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~ ) Pergamon I n t . J . H e a t M a s s T r a n s f e r . Vol. 40, No. 15, pp. 3703 3716, 1997 1997 Elsevier Science Ltd. A ll fights reservedPrinted in Great Britain0017-9310/97 $17.00+0.00PII: S 0 0 1 7 - 9 3 1 0 ( 9 6 ) 0 0 3 1 1 - 0

A n e x p e r i m e n t a l s t u d y o f h e a t t ra n s f e r in as i m u l a t e d t u r b i n e b l a d e c o o l i n g p a s s a g eW . D . M O R R I S a nd S . W . C H A N G

De partm ent o f Mechanical Engineering , Universi ty of Wales , Swansea , Single ton Park , SwanseaSA2 8 PP, U.K .(Received in f inal for m 9 September 1996)

Ab st rac t --T his paper describes a n experimenta l invest iga t ion of heat t ransfer inside a s imula ted cool ingchannel for a gas tu rb ine ro tor b lade . The channel i s c i rcu lar in cross-sect ion and ro ta tes about an ax iswhich i s o rthogonal to i t s cen t re l ine . The s tudy i s a imed a t the dev elopm ent of an experim enta l p rocedurea n d me t h o d o f d a t a p ro c e ssi n g wh i c h p e rmi t s t h e d e t e rmi n a t io n o f fu l l a x i a l a n d c i rc u mfe re n ti a l h e a tt ransfer da ta over the tube ' s inne r surface . Th is i s referred to as fu l l f ie ld heat t ransfer da ta . In th is respecta s tudy of the combined effec t o f Corio l i s and cen t r ipe ta l buoyancy forces on the forced convect ionmechanism inside the tube i s the s t ra teg ic a im. The experimenta l technique involved the determinat ion ofthe inside surface temperature and heat f lux d is t r ibu t ion using a so lu t ion of the channel w al l heat conduct ionequat ion . A series of wal l tempe rature m easurem ents on the lead ing and t ra i l ing edges of the channel ,together w i th a prescribed e lec t r ica l ly genera ted h eat f lux on the ex ternal surface , were used as b ounda rycondi t ions wi th which to so lve the channel wal l heat conduct ion equat ion . The resu l t ing in ternal heat f luxdis t r ibu t ion ov er the fu l l inner surface was subsequent ly used to determine the local varia t ion of heat t ransfercoefficient . The me thod w as val ida ted using data avai lab le in the technical l i t e ra ture and subsequent ly usedto s tudy the ind iv idual effec ts o f Corio l i s induced secondary f low and cen t r ipe ta l buoyancy using new datagenera ted fo r the invest iga t ion . 1997 Elsev ier Science Ltd .

1 . I N T R O D U C T I O N

A i r c r a f t g a s t u r b i : a e e n g i n e s r e q u i r e h i g h c y c l e p r e s -s u r e r a t i o s a n d h i g h t u r b i n e e n t r y t e m p e r a t u r e s t oa c h i e v e h i g h t h r u s t t o w e i g h t r a t i o s w i t h l o w s p e ci f icf u e l c o n s u m p t i o n . A s u i t a b l y h i g h g a s t e m p e r a t u r e i nt h e h o t s e c t i o n o f th e e n g i n e ( i .e . t h e c o m b u s t o r , t u r -b i n e d i sc s , a n d t u : rb i n e s t a t o r / r o t o r b l a d e s ) c a n o n l yb e a c h i e v e d i f s o m e f o r m o f c o o l i n g is u s e d f o r t h e s ec o m p o n e n t s a n d h i g h p r e s s u r e a i r, b l e d f r o m t h e c o m -p r e s s o r , i s t h e c u s t o m a r y c o o l a n t u s e d i n p r ac t i c e .

H i g h p r e s s u r e t u r b i n e r o t o r b l a d e s a r e p a r t i c u l a r l yp r o b l e m a t i c . T h e a i r f o il s e c ti o n s m u s t b e a b l e to s u s -t a i n t h e r m a l a n d m e c h a n i c a l s t r e s s e s , f a t i g u e , c r e e pa n d c h e m i c a l d e t e r i o r a t i o n e x p e r i e n c e d w h i l s t m a i n -t a i n i n g a n a c c e p t a b l e o p e r a t i n g l if e .

T h e a c c u r a t e p r e d i c t i o n o f t h e a ir f o i l m e t a l t e m -p e r a t u r e d i s t r i b u t i o n i s t h e fi r s t s t e p i n t h e a s s e s s m e n to f b l a d e l i fe . T h i s r e q u i r e s t h e e x t e r n a l h e a t t r a n s f e rc o e f f i c i e n t d i s t r i b u t i o n o v e r t h e a i r f o i l s u r f a c e a n dt h e i n t e r n a l c o o l a n t p a s s a g e h e a t t r a n s f e r c o e f f i c i e n td i s t r i b u ti o n s a s b o u n d a r y c o n d i t i o n s f o r t h e s o l u t io no f th e c o n d u c t i o n , e q u at i on i n t h e c o m p o n e n t d o m a i n .I t h a s b e e n d e m o n s t r a t e d b y T a y l o r [1 ] t h a t a n u n c e r -t a i n t y o f + 1 0 % il n t h e h e a t t r a n s f e r c o e f f i c i e n t d is -t r i b u t i o n s r e su l t s i n a n u n c e r t a i n t y o f + 2 % i n t h em e t a l t e m p e r a t u r e . T h i s p r o d u c e s a n u n c e r t a i n t y o f___ 5 0 % o n p r e d i c t e d b l a d e l if e . T h e p r e s e n t p a p e r i sc o n c e r n e d w i t h th e d e t e r m i n a t i o n o f t h e i n te r n a l c o o -

l i n g h o l e h e a t t r a n s f e r c o e f f i ci e n t d i s t r i b u t i o n u n d e rd i f f e r e nt o p e r a t i n g c o n d i t i o n s .

F i g u r e 1 i l lu s t r a te s t h e c o m p l e x i t y o f t h e c o o l a n tp a s s a g e s i n s id e a r o t o r b l a d e . A c o m b i n a t i o n o f f il mc o o l i n g , i m p i n g e m e n t c o o l i n g a n d c o n v e c t i o n c o o l i n g( u t il i si n g s m o o t h - w a l l e d a n d a r t if i c ia l l y r o u g h e n e dc h a n n e l s u r f a c e s ) i s u s e d t o s a t i s f y t h e d e s i g n r e q u i r e -m e n t s . C o n v e c t i o n c o o l i n g c h a n n e l s a r e m a i n l yo r t h o g o n a l t o t h e a x is o f th e t u r b i n e w i t h t h e c o o l a n tf l o w i n g in a r o o t t o t i p o r t i p t o r o o t d i r e c t i o n a ss h o w n i n t h e f ig u r e . P r e d i c t i o n o f t h e c o o l a n t f l o wf i e l d a n d t h e h e a t t r a n s f e r c o e f f i c i e n t d i s t r i b u t i o n i nt h e s e p a s s a g e s i s e x t r e m e l y d i f f i c u lt d u e t o t h e g e o -m e t r i c f e a t u r e s o f t h e p a s s a g e s a n d a l s o b e c a u s e t h ec o o l a n t r o t a t e s w i t h t h e b l a d e .

I t h a s b e e n w e l l e s t a b l is h e d t h a t c o o l a n t f l o w i n g i nt h e s e r a d i a l l y r o t a t i n g c h a n n e l s i s s u b j e c t e d t o t h ec o m b i n e d e f f e ct o f C o r i o l i s f o r c e a n d c e n t r i p e t a l b u o y -a n c y w h i c h s e v e r e l y a l te r s t h e h e a t t r a n s f e r o b t a i n e dw i t h f o r c e d c o n v e c t i o n d u c t e d f l o w , s e e M o r r i s a n dA y h a n [ 2 ] , W a g n e r et a l . [ 3 ] a n d M o r r i s a n d S a l e m i[ 4] , f o r s m o o t h w a l l e d s u r f a ce s a n d T a s l i m e t a L [5],W a g n e r e t a L [ 6] a n d M o r r i s a n d S a l e m i [ 7] f o r r o u g h -e n e d s u r f a c e s .

W i t h c i r c u l a r - s e c t io n e d c h a n n e l s , f o r e x a m p l e ,C o r i o l i s f o r c e c r e a t e s a s y m m e t r i c a l c r o s s s t r e a m s e c -o n d a r y f l o w , s e e F i g . 2 , w i t h f l u i d f r o m t h e c e n t r a lr e g i o n m o v i n g t o w a r d s t h e t ra i l i n g e d g e a n d a r e t u r nf l o w a l o n g t h e c ir c u m f e r e n t i a l w a l l r e g i o n t o w a r d s t h e

3703

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3704 W.D. MORRIS and S. W. CHANG

NOMENC L A T U R EA con stant coefficients VB. con stant coefficients wB u fluid buoyan cy parameter Y.C axial location dependent coefficient zCp fluid con stant pressure specific heat ZD~, D2 co nst an t coefficientsd diameter of test sectionE consta nt coefficientF functio n of r flG functio n of 0 Pk fluid thermal conductivity /~vkw thermal conductivity of test sectionmaterial toflh heat transf er coefficientL function of z t/n solution parameterN angular velocity of rotating frame 0N u local Nusselt numberp flow pressureP dimensionless pressure SubscriptsP r fluid Prandtl number iq heat flux Lr radial coordina te mr position vector oR dimensionless position vector 0R e Reynolds number RR o Rossby number Tt time wT temperature zv velocity vector 0

dimensionless velocity vectormean axial flow velocityBessel functio n solutionaxial coordinatenon-d imensi onal axial coordinate.

Greek symbolsfluid volume expansion coefficientfluid densityfluid dynami c viscosityfluid kinematic viscosityangula r velocity vectordimensionless angular velocity vectordimensionless fluid tempera tureunknown functionunknown functionangular coordinate.

inner tube surfaceleading edgemeasurement locationouter tube surfacezero rotati onal speedreference conditi ontrailing edgetest section wallaxial locationangular location.

leading edge. The net result of the pressure-drivenflow in the root to tip direction and this secondaryflow is a double helical flow pattern. This tends toimprove heat transfer on the trailing edge of the chan-nel relative to that on the leading edge. With a drivenflow in the tip to root direction the direction of thesecond ary flow reverses.

With outward radial flow the axial velocity profileis asymmetrical with the maxi mum velocity locationdeflected towards the trailing edge as shown in Fig. 2.Centripetal b uoyan cy further modifies the flow fieldin a mann er analogous to combined forced and gravi-tational free convection in vertical tubes where theradially outward orthogonal-mode rotation cor-responds to a vertical tube with a d ownward flow.

The proble m of predicting the flow and he at transferinside this class of rota ting tube is further complicateddue to the fact that the rotation can alter the basicflow stability and turbulence structure, see Johnson[8], Johnson e t a l . [9], Koyama e t a l . [10, 11] andRothe and Johnson [12].

The present paper presents the results of an exper-

imental investigation which attempts to assess theeffect of the combined Coriolis and buoyant inter-actions on the heat transfer mecha nism over the entiresurface of a smooth-walled circular tube which rotatesin this orthogo nal mode, thus simulatin g a rotat ingturbine airfoil cooling passage. The range of exper-imental variables over which tests have been con-ducted is extended beyond earlier published work bythis research group and approaches the lower rangeof real engine conditions.

The full field heat transfer distribution over theentire tube surface is determined by solving the con-duction equation in the tube wall. Thermocouplemeasurements of wall temperature at selectedlocations on the lea ding and trailing edge of the tube,together with a prescription of the external surfaceheat flux, give the required boundary conditions.Details of local heat transfer on the leading and trai-ling edges of the bla de are treated initially and this isfollowed by an analysis of the full axial and cir-cumferential heat transfe r behaviour. This is referredto as full field data.

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Heat transfer in a simulated turbine blade cooling passage 3705tion. The centripeta l term, to ^ to ^ r, is conservativeand only contribute s to a hydrostatic effect in the flow.The net effect is that the Coriolis term is exclusivelyresponsible for deviations in the flow field from thestationary duct situation.

With heated flow where the density is temperaturedependen t the centripetal term creates addit ional sec-ondar y effects via a buo yant interaction. This may bedemonstrated as follows. Suppose the density of thefluid obeys the equa tion of state

p = pr(1 "Ff l (T-- TR)) (2)Substituti on of eqn (2) into eqn (1) and inc orpora tingthe Boussinesq approxima tion yieldsO~+ 2t o ^ v+(to ^ to ^ r)fl(T-- Tr)

1= --Vp+/ ~V2v. (3)PThe so-called buoyancy force, (to A to A r)fl(T-- TR),now acts as an additional source term for modifyingthe s tati onar y flow field.

In order to identify non-dimensionalgroups whichdescribe this flow problem parametrically, the fol-lowing variable transfo rmation s are made.

V = - - (4)Wm

Fig. 1. Typical turbine rotor blade cooling channel network.tot ~ = ~ ( 5 )

(T- TR)t / - - ( T w - T R ) ( 6 )

2 . T H E P H Y S I C A L P R O B L E MThe flow field in the channel is controlled by the

moment um conservation equations, but the inertialterm has to be modified to account for Coriolis andcentripetal accelerations because the flow is referredto a r otatin g reference frame. The influence of rotatio nmay be explained by considering a laminar flow asfollows. Consider a circular-sectioned tube r otating inthe orthogo nal mode as indica ted in Fig. 2. The vectorform of the mome ntum conservation equation is

Dv 1D--t +2 to A v+,~ A ~ A r = - - Vp + # V2 v . (1)PAll symbols are described in the Nomenc lature.

The second two terms on the le ft-hand side of eqn(1) are the Coriolis and centripeta l corrections, respec-tively. Implicit in eqn (1) is the fact that the fluiddensity and viscosity are considered to be invariant,at this stage. Exami:aatio n of eqn (1), see Morr is [13]for details, demonstrates that the Coriolis term,2 to A v, generate s a cross st ream secon dary flowwhich causes the fluid to spiral in the streamwise direc-

PP = (7)pR w2rR = ~ . ( 8 )

Substitution of the transformed variables given byeqns (4)-(8) into eqn (3) yields, after some algebraicmanipulationsDV (2~ ^ V) VzvD t + R ~ + Bu ( ~ A ~ A R) = -V P+ R~

(9)where

wdpRRe - (Reynolds numbe r) (10)#wRo = ~ (Rossby number ) (11)

o ( T ~ - T ~ )Bu = p ~0 2 (Buoyancy parameter). (12)

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3 70 6 W . D . M O R R I S a n d S . W . C H A N G

g = c .

D i r e c t i o n o f f l o w

==c~> @

" O l , o~ 1 A =b ~ l | C

L o w r o ta t io n a l \ / H ig h r o ta t io n a ls p e e d p a t t e rn ~ / s p e e d p a t t e rnC o r io J is

s y m m e t r i c a l p l a n eSectional view of plane AA'

Fig. 2 . Heat co nduction mechanism tak ing place inside a rotating tube w all .

I n a d d i t i o n t o t h e m o m e n t u m c o n s e r v a t i o n e q u a t i o n ,t h e e n e r g y c o n s e r v a t i o n e q u a t i o n m a y b e e x p r e s s e d i nn o n - d i m e n s i o n a l f o r m a s

Dr/ V2r/D t R e P r

w h e r e/~CpP r = ~ - ( P r a n d t l n u m b e r ) . ( 14 )

T h e d e t a i l e d fl o w a n d t e m p e r a t u r e f i e l ds c r e a te d i na p a r t i c u l a r s i t u a t i o n w il l a l s o d e p e n d o n b o u n d a r yc o n d i t i o n s i m p o s e d . F o r e x a m p l e , t h e f l ow a n d t h e r -m a l f i el d s m a y d e v e l o p t o g e t h e r o r i n o t h e r c a s e s t h ef l o w m a y b e a l l o w e d t o d e v e l o p p r i o r t o t h e i n i t i a t i o no f h e a t i n g . I r r e s p ec t i v e o f b o u n d a r y c o n d i t i o n s , t h en o n - d i m e n s i o n a l g r o u p s i d e n t i f ie d b y t h e a b o v e a n a l y -s i s w i l l a l w a y s b e i m p o r t a n t . I f l o c a l h e a t f l u x a t a n yp o s i t i o n o n t h e s u r f a c e o f t h e d u c t i s e x p re s s e d v i a al o c a l h e a t t r a n s f e r c o e f f i c i e n t , hz , o , t h e n w e m i g h te x p e c t a l o c a l N u s s e l t n u m b e r t o h a v e t h e f o l l o w i n gs t r u c t u r e f o r a s p e c if i ed s et o f b o u n d a r y c o n d i t i o n s .

N,o,z = ~ ' ( R e , P r , R o , B u , Z , O ) (15)w h e r e

ho,zdN".~ = k ( l o c a l N u s s e l t n u m b e r ) .T h i s e q u a t i o n m a y b e u s e d a s a g u i d e t o d e v i s e a ne x p e r i m e n t a l p r o g r a m m e t o s t u d y t h e e f fe c ts o f C o r i -o l is a n d c e n t r i p e t a l b u o y a n c y f o rc e s .

3 . A P P A R A T U ST h e r o t a t i n g f a c i l it y u se d f o r t h i s i n v e s t i g a t i o n h a s

b e e n p r e v i o u s l y d e s c r i b e d , s e e M o r r i s a n d S a l e m i [ 4 ].N e v e r t h e l e s s , i n t h e i n t e r e s t o f c o m p l e t e n e s s a b r i e f

( 1 3 ) d e s c r i p t i o n o f th i s f a c i l i t y w i l l b e p r e s e n t e d h e r e t o g -e t h e r w i t h a d e t a i l e d d e s c r i p t i o n o f t h e n e w t e s t s e c t io nw h i c h h a s b e e n t h e s u b j e c t o f t he p r e s e n t i n v e s t i g a t io n .

T h e s c h e m a t i c s o f th e f a c i l i t y a r e s h o w n i n F i g . 3 .A h o l l o w p l e n u m c h a m b e r ( 1 ) c o n t a i n e d t h e i n s t r u -m e n t e d t e s t s e c t i o n ( 2 ) a n d t h i s w a s s u p p o r t e d o n as h a f t ( 3 ) . T h e s h a f t w a s m o u n t e d b e t w e e n b e a r i n g s( 4 ) a n d t h e a s s e m b l y w a s d r iv e n b y a D C m o t o r ( 5 )v i a a t o o t h e d b e l t d r i v e ( 6 ) . P r e s s u r i s e d , o i l - f r e e a i rw a s f e d t o t h e p l e n u m c h a m b e r v i a a r o t a t i n g s e a l( 7 ). T h i s s e a l i n c o r p o r a t e d t w o m a g n e t i c a l l y h e l d f a c es e a ls w h i c h w e r e l u b r i c a t e d w i t h a n a i r / o i l f o g m i x t u re .T h e c o o l i n g a i r e n t e r e d t h e t e s t s e c t i o n v ia t h e p l e n u ma n d v e n t e d t o a t m o s p h e r e . T o i n c r e a s e t h e r a n g e o fi n v e r s e R o s s b y n u m b e r s o v e r w h i c h t e s t s c o u l d b eu n d e r t a k e n , t h e e x i t e n d o f t h e t e s t s e c t i o n w a s f i tt e dw i t h n o z z l e s ( 8 ) o f v a r y i n g d i a m e t e r . I n t h i s w a y t h ep l e n u m c h a m b e r p r e s s u re c o u l d b e m a i n t a i n e d a b o v et h e c r i t ic a l c h o k e d f l o w c o n d i t i o n p e r m i t t i n g t h e a i rd e n s i t y i n c re a s e t o l o w e r t h e m e a n a x i a l v e l o ci t y f o r as p e c if i ed R e y n o l d s n u m b e r v a l u e .

A t w o c h a n n e l p o w e r s l i p r i n g ( 9 ) p e r m i t t e d c u r r e n t( 16 ) f r o m a t r a n s f o r m e r t o b e f e d t o a h e a t e r w r a p p e d o v e r

t h e o u t e r s u r f a c e o f t h e t e s t s e c t i o n w h i c h i s d e s c r i b e db e l o w . E l e c t ri c a l p o w e r c o n s u m e d b y t h e h e a t e r w a sm e a s u r e d u s i n g a n a m m e t e r a n d v o l t m e t e r .

T h e r m o c o u p l e s i g n a l s f r o m t h e t e s t s e c t i o n w e r e

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Heat transfer in a simulated turbine blade cooling passage 3707

1 P l e n u m c h a m b e r2 T e s t m o d u l e3 S h a f t4 P I u m m e r b l o c k

b e a r i n g s5 E l e c t r i c m o t o r6 B e l t d r i v e p u l l e ys y s t e m

7 R o t a t i n g a i r s e a l8 E x i t n o z z l e p l a te9 P o w e r s l ip r i n g1 0 M u l t i - c h a n n e l s l ip r i n g1 1 M a g n e t i c p i c k - u pt r a n s d u c e r

A i r e x i t

A i r / o i l f o g

I o J L ~C o n c r e t e b a s e

F ig . 3 . Me ch a n i ca l l a yo u t o f r o ta t i n g te s t fa c i l it y .

m < C = L ~ _ I < C = A i r i n le t

taken from the ro toI thro ugh a multi- channel slip ring(10) inv olvin g a series of shaf t moun ted silver ringsrotating in contact with stationary silver/graphitebrushes. The voltages from the thermocouples werefed to a Sc lumberg er Type S 13535D data acqu isiti onsystem and recorded on a Dell system 210 PC forsubsequent data processing. Other required data weremanua lly typed into the rec ording system as required.

The speed of the rotor was measured using astationar y magnetic encoder which detected the pass-age of iron inserts embedded into the periphery of atufn ol disc (11) wl~:ich was at tached t o the instr u-ment ation slip ring rotor shaft. The e ncoder was usedin c onjunc tion wit~L a timer co unter which gave adirect rotor speed read out.

The test section details are shown in Fig. 4. Theheated tube (1) was ',made from stainless steel and h ada bore diame ter of 15 mm, a wall thickness of 1.5 mmand a nomi nal length of 253 mm. This tube was heldbetween insulating bushes (2) and (3) to give a heatedlength of 225 mm between the faces of the su pportingbushes. The bush/tube assembly was enclosed in analum inium tube (4) fitted with external 'O' ring seals(5) at both ends.

A pair of twin start threads (see inset (6) in thefigure), ha ving the :same pitch but differing depths,was machined onto the outer surface of the test tubeto facilitate the insta ]lation of thermocouples and elec-trical heating wire. The deeper of the two grooves wasused for embedding Type K ther mocouples along the

leading and trailing edges of the tube. The depth ofthis groove was arranged so that each thermocouplesensing jun ctio n was located 0.5 mm from the innersurface of the tube. Twelve thermocouples wereattached to the leading and trailing edges as indicatedin the figure. Thermocouples were also used to mea-sure the temperature of the air upstream of the testsection entry and located in the plenum chamber.Similarly, the temperature of the air leaving the testsection was measured in the central region of bush(3). Nichrome resistance wire was used for the heatin gwire.

A cover plate (7) at exit from the test section incor-porated a conver gent nozzle which permitted controlof the air density in the test section, as mentionedearlier. Grooves machined in this cover plate per-mitted access of the heater and thermocouple cablesto their electrical circuitry. The space between theouter surface of the heated test section and the alu-minium tube was back filled with thermal insulatingfoam to minimise external heat loss.

The test assembly was mounted in the air deliveryplenum chamber (see item (1) in Fig. 3) an d bolted inposition with bolts passing through the matchingholes on the cover plate (7) and a correspon ding flangemachine d at the outer end o f the plenum. The eccen-tricity of the entry station of the test section was 150mm and this was set by means of an internal spacerbush fitted inside the plenum. Two diametricallyopposed tubes (shown as (12) in Fig. 3) permitted all

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3 70 8 W . D . M O R R I S a n d S . W . C H A N G

= = = = = = = = = = = = = = = = = = = = = = = = = = = = =

G r o o v eW i d t h - O . 6 m mD e p t h - 0 . 6 m mC u r v e d s h a p e f o rh e s t l n g c o l l

1 T e s t s e c t i o n2 I n l e t s u p p o r t b a h3 E x i t s u p p o r t b u s h4 A l u m l n l u m t u b e

G r o o v eWi d th - 1 . 0 ra mD e p th - 1 . 0 r amS q u a r e s h a p e f o rt h e r m a l c o u p l e

Fig. 4. Test section construction.

5 ' 0 ' r i n g s e a l s6 H e a t i n g w i r e7 N o z z l e p l a t e8 T h e r m a l I n s u l a t io n

h e a t e r a n d t h e r m o c o u p l e w i r e s t o b e b r o u g h t t o t h em a i n r o t o r s h a f t a n d h e n c e t o t h e r e s p e c ti v e p o w e ra n d i n s t r u m e n t a t i o n s l i p r i n g s v ia c a b l e s m o u n t e d i nk e y w a y s m a c h i n e d o n t h e m a i n s h a f t . S i l ic o n ru b b e rw a s i n j e c t e d i n t o t h e s e w i r e - c a r r y i n g p a s s a g e s t oi n h i b i t c ha f f in g m o v e m e n t a n d a i d s e a l in g .

4 . M E T H O D O F D A T A E V A L U A T I O NT e m p e r a t u r e m e a s u r e m e n t s w e r e t a k e n a l o n g t h e

l e a d i n g a n d t r a i l i n g e d ge s o f t h e t u b e v i a w a l l - e m b e d -d e d t h e r m o c o u p l e s a t t w e l v e d o w n s t r e a m a x i a ll o c a t i o n s re l a t iv e t o t h e t u b e e n t r y p l a n e a n d t h e r a d i a lp o s i t i o n o f t h e s e t h e r m o c o u p l e s w a s s p e c i fi e d . B yi n d e p e n d e n t l y v a r y i n g t h e f l o w r a t e o f c o o l a n t , t h eh e a t i n g r a t e a n d t h e r o t a t i o n a l s p e e d i t w a s p o s s i b l et o g e n e r a t e a s e r i e s o f e x p e r i m e n t a l r e s u l t s o v e r ar a n g e o f R e y n o l d s a n d R o s s b y n u m b e r s t o g e t h e r w i t ha s y s t e m a t i c s t u d y o f th e b u o y a n c y p a r a m e t e r a t e a c hR e y n o l d s / R o s s b y p a i r . U s i n g m e a s u r e m e n t s o f t h el e a d i n g a n d t r a i l i n g e d g e w a l l t e m p e r a t u r e s , t h e c o o -l i n g a ir i n l e t t e m p e r a t u r e a n d t h e e x t e m a l h e a t f l u x o nt h e o u t e r r a d i u s o f t h e t u b e , i t w a s p o s s i b l e t o g e n e r a t ea f u l l s u rf a c e m a p o f t h e h e a t t r a n s f e r r e s p o n s e i n d i -c a t e d b y t h e f u n c t i o n a l f o r m o f e q n ( 1 5) .

T h e f u l l f i el d v a r i a t i o n o f h e a t t r a n s f e r i n t h e c i r -c u m f e r e n t i a l a n d a x i a l d i r e c t i o n s w a s d e t e r m i n e du s i n g a c o m b i n a t i o n o f in s i g h t, m e a s u r e d b o u n d a r yc o n d i t i o n s a n d a n a n a l y t i c a l s o l u t io n o f t h e h e a t c o n -d u c t i o n e q u a t i o n i n t h e t u b e w a l l . I t i s a c c e p t e d t h a tt h e m e t h o d i s a p p r o x i m a t e a t t h i s s ta g e b u t , e v e n s o ,

s o m e i n t e r e s t i n g r e s u l t s h a v e e m e r g e d . T h e m e t h o da d o p t e d i s n o w d e s c r i b ed .

T h e s t e a d y s t a t e h e a t c o n d u c t i o n w i t h i n t h e w a l lm a t e r i a l i s g i v e n b y ,

0 2 T w _ --IOT~+ 0 2 T w c 3 2 T ,, ,WTw - - ( ~r + r ~ r ~ ~ + Oz- ~ - = O .(17)

D u r i n g a r o t a t i n g e x p e r i m e n t t h e r e i s a s y m m e t r yd i a m e t e r a c r o s s t h e l e a d i n g a n d t r a i l i n g e d g es d u e t ot h e s y m m e t r y o f th e s e c o n d a r y f lo w g e n e r a t e d . T h u st h e r e w i ll b e n o c o n d u c t i o n t h r o u g h t h e w a l l s a t t h el e a d i n g a n d t r a i l i n g e d g e s. T h u s t h e s o l u t i o n o f e q n( 1 7) i s s u b j e c t to t h e b o u n d a r y c o n d i t i o n

OTw0 - - 0 - = 0 a t 0 = 0 a n d a t 0 = n . (1 8)A l s o , t h e h e a t f l u x a t t h e o u t e r r a d i u s , r e , is s p e c if i e da s b e i n g u n i f o r m i n t h e c i r c u m f e r e n t i a l d i r e c ti o n b u tv a r i a b l e i n t h e a x i a l d i r e c t i o n d u e t o e x t e r n a l h e a tl o ss . T h e r e s u l t is t h a t t h i s e x t e r n a ll y i m p o s e d h e a tf l u x , q o (Z ), i s k n o w n a n d t h i s g i v e s a n e x t e r n a l s u r f a c eb o u n d a r y c o n d i t io n a s

k O T w- ~ = q o ( Z ) a t r = r o . ( 1 9 )A l s o w e k n o w t h e w a l l t e m p e r a t u r e a t t w o m e a s u r e dl o c a t i o n s c o r r e s p o n d i n g t o t h e l e a d i n g a n d t r a i l in g

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Heat transfer in a simulated turbine blade cooling passage 3709edge at a number of axial locations along the testsection. Th us

Tw (rm, O, zm) := TL and T ~ ( r ~ , n , Z m ) = T T .(20)

Assume the wall temperature distribution may beexpressed in separable variab le form as] w = F ( r ) G ( O ) L ( z ) (21)

where the three functions F ( r ) , G ( O ) an d L ( z ) are onlyfunctions of one variable.

At any axial location, Zm,a 2T

- F(r)a(o)L(z)". ( 2 2 )0g 2Suppose we seek: a solut ion to eqn (17) at a specified

z-location, L(zm) and treat the u nkn own L(zm) and itssecond derivative, L " ( z m ) , as known constants. Wemay hence re-write eqn (17), using eqns (10) and (21)as

r 2F ' ( r ) q - r F ( r ) + C r 2F ( r ) - G " ( 0 )F ( r ) G ( O )

(23)-- ___n 2

where n is some appropriate constant which satisfiesthe bounda ry conditions and

L " ( Z m )C - (24)L (gm) "Solutions exist for the two cases n 2 = 0 and +n 2 andthese result in a superimposed solution

Tw = ~ , B , Y , ( C r ) c o s n O + A l n r + E (25)n= l

where B,, A and E are c onstants and Y, is the secondtype of Bessel function.

The two measured wall temperatures permit us toevaluate one term of the Bessel function and this isthe accepted approximation in this first attempt atusing this method. T hus

T = I ~ - + D 2 r ] c o s n O + A l n r + E (26)

where Di and D2 are constants. The boundary con-ditions permit the specification of the four constantsin eqn (26) as

l [ 1 r m l - ' ( T ~ _ T T )D, =~i ~m + roSA (27)

2 _ [ 2 + r _ l - 1 r T ) (28)Dz = 2r0ZLrm r Jqo(zm)roA - k (29)

E = q o ( zm ) r O ln r m + 1kw ~( TL + TT). (30)Note that additional temperature measurements at

other angular positions would permit higher orderterms to be included in future. Here the approxi-mation made is that the two temperature measure-ments permit a reasonable approxima tion to the deter-mination of the full field temperature variation overthe inne r surface of the tube.

The circumferential heat flux variation, qt (O,Zm) ata specified zm on the inner surface of the tube is cal-culated via

q i(O , Z m ) = - k w O zw a t r = r i . (31)O rKnowing the heat flux on the inner tube surface, viaeqn (31) and the corresponding inner surface tem-perature distribution, via eqn (26), permits the cir-cumferential distr ibution of heat transfer coefficient,h(O, z ) to be calculated at each axial location wheremeasuremen ts were made using the definition

h ( O , z m ) = q iT w ( r, , 0 , Z m ) - T B Z m ) " ( 3 2 )In eqn (32) TB(Zm) s the fluid bulk tem pera ture a t

location Zm. This is determine d by an enthalpy balanceon the fluid commencing at the inlet where the entrybulk temperature is known from direct measurementand using eqn (31) to specify the integrate d heat fluxbetween axial measuring stations. Finall y a full fieldNusselt numbe r, N u ( O , z m ) , distribut ion could be gen-erated from the evaluation of the above equations ateach of the twelve axial measureme nt locations usingeqn (16).

The circumferential variations of heat transfercoefficient which resulted were finally mapped usingstandard plotting software over the entire inner sur-face of the tube.

5 . E X P E R I M E N T A L P R O G R A M M EInitially a series of tests was unde rtak en at zero

rotatio nal speed to give a comparative reference basewith which to assess the effect of rotation. Experi-ments were conducted over the Reynolds numberrange 10 000-35 000. For each Reyno lds number , fivedifferent heat flux conditions were used. The heaterpowers were actually selected to give maxim um walltemperatures of the test section of 50, 75, 100, 125and 155C, respectively. After completion of the zerorotati onal speed tests the programme was repeated atrotati onal speeds up to a maximu m of 2000 rev rain 1.

For these tests the inverse Rossby number at theentry region to the test section was controlled to beconstant at various rotational speeds by altering theflow rate. This me ant tha t the influence of Reynoldsnumb er could be examined at a fixed inverse Rossby

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3710 W.D. MORRIS and S. W. CHANGnumber. The actual range of entry plane inverseRossby numbe rs covered was 0-0.54.

Variation of the heater power permitted a sys-tematic variation of the buoyancy effect at fixedinverse Rossby number/Reynolds number combi-nations. Because the Buoyancy parameter given byeqn (12) includes the Rossby numbe r, the compon entfl( Tw-T s) will be used as a convenient charac-terisation of buoyancy at the present stage of devel-opment o f the method. Thus the difference in wall tobulk coolant temperature is used to describe buoy-ancy. The range of the fl(Tw- TB) component of theBuoyancy parameter covered was 0.0~.4.

The power dissipated by the heater is not entirelytransferred to the coolant owing to external losses andconduction through the metallic walls. The externalheat loss is dependent on the r otatio nal speed and thiswas determined by means of heat loss experimentscarried out prior to the main experimentalprogramme. For these tests the coolant passage wasfilled with insulation and measurements of the walltemperature distri butions taken over a range of heatersettings at each of the rotational speeds used for themain test programme. In the steady operating statethe heat generated by the heater was deemed to belost to the en viron ment via the difference in mean walltemperature and the ambient temperature. A plot ofthe heat loss against this temperature difference wasfound to be li near at all speeds. Consequently the localheat loss at a ny axial location during the ma in exper-iments could be assessed by assuming that the losswas directly proportional to the locally prevailingdifference in wall to ambient temperature. For rot-ating experiments where the leading and trailing edgesoperate at different temperatures the average wall tem-perature was used to estimate the heat losses. In thisway the axial variation of heat flux along the testsection was prescribed being assumed to be girthwiseuniform.

The wall temperature measuremen ts on the leadingand trailing edges of the tube and the axial heat fluxwere used to determine the circumferential variationof wall temperature at the in ner bore of the tube usingthe results of Section 4. Also the actual circumferentialheat flux transferred to the fluid at each measuringstation was determined. The circumferential variationof the local Nusselt number could then be evaluatedusing the inner bore wall temperature, the heat fluxand the fluid bulk temperature. A post processinggraph plotting routine was used to plot the Nusseltnumber contours using the calculations from eachaxial measuring position.

involved a detailed measurement of the cir-cumferential wall variation in a radially rotating tubeat numerous axial locations using the same rotatingfacility as that o f the present investigation. The dataavailable were taken with a stainless steel test sectionof 10.0 mm bore, a wall thickness of 1.5 mm an d alength of 130 mm. The eccentricity of the entry planeof the test section was 389 ram. For a range ofrotati onal speeds, heat fluxes and Reynolds numbersthe measured leading and trailing edge temperatureswere used as input data for the prediction of the cir-cumferential temperature variation using eqn (26).Figure 5 shows a selection of the comparisons betweenthe detailed measurements and the cu rrent predictionmethod. The agreement is very agreeable and gaveconfidence in the use of the method with the newexperiments presented in the p resent paper.6.2. Leading and trai l ing edge results

For all non-rot ating tests there was no systematicvariation in the measured wall temperatures on theleading and trailing edges. This was consistent with theaxisymmetric nature of the flow an d forced conve ctionheat transfer mechanism present. An examination ofthe local variation of Nusselt number demonstratedthe well known approach to a terminal or fullydeveloped Nusselt number which occurs as the flowand thermal boun dary layers develop along the tube.At axial locations of about 10 diameters downstreamof entry the Nusselt number was in good agreementwith the Dittus and Boelter [15] correlation.

When the experiments were repeated with rotationit was immediately found tha t the tempe ratures mea-sured on the leading and trailing edges were sig-nificantly different. Consistent with other studies oforthogo nal-mode rotation, see Morris and Salemi [4],the leading edge was consistently at higher tem-perature than the trailing edge at any axial location.This was due to the Coriolis driven secondary flow,described earlier, causing relatively cool core regionflow to move towa rds the tra iling edge. Figu re 6 illus-trates the typical differences in wall temperature mea-sured for a Reynolds numb er of 25 000, a r otationa lspeed of 2000 rev min-~ and a variety of heat fluxsettings.

At zero rotational speed eqn (15) must reduce tothe case of ducted flow forced convection. If Nuo,zrepresents the zero speed Nusselt numbe r at any axiallocation, then we expected that for turb ulent flow

No,z = Re S Pr33 dp(Z). (33)This means that eqn (15) will have the form

6 . R E S U L T S A N D D I S C U S S I O N6. l . Validation of the data processing me tho d

The approximate metho d for determin ing the fullfield heat transfer di stributi on described in Section 3was validated using experimental data ob tained withinthe research gr oup by A1 Merri [14]. This work

No.z _ 1 + p(Re, Pr, Ro, Bu, Z, O) (34)No,z Re-S er33 ~( Z)Based on empirical data Morris et al. [16] have

suggested that the overall effect of Reynolds nu mbe ron the second term on the right-hand side of eqn(34) vanishes or, at least, is small. This means that

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Hea t transfer in a simulated turb ine blade cooling passage 3711R e - < 3 0 . 0 0 0 . z / d , 7 . 8 a t 2 0 0 0 r . p . m .

1 6 0~ ; 1 4 0

",i"o~- o oIDID .~ , eo

4 02 0 I I I

0 1 0 0 2 0 0 3 0 0

Re o ~ .0 ,000 . z /d - 7 .8 a t 270 0 r .p .m.1 6 014.012o ] ~1 0 0 ,I8o ~6 O

4 0 -

2 O I I I0 1 0 0 2 0 0 3 0 o

C i r c u m f e r e n t i a l l o c a t i o n ( d e g r e e s )

R e - 3 0 , 0 0 0 . z / d - 1 0 .4 a t 2 0 0 0 r . p . m .1 6 01 4 0

- ~ 1 2 0iDi . 1 o oI D

~3., 8 0~ E~D~ = 6 0": ~ 4 0

2 0

R e , 3 0 , 0 0 0 . Z/d - 1 0 . 4 a t 2 7 0 0 r . p . m .

"'3( ~3c

- - } C -

- $ ( -

[ I I . ~ C L I I0 1 0 0 2 0 0 3 0 0 0 1 0 0 2 0 0 3 0 0

C i r c u m f e r e n t i a l l o c a t i o n ( d e g r e e s )Fig. 5 . Com parison o f theoretical predictions with experimental results obtaine d from various rotatio nal tests [14].

r o t a t i o n a l d a t a p l o t t e d i n t e rm s o f th e N u s s e l t n u m b e rr a t i o Nuo,z/Nuo,z w o u l d b e i n d e p e n d e n t o f R e y n o l d sn u m b e r . I n o t h e r w o r d s t h e e f fe c t o f R e y n o l d s n u m b e ri s f ul l y a c c o u n t e d f i ) r v i a t h e 0 . 8 e x p o n e n t a p p r o p r i a t et o f o r c e d c o n v e c t i o n . T h i s h y p o t h e s i s w a s e x p l o r e d i nt h e p r e s e n t s t u d y b y c o n d u c t i n g e x p e r i m e n t s w i t hf i x ed v a lu e s o f t h e i n v e rs e R o s s b y n u m b e r b u t m a d eu p w i t h d i f fe r e n t c o m b i n a t i o n s o f f l o w a n d r o t a t i o n a ls p e e d . T h e r e s u l t s f o r t h e l e a d i n g a n d t r a i l i n g e d g e sw e r e u s e d f o r t h e a s s e s s m e n t o f t h i s h y p o t h e s i s . As a m p l e o f t h e t y p i c a l r e s u lt s o b t a i n e d a r e g i v e n i n F i g .7 . H e r e t h e r a t i o o f t h e l o c a l N u s s e l t n u m b e r o n t h el e a d i n g a n d t r a i l i n g ed g e s s c a l e d w i t h t h e 0 .8 p o w e ro f t h e a c t u a l R e y n o l d s n u m b e r t e s t e d i s p l o t t e da g a i n s t a x i a l l o c a t i o n . A x i a l l o c a t i o n i s e x p r e s s e d i nt e r m s o f t h e e f fe c ti v e d i a m e t e r s d o w n s t r e a m o f e n t r y .I t s h o u l d b e n o t e d t h a t t h e e x p e r i m e n t s w h i c h p r o -d u c e d t h e d a t a p o i n t s i n F i g . 7 w e r e c o n t r o l l e d t oa p p r o x i m a t e l y c r e a t e t h e s a m e a v e r a g e v a l u e o ff l ( T w - T B ) s o t h a t t h e e f f e c t o f c e n t r i p e t a l b u o y a n c yw a s b e i n g m a i n t a i n e d c o n s t a n t d u r i n g t h e s e r i e s o ft e s t s . T h e r e i s a s t r o n g t e n d e n c y f o r d a t a a t f i x e di n v e r s e R o s s b y n u m b e r a n d v a r i a b l e R e y n o l d s n u m -b e r t o c o l l a p s e o n t o a s i n g l e l i n e . T h i s s e e m s t o s u b -

s t a n t i a t e t h e t h e o r y t h a t R e y n o l d s n u m b e r e f fe c ts m a yb e t a k e n i n t o a c c o u n t t h r o u g h a n o r m a l s t a t i o n a r yt u b e f o r c e d c o n v e c t i o n m e c h a n i s m .

T h e i n f l u e n c e o f c e n t r i p e t a l b u o y a n c y w a s e x a m -i n e d b y c o n d u c t i n g t e s t s a t f i x e d v a l u e s o f t h e R e y -n o l d s n u m b e r a n d i n v e r s e R o s s b y n u m b e r a n d v a r y i n gt h e h e a t f l u x t o g i v e a v a r i e t y o f w a l l t o f l u i d t e m -p e r a t u r e d i f f e r e n c e s . T h u s f o r c e d c o n v e c t i o n a n dC o r i o l i s s e c o n d a r y f l o w e f fe c ts w e r e m a i n t a i n e d c o n -s t a n t w h i l s t t h e B u o y a n c y p a r a m e t e r w a s v a r i e d . F i g -u r e 8 i l l u s t r a t e s t h e c e n t r i p e t a l e f f e c t s w h i c h w e r eo b s e r v e d . A s t h e h e a t f l u x i s i n c r e a s e d t h e l o c a l N u s -s e lt n u m b e r o n t h e l e a d i n g a n d t r a i l in g e d g e s a ls oi n c r e a s e d s i g n i f i c a n t l y . I n a l l c a s e s t h e t r a i l i n g e d g e i sb e t t e r c o o l e d t h a n t h e l e a d i n g e d g e. A d d i t i o n a l l y i t i sn o t e d t h a t h e a t t r a n s f e r o n t h e l e a d i n g e d g e c a n f a l lb e l o w t h e s t a t i o n a r y t u b e l e ve l u n d e r c e r t a i n o p e r a t i n gc o n d i t i o n s .6.3. Full surface heat transfer resultsF i n a l l y , t h e r e v e r s e e n g i n e e r i n g d a t a p r o c e s s i n gm e t h o d w a s u s e d t o s t u d y t h e i n d i v i d u a l ef f ec t s o fC o r i o l i s s e c o n d a r y f lo w a n d c e n t r i p e t a l b u o y a n c y . As e t o f t es t s w a s u n d e r t a k e n w i t h t h e a v e r a g e v a l u e o f

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3712 W.D. MORRIS and S. W. CHANG

L L e a d l n g e d g e2 0 0 T r a il in g e d g eE

, v0o . . 1 0 0Q~ a o ~ i

6 0 q

m

I A I 3 " 4o , 2 o l - - B I s . 4J C J T . gI D I l l . a

0 2 4 6 8 1 0 1 2 1 4A x i a l l o c a t i o n ( z / d )

F i g . 6 . T y p i c a l a x i a l w a l l te m p e r a t u r e d i s t r i b u t i o n w i t h s y s -t e m r o t a t i o n : r o t a t i o n a l s p e e d = 2 0 0 0 r . p . m . ; R e y n o l d s

n u m b e r = 2 5 0 0 0 ; I n v e r s e R o s s b y n u m b e r = 0 . 1 6 6 .

the term fl (T w- TB) mainta ined at approxima tely 0.3in order to standardise the buoy ancy effect. Wi th thisseries of tests experiments were undertaken withinverse Rossby n umb er values of 0.08, 0.15, 0.20, 0.40and 0.47. The data process model was used to evaluatethe circumferential variation in Nusselt number ateach of the twelve axial measuring stations and thepost processing interpolation software used to gen-erate the full field variation. Figure 9 shows thederived Nusselt numbe r variations which resulted.

The development of the three dimensional flow isclearly complex but certain features are worthy ofnote. As the inverse Rossby number increases thecommencement of the circumferential variation ofNusselt numbe r seems to occur near er to the plane ofentry. This indicates that the Coriolis induced sec-ondary flow is makin g its presence felt almost immedi-ately at the higher rotati onal speeds. For inverseRossby numbers up to 0.4 there was some evidencethat a fully developed Nusselt number distributionwas forming at axial locations about 7-9 diametersdownstrea m of entry. This is indicated by the tendencyfor the Nusselt number bands to be parallel in thecircumferential direction. At higher values of inverseRossby number the Nusselt number patterns are morecomplicated with a possibly stronger interaction withthe exit region may be occurring.Experiments were also conducted at a fixed valueof the inverse Rossby n umber, namely 0.52, with fivedifferent external hea t fluxes. I n this way the effect ofcentripetal buoyanc y was the subject of a systematic

investig ation. The heat flux levels gave average valuesof the fl(Tw-TB) term of 0.07, 0.12, 0.17, 0.22 and0.27. It was no t possible to co ntrol the detailed vari-ation of the wall to fluid temperature differences alongthe tube for all inverse Rossby numb er values testedand this is why an average value was used as a buoy-ancy variable.Figure l0 shows the effect of buoyancy whichresulted from the tests. The Nusselt number at allcircumferential locations tended to increase withincreases in the strength of b uoyan cy as already dem-onstrated above for the leading and tra iling edges. Atthe relatively high value of inverse Rossby num ber inFig. l0 there was no clear evidence of the format ionof a traditional developed flow region. What is veryevident from these systematic plots is the fact thatbuoyancy was a strong effect even with high levels ofCoriolis cross flow. This is particular ly import ant forthe dev elopment of prediction methods for this effectmust be includ ed in the analysis. Indeed the Coriolisand buoyant forces are also likely to influence theturbulent flow structure itself.

7 . C O N C L U S I O N SThe purpose of this paper has been to illustrate the

use of a reverse engineered method for determiningthe full field heat transfer distribution in a radiallyrotating circular-sectioned tube, with application tothe design of cooled gas turbine rotor blades. Themethod involves a solution of the heat conductionequati on in the wall of the test section using measuredtemperatures and an externally prescribed heat fluxas required boundary conditions. The salient pointswhich have been demonstr ated are as follows.

(1) The method has been validated against a set ofexperimental data which indepe ndently measured thecircumferential variati on in wall temperature in a radi-ally rotating tube. The prediction of circumferentialwall temperatures using the measured temperatureson the leading and trailing edges gave very good agree-ment with the subsequent girthwise predictions andthe actual independ ently measured values.

(2) Experiments designed to test the validity of anempirical suggestion that the use of a forced con-vection Reynolds num ber effect, in the for m of a 0.8exponent of Reynolds number, has demonstrated thatthis is a strongly valid assumption. This will be auseful result for future attempts to derive empiricalcorrelations for design purposes.

(3) The method has bee n shown to be able to dis-cern systematic changes in the strength o f the Coriolisdriven secondary flow and also the centripetal buoy-ancy. The full interpret ation of these interactive effectsstill requires additional experimental work to beundertaken, notably at near engine conditions. Themethod gives an ad ditional useful tool for deter miningfull field data from limited thermocouple spotmeasurements. The method may be refined if a hum -

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F i g . 8 . N o r m a l i s e d a x i a l h e a t t r a n s f e r d i s t r i b u t i o n w i t h v a r i o u s c e n t r i p e t a l b u o y a n c y l e v e ls a t f i x e d i n v e r s eR o s s b y n u m b e r s .

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