 Inside d ia m e te r 5 .9 5 0 ±0. 001 ± 0 .0 0 0 2 Inside length 1 1 .4 9 0 - ± 0 .0 0 8 A v e ra g e thickness of glass end plates 0. 1841 ± 0 .0 0 2 ± 0 .0 0 0 1 A v erag e p late flatn e s s (m ounted in position ) - ± 0 .0 0 0 2 5 ± 0 .0 0 0 0 5 R e la tiv e fla tn e s s of c y lin d e r end su rface s - ± 0 .0 0 0 7 ± 0 .0 0 0 0 5 E c c e n tric ity o f ro ta tio n (m e asu red a t end oppoSüte the d riv e m o to r) ± 0 .0 0 1 2 ± 0 .0 0 0 0 5 T a b le IX . 2 . C y lin d e r dim ensions and to le ra n c e . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p te r X F IG U R E S 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 S P I N U P A T C O N S T A N T M = 0 . 2 5 0 . 9 A C C E L E R A T I O N ,-6 a = 2 . 0 R A D / S E C ' = 0 . 0 1 2 C M V S E C 0. 8 0 . 7 0.6 o o w 0 . 5 / > 0 . 4 t = 1 6 9 0 . 3 = 8 1 0.2 t = 4 9 t = 3 6 0. 1 t = 1 6 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 r = r / a F ig u re X . 1. V e lo c ity p ro file s fo r sp in -u p a t constant a c c e le ra tio n as com puted fro m equations (11.60) and (11.61). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD " O O Q . C g Q . " O CD C/) W o" 3 O 8 " O ( O ' 3 " i 3 CD 3. 3" CD CD " O O Q . C a O 3 " O O CD Q . " O CD t =t/t„ 3 (/) W o" 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 0 . 9 0.8 a = 0 . 2 7 4 R A D / S E C " .0 = 0 . 4 2 8 R A D / S E C ^ ^ ,0 = 1 . 7 1 3 R A D / S E C 0 . 7 C\J 0.6 '■ " 0 . 5 3 II 13 S P I N D O W N A T C O N S T A N T 0 . 4 D E C E L E R A T I O N 0 . 3 / R = 0 . 2 5 A ; = 1 5 7 . 1 R A D / S E C 1 / = 0.012 CmVsEC 0.2 F ig u re X . 2, A n g u lar v e lo c ity of the in te rio r flu id fo r spin-dow n a t constant d e c e le ra tio n as com puted fro m equations (11.90) and (II. 91). CD " O O Q . C g Q . " O CD C/) W o" 3 0 3 CD 8 ci' 3 " 1 3 CD "n c 3. 3" CD CD " O O Q . C a O 3 " O O CD Q . " O CD C /) C /) (a) O scillosco pe @ 20 s e c /c m sweep. ■ ■ ■ ■ I (c) S p ectru m a n a ly z e r @ 1 K H z /c jn . !Î'! I ‘ m ^ î / ' i . (b) O scillosco pe @ 100 s e c /c m sweep. -J (d) S p ectru m a n a ly z e r; upper d is p la y is D o p p ler signal; lo w e r d is p la y is sig n al g e n e ra to r. F ig u re X . 3. O scillosco pe and s p e c tru m a n a ly z e r d isp lays of the D o p p le r signal (V^ a; 60 K H z ). 7 3 C D ■ D 8 Q . C g Q . ■ D C D % O 3 O 8 ■ D C 5 - C D C D " O 8 û . C O ■ D 8 C D Q . O C " O C D LUCITE WHEEL VELOCITY MEASUREMENT û L U L U §5 120 ce L U 0_ C L O O + 0.0096 £r 80 L U (/) < _J dV + 0.01 X o > ' 40 120 160 80 40 V w '" WHEEL SPEED (CM/SEC) F ig u re X . 4. V e lo c ity m easu rem en ts of the ro ta tin g lu c ite w heel. -0 00 CD ■ D O Q . C g Q . ■ D CD C/) W o" 3 O 8 ■ D CD 3. 3" CD CD ■ D O Q . C a O 3 ■ D O 3" C T I —H CD C /) C /) L A S E R O P TIC S © MOTOR D R IV E SYSTEM C O N STA N T - A CCELERATIO N CO NTRO L U N IT --------------------- »• & C Y L IN D E R 0 f ® ■ ^ © © Q . $ X - Y F R E Q U E N C Y 1 — H 3" O P L O T T E R M E T E R 5- ■ D I R .P .M . INDICATO R SYSTEM © S IG N A L C O N D ITIO N IN G ELECTR O N IC S TA P E HEAD + I5 V © C O U N TE R ® BAND PASS F IL T E R SIGNAL GENERATOR © A M P L IF IE R ' © SPECTRUM A N A LY ZE R © SCOPE D O PP LER S IG N A L E LE C TR O N IC S F ig u re X . 5. S chem atic d ia g ra m of the apparatus fo r flu id v e lo c ity m easu rem en ts inside the ro tatin g c y lin d e r including the L a s e r O p tics, the M o to r D riv e S ystem , the R . P . M . In d ic a to r System , and the D o p p ler S ignal E le c tro n ic s . The lis t of com ponent p a rts is given in the fo llo w in g two pages. -J : v o : 180 E q u ip m e n t L is t fo r F ig u re X . 5 L a s e r O ptics 1. P o w e r Supply fo r O ptics T echnology M o d e l 250 H e -N e CW L a s e r . 2. O ptics Tehhnology M o d e l 250 H e -N e CW L a s e r; n o m in al pow er 15 m w . 3. P a r a lle l S u rface F la t; su rface s f la t to w ith in 1 /2 0 X and p a r a lle l vto w ith in 0. 10 a rcs eco n d s . ° 4. Focusing lens (f ~ 250 m m ) m ounted on a 12-in c h U n is lid e T r a v e r s e . 5. 1 /7 0 H P Bodine V a ria b le Speed M o to r (M o d el N S H 12R) w ith V a ria b le Speed C o n tro l (M o d el SH 14). 6. V e e d e r R oot C o u n te r, M o d el E -1 6 6 9 1 6 -1 . 7. D efocusing L en s. 8. R C A M o d el 8645 P h o to m u ltip lie r T u b e , S -2 0 resp o n se, 5% quantum e ffic ie n c y a t 6328 an g stro m s ; M a x im u m Supply V o lta g e 1800 v o lts . 9. K e ith le y M o d el 245 H ig h V o lta g e Supply; 0 -2 1 0 0 vo lts dc in 10-v o lt steps. M o to r D riv e S ystem 1. \ H P Bodine V a ria b le Speed M o to r (M o d el N S H 55) 2 . V a ria b le Speed C o n tro l U n it (M o d el S H 12F B ) 3. 1 /7 0 H P V a ria b le Speed Bodine M o to r (M o d el N SH 12R) 4. 1 0 -tu rn B eckm an In s tru m e n ts H e lio p o t (M o d el 7246, 25 KQ) 5. V a ria b le Speed ContrM . (M o d el M 26U ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 R . P . M , In d ic a to r S ystem 1. R otating sh aft m ounted in p illo w blocks. 2. P la s tic D is c holding 12 m ag n ets. 3. M ag n etic Tape H ead , N o rtro n ic s M o d el 3600. 4. S ignal C onditioning E le c tro n ic s , see F ig u re X . 4. 5. H e w le tt-P a c k a rd F re q u e n c y M e te r, M o d e l 510A . RcPj6i/I. H e w le tt-P a c k a rd M o d el 7004B X - Y Pen R e c o rd e r including the fo llo w in g p lu g -in u n its: M odel 17172A T im e B ase Input M odule; M o d el 17170A D . C . C o u p ler; M o d el 17178A D .C . A tten u ato r ; M o d e l 1 7 1 7 1 A D , C . P r e a m p lifie r . D o p p ler S ignal E le c tro n ic s 1. A m m e te r, M a rio n E le c tro n ic s M odel H S3, 0 -1 m illia m p s d. c. 2. F ie ld E ffe c t T ra n s is to r, A nalog D evices M o d el 4 0 - J V ; input im pedance l O ^ ohm s. F E T pow ered by two H e w le tt- P a c k a rd M o d el 6 2 IS A 0 -3 0 v o lt d. c. pow er s u p p lies. 3. K h ro n -H ite M o d e l 3103(R ) Solid State Band Pass F ilt e r ; 1 0 H z -3 M H z . 4. H e w le tt-P a c k a rd M o d el 465A A m p lifie r; less than 2 db down a t 5 H z and 1 M H z . 5. T e k tro n ix Type 502A O scillo sco p e w ith D u a l B e am D is p la y . 6. T e k tro n ix Type 564 O scillo sco p e w ith Type 3L 5 S p e c tru m A n a ly z e r P lu g In U n it and Type 2B 67 T im e Base P lu g In U n it; w ith upper and lo w e r storage d is p la y s c re e n s . 7. H e w le tt-P a c k a rd M o d e l 3310A F u n ctio n G e n e ra to r; 0 .0 0 0 5 H z to 5 M H z . 8. H e w le tt-P a c k a rd M o d el 5326B 5/0 M H z T T im e r-C o u n te r- D ig ita l V o ltm e te r. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD ■ D O Q . C g Q . ■ D CD C/) W o" 3 O INPUT AMPLIFIER (40:1) OUTPUT AMPLIFIER (1:5) 8 ■ D CD 3. 3" CD CD ■ D O Q . C a O 3 ■ D O CD Q . ■ D CD C /) C /) JLZ.A741C OP. AMP. (TYP.) R, ->AAAAAA MAGNETIC SIGNAL INPUT R, = IKA i R a = 3 9 K X 1 R ^ z l O K A R4 = 5KÜ R g H . S K i i R 6 = I , 5 K A — \AA/VW'— "GND DUAL NAND SCHMITT TRIGGER SN74I3N INVERTER (Unit Gain) W J tF b - OUTPUT (IV Peok-to-Peak)! F ig u re X . 6. S ignal processing e le c tro n ic s fo r the R . P . M . In d ic a to r System . 0 0 N CD ■ D O Q . C g Q . ■ D CD C/) W o" 3 O 8 ci' 3 3" CD CD ■ D O O . C a O 3 ■ D O CD Q . ■ D CD C /) C /) C Y L I N D E R P A R A L L E L F L A T S I L V E R D E P O S I T L E N S E P H O T O M U L T I P L I E R L A S E R ( X j F I L T E R T O S P E C T R U M A N A L Y Z E R ( % / n ) ^ W A T E R ( n ^ ) P L E X I G L A S S A I R D O P P L E R F R E Q U E N C Y : _ 2 n w V S IN ( ^ /2 ) X q = l a s e r r a d ia t io n w a v e l e n g t h n ^ = R E F R A C T I V E I N D E X O F W A T E R V = A Z I M U T H A L V E L O C I T Y LASER DOPPLER MEASUREMENT SYSTEM F ig u re X . 7, S elf-fo c u s in g la s e r o p t if ^ f c r the flu id v e lo c ity m eas u rem en ts. 0 0 00 CD ■ D O Q . C g Q . ■ D CD C/) W o" 3 O 3 CD 8 ■ D ( O ' 3. 3" CD CD ■ D O Q . C a O 3 ■ D O CD Q . ■ D CD (/) (/) 9 0 0 O U T S I D E S O L I D B O D Y R O T A T I O N C A L I B R A T I O N I N S I D E a = 9 3 . 0 R A D / S E C 8 7 1 K H Z (q) o u ts id e > ; 6 0 0 1 V .R . U N I T = 0 . 0 0 4 1 6 6 7 I N o 5 0 0 A I R W A T E R P L E X I G L A S S o 3 0 0 100 100 200 4 0 0 6 0 0 8 0 0 L - L E N S P O S I T I O N ( V . R . U N I T S ) 1000 1200 F ig u re X . 8. P lo t of the D o p p ler fre q u en c y as a function o f the lens position L fo r so lid body ro ta tio n of the flu id . 00 4 ^ 185 6 C O d II o h- lO ro ro d d I I I I a \ V . - CJ o (/) * U D /A A il 3 0 1 3 A eu d I V H i n i A I Z V n O I o I— t X .-o C r N N M o C Q 0 ) M a 0 k « H 0 4 0 1 d a w o ; % : | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 rO rO lO 00 C J C O L U lU CJ L U CM o o LU L U C O C O Q Q 00 C M UD/A À 1I3013A IV H iniA IIZV n O I in r- M m < D 0 0 1 3 I • S a. M % a > u & Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 8 7 1 ; CJ üü L U v O § < OJ 00 a >- LU C V I O O LU LU C O (/) Q O Ô ^ 00 OJ a LU 00 C O ^ < M O O d d U D / A ^ A 1I3013A IV H ID M IZ V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 % o o o t y o /A _ A 1I3013A IV H IA M IZ V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 o I o r— 4 X T h in ra W (0 ■ ( U h d g 0 1 3 1 d •t-< e u C Q m . % d k d O J O . M • y j / A C D . o A 1ID 013A ^ CJ O d “IVHiniAIIZV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 > - 1 — Ô o _ l o L U o q _1 < o > _1 II Û - _J > - < V . i~ § S CJ d vO I X O' O ' W h (0 ( U h d I 'O I d •■-I a c o % ( U h & . M h •Ü J /A - A 1 I 0 0 1 3 A n V H i n i A I I Z V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C M O UJ ÜJ (O œ V. ro C M o ro L U o 00 o CL LU IT> C M 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lO lO m i n O i n 'U J/A A 1 I3 0 1 3 A nV H iniA llZV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 S § § o I X GO PO w h s % « u î I •S L O X tu h 3013A IVHiniAllZV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o > p v n lO w U i LÜ h - C P Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cl % O ' o c P A 1 1 3 0 1 3 A I V H l O W Reproduced with permission of the copyright owner . Further reproduction prohibited without permission. 192 R I D 0 0 O O & 0 » o a H - to C E tu t- z tn n O I N O' O' I I u O to 0 ) k I rO I .5 e g - X 0 ) & £ k 1 0 0 1 3 A I V H i n i A I I Z V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD ■ D O Q . C S Q . ■ D CD 0 0 C/) CD 8 ■ D C Q ' 3. 3 " CD CD " O O Q . C a O 3 " O O CD Û . ■ D CD C/) C/) 1 .0 G a 0.8 > t 0.6 CJ 3 L Ü ^ 0 . 4 < X h- 3 S N 0.2 < n. = 108.9 RAD/SEC = -0 .5 3 5 RAD/SEC î = 0 0 INVISCID STABILITY LIMIT (T = 0.192) FIRST APPEARANCE 0 6 OF INSTABILITY BY FLOW VISUALIZATION 0 7 ( t = 0.52 5) •THEORETICAL PROFILES 0.4 0.6 0.8 RADIUS ^ r/Q F ig u re X . 17. V e lo c ity p ro file s eorresp onding to F ig u re X . 15. s O W CD ■ D O Q . C g Q . ■ D CD C/) o" 3 8 " O i 3 CD 3. 3" CD CD ■ D O Q . C a O 3 ■ D O CD Q . ■ D CD C /) C /) l.Oh n. = 109.0 RAD/SEC a =-1.083 RAD/SEC E g = 9 . 9 2 X 10 t o . 6 t= 0 0 . 1 0.2 0.8 1. 0 INVISCID STABILITY LIMIT (T = 0.066) FIRST APPEARANCE 0 3 OF INSTABILITY BY FLOW VISUALIZATION Q ^ ( t = 0.238) 0.5 0.6 THEORETICAL PROFILES •0.2 0.4 0.6 0.8 * 1.0 RADIUS r/Q F ig u re X . 18. V e lo c ity p ro file s co rresponding to F ig u re X . 16. v O CD ■ D O Q . C g Q . ■ D CD C/) C/) 8 ■ D CD 3. 3" CD CD ■ D O Q . C a O 3 " O O CD Q . ■ D CD C /) C /) 1.0 n, = 39.0 RAD/SEC n, = 104.0 RAD/SEC a =0.433 RAD/SEC^ r/a = 1.000 TYPICAL WALL VELOCITY) r/a = 0.807 = 16.2 X 10 CHARACTERISTICS SOLUTION r/o =0.517 t=0.474 LÜ 0.4 r/a = 0.334 0 LIMITING EXTENT OF CHARACTERISTICS: 3 0.2 0.6 0.8 TIME - t/to Figu:fe X . 19. A c c e le ra tio n fro m one a n g u la r v e lo c ity to another a t s m a ll Robb&y n um ber fo r = 16.2 X 10-6. C t ' s O < J l CD ■ D O Q . C g Q . ■ D CD C/) W o" 3 0 5 CD 8 ■ D C Q ' 3 " 1 3 CD 3. 3" CD CD ■ D O Q . C a O 3 ■ D O CD Q . ■ D CD C/) C/) A. = 104.4 RAD/SEC 39.8 RAD/SEC a =-0.423 RAD/SEC E =15.75 xlO"® • 't: 0 .8 o 0 .6 r/a = 1.000 (TYPICAL WALL VELOCITY) LÜ 0.4 r/o = 0.807 r / a = 0 . 5 1 7 1!^ < 0.2 r/o =0.334 CLOSED FORM SOLUTION OF RICATTI EQUATION 0 0 .2 0.4 0 .6 0.8 1.0 1 .2 1.4 TIME ^ t/tg F ig u re X . 20. D e c e le ra tio n fro m one an g u la r v e lo c ity to another a t s m a ll Rossby num ber fo r ; E a = 15. 75 X 1 0 -6 . . \0 O' CD ■ D O a. c s a. ■ D CD c /) c o 8 ■ D C Q ' 3. 3 " CD CD ■ D O Q . C O ■ D O CD Q . ■ D CD C /) C /) F ra m e # 1-6 7-12 1 3 -1 8 1 9 -2 4 2 5 -3 0 3 1 -3 6 v O F ig u re X . 21. Photographs of the E k m a n in s ta b ilitie s fo r spin -u p fro m re s t a t =_7. 64 X 10 T h e re a re th re e -s e c o n d in te rv a ls betw een fra m e s ; fra m e 2 corresponds to t c: 0 and fra m e 31 corresponds to t cr 1. 0. CD ■ D O Q . C g Q . ■ D CD C/) C/) 8 ■ D 3. 3" CD CD ■ D O Q . C a O 3 " O O CD Q . ■ D CD C /) C /) (a) F ra m e N o . 7 (c) F ra m e N o . 23 (b) F ra m e N o . 13 (d) F ra m e N o . 32 vO 0 0 F ig u re X . 22. P rin ts of the d e ta ile d E k m a n flo w at (a) t ~ 0. 18, (b) t oi 0. 4 3 , (c) t rü 0. 79, and (d) t ~ 1. 04. 199 1 I T ❖ T Y P E I L I M I T 1 F A L L E R 8 e T Y P E I L I M I T J K A Y L O R [3 9 ] o ® S M I T H [ 3 2 ] e B G R E G O R Y , S T U A R T 8 W A L K E R [ 3 3 ] T A T R O 8 M O L L O - C H R I S T E N S E N [ 3 5 ] _ C A L D W E L L 8 V A N A T T A [ 3 6 ] 0 a" \ \ \ \ Q Q }t Y P E I S T A B I L I T Y L I M I T A A & A - 0 . 5 6 0 0 - 4 0 0 - S T A B I L I T Y B O U N D A R Y O N S E T O F T U R B U L E N C E 200- T Y P E I % } 1 0 8 < X < 1 6 8 T Y P E Ï Ï A L A M I N A R S T A B L E • M A R G I N A L S T A B I L I T Y \ X = 2 0 . 4 8 o U N S T A B L E Jo-= 2 . 4 5 ♦ M A R G I N A L S T A B I L I T Y o U N S T A B L E X T U R B U L E N T - 0 .4 - 0 . 3 - 0 . 2 - 0 . 1 R O S S B Y N U M B E R V g / 2 A r F ig u re X . 2 3 . O bservatio n s of E k m a n s ta b ility and tra n s itio n to tu rb u le n c e . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD ■ D O Q . C g Q . ■ D CD C/) o" 3 O 8 ■ D CD 3. 3" CD CD ■ D O Q . C a O 3 ■ D O CD Q . ■ D CD C /) C /) 80 4 — 1 x 1 2 A .= 106.8 RAD/SEC h “ 4 0 L h- if) Z UNSTABLE STABLE 0 4 6 8 10 EKMAN NUMBER -E ^ lx lO ® ) 12 14 F ig u re X . 24. T im e fo r the f ir s t appearance of G b rtle r v o rtic e s as d eterm in ed fro m the flow v is u a liz a tio n o b servatio n s. N o o CD T D O Q . C g Q . T D CD C/) C/) 8 T D C Q ' 3. 3 " CD CD ■ Q O Q . O 3 T D O CD Q . ■ Q CD C /) C /) $ I4 0 r c 1 2 0- (T I C C U J 1 0 0 - l — U J 8 0 - < £r < 6 0 - C L >- 4 0 - H _ J C Û r < H K i_ (/) 0 O □ X 1 1 MAXWORTHY [4 2 ] TILLMANN [4 l] MAXWORTHY-USING GROHNE'S MOMENTUM THICKNESS UNSTABLE A : = 106.8 RAD/SEC 4 6 EKMAN NUMBER 8 r s j a 10 (xIO®) 12 14 F ig u re X . 25. P lo t of the c r itic a l G b r lte r s ta b ility p a ra m e te r as a function of the E k m a n num ber fo r the flow v is u a liz a tio n observation s given in F ig u re X , 24. CD T3 S Q . C g Q . T3 CD g O = J 8 T3 C 5 - = J CD C 3. = r CD c B T3 3 Q . C a O = J T3 3 CD Q . O C T3 CD 8 O = J g 0.08 80 Ü J 4 0 >- y 0.04 E =9.92 X 10 2 0 < E^2=I3.8 xIO 0.3 0.4 03 TIME t = t/t„ F ig u re X . 26. T im e develo pm ent of the m om entum thickness and G 'd rtle r s ta b ility p a ra m e te r " corresponding to the v e lo c ity p ro file s of F ig u re s X . 17 and X . 18. si -J 203 I 1.0 A F L O W V I S U A L I Z A T I O N D A T A - N U M E R I C A L C O M P U T A T I O N - 0 . 8 - \ 0.6 0 . 4 .-6 0 .2 0 0 . 0 5 0 . 1 0 0 . 1 5 0. 20 0 . 2 5 t = t /Î Q F ig u re X . 27. O bservatio n s o f the p o s itio n of the w ave fro n t d u rin g spin-up fo r = 1 9 .6 7 X 10T&. ^ F L O W V I S U A L I Z A T I O N D A T A - N U M E R I C A L C O M P U T A T I O N • V E L O C I T Y P R O F I L E D A T A 0 .8 P 0 . 6 0 . 4 .-G . 1 8 x I O 0 .2 0. 10 0.20 0 . 3 0 0 . 4 0 0 . 5 0 F ig u re X . 28. O bservations fii Qie wave fro n t during sp in -u p fo r E = 1 * . 4 8 X 1 0 “®. a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 . 0 ^ F L O W V I S U A L I Z A T I O N D A T A - N U M E R I C A L C O M P U T A T I O N - • V E L O C I T Y P R O F I L E D A T A p 0.6 -6 0 .2 0 0 .2 0 . 4 0 .6 0.8 1 . 0 t = t/t^ F ig u re X . 29. O b servatio n s of the p o s itio n of th e w ave fro n t d u rin g sp in -u p fo r = 7. 64 X 1 0 ”^. ^ F L O W V I S U A L I Z A T I O N D A T A - N U M E R I C A L C O M P U T A T I O N • V E L O C I T Y P R O F I L E D A T A 0.8 o 0.6 0 . 4 -6 0.2 0 1 . 0 2.0 3 . 0 4 . 0 5 . 0 t = t / t c F ig u re X . 30. O bservations of the position^of the wave fro n t during s p in -u p fo r = 2. 12 X 10"^. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD ■ D O Q . C g Q . ■ D CD C/) C/) 8 C Q ' 3 3" CD CD ■ D O Q . C a O 3 " O O CD Q . ■ D CD C /) C /) ^ 1 0 0 _ A:= 106.8 RAD/SEC STABLE UNSTABLE o 40 Cüj ~ X I X FLOW VISUALIZATION DATA o VELOCITY PROFILE CRITERION 0 20 40 60 80 100 WALL ROTATION RATE - A (RAD/SEC) F ig u re X . 3 1 . The in te rio r flu id ro ta tio n a l v e lo c ity U ü as a function of the w a ll ro ta tio n a l v e lo c ity 0 a t the f ir s t appearance of the G b rtle r c e lls . N o un CD ■ D O Q . C g Q . ■ D CD C/) C/) 8 ■ D CD 3. 3" CD CD ■ D O Q . C a O 3 ■ D O CD Q . ■ D CD C/) (/i 1.16 H 1.20 t e u 1 M 2 i g 5 ca > t § 1.08 ÜJ > i 1.04 o 1 . 1 2 a: u 1 .00, A FLOW VISUALIZATION DATA o VELOCITY PROFILE CRITERION UNSTABLE . a A STABLE (Q)E^ = 0. (VXi)cRiT- 1.06 A. = 106.8 RAD/SEC 0 4 6 EKMAN NUMBER 8 a 10 (xlO®) 12 14 F ig u re X . 32. The c r itic a l v e lo c ity ra tio (<J^>j/^)crit the f ir s t appearance of the G b r tle r c e lls as a function of the E k m a n n u m b e r. CD ■ D O Q . C g Q . ■ D CD C/) W o' 3 0 3 CD 8 ■ D ( O ' 3 " 1 3 CD 3. 3" CD CD ■ D O Û . C a O 3 ■ D O CD Q . " O CD (/) (/) 3.2 NUMERICAL COMPUTATIONS = 108.73 RAD/SEC p- _ . m -6 2.8 r. = 0.334 2.4 I 2.0 r,= 0.807 (A 4— L U IMPULSIVE SPIN UP VALUES : 0.8 3.194 2.691 1.943 0.334 0.517 0.807 0.4- 0 20 40 60 80 100 120 140 160 180 200 220 F4gure X . 33. The re s id u a l sp in -u p tim e OjgEq(tg-t^) as a function of the nondim ensional tim e re q u ire d fo r the c y lin d e r to a tta in its fin a l an g u la r v e lo c ity . C O o -o '208 o N - CO l O d d I I I I O O O O O lO o o Cvi O O g r O o ü 'a Üj O c v j in h « I I W 0) b O k u o « H 0 ) s a a I d a CQ I m d o ( U k h O ü e n X o h & D /A _ 3MI1 dfi N id s nvnais3d Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ■a I I % C/) (g o' 3 CD 8 ë' 3 CD C p. 3" CD CD ■ o I C a o 3 & O c % ( / > (g o' 3 iOOOr 5 0 0 200 100 5 0 - c o g X 20 ( V I a lu 10- 5 2 - N U M E R I C A L C A L C U L A T I O N 1 0 8 . 7 3 R A D /S E C E ^ = 9 . 2 7 x I O r , = 0 . 3 3 4 Tz = 0 . 5 1 7 0 . 5 73 = 0 .8 0 7 F O R E a < 5 x l0 '® : = ----------- K (r,) = 3 . 0 0 x 1 0 K (rJ = 2 . 1 5 x 1 0 K ( 7 3 ) = 1 . 5 9 x 1 0 ' ° t^ (r,) = 9 5 . 8 8 S E C _ t^ (r 2) = 8 0 . 7 7 S E C 7 3) = 5 8 . 3 4 S E C 1 0 20 _L 5 0 0 . 0 1 0 . 0 2 0 . 0 5 0 .1 0 . 2 0 . 5 . I R E S I D U A L S P I N U P T I M E F ig u re X . 35. C o rre la tio n s of the sp in -u p tim e fo r s m a ll E km an num ber. I 35 w h e re C = a-nd the in te g ra l is d efin ed by I(x) = 3 exp(§ )d : . ( II. 57) Thus w e see th a t as in the case o f im p u ls iv e s p in -u p , th e flu id in 4 R eg io n I has an a z im u th a l v e lo c ity p ro file p ro p o rtio n a l to r . In R egio n I I w e m u st d e te rm in e the c h a ra c te ris tic paths fo r (II. 50) fro m the d iffe re n tia l eq u atio n If = T , < “ • = » > w h e re T is the va lu e of the c irc u la tio n along the c h a ra c te ris tic . W e 2 can s e p a ra te v a ria b le s b y su b stitu tin g u = r . E q u atio n (11.58) then becom es I f + ( II. 59) w h ic h is id e n tic a l in fo rm to eq u atio n (II. 55) and can be re a d ily solved along w ith the boundary co n d itio n r = t J • u = 1 , T = T . . (11.60) o i 1 The so lu tio n is then r^(T) = exp(-C T^{exp(C T^) + 2B t ^ C '^ /^ [ i( C " /^ t ) - I( C ^ ''\) ] 3 (11.61) 4 W e note th at th is does not re p re s e n t solid body ro ta tio n sin ce, b e cause of the secondary m o tio n , th e flu id p a rtic le s a re being con tin u a lly re d is trib u te d in the m e rid io n a l plane. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed a s received. Xerox University Microfilms 300 North Zeeb Road Ann Art)or, Michigan 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74-11,718 WEIEMAN, Patrick Dan, 1941- ON TEE SPIN-UP AND SPIN-DOWN OF A CQNTAINED FUJID. University of Southern California, Ph.D., 1973 Engineering, aeronautical University Microfilms, A XERQ\ Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Asset Metadata

Creator WEIDMAN, PATRICK DAN (author)

Core Title On the spin-up and spin-down of a contained fluid

School Graduate School

Degree Doctor of Philosophy

Degree Program Aerospace Engineering

Degree Conferral Date 1973-06

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, aerospace,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-447673

Unique identifier UC11352687

Identifier 7411718.pdf (filename),usctheses-c17-447673 (legacy record id)

Legacy Identifier 7411718-0.pdf

Dmrecord 447673

Document Type Dissertation

Rights WEIDMAN, PATRICK DAN

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA