applied scientific research volume 44 issue 1-2 1987 [doi 10.1007_bf00412016] p. a. davidson_ f....

8/13/2019 Applied Scientific Research Volume 44 Issue 1-2 1987 [Doi 10.1007_bf00412016] P. a. Davidson_ F. Boysan -- The

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Applied Scientific Research 44:241-259 (1987) 24 Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands

he importance of secondary f low in the rotary e lectromagneticstirring of steel during continuous casting

P.A. DAV IDS ON 1 F . BOYS AN 21 Department of Engineering, Cambridge University, Cambridge, UK;2 Depa rtment o f Chem ical Engineering and Fuel Technology , Sheffield University, Sheffield, UK

Abstract. This paper considers some aspects of the flow generated in a circular strand by a rotaryelectromagnetic stirrer. A review is given of one-dimensional models of stirring in which the axialvariation in the stirring force is ignored. In these models the magnetic body force is balanced byshear, all the inertial forces being zero (except for the centripetal acceleration).

In practice, the magnetic torque occurs only over a relatively short length of the strand. Theeffect of this axial dependence in driving force is an axial variation in swirl, which in turn drives asecondary poloidal flow. Dimensional analysis shows that the poloidal motion is as strong as theprimary swirl flow.

The principle force balance in the forced region is now between the magnetic body force andinertial. The secondary flow sweeps the angular momentum out of the forced region, so that theforced vortex penetrates some distance from the magnetic stirrer. The length of the recirculatingeddy is controlled by wall shear. This acts, predominantly in the unforced region, to diffuse anddissipate the angular momentum and energy created by the body force.

o t a t i o nB magnetic field strength F angular momentum uorE electric field 8 boundary layer thickness

unit vector ~ viscous dissipat ion rateF8 force 0 angular coordinatef z / R ) dimensionless variation of force v viscosity

with depth v eddy viscosityJ current density p densityk turbulence kinetic energy o conductivityk' wall roughness ~j shear stressL axial length scale in ~2 angular velocity

unforced region w vorticityp pressure w frequencyR radiusRe Reynolds number Subscriptsr radial coordinate R wallT torque r radialt time 0 azimuthalu_ velocity z axialV characteristic velocity B ~ o R ~ 0 coreV. shear velocity p poloidalv fluctuating velocityz axial coordinate


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242 P A Davidson and F Boysan1 I n t r o d u c t i o n

C o n t i n u o u s c a st in g h a s b e c o m e a n i n cr e as in g l y c o m m o n m e a n s o f p r o d u c in gs tee l ingots . The process i s shown schemat ica l ly in F ig . 1 . There i s a me ta l -lurg ica l r equi rement to s t i r the mel t a s i t so l id i f ie s , and th is has led to the useo f e l e c t rom a gne t i c s t ir ri ng [ 3 ]. S t ir re r s ha ve be e n p l a c e d a r oun d the m ou ld ,be low the mou ld a nd ne a r t he po in t o f f i na l so l id i f i c a t ion . Typ ic a l ly the ses t i r re r s re semble the s ta tor of an induc t ion motor , p roduc ing a t r ave l l ing orr o t a t ing ma gn e t i c f i e ld . The f ie ld induc e s m o t ion in the me l t w i th p e r iphe ra lve loci ti e s o f t he o r de r o f 20 c m /s . W e shaU be c onc e r ne d w i th r o t a r y s ti rr e rs ,wh ose p r ima r y p u r po se i s t o i nduc e sw i rl i n t he m e lt .

The c os t o f imp le me n ta t ion o f ma gne t i c s t i r r i ng i s c ons ide r a b le ; ye t t heopt imum conf igura t ion for s t i r r ing i s o f ten assessed on an empir ica l bas is [4] .The que s t ion o f how ma ny s t i r r e r s a r e r e qu i r e d , a nd whe r e the y shou ld beplaced , f requent ly a r i se in the l i te ra ture [3 ,4] . In orde r to g ive genera l answersto these , i t i s necessa ry to unders tand not on ly the meta l lurg ica l p rocesses a two rk , bu t a l so the na ture of the ve loc i ty f ie ld indu ced b y s t ir ring . In pa r t icu-la r ; the fo l lowing hydrodynamic ques t ions a r i se .

i ) How doe s the ma gn i tude o f t he induc e d swi r l s c a l e on ma gne t i c f i e lds t rength , mould s ize and mel t p roper t ie s?

i_i) H ow f a r be y on d the s t ir re r d oe s the ind uc e d vo r t e x e x te nd?ii i) D o se c o nda r y f lows de ve lop a nd a r e the y imp or t a n t?

Incomin9 ste el 1,

Coppe r mould

Mel tSol id s teel~i i~, m

F i g 1 D i a g r a m m a t i c r e p r e s e n ta t i o n o f t h e c o n t i n u o u s c a s t in g p r o ce s s .


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he importance of secondary low 243ould stirring

surface~ - _ _orced

. . . . 7 - - [J . . . .. ~ Magnetic force

Z

Sub-mould s t i r ring

Line~ ~of symmetly_ lreg ioni

Magnetic

Fig 2 Typicalmagnetic orcedistributions

force

In or der to s impl i fy the prob lem a t t en t ion i s res t r i c ted to flow in a c i rcu lars t rand and en t ran ce ef fec t s o f the mel t in the mould are ignored . Typica lmag net ic fo rce d i s t r ibu t ions are shown in Fig . 2. Of ten ro ta t ion ra tes a resuf f ic ien t ly low tha t the sur face of the me l t remains f l at . In th is s i tua t ion thesu r face m a y be t r ea t ed a s a p l ane o f sym m et ry and t he p rob lem s o f m ou l d andsub -m ou l d s t ir r ing becom e hy d rodynam i ca l l y i den ti cal .

We shal l cons ider two models o f s t ir r ing . Fi rst ly a one-d imen s ional modeli s rev iewed. In th i s ana lysi s the ax ial var ia t ion in the magne t ic fo rce andhenc e swir l i s ignored . In such a s i tua t ion the second ary flows are bydefini t io n zero. This f low has bee n analysed several t imes [1 5 7 9] and is apop ular m odel o f s ti r ring . We shall show however tha t i t i s mis lead ing in thecontex t o f con t inuous cas t ing . A more rea l i s t i c two-d imens ional ax i symmet r icmod el is then cons idered in which a ll th ree ve loc i ty com pon ents a re non-zero .The co ord in a te sys tem used is shown in Fig . 2 and no ta t ion i s g iven a t thes tar t o f th is paper . Th e rad ius R shown in Fig . 2 re fers to the ou ter rad ius o fthe mel t. I t i s assumed to be cons tan t the t aper resu l ting f rom the increas ingshel l thickness being ignored.


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2 4 4 P.A. Davidson and F. BoysanI I . T h e m a g n e t i c b o d y f o r c eT h e m a g n e t i c f ie l d w i t h i n t h e m e l t is g o v e r n e d b y t h e a d v e c t i o n d i f f u s i o ne q u a t i o n ,

3 B7=vT h e r e l a t i v e s i z e o f t h e a d v e c t i o n t o d i f f u s i o n t e r m s i s g i v e n b y t h e

m a g n e t i c R e y n o l d s n u m b e r ,R = uRl~ .W e s h al l a s s u m e t h a t R e m i s sm a l l. T h i s is g e n e r a l l y t ru e i n b o t h l a b o r a t o r y

a n d i n d u s t r i a l s i t u a t i o n s , a n d a l l o w s u s t o i g n o r e a d v e c t i o n o f t h e m a g n e t i cf i e l d . I n t h i s a p p r o x i m a t i o n t h e m e l t i s t r e a t e d a s a s o l i d c o n d u c t o r , a n d t h em a g n e t i c fi e ld d e t e r m i n e d b y t h e s t a n d a r d e d d y - c u r r e n t e q u a ti o n ,

3 B 1= ~ 7 2 B .3t /~rrI t i s w o r t h n o t i n g t h a t , i n t h i s a p p r o x i m a t i o n , t h e r e i s n o c h a r g e d i s t r i b u -

t i o n w i t h i n t h e m e l t , a n d t h a t c h a r g e s w i ll n o t b e d e p o s i t e d o n t h e f r e e s u r f a c eo f t h e m e l t b y e d d y c u r r e n ts , s in c e t h e r e is n o c o m p o n e n t o f c u r r e n t n o r m a l t othe sur f ace [2] .

T h e r a t i o o f t h e t i m e d e r i v a t i v e t e r m t o t h e d i f f u s i o n t e r m i n t h e e q u a t i o nf o r B i s g i v e n b y t h e s k i n d e p t h p a r a m e t e r ,

A = R2~oI~ = 2(R /3 ) 2w h e r e

= 2///~oo) 1/2,O u r s e c o n d a s s u m p t i o n i s t h a t A i s a l so s m a ll , a l t h o u g h l a r g e r t h a n R e m.R em


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The importance of secondary f low 245The assumption Rem


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246 P . A . D a v i d s o n a n d F . B o y s a nf ie ld . T h e f i rs t o r d e r c u r r e n t d e n s i t y i s t h e n o b t a i n e d b y s u b s t i t u t in g t h i s f i el di n t o F a r a d a y s e q u a t i o n , a n d t h e f ir s t o r d e r f o r c e d i s t r ib u t i o n f o l lo w s [2 ].

0 BX E = - - - a n d J = o EO tJ o o ~ B R .

A l s o ,F = J x B

F o ~ o o ~ B 2 R .W e m a y t h e r e f o r e w r i t e F o i n t he f o r m ,F o = [ a ~ B 2 R ] F r / R , z / R ) .T h e f u n c t i o n F r / R , z / R ) m a y b e e x p a n d e d a s a T a y l o r se ri es a b o u t t h e

axis r = 0.

M a n y f i e l d s h a v e s y m m e t r y a b o u t t h e a x is , i n w h i c h c a s e th e f ir s t te r m i nt h e e x p a n s i o n m u s t b e z e r o . I t fo l lo w s t h a t, t o f i r st o r d er , t h e b o d y f o r c e m a yb e a p p r o x i m a t e d b y ,

F o = [ o w B Z r ] f , z / R ) .T h i s is th e s a m e a s e q u a t i o n 1 ), w h e r e f l = f . I t is c o n v e n i e n t t o i n t r o d u c e

a c h a r a c t e r i s t ic ve l oc i t y , de f i ne d a s ,B / - - 6 - al = V ~ - - ~ R . 2 )

T h e i d e a li s ed f o r c e d i s t r i b u t i o n t h e n b e c o m e s ,F o = [ p V Z r / R 2 ] f z / R ) . 3)S i nc e t h i s f o r c e d i s t r i bu t i on i s a x i s ym m e t r i c , a nd on l y c i r c u l a r s t r a nds a r ec ons i de r e d , t he r e s u l t i ng f l ow i s i t s e lf a x i s ym m e t r i c .

I I I . A r e v i e w o f o n e - d i m e n s i o n a l m o d e l s o f s t i r r i n gW e n o w c o n s i d e r t h e f lo w w h i c h re s u lt s w h e n t h e a x i a l v a r i a t i o n in t h e b o d yf o r c e i s ne g l e c t e d . T he body f o r c e i s ,

F 0 = p V 2 r / R 2 .T hi s f o r c e d r i ve s a n a x i s ym m e t r i c , s t e a dy , one - d i m e ns i ona l , a z i m u t ha l f l ow ,

t he r a d i a l ve l oc i t y be i ng z e r o f o r r e a s ons o f c on t i nu i t y .u= uo r )~o.


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The importance of secondary f low 247The r a d ia l a nd a z imf i tha l c ompone n t s o f t he t ime a ve r a ge d Na v ie r - S toke s

equa t ions for ax isymmetr ic swir l f low a re ,u ~ d p 1 d 1P r d r + r - d ~ r [ r - p - ~ r ) ] - 7 - o r 2 )

dF o - r 2 d r r 2 rr )w h e r e

d(v - f luc tua t ing ve loc i ty , u - t ime me an ve loc i ty).

The f i r s t o f these shows tha t the rad ia l g rad ien t of the to ta l p ressurep + PVZrba la nc e s the c e n t ripe t a l a c c e le r at ion . T he se c on d ma y be in t e g r a t e d tog ive the shea r s t re ss d is t r ibu t ion . Subs t i tu t ing for Fo a nd in t e g r a t ing weobta in ,

d [ Uo 1 - 2 / r ~ r 0 / 0 = - v r v 0 + = - 4 )

The f low cons ide red h e re is s imi la r to tha t o f ax ia l f low in a p ipe . In b othc a se s we w i sh to de t e r m ine the f low r e su lt ing f r om a know n im pose d she a rs t ress . In the case of laminar f low, equa t ions (4) may be in tegra ted to g ive thewel l known resu l t [5] ,

1 ~2 [1 r / R ) 2 ] .ru 16 v

In th is case the ve loc i ty sca les on V 2 R / v . F or tu r bu le n t f l ow , howe ve r , wee xpe c t u o to sca le on the shea r ve loc i ty , and hence V. Solu t ion of equa t ion (4)in the c a se o f t u r bu le n t f l ow r e qu i r e s some e s t ima te o f t he R e yno lds s t r e s st e r m. Two e qua t ion c lo su r e mode l s o f t u r bu le nc e ha ve be e n a pp l i e d to th i sp r ob le m [ 9 ]. Th i s i nvo lve s so lv ing se ve r al subs id i a r y e qua t ions wh ic h r e qu i r eextens ive com puta t ion . I t i s sugges ted in [1] , how ever , tha t th is shea r f low isla rge ly cont ro l led by events nea r the wa l l , a s in ax ia l p ipe f low. Consequent ly ,the c o m pu te d f low is in se ns it ive to the tu r bu le nc e m ode l u se d , p r ov ide d tha t i tc o r r e c t ly mode l s t he wa l l r e g ion . A t h igh R e yno lds numbe r , c u r va tu r e e f f e c t sa r e ne g lig ib l e ne a r t he wa l l a nd a s imp le mix ing l e ng th m ode l m a y b e a pp l i e d .This re su l t s in an expl ic i t equa t ion for the core ang ula r v e loc i ty [1] .

f ~ = ; { 0 8 8 1 n V R + 1 0 )v (5 )Th i s e qua t ion i s c ons i s t e n t w i th the r e su l t s o f c ompu ta t ions u s ing h ighe r

o r de r t u r bu le nc e mode l s , a nd w i th e xpe r ime n ta l da t a . I t i s de r ive d a s suming asmoo th wa l l . I f t he wa l l i s r ough , t he n e qua t ion ( 5 ) mus t be r e p la c e d by ,f~0 = ~ {0 .88 In ~---z}. (6 )


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248 P A D a v i d s on a n d F B o y s a nF o r a R e y n o l d s n u m b e r o f 10 5, a ra d i u s o f 1 00 m m , a n d a r o u g h n e s s k ' o f

I m m , w h i c h i s ty p i c a l o f a c o n t i n u o u s c a s t in g a p p l i c a t io n , e q u a t i o n ( 6)p r e d i c t s o n l y a ha l f o f t he ve l oc i t y g i ve n by e q ua t i o n (5 ). T h us r ough ne s s i s a ni m p o r t a n t p a r a m e t e r i n t h i s f l o w .T h e e s s e n t i a l f e a t u r e o f t h i s t ype o f f l ow is t ha t t he i ne r t i a l f o r c e s a r e z e r o ,e xc e p t f o r t he c e n t r i pe t a l a c c e l e r a t ion . I t f o l l ow s t ha t t he on l y f o r c e s a va i l a b l et o ba l a nc e t he m a gne t i c t o r que a r e s he a r s t r e s s e s . I n a ny r e a l s t i r r e r , how e ve r ,t he m a gn e t i c f o r c i ng oc c u r s w i t h i n a r e l a ti ve l y s ho r t l e ng t h o f t he c y l i nd e r (s e eF i g . 2 ). T h i s r e s u lt s i n d i f f e r e n t ia l r o t a t i o n b e t w e e n f o r c e d a n d u n f o r c e dr e g i ons , w h i c h , i n t u r n , p r oduc e s s e c onda r y f l ow . T he i ne r t i a l f o r c e s a r e t he nn o n - z e r o a n d a r e a v a i l a b l e f o r b a l a n c i n g t h e m a g n e t i c b o d y f o r c e . T h i s l e a d st o a qu i t e d i f f e r e n t t ype o f f l ow , a nd a d i f f e r e n t s c a r i ng l a w f o r t he ve l oc i t y .W e s h a l l s e e t h a t t h e m a g n i t u d e o f t h e s w i r l i s l a r g e l y d e t e r m i n e d b y i n e r t i aa n d n o t s h e a r .

I V . A t w o d i m e n s i o n a l a x i s y m m e t r i e m o d e l o f s ti r ri n g s o m e q u a l i ta t iv ef e a t u r e s o f t h e f l o w

I n t h i s s e c t i o n w e s h a l l e x a m i n e t h e l a m i n a r e q u a t i o n s o f m o t i o n f o r a x i s y m -m e t r i c f l ow . I n p r a c t i c e , how e ve r , a l l r e a l f l ow s o f i n t e r e s t a r e t u r bu l e n t . I no r de r t o i n t e r p r e t t he d i s c us s i on i n t e r m s o f a t u r bu l e n t f l ow , t he v i s c os i t y 1 ,m u s t b e c o n s i d e r e d a s a m e a n ' e d d y v i sc o s it y '. S u c h a p r o c e d u r e i s r e a s o n a b l ei n t h i s c a s e a s m a n y f e a t u r e s o f t h e f l o w a r e c o n t r o l l e d b y t h e i n e r t i a o f t h em e a n f l ow , a nd a r e no t s e ns i t i ve t o t he de t a i l s o f t he s he a r .

T o d e t e r m i n e w h y a n a x i al v a r ia t i o n i n t h e a z i m u t h a l b o d y f o r c e g iv e s r is et o po l o i da l m o t i on , i t i s c onve n i e n t t o s p r i t t he ve l oc i t y f i e l d i n t o po l o i da l a nda z i m u t h a l p a r t s , a n d e x a m i n e t h e i n t e r a c t i o n b e t w e e n t h e m . ( P o l o i d a l m o t i o ni s tha t in the r - z p lane . )

g = l i p - ~ U ow h e r e

U p ~ U r e r ~ U z L .T h e v o r t i c i ty m a y b e s i m i la r ly d i v i d e d~ = + ~ o

w h e r eO~ = V X UpO~p = ~7 u o

T h e N a v i e r - S to k e s e q u a t i o n m a y i ts e l f b e s p l i t i n t o a x i m u t h a l a n d p o l o i d a lp a r t s . T o e l i m i n a t e t h e p r e s s u r e f r o m t h e p o l o i d a l e q u a t i o n w e m a y t a k e i t s


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T h e i m p o r t a n c e o f s e c o n d a ry f l o w 249c ur l a nd ob ta in a vo r t i c i ty a dve c t ion - d i f f u s ion e qua t ion . The r e su l t i ng e qua -tions are,

U p = l , V 2 u o + 1 Fo ~ . o

v x u p x , o 0 ) + 2 + = _ t s 0 . 7 )Th e f ir s t equ a t ion represen ts the az imutha l force ba lance . I t is th roug h the

ine r t ia l te rm ~0p X Up tha t the az imu tha l m ot io n i s coup led to the p olo id a lmo t ion . F o r l a r ge R e yn o lds num be r s the iner t ia l f o r c e w il l be m uc h l a rge rtha n the she a r t e r m, a nd we e xpe c t t he p r ima r y f o r c e ba l a nc e to be be twe e nJ x B and iner t ia .

W e sha l l show tha t t h i s i ne r t i a l t e r m r e p r e se n t s a f l ux o f a ngu la r mome m-turn out of the forced reg ion .

The se c ond e qu a t ion i s the s t e a dy - s t a t e a dve c t ion - d i f fu s ion e qua t ion f o r t hea z imu tha l vo r t i c i ty a nd c o r r e spond ing po lo ida l ve loc i ty . I t i s c oup le d to theaz imu tha l v e loc i ty throug h a source te rm w hich is the ax ia l g rad ien t of thecent r ipe ta l acce le ra t ion . This addi t io na l te rm der ives f rom V x (u 0 x ) andr e p r e se n ts swe e p ing o f po lo ida l vo r t i c i ty th r ough a n a x ia l va r i a t ion in theaz imutha l ve loc i ty , genera t ing az imutha l vor t ic i ty . This i s i l lus t ra ted in F ig . 3 .N o t e t h a t i f u o i s independent of z , then equa t ion (7) impl ies tha t no f ie ldsweep ing of ~0p occurs . This i s to be expec ted s ince in such a s i tua t ion thepolo ida l vor tex l ines l ie pa ra l le l to the ax is and each poin t on a vor tex l ineexper iences the same az imutha l ve loc i ty .

The se e qua t ions o f mo t ion ma y be r e wr i t t e n a s s c a l a r t r a nspo r t e qua t ionsf o r a n g u la r m o m e n t u m F a n d ~ / r .

F = Uor

The a z imu tha l f o r c e ba l a nc e be c om e s a tr a nspo r t e qua t ion f o r F , a nd thea dve c t ion - d i f f u s ion e qua t ion a t r a nspo r t e qua t ion f o r ~oo/r.

u . V F = v { V 2 F 2 0 F } r E7 Or + ; o (8)2 0 ( 0~0 0u - v ( + ) : p ( V 2 ( ~ ) + r - ~ r , r ) l + -O-~z r ~ ) . ( 9)

F r o m e qua t ion (8 ) we see tha t, i n t he a bse nc e o f she a r a nd m a gne t i c f o r ce s ,the a ngu la r mome n tum i s a dve c te d unc ha nge d a long a s t r e a ml ine . Th i s i s i na c c o r da nc e w i th Ke lv ins c i r c u la t ion the o r e m a pp l i e d to a ma te r i a l hoopcent red on the z ax is .

A n e xa m ina t ion o f e qua t ion (9 ), o r a c ons ide r a tion o f vo r te x swe e p ing ,sugge st s tha t t he s e c on da r y po lo ida l f l ow m us t b e o f t he f o r m show n in F ig . 4.The de c r e a se o f a ngu la r v e loc ity w i th de p th a c t s a s a sou r c e o f ne ga tiveaz imutha l vor t ic i ty . This requi res ro ta t ion in an ax ia l p lane as shown in thefigure.


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250 P A Davidson and F Boysan

i ) g 6 , ~ pi i ) ~ P

C i i i ) - G 3 0 ~ Y P

F i g 3 Sw eeping f poloidal or t ici ty y an ax ialgradient n azim uthal eloci ty o produceazim uthal orticity.The primary effect of the body force is to spin up the fluid as it passes

through the forced region, in accordance with equation (8). The poloidal flowthen sweeps this angular momentum into the unforced region.The viscous terms in equations (8) and (9) are not purely diffusive, they alsocontain source terms. However, their primary role is to allow the vortex sheet,created at the wall by the no-slip condition, to diffuse into the flow. Since thisis a relatively slow process, the Reynolds number being assumed to be large,the diffusion occurs primarily in the unforced region.

The length scales for r and z in the forced region are dictated by the spatialvariation of the body force (except in the boundary layer adjacent to the wall),and are of order R. It follows from equation (8) that the magnetic force islocally balanced by inertia, assuming the Reynolds number is large. However,there is also a well known integral requirement on closed streamline flows. Itmay be derived from the Navier-Stokes equation,

,o . = - v p / p + u 2 / 2 ) + v2u + r / p .Integrating around a closed streamline we deduce,

~F dr + p~x7 2u. dr = 0. (10)


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Free surface or q[Line of symm,

Forc~regio

he importance of secondary flow 251

]tion poFnty layer;s

vor tex

Unfregi .~ngt h sc al e ed region

Fig 4 Shape of poloidal eddy

This is a form of Bernoulli's equation showing that the work done on afluid element by the body force must be dissipated by shear. This hasimplications regarding the length scale for z in the unforced region, and theaxial position of the eye of the vortex.

The eye of the vortex cannot lie in the forced region since this would implythat closed streamlines exist entirely within that region, and since the flowthere is essentially inviscid, equation (10) could not be satisfied. On the otherhand, if the eye of the vortex lay some distance below the forced region, thenequation (10) would require the velocity field in the region of the eye of thevortex to satisfy

fVau dr = 0.In general the velocity field will not satisfy this constraint. Instead, the eyeof the vortex positions itself at the edge of the forced region, with each

streamline passing through both the forced and unforced regions. This isshown in Fig. 4.


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252 P . A . D a v i d s o n a n d F . B o y s a nThe boundary layer on the wall grows at a rate,

Equation (10) requires all streamlines to pass through this boundary layer.This suggests that the axial length scale in the unforced region, L, satisfies,

R - ~ / VLu~ L - R e . R .This result will be deduced later on dimensional grounds. It implies that the

penetration of the swirl beyond the forced region is extensive.It is useful to consider the angular momentum integral applied to theproblem.T=~(prXu)u ds.If the control volume is taken as the flow field as a whole, then the appfied

magnetic torque must be balanced by shear on the walls.f S P j2 dr dz = R 2 [r,0[R dz.If the momentum integral is applied from the free surface to the bottom of

the forced region, and the wall shear in the forced region neglected incomparison with the total wall shear, then,oo R R 2f o f o F o r 2 d r d z = P f o r U o U z d r .

The bars over the velocity terms indicate values at the bottom of the forcedregion. Combining these equations and substituting for F o from equation (3)we deduce,

R 2 ~ ~ I oIR dz .1V2R2f0 f z / R ) d z = f o r fi0fizdr = R 2 j o1 t 1Total applied = Flux of angular = Torque due tomagnetic torque momentum out of wall shearforced region

1 1 )

It follows from equation (11) that in the forced region,U o U z ~ 7 2

By comparing inertial terms in equation (9), and invoking continuity, it maybe concluded that,

U r ~ U z ~ U 0

Combining these two estimates we deduce that in the forced region,U r ~ b l z ~ U 0 ~ V


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The im portance of secondary low 253T h u s t h e s e c o n d a r y p o l o i d a l f l o w is a s la r g e a s t h e p r i m a r y a z i m u t h a l f lo w ,

a ll t h r e e v e l o c it y c o m p o n e n t s b e i n g o f o r d e r V .I n t he un f o r c e d r e g i on , how e ve r , w e e xpe c t t he a x i a l l e ng t h s c a l e t o be

l a r ge . Com pa r i ng t he f i r s t a nd l a s t t e r m s i n e qua t i on ( 11 ) w e de duc e ,V 2 R 3 - R2 o~ rr'Rp d z - R2p -~ LT a k i n g t h e b o u n d a r y l a y e r t h i c k n e ss i n th e u n f o r c e d r e g i o n t o b e o f o r d e r R

w e d e d u c e ,L - R e . R . ( 1 2 )T h i s i s t h e s a m e e s t i m a t e f o r L t h a t w a s o b t a i n e d f r o m a c o n s i d e r a t i o n o f

t h e f o r c e i n t e g r a l ~ F . d r . N o t e t h a t t h e l a r g e a x ia l l e n g t h s c a le in t h e u n f o r c e dr e g i on r e qu i r e s t ha t ,

u r - 17 / Re i n t he un f o r c e d r e g i on .I n s u m m a r y , w e h a v e t h e f o l l o w i n g sc a ri n g re l a ti o n s h i p s ,

F o r c e d r e g i o n :U r ~ IX 0 ~ bl zr - z - R

U n f o r c e d r e gi o n:/ gO - - U z - - V / / r - V / R er - R , L - R . R e .I n t h e p r e v i o u s s e c t i o n w e s a w t h a t t h e c o r e a n g u l a r v e l o c i t y i s p r e d i c t e d b y

a o n e - d i m e n s i o n a l m o d e l a s ,

I n a t y p i c a l c o n t i n u o u s c a s ti n g p l a n t t h e R e y n o l d s n u m b e r i s o f o r d e r 1 0 5.T h i s e q u a t i o n t h e n g iv e s,

a 0 - 1 0 ~ .T h e d i m e n s i o n a l a n a l y s i s g i v e n a b o v e s u g g e s t s t h a t i n a t w o - d i m e n s i o n a l

a x i s y m m e t r ic fl ow , f ~ 0 - V / R . I n f a c t , w e s ha l l s e e i n t he ne x t s e c t i on t ha ttypical ly , ~20 - 2 V / R . T h i s is a f a ct o r o f f iv e s m a l le r th a n t h a t p r e d i c te d b yt h e o n e - d i m e n s i o n a l a n a l y s is . T h e r e a s o n f o r t h e d i f f e r e n c e i s, o f c o u rs e , t h a ti n o n e c a s e t h e m a g n e t i c b o d y f o r c e i s b a l a n c e d b y s h e ar , w h i l e i n t h e o t h e r i ti s b a l a n c e d b y i n e rt ia .

V . N u m e r i c a l e x p e r i m e n t s o f t w o d i m e n s i o n a l s t i r r i n gT h e a r g u m e n t s p r e s e n t e d i n t h e p r e c e d i n g s e c t i o n c a n b e s u b s t a n t i a t e d b ym e a n s o f f i n it e d i f fe r e n c e c o m p u t a t i o n s o f t w o - d i m e n s i o n a l a x is y m m e t r ic


14/19

2 5 4 P.A. Davidson and F. Boysans ti rr in g . T h e t u rb u l e n c e is m o d e l l e d v i a t h e w i d e l y u s e d k - ~ m o d e l w h i c he n t a i l s t h e s o l u t i o n o f t r a n s p o r t e q u a t i o n s f o r th e k i n e t i c e n e r g y o f t u r b u l e n c ea n d i t s d i s s i p a t i o n r a t e . T h e R e y n o l d s s t r e s s e s a r e r e l a t e d t o t h e r a t e o f s t r a i nu s i n g a n e d d y v i s c o s i t y h y p o t h e s i s [ 8].

T h e e d d y v i s c o s it y v i s t h e n r e l a t e d t o t h e t u r b u l e n c e k i n e t i c e n e r g y k a n dv i s c o u s d i s s i p a t i o n , c .

w h e r e c , i s a c o n s t a n t .T h e t r a n s p o r t e q u a t i o n f o r k is b a s e d o n t h e t u r b u l e n c e k i n e ti c e n e r g ye q u a t i o n , w h i l e t h e t r a n s p o r t e q u a t i o n f o r e i s s o m e w h a t m o r e e m p i r i c a l l yb a s e d .

W e h a v e u s e d t h e st a n d a r d f o r m o f t h e t r a n s p o r t e q u a t i o n s f o r k a n d ew h i c h m a y b e f o u n d i n [ 8 ] . S o m e w o r k e r s i n s w i r l i n g f l o w s s u g g e s t t h a t t h e s es t a n d a r d e q u a t i o n s n e e d t o b e a l t e re d t o t a k e i n t o a c c o u n t t h e e f f ec t o fs t r e a m l i n e c u r v a t u r e o n t h e t u r b u l e n c e . T y p i c a l l y , b y a n a l o g y w i t h t h e c o r r e c -t i o n f o r b u o y a n c y [8 ], th e s o u r c e t e r m i n t h e c e q u a t i o n m a y b e a l t e re d t o t a k ei n t o a c c o u n t t h e s t ab i li s in g o r d e s ta b i l is i n g e f f e c t o f r o t a t i o n . ( W e w o u l de x p e c t t h e t u r b u l e n c e t o b e d e s t a b i l is e d f o r 0 F / D r < 0 , a n d s t a b i li s e d f o r

3 F / O r > 0 . ) H o w e v e r , th e r e is n o u n i v e r sa l ly a c c e p t e d c o r r e c t io n f o r c u r v a t u r ea n d w e h a v e n o t u s e d a n y s u c h c o r r e c t io n i n t h e c o m p u t a t i o n s .

T h e r e is s o m e d e b a t e a s to t h e g e n e r a l i t y o f d i f f e r e n t t u r b u l e n c e m o d e l s .H o w e v e r , w e h a v e s h o w n t h a t t h e f l o w i n t h e f o r c e d r e g i o n i s l a r g e l yc o n t r o l l e d b y t h e i n e r ti a o f t h e m e a n f lo w a n d t h a t t h e p r i m a r y e f f e c t o f s h e a ri s t o c o n t r o l t h e l e n g t h o f t h e d i f fu s i v e r e g io n . W e e x p e c t , t h e r e f o r e , t h a t m a n yo f t h e b r o a d f e a t u r e s o f th e f l o w a r e in s e n s i t iv e t o t h e d e t a i l s o f t h e t u r b u l e n c em o d e l l i n g .

C o m p u t a t i o n s w e r e p e r f o r m e d u s i ng a f i n i te d i f fe r e n c e c o d e e m p l o y i n g ap o w e r - l a w d i f f er e n c i n g s c he m e . T h e d e t ai ls o f t h e s o l u t io n m e t h o d o l o g y c a nb e f o u n d e l s e wh e r e [ 6 ] .

T h e e l e c t ro m a g n e t i c b o d y f o r c e u s e d i n t h e c o m p u t a t i o n s is t h a t g i v e n b ye q u a t i o n ( 3) , w i t h a n i n v e rs e f o u r t h p o w e r l a w d i s t r ib u t i o n f o r f z / R ) .

T h i s f o r c e d r o p s t o 5 o f i ts i n i ti a l v a l u e a f t e r o n e d i a m e t e r d e p t h ( z = 2 R ) .T w o c a se s w e re a n a l y s e d c o r r e s p o n d i n g t o d i f f e re n t m a g n i t u d e s o f t h e b o d yf o rc e , a n d h e n c e 17. T h e c a s es c o n s i d e r e d c o r r e s p o n d t o V = 1 c m / s a n d

V = 1 0 c m / s . I f t h e a n al y s is p r e s e n t e d i n S e c t i o n I V is c o r re c t , t h e n u s h o u l ds c a l e o n V . W e s h a l l s e e t h a t t h i s i s i n d e e d t h e c a s e . T h e r e l e v a n t p h y s i c a lp r o p e r t i e s u s e d i n t h e c o m p u t a t i o n s a r e g iv e n i n T a b l e 1 .


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T h e i m p o r t a n c e o f s e c o n d a r y f l o wTab le 1. Physical properties use d in com putations.

255

Case p p R V Roughness parameter( k g / m3) (m2/s) (m) (m/s) E1 7 103 10 - 6 0.1 0.01 Smo oth2 7xl O 3 10 6 0.1 0.1 0.1(k - 1/2 ram)

i I : ~ j l

i : i i

Fig . 5. Com puted poloidal flow pattern (Case 2).

I n o r d e r t o i n v e s t i g a t e t h e e f f e c t o f w a l l r o u g h n e s s o n t h e f l o w , a r o u g h w a l lw a s u s e d i n t h e s e c o n d c a se . T h e r o u g h n e s s i s s p e ci f ie d i n t e r m s o f a r o u g h n e s sp a r a m e t e r , E . T h i s a p p e a r s i n a w a l l f u n c t i o n u s e d i n t h e c o m p u t a t i o n s t or e l a t e t h e w a l l s l i p v e l o c i t y , U s , t o t h e l o c a l s h e a r v e l o c i t y , V , [ 8 ] .

u s / V , = 2.5 l n ( E V , y / p ) .F o r a s m o o t h w a l l E = 9 , a n d f o r a r o u g h w a l l E = 1 , / V , k , w h e r e k is t h e

r o u g h n e s s h e i g h t . W e c h o s e E = 0 .1 f o r t h e r o u g h w a l l, w h i c h , f o r t h eR e y n o l d s n u m b e r us ed , c o r r e s p o n d s a p p r o x i m a t e l y t o k = 1 / 2 m m .F i g u r e 5 s h o w s t h e c o m p u t e d p o l o i d a l f l o w f o r t h e r o u g h w a l l c a s e . T h ed i f f e r e n c e b e t w e e n t h e a x i a l a n d r a d i a l l e n g t h s c a l e s i s c l e a r . T h e e y e o f t h ep o l o i d a l v o r t e x li e s a t z = 2 .0 R , w h i c h c o r r e s p o n d s , a p p r o x i m a t e l y , t o t h eb o t t o m o f th e f o r c e d r e g io n .


16/19

56 P A Davidson and F Boysan

i / ~ III3.0 I

2 . 0 -

1 . 0

S m o o t h wall I

Roug h all

0.25 0~5 0.75 1.0F i g 6 Com puted az imutha l s u r f ace ve loc i ty.

r R

0.6

0 . 4 -

0 .2 -

- 0 , 2 -

- 0 . 4 -

- 0 . 6 -

u ~ 9{ z =R

~ r l R.0

Rough allF i g 7. Com puted ax ia l ve loc i ty z = R ) .

F igu r e s 6 a nd 7 show the c om pu te d a z imu tha l ve loc i ty p r o f i le a t t he f r e e


17/19

2~0

~ 0R

The importance of secondary flow 257

1.5

1 0

0.5

woll

Rough

Forced region

2 ~Fig 8 Variation of core angular ve locity with depth.

z~R

sur face z = 0) and ax ia l ve loc i ty p rof i l e a t z = R. B oth the smooth a nd rou ghwal l resul ts are given. I t i s clear that the axial and azimuthal veloci t ies are ofthe same order , and are insensi t ive to the wal l shear .

Figure 8 shows the v ar ia t ion of core angular ve loc i ty wi th dep th . Th e swi r lpenet ra tes w el l ou t o f the forced reg ion , be ing car r i ed by the se cond ary f low.

The com pute d edd y v iscos i ty var ies th rou gho ut the f low f ield . Typic a l ly thee f f ec ti ve Reyno l ds n um b er Re = VR/I t based on the eddy v i scos i ty , has avalue of - 30 in bo th cases . The s imi lar i ty in Re for the smoo th and roug hwal l f lows explains why the rates of decay of , shown in Fig. 8 , are s imilar .

I t i s c lear tha t the resu l t s o f the computa t ions are in accordance wi th thed i scuss ion of Sec t ion IV.


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2 5 8 P A D a v i d s o n a n d F B o y s a nVI ConclusionsO n i g n o r i n g t h e a x i a l v a r i a t i o n o f t h e e l e c t r o m a g n e t i c b o d y f o r c e , a s i m p l e,o n e - d i m e n s i o n a l , s w i rl f lo w i s p r o d u c e d . T h e r e a r e n o i n e r t i a l e ff e c t s e x c e p tf o r t h e c e n t r i p e t a l a c c e l e r a t i o n , w h i c h i s b a l a n c e d b y a r a d i a l p r e s s u r e g r a d i -e n t . I t fo l l o w s t h a t t h e o n l y f o r c e s a v a i l a b le t o b a l a n c e t h e m a g n e t i c t o r q u e a r es h e a r s t r e s s e s . I n f o r m a t i o n a b o u t t h e r e l a t i o n s h i p b e t w e e n s h e a r s t r e s s a n dv e l o c i t y g ra d i e n t s i n t h e f l u i d i s th e n s u f f i c ie n t t o a l l o w t h e v e l o c i t y p r o f i l e t ob e c a l c u l a t e d .

T h e s e o n e d i m e n s i o n a l m o d e l s a r e m i s l e a d i n g i n t h e c o n t e x t o f m a g n e t i cs t i r r i n g . I n a r e a l s t i r r e r , t h e m a g n e t i c f o r c i n g o c c u r s o n l y o v e r a r e l a t i v e l ys h o r t l e n g t h o f t h e c y l i n d e r . T h i s r e s u lt s i n d i f fe r e n t i a l r o t a t i o n b e t w e e n f o r c e da n d u n f o r c e d re g i o n s . T h i s d i f f e r e n t i a l r o t a t i o n d r i v e s a s e c o n d a r y p o l o i d a lf lo w , w h e r e f l u i d p a r t i c l e s n o w f o l l o w h e l ic a l p a t h s .

D i m e n s i o n a l a n a ly s is a n d n u m e r i c a l e x p e r i m e n t s s u g ge s t t h a t t h e s e c o n d a r yf l o w is a s l a r g e a s t h e p r i m a r y s w i rl f lo w . T h i s s e c o n d a r y f l o w s w e e p s a n g u l a rm o m e n t u m o u t o f t h e f o r c e d r e g io n s o th a t t h e f o r c e d v o r t e x u l t i m a t e l yp e n e t r a t e s s o m e d i s t a n c e b e y o n d t h e s t i rr e r. S i n c e i n e r t i a l f o r c e s a re n o wn o n - z e r o , t h e m a g n e t i c b o d y f o r c e is l o c a ll y b a l a n c e d b y i n e rt ia , r a t h e r t h a ns h e a r . T h i s r e s u l ts i n a q u i t e d i f f e r e n t m a g n i t u d e o f sw i r l t h a n t h a t p r e d i c t e db y t h e o n e - d i m e n s i o n a l m o d e l .

T h e p r i m a r y r o l e o f s h e a r is t o d i s si p a t e t h e e n e r g y c r e a t e d b y J x B . T h i so c c u r s p r e d o m i n a n t l y i n t h e u n f o r c e d r e g i o n , i n w h i c h t h e a x i a l l e n g t h s c a l e i sc o n t r o l l e d b y t h e s h e a r .

T h e e x i st e n c e o f s tr o n g s e c o n d a r y f l o w s d u r i n g r o t a r y m a g n e t i c s t ir ri n g h a sb e e n l a r g e l y i g n o r e d i n t h e l i t e ra t u r e . T h e i r e f f e c t i n e x t e n d i n g t h e f o r c e dv o r t e x w e l l b e y o n d t h e i m m e d i a t e v i c i n i t y o f t h e m a g n e t i c s t i r r e r i s c l e a r l yi m p o r t a n t w i t h r e g a r d t o i n t e r p r e t i n g t h e m e t a l l u r g i c a l i n f l u e n c e o f s t i r r i n g .

AcknowledgementsT h e a u t h o r s w o u l d l ik e t o t h a n k T I R e s e a r c h L ab s ., a n d D r J .C . R . H u n t o fC a m b r i d g e U n i v e r s i t y f o r t h e ir a d v i c e a n d s u p p o r t i n t h i s s t u d y . T h e c o m -p u t e r c o d e w a s a m o d i f ie d v er si o n o f F L U E N T m a d e a v ai la b le b y C r e a r eR D .

References1. P.A. Davidson: Estimation of turbu lent velocities nduc ed by mag netic stirring during continu-ous casting, Ma terials Sc and Technology 1 (1985) 994-9992. L.D . Landau and E.M. Lifshitz: Electrodynamics of Continuous Med ia Pergam on Press (1960)Chap . VII.3. H .S. M art: Electromagneticstirring in continu ous casting of steel. In: M .R.E. Proctor andH.K. M offatt (eds) Proceedings of Metallurgical Applications of M HD IUTAM, Cambridge(1982) pp 143-153.


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The importance of secondary flow 2 5 94 . D .A . M e l fo rd a nd P .A. Da v i dson ; Th e so l i d i f i ca t i on o f c on t i nuou s l y c a s t s te e l un de r c on d i -

t i ons o f fo rc e d f l ow. In : C onf . Modelling of Casting and Welding Processes. N e w E n g l a n dC o ll ege, U S A 1983) pp 101 -104 .

5 . H .K. M of fa t t : R o t a t i o n o f a fi qu i d me t a l unde r t he a c t i on o f a ro t a t i ng m a gne t i c fi el d. In : H .B r a n o v e r a n d A . Y a k h o t e d s ) MHD Flows and Turbulence H. I s r a d U n i v e r s i ty P r e ss 1 9 7 8 )p p 4 5 - 6 2 .

6 . S .V. Pa ta nk ar ; Numerical Heat Transfer and Fluid Flow. He mi sphe re 1980) .7 . T . R o b i n s o n a n d K . L a r s s o n ; A n e x p e r i m e n t a l i n v e s ti g a ti o n o f a m a g n e t i c a l ly d r i v en r o t a t i n g

l iquid -m eta l f low. J . Fluid Mech. 6 0 1 9 7 3) 6 4 1 - 6 6 48 . W . R o d i : T u r b u l e n c e m o d d s a n d t h e i r a p p l i c at i o n s i n h y d ra u l ic s . S t a te - o f - th e - a r t p a p e r I A H R

1984).9 . K .H. Ta c ke a nd K. S c hwe rd t fe ge r : S t i r r i ng ve l oc it i es in c on t i nu ous l y c a s t roun d b i l l e t s a s

i ndu c e d b y r o t a t i ng e l e c t roma gne t i c f ie ld s. Stahl und Eisen 99 1979) 7 -12 .

applied scientific research volume 44 issue 1-2 1987 [doi 10.1007_bf00412016] p. a. davidson_ f....

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