turbulence and inertia effects in bearings

8/12/2019 Turbulence and Inertia Effects in Bearings

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u r b u l e n c e a n d i n e r t i a e f f e c t si n b e a r i n g sN. A. Macken and E. A. Saibel t

T h is a r t i c le s u rv e y s t u r b u le n c e a n d in e r t i a e f f e c t s in b e ar in g s . T h e e f f e c t s o c c u r f o r t w o m a inr e as o n s h igh s p e ed o p e r a t io n a n d th e u se o f u n c o n v e n t io n a l lu b r ic a n t s . E x p e r im e n t a l a n dt h e o r e t ic a l s t u d ie s a r e c o n s id e r e d a n d r e c o m m e n d a t io n s a r e g iv en f o r f u t u r e in v e s t ig a t io n .

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Clea ranceF r i c ti o n f a c t o r = 2 r w / p U 2Mixing l eng th cons tan tMix ing l eng thBear ing l eng thAngula r ve loc i tyPressureMean radiusR e y n o l d s n u m b e r , UC/vR e d u c e d R e y n o l d s n u m b e r , UC/v) C/L)T o r q u eT a n g e n t i al v e l o c i t y o f b e a r i n gC o r e r e f e r e n c e v e l o c i t y u s e d b y R e y n o l d s 2 sMean ve loc i ty in t angen t ia l d i rec t io nM a i n s tr e a m v e l o c i t y ( f r o m b o u n d a r y l a y e rt h e o r y )F l u c t u a t i n g c o m p o n e n t o f v e l o c i t y i nt a n g e n t i a l d i r e c t i o nA v e r a ge f l u c t u a t i n g c o m p o n e n t sAxia l ve loc i tyM e a n v e l o c i t y n o r m a l t o t a n g e n ti a l v e l o c i t yF l u c t u a t i n g c o m p o n e n t o f v e l o c it y i nn o r m a l d i r e c t i o nTangent ia l . d i rec t ionD i r e c t i o n n o r m a l t o t a n g e n ti a lDim ens ion les s d i s t ance y/OV~w/PC o n s t a n t i n R e i c h a r d t s l a w o f t h e w a llE c c e n t r i c i t y r a t i o = e c c e n t r i c i t y / c le a r a n c eCore eddy v i s cos i tyEddy v i s cos i tyAngleAbso lu te v i s cos i tyD e n s i t yBearing shear s t ressS ca le shea r s t re ss u sed b y Rey nolds 2sKinem at ic v i s cos i ty

* A s s i s t a n t P r o f e s s o r o f M e c h a n i c a l E n g i n e e r i n g , ~ P r o f e s s o r o fA p p l i e d M e c h a n i c s , C a r n e g i e - M e l l o n U n i v e r s i t y , P i tt s b u r g h ,P e n n s y l v a n ia 1 5 2 1 3 , U S A

In rec en t yea rs the re has bee n a g rea t dea l o f in te re s t intu rbu len ce in bea r ings . P rev ious to th i s , f lows tha t o ccur r edin lubr ican t f i lm s were lam ina r in na tu re . Th i s was borne ou tb y e x p e r i m e n t a n d a l s o b y t h e a g r e e m e n t b e t w e e n l a m i n a rt h e o r y a n d e x p e r i m e n t .Non - lam ina r f low occurs in bea r ings fo r two m a in reasons .The f i r s t i s h igh-speed ope ra t io n . The s eco nd i s the use o funco nven t iona l lubr ican t s such a s wa te r o r l iqu id m e ta l s .These m a te r ia l s a re f requ en t ly used a s p roces s f lu ids . Then e e d t o s i m p l if y e q u i p m e n t d e s i g n o r t h e d i f f i c u l t y o f o v e r -com in g sha f t s ea ling p rob lem s lead to the i r use as lubr ican ta s w el l. A h i g h v e l o c i t y c o m b i n e d w i t h a l o w k i n e m a t i cv i s co s i ty l ea d s t o h i g h R e y n o l d s n u m b e r s a n d d e p a r t u r ef r o m l a m i n a r f l o w .F o r t h e c a s e o f jo u r n a l t y p e b e a r i n g s, w h i c h a r e t h e o n e sm o s t a t t e n t i o n h a s b e e n g i v e n t o , a b o v e t h e l a m i n a r r a n g ethe re i s a lm os t a lways a f low reg im e in which a vo r texs t ruc ture ex i s t s . As the R eyno lds nu m b er inc reases , de -p a r t u r e f r o m l a m i n a r f l o w i s f i r st r e c o g n i z e d b y t h e f o r m a -t i o n o f T a y l o r v o r t i ce s , n a m e d a f t e r G . I . T a y l o r w h o f i rs to b s e r v e d t h is p h e n o m e n o n . W i t h a f u r t h e r i n c r e a s e ,t u r b u l e n c e a p p e a r s t o o c c u r b u t s o m e v o r t e x s t r u c t u r ea lm os t a lways ex i s t s, even wi th h igh e r Reyn olds num bers .T h u s t h e f l o w i s i n h e r e n t l y d i f f e r e n t f r o m t h e t u r b u l e n tf low in p ipes , bou nda ry l aye rs , o r j e t s .U n f o r t u n a t e l y , t h e e x p e r i m e n t a l w o r k t h a t h a s t a k e np l ac e h a s m o s t l y c o n c e r n e d i t s e l f w i t h g r o s s m e a s u r e m e n t ss u c h as t o r q u e a n d l o a d c a p a c i t y . N o t m u c h h a s b e e n d o n et o m e a s u r e q u a n t i t i e s s u c h a s f l u c t u a ti n g f l o w c o m p o n e n t s .T h e o r e t i c a l d e s c r ip t i o n s o f t u r b u l e n t f l o w i n b e a ri n g sh a v e d ra w n u p o n m i x i n g l e n g t h t h e o r y a n d l aw s d e v e l o p e dfor fu l ly deve lope d f low in p ipes . I t i s obv ious f rom th i s

br ie f d i scus s ion tha t such m od e ls canno t adequ a te ly des-c r ibe the f low f i e ld . Also , the co nvec t ive ine r t i a t e rm s havea l m o s t a l w a y s b e e n n e g l e c t e d i n t h e e q u a t i o n s o f m o t i o n .T h e s e t e r m s a r e i m p o r t a n t i n h i g h s p e e d f l o w a n d i t ist h o u g h t t h a t s o m e e f f e c t s a t t r i b u t e d t o t u r b u l e n c e m a y b ecaused by ine r t i a . S upr i s ing ly enou gh , these theore t i ca lana lyses have been reasonab ly succes s fu l in p red ic t ingcha rac te r i s t i c s o f bea r ings ope ra t ing in th i s reg im e . P e rhapsth i s i s because th ey con ta in ad jus tab le cons tan t s tha t canbe m ade to f i t ex i s t ing expe r im enta l da ta .

154 TRIBOLOGY August 1972


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E x p e r i m e n t a l i n v e s t i g a t i o n sT h e e a r l ie s t w o r k c o n n e c t e d w i t h b e a ri n g s o u t s i d e t h elam ina r reg im e was expe r im enta l in na tu re . Many inves t i -ga t ions pa r t i cu la r ly the ea r ly ones have been v isual inn a t u r e . G r o s s m e a s u r e m e n t s o f s u c h q u a n t i t ie s a s t o r q u eh a v e b e e n u s e d t o i n d i c a t e n o n - l a m i n a r b e h a v i o r . A f e ws t u d ie s h a v e u s e d a n e m o m e t r y . L i m i t e d i n - d e p t h s t u d i e sh a v e b e e n m a d e o f t u r b u l e n t q u a n t i t i e s.T h e t y p e s o f b e a ri n g s s t u d ie d h a v e m a i n l y b e e n p l a i njourn a l bea r ings . Th i s type o f f i lm is m o s t l ike ly to gob e y o n d t h e l a m i n a r r e g im e . H o w e v e r i n t e r e s ti n g v a r ia t i o n sof the p la in journ a l such a s s t ep journ a l he l i ca l -g roovedjourn a l bea r ings o r v i scosea ls have rece ived som e a t t en t ion .The ea r l i e s t exp e r im ents invo lved long conc en t r i c cy l inde rs .L a t e r r e f i n e m e n t s i n c l u d e d e n d e f f e c t s a n d e c c e n t r i c i t y .S o m e r e c e n t w o r k h a s a l s o i n c lu d e d b l o c k a g e a n d t i l te d - p a de f fec t s .Non- lam ina r f low in bea r ings has s eve ra l cha rac te r i s t i cf e a t u r e s. S o m e o f t h e s e a r e u n i q u e t o b e a r i n g s a n d s o m e a r ef o u n d i n o t h e r a r e a s o f f l u i d m e c h a n ic s . T h e s e f e a t u r e si n c l u d e :1 A d e v e l o p m e n t o f s e c o n d a ry v o r t e x f l o w w h e n t h e

f l o w b e c o m e s n o n - l a m i n a r . P o ss i bl e e x c e p t i o n s t o t h isa re f lows whe re ~C/R < 1 / 2 5 0 0

Gen era l ly fo r bea r ings th i s va lue i s a rou nd 0 . 001 .A n o t h e r e x c e p t i o n i s p o s si b l y i n t h e l a rg e s t c le a r a n c eof an eccen t r i c bea r ing 2 .

2 S e c o n d a r y f l o w s w h i c h a r e o f t e n l a m i n a r f o r a l ar ger a n g e o f p a r a m e t e r s d e s p i te t h e i r c o m p l e x n a t u r e .S ince they a l so a re o r ig ina l ly s t ab le the re i s no reasont h a t t h e y w o u l d d i s a p p e a r f r o m t h e f l o w f i e ldc o m p l e t e l y b y t h e p r e s e n c e o f t u r b u l e n c e . S o m e m o d i -f i c a t io n h o w e v e r i s a p p a r e n t l y e v i d e n t 3 .3 A th ree -d im en s iona l f low f i e ld which due to end l eak-a g e i s c o m p o s e d o f a n a xi a l f l o w in a d d i t i o n t o t h et a n g e n t ia l f lo w . A s p o i n t e d o u t b y B u r t o n 4 t h e c i r-c u m f e r e n t i a l t y p e f l o w i s a C o u e t t e t y p e f l o w w h e r e a sthe long i tud ina l f low is a p res sure type f low . The twotypes can not be s epa ra ted s ince the f low f i e ld i s rea l lya c o m b i n a t i o n o f b o t h .4 G e o m e t r i c f a c t o r s s u c h a s e x i s t e n c e o f s t e ps l e n g t h o fb e a r in g s a n d t h e w a y t h e l u b r i c a n t e n t e r s t h e b e a r i n gwhich a l l have s t rong in f luence .

T h e t e r m t u r b u l e n t f l o w is f r e q u e n t l y a p p li e d i n b e a r in gl i t e ra t u r e t o a n y f l o w w h i c h i s n o n - l a m i n a r. F r o m t h e a b o v es ta tem en ts th i s i s pe rhaps a m isn om e r .The fo l lowing pa ragraphs wi l l d is cus s s ign i f i can t expe r -im enta l dev e lopm ents in m ore de ta i l . Tab le 1 a lso sum m ar izesw o r k t h a t h a s b e e n d o n e .oncentric cyl indersG . I . T a y l o rs was the f i r s t to d i s cove r the non- lam ina r na tu reo f f l o w b e t w e e n c o n c e n t r i c c y l in d e r s . L o n g c o n c e n t r i cc y l i n d e rs w i t h t h e i n n e r c y l i n d e r r o t a t i n g a n d t h e o u t e rc y l i n d e r s t a t io n a r y m o d e l f u l l j o u r n a l b e a r in g s w i t h t w o i m -p o r t a n t e x c e p t i o n s . T h e y d o n o t s h o w e n d e f f e c t s a n d t h e

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T a y l o rs C o n c e n t r i ccy l inde rsT a y l o r6 C o n c e n t r i ccy l inde rsS c h u l t z - G r u n o w C o n c e n t r i cand He in 7 cy l inde rsWi lcock 9 J o u r n a l b e a r i n g

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Tor que aga ins t speed c r i t i ca l speede f r i c t ion fac to r agains t Re visualo b s e r v a t i o nVisua l obse rva t ion IT o r q u e a g a i n st s p e e dVisua l obse rva t ionP ow er los s aga ins t speed bea r ingtem p era tu re aga ins t speed o i l f lowaga ins t speed ecce n t r i c i ty ra t ioaga inst speed f r i c t ion coe f f i c ien tagains t speedF r ic t ion c oe f f i c ien t aga ins t speed

F r i c t i o n c o e f f i c i e n t a g a i n st Repres sure aga ins t 0 ecce n t r i c i tyaga ins t loadF l o w f l u c t u a t i o n s b y h o t w i r eT o r q u e a n d p r e s s u r eT o r q u e a n d p r e ss u r e

V e l o c i t y p r o f i l e s t u r b u l e n tin tens i t i e s p res sure d i s t r ibu t ion

D e t e r m i n a t i o n o f c r it i c al s p e e d b y t h em e a s u r e m e n t s m a d eM a t h e m a t i c a l d e s c r i p t i o n o f s t a b i li t y a n dconf i rm a t ion by v i sua l obse rva t ionsF u r t h e r c o n f i r m a t i o n o f t h e s t a b il i tyc r i t e r i o nE x a m i n a t i o n o f t h e f l o w s t a b il i t y b e f o r eand a f t e r the c r i t i ca l speedC o n f i r m a t i o n o f t h e s t a b i l i ty c r i t e r io n b ythe d i s t ingu i shed happenings a t c r i t ica lspeed

F r i c t i o n t o r q u e i n c r e a s e d a b n o r m a l l y w h e ns p e e d e x c e e d e d w h a t a p p e a r e d t o b e acri t ical valueC o m p a r i s o n s b e t w e e n l a m i n a r f l o w a n d t h ef l o w w i t h a s p e e d o f f iv e t im e s t h a t o fcri t ical speed.D i v is i on o f f l o w p a t t e r n s w i t h a x i al f l o wS h o w e d e f f e c t o f ax i a l f l o w a g r e e m e n tw i t h R e f 1 2Ex tend re su l t s in Refs 13 14 to g rooveg e o m e t r yD i s cu s s ed i n t e n s i t y re s u lt s c o m p u t a t i o nof wa l l shea r p res sure jum ps a t d is -c o n t i n u i t i e s n o t e d

T RI B O L O G Y Au g u s t 9 7 2 5 5


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e f f e c t s o f e c c e n t r ic i t y . T a y l o r a n a l y s ed m a t h e m a t i c a l l y t h es t a b i l i t y o f i nc ompre ss ib l e v i sc ous f l ow in a na r row a nnu lusbe tw e e n ro t a t i ng c on c e n t r i c c y l inde r s o f i n f in it e l e ng th . H i sa na ly s is p re d ic t e d the i n i t i a l i n s t a b i l it y fo r t he c a se o fro t a t i ng inne r c y l inde r a nd f ixe d ou te r c y l inde r t o be i n t hef o r m o f u n i f o r m s y m m e t r i c a l v o r t ic e s o r e d d ie s . H e a l s oc on f i rm e d bo th the va lue o f t he c r i t ic a l spe e d a nd thee x i s t e nc e o f t he pa i r s o f vo t i c e s by me a ns o f f l ow v i sua l iz a -t i on , u s ing the f l ow o f w a te r be tw e e n ve r t i c a l c y l inde r s w i thz e ro a x i a l ve loc i ty . La t e r on , Ta y lo r6 w a s a b l e t o ve r i fy h i ss t a b i li t y c ri t e ri o n b y a n u m b e r o f t o r q u e m e a s u r e m e n te xpe r im e n t s . F ig 1 is one o f h i s r e su l ts show ing the r e l a t i on -sh ip be tw e e n log T/pN 2 a nd log Np/ . L a m i n a r f l o w w o u l dbe ind ic a t e d f rom the so l id l ine . D e v ia t ions f rom th i s ind i -c a t e t he onse t o f vo r t e x f low .

S c h u l t z - G r u n o w a n d H e i n 7 a m o n g o t h e r s h a v e a l s o m a d ev i su a l o b s e r v a ti o n s o f t h e f o r m a t i o n o f T a y l o r v o r t ic e s inc onc e n t r i c c y l inde r s . The vo r t e x na tu re o f t he f l ow i s s t il le v ide n t , e ve n a t a R e y no lds num be r o f 2 .13 X 10 4 . Pa i sa l so s tud ie d the vo r t i c e s be tw e e n c o nc e n t r i c c y l inde r s u s ingh o t w i r e a n e m o m e t r y a n d f o u n d t h a t t h e v o r ti c e s e x is t e d atse ve ra l hund re d t ime s the c r i t ic a l spe e d .Eccentr ic bear ingsW ilc oc k 9 a nd A b ra m ov i t z t w e re a mo ng the f i r s t t o 9bse rve .non - l a m ina r c ha ra c t e r i s t i c s i n r e a l j ou rn a l be a t ings . W i l c oc kp e r f o r m e d s o m e e x p e r i m e n t s w i t h n o n - c o n c e n t r i c j o u r n a lb e a r in g s u n d e r l o a d s o p e r a t in g a b o v e a n d b e l o w t h eTa y lo r t r a ns i t i on . H e a lso c he c ke d the c r i t i c al ,~pe ed by thet o r q u e m e a s u r e m e n t t e c h n i q u e w h i c h T a y l o r6 e m p l o y e dear l ie r .

Sm i th a nd Fu l l e r l~ i nve s t iga t e d a n un loa de d fu l l j ou rna lbe a r ing a nd a ga in ve r i f i e d tha t Ta y lo r s c r i t ic a l R e yn o ldsn u m b e r w a s a c t u a l l y v e r y cl o se t o t h e p o i n t o f d e p a r t u r ef r o m l a m i n a r b e h a v i o r (s e e F i g 2 ) . T h e y w e r e o p e r a t i n g i nt h e v o r t e x r e g io n w i t h p r o b a b l y l i t tl e s u p e r i m p o s e dtu rbu le nc e .T h e e f f e c t o f e c c e n t r i c i t y w a s d is c u s se d m o s t c o m p l e t e l yb y V o h r 2 . H e d e m o n s t r a t e d t h a t t h e t r a n s i t i o n s p e e d f o ronse t o f vo r t i c e s i nc re ase s s ign i f i c a n t ly w i th e c c e n t r i c i t yr a t io a n d t h a t e c c e n t r i c i t y h a s m a r k e d e f f e c t o n f r i c t i o nfa c to r . V i sua l r e su l t s a l so show e d a b re a kdow n in to l a rge ,a ppa re n t ly i r r e gu la r e dd ie s in t he w ide c l e a ra nc e r e g ion .The vo r t e x s t ruc tu re i s i n t a c t i n t he na r row re g ion . Thee d d ie s e v e n t u a l l y b e c o m e f i n e r a n d a v o r t e x s t r u c t u r ea ppe a re d a ga in a nd pe r s i s t e d up to t he h ighe s t R e yno ldsn u m b e r s t u d i e d .

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3 .5 ~ d o -a --- 1000, an ex t rem ely h igh leakage ra te fo r bear ings .For no leakage no tu rbu len ce was observed.Yamada13,14, is d id considerable w or k in th is area. Het o o k t o r q u e m e a su re m e n ts o f c o n c e n t r ic c y l i n d e r s w i t hax ia l f l ow . The e f fec t o f rec tangu lar g rooves was a lsostudied Is ,S inc e mo s t j ou rna l be a r ings ha ve some s ide l e a ka ge , ac o m b i n a t i o n o f t u r b u l e n t a n d v o r t e x f l o w i s i n d i c at e dfo r h igh ro t a t i ona l spe e ds .

Measurement o f t u rb u len t i n tens i t iesQu ant i ta t i ve measurements o f tu rbu len ce have beenmeasured b y Bu r ton and Carper 16 . Exper im ents werec a r r i e d ou t u s ing a ho t w i re a ne m om e te r on a l a rge sc a lemod e l o f a j ou rna l b e a r ing . V e loc i ty p ro f i l e s show e d the f l owto be tu rbu le n t i n na tu re . Me a su re me n t o f t he f l uc tua t ingc o m p o n e n t u i n d i c at e d a g r e e m e n t w i t h p ip e f l o w n e a r t h ew a l l . A w a y f rom the w a l l t he re w a s d i sa g re e me n t , poss ib lydue to t he p re se nc e o f mids t r e a m she a r s tr e ss , w h ic h i sa bse n t i n p ipe f l ow .Tra u go t t ~7 ha s pe r fo rm e d e xp e r ime n t s i n i nduc e d sw i r li ng

1 5 6 T R I B O L O G Y A u g u s t 1 9 72


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f lows wi th b o th t angen t ia l and ax ia l f low f i elds . Measurem entso f f l u c t u a t i n g v e l o c i t y c o m p o n e n t s a n d t u r b u l e n t s t r es s eswere t aken .h e o r e t i c a l i n v e s t i g a t i o n s

The ore t i ca l ly speak ing , the re a re s eve ral p rob lem s as soc ia tedwi th n on- lam ina r f low in bea r ings .F i r s t t h e r e is t h e p r o b l e m o f m a t h e m a t i c a l l y d e t e rm i n i n gthe v i s cous t e rm s th a t ap pea r in the Navie r -S tokes equa t ions .T h e s e m a t h e m a t i c a l d e s c r i p t io n s a r e b a s i ca l ly o f t w o t y p e sa s d e s c r ib e d b e l o w . B o t h t y p e s h a v e t h e o r e t i c a l b a s e s i nf u l l y d e v e l o p e d t u r b u l e n t f l o w . O n e m e t h o d i s d er i v e d f r o mm i x i n g l e n g t h t h e o r i e s , a n d t h e o t h e r i s d e r iv e d f r o ml o g a r it h m i c p r o f i l e s o f f u l l y d e v e l o p e d t u r b u l e n t p i p e f l o w .

A s p o i n t e d o u t e a r li e r , t h e f l o w i n b e a r i n g s is m u c h d i f f e r e n tt h a n i n p i p e s o r b o u n d a r y l a y e r s , w h e r e t h e s e t h e o r i e s m a ybe val id.I n c o n v e n t i o n a l l u b r ic a t i o n , t h e i n e r t i a t e r m s i n t h eequa t ions a re neg l ig ible to the v i s cous t e rm s . A s d i s cus sedb e l o w , a t h i g h s p e e d s , b o t h i n e r t ia a n d t u r b u l e n c e e f f e c t sp lay a ro le and i t is d i f f i cu l t to d i s tingu i sh be twe en the two .A t h i r d d i f f i c u l t y w i t h t h e t h e o r i e s i s t h a t m a n y o f t h e ma r e t w o d i m e n s i o n a l. F r o m t h e e x p e r i m e n t a l d i s c us s io n , i t

shou ld be c lea r tha t a th ree -d im ens ion a l m o de l i s ca l l ed fo r .Turbu lence m ode l s and ine r t i a a re d i s cus sed in m orede ta i l in the fo l lowing pa ragraphs . Tab le 2 sum m ar izes im -p o r t a n t d e v e l o p m e n t s .Table 2 Theo ret ical s tudies of turbu lence and iner t ia ef fects in bearingsA u t h o r T u r b u l e n c e m o d e l T y p e o f B e a r in g R e s u l tsTa o ~s

Tao 9

Chow and S a ibe l 2Cons tan t inescu 2 t

Con s tan t inescu 22Arwas , S te rn l i ch ta n d W e r n i ck 2aRey nolds 2s

N g~s

Ng and pan27

El rod and Ng 2s

C h o w a n d V o h r 29V o h r a n d C h o w a K e t t l e b o r o u g ha9

Con s tan t inescu a s

1 /7 ve loc i ty power l aw. B lasius l aw off r i c t ionBlas ius l aw of f r i c t io nMixing l eng th :~2 = K 2 y C _ y)Mixing l eng th : = Ky O


5/7

T u r b u l e n c e m o d e l s b a s e d o n m i x i n g l e n g t hP e r h a p s th e s i m p l e s t w a y t o d e a l w i t h t h e t u r b u l e n t m o d e li s to m a k e u s e o f m i x i n g le n g t h t h e o r y . A s s u m i n g n o s i d el e a ka ge a nd d ro pp ing ine r t i a t e rms , t he e qua t ion o f in t e re s tr e duc e s t o :OZu ~dig = la + (-- p u v )

The tu rbu le n t s t r e ss t e rm c a n be w r i t t e n a s :- p u v p ~ a u g uay ay

w he re ~ is t he m ix ing l e ng th P ra nd t l u se d fo r p ipe f l ow , i . e.= K y

w i t h K = 0 . 4 . A n a l t e rn a t e a p p r o a c h , p r e s e n t e d b y T a o i s a 9de a l s w i th t u rbu le nc e in a n a pp rox ima te f a sh ion fo r t he c a seo f sho r t j ou rna l be a r ings a nd s l i der be a r ings . A sh o r t j ou rna lbe a r ing is one in w h ic h the a x i a l le ng th i s sma l l , a nd thep re ssu re t e rm in t he x d i r e c t ion c a n be ne g le c t e d . T a oc om pu t e d the a ve ra ge f low s by u s ing the B la sius l a w o ff r i c t i on a nd the 1 ]7 ve loc i ty p ro f i l e .C how a nd Sa ibe l 2e so lve d e qua t ions fo r a s l i de r be a r ing .The y ne g le c t e d the l a mina r f r i c t i on t e rm a nd a l somo d i f i e d the m ix ing l e ng th mode l . C ons t a n t ine sc u 21 in a ne a r ly w o r k e m p l o y e d a s i m i l a r m i x i n g l en g t h m o d e l w h e r e :

= K y O < y < C / 2= K ( C - y ) C /2 ~ y


6/7

C l a u se r . T h e i r c o m p u t a t i o n s a l so d i d n o t u s e t h e l i n e a r i ze de q u a t i o n s . R e s u l t s a re c o m p a r e d t o t h e l i n e a r i z ed c a s e .

C h o w a n d V o h r 2 9 3 h a v e a p p l i ed t h e l i n e a r i z e d a n d n o n -l i n e a ri z e d a n a l y s e s d e s c r i b e d a b o v e t o h e l i c a l - g ro o v e d a n ds p i r a l- g r o o v e d b e ar i n g s. R e a s o n a b l e a g r e e m e r it w a s s h o w n w i t hr e s u l ts o f P a p e a n d V r a k k i n g 31, a s w e l l a s K e t o l a a n dM c G r e w 32 . T h e b e a r i n g s s t u d i e d h a v e s t e p c o n f i g u r a t i o n s ,a n d t h e r e s u lt s a r e s u rp r i s in g i n t e r m s o f w h a t i s m e n t i o n e db e l o w a b o u t i n e r t i a e f f e c t s .nertia effects

I n t h e e a rl y e x p e r i m e n t a l w o r k w i t h n o n - l a m i n a r f l o w inb e a r i n g s t h e r e a p p e a r e d i n t h e l i t e r a t u r e m u c h d i s c u ss i o n a st o w h y a d e p a r t u r e f r o m l a m i n a r f l o w o c c u r e d 9 . I t w a ss u g g e s te d t h a t t h e p r e s e n c e o f t h e T a y l o r v o r t i c e s i n d i c a t e da n in c re a s e in in e r t i a fo rc e s in th e f lu id s . A s in d ic a t e d , t h ea n a ly t i c a l i n v e s t ig a t io n s h a v e b y a n d l a rg e ig n o re d th i s f a c ta n d h a v e c o n s i d e r e d o n l y t u r b u l e n c e .

A r g u m e n t s h a v e b e e n m a d e t h a t t h e i n e r t ia f o r c e s a r e s ti lli n s i g n i fi c a n t c o m p a r e d t o t h e v i s co u s f o r c e s i s . K u l i n s k i a n d

j o u r n a l b e a r i n gs , s t e p j o u r n a l b e a r in g s , h e l i ca l - g r o o v e db e a t i n gs a n d s p i r a l- g r o o v e d b e a ri n g s . T h e e x p e r i m e n t ss h o u l d a l s o s t u d y t h r e e - d i m e n s i o n a l e f f e c t s o f s id e l e a ka g e ,a n d d i f f e r e n t m e t h o d s f o r in t r o d u c i n g t h e l u b r ic a n t i n t o t h eb e a r in g s .

O n t h e t h e o r e t i c a l s i d e , it s e e m s t h a t i n e r t i a e f f e c t s m u s tb e m o r e c a r e f u l l y d e a l t w i t h a n d i n c l u d e d i n th e a n a l y se s .T h e r e s e e m s l i t t l e n e e d t o c o n t i n u e t o s o l v e p r o b l e m s o fm o r e a n d m o r e c o m p l e x i t y w h i l e st i ll n e g l ec t i n g t h es et e r m s . A l s o , t h e t u r b u l e n c e m o d e l s d o n o t a d e q u a t e l y d e s-c r i b e t h e p h e n o m e n o n t h a t i s o c c u r i n g i n t h e n o n - l a m i n a rr a n g e o f m o s t b e a r i ng s . A n e w m o d e l w h i c h t a k e s i n t oa c c o u n t t h e v o r t e x n a t u r e o f t h e f l o w a n d t h r e e - d i m e n s i o n a le f f e c t s is n e e d e d . T h e m e a s u r e m e n t s s u g g e s t e d a b o v e c a nb e u s e d t o d e v e l o p t h e s e n e w t h e o r i e s.

cknowledgementsT h e s u r v e y a b o v e w a s p a r t i a l l y s u p p o r t e d b y N S F g r a n tG K - 2 7 8 3 1 . T h i s s u p p o r t i s g r a te f u l l y a c k n o w l e d g e d .

O s t r a c h 39 h a ve p e r f o r m e d a c a r e f u l o r d e r o f m a g n i t u d ea n a l ys i s. T h e y h a v e s h o w n t h a t i f t h e r e d u c e d R e y n o l d s n u m b e r e f e r e n c e s

Re* = UC/o( C/L) ~ 0 ( 1 )t h e n t h e i n e r t i a t e r m s s h o u l d b e r e t a i n e d i n t h e e q u a t i o n s .O n l y w h e n R e *< < 1 c a n t h e i n e rt i a t e r m s b e o m i t t e d f r o mth e e q u a t io n s . Z u k e t a 1 34 a l s o d i s c u s s e d in e r t i a e f f e c t s a n dc a m e t o t h e s a m e c o n c l u s i o n s . I n a d d i t i o n , t h e y c o m p u t e dt h e m o d i f i e d R e y n o l d s n u m b e r f o r o n s e t o f tu r b u l e n c e f o rv a r i o u s e x p e r i m e n t s . T h e v a l u e s w e r e o f o r d e r o n e w h i c hs u g g e s t s t h a t i n e r t i a e f f e c t s m a y b e p l a y in g a s ig n i f i c a n t ro l e .I t w o u l d a p p e a r t h e r e f o r e , t h a t a n y t h e o r e t i c a l s t u d y o ft u r b u l e n c e w o u l d h a v e t o i n c l u d e i n e r t i a a s w e l l .

In a d d i t io n , Z u k e t a l a 4 , as w e l l a s B u r t o n ( s e e d i s c u s s io ni n R e f 3 5 ) h a ve p o i n t e d o u t t h a t n e a r s t e p s o r o t h e r d is -c o n t in u i t i e s s ig n i f i c a n t i n e r t i a e f f e c t s a re f e l t . B u r to n 3 s a l s oi n d i c a t e s t h a t i n e r t i a m a y b e i m p o r t a n t i n s h o r t b e a r i n g s a swel l .

P in k u s a n d S te rn l i c h t 3 6 h a v e d i s c u s s e d th e tw o c l a s s i c ala p p r o a c h e s t o t h e p r o b l e m o f h o w t o d e a l w i t h i n e r ti a . T h ea v e r a gi n g m e t h o d w a s f i r s t f o r m u l a t e d b y O s t e r l e a n dS a ib e1 3 7 . T h e i t e ra t io n m e th o d w a s s u g g e s t e d b y K a h le r t 3 s ,

T h e d r a w b a c k t o t h e s e a b o v e a p p r o a c h e s i s t h a t t h e y a r eb o t h a p p r o x i m a t e i n t e c h n i q u e . T u r b u l e n c e i s a l s o n o ti n c l u d e d . K e t t l e b o r o u g h 3 9 c o n s i d e r e d b o t h i n e r t i a a n dtu rb u le n c e in h i s d i s c u s s io n o f a s l i d e r b e a r in g . H e u s e dC h o w a n d S a i b e l s 2 d e f i n i t i o n o f m i x i n g l e n g t h . H i s r es u l t ss h o w e d t h a t t h e i n c l u s io n o f t u r b u l e n c e h a d a c o m p a r a t i v e l ys m a l l e f f e c t a n d c a u s e d a s l ig h t d e c re a s e in lo a d c a p a c i ty , i nc o n t r a d i c t i o n t o o t h e r t h e o r e t i c a l a n a l y s e s . T h e i n c r e a s e i nl o a d - c a r r y i n g c a p a c i t y w a s s h o w n t o b e d u e t o i n e r t ia a n dn o t t o t u r b u l e n c e e f f e c t s . W i t h i n e r t i a , n o t t u r b u l e n c e ,i n c l u d e d i n t h e a n a l y s i s, q u a l i t at i v e a g r e e m e n t w a s s h o w nw i t h t h e r e s u lt s o f A b r a m o v i t z 1 .

C o n s t a n t i n e s c u i n a r e c e n t p a p e r 3 s h a s d i s c u s s ed a n e wa p p r o a c h t o t u r b u l e n c e a n d i n e r t i a e f f e c t s . T h e i n t e g r a le q u a t i o n s a r e u s e d r a t h e r t h a n t h e d i f f e r e n t i a l e q u a t i o n s o fm o t i o n . A p p r o x i m a t e c o n s t a n t s a r e u s e d i n t h e v e l o c i t yp r o f i le s . T h e s k i n f r i c t i o n v a r i a t i o n c a n b e a r r i v e d a t f r o mm i x i n g l e n g t h o r E l r o d a n d N g s 2 s t h e o r y . T h e m e t h o d i sa p p l i e d t o a s l i de r a n d s t e p b e a r i n g . E q u a t i o n s a r e p r e s e n t e dfo r th re e -d im e n s io n a l f lo w s a s w e l l .ConclusionsE x p e r i m e n t a l i n f o r m a t i o n i s s o r e l y n e e d e d t o b e t t e r d e s-c r i be t h e n a t u r e o f t u r b u l e n t f l o w i n b e ar i n g s. S u c h e x p e r -i m e n t s s h o u l d m e a su r e m e a n a n d f l u c t u a ti n g c o m p o n e n t s

1 D i P r i m a ,R. C. A note on the s tability o f flow in loadedjournal bearings , TransAm Soc Lubric Engrs, Vol 6 , No 3(1963) pp 249-2532 Voh r, J . H. A n experimental s tudy of Taylor vortices andturbulence in flow betwee n eccentric rotating cylinders ,Trans Am Soc mech Engrs, J Lubric Technol, Ser F, Vo l 90,No 1 (Jan 1968) pp 285-2963 Coles, D. Transition in circular Co uett e flow , J Fluid Mech,V.ol 21, Pt 3 (1 965) pp 385-4254 Bu rton, R. A. ~rurbulent film bearings unde r small dis-placements , Trans Am Soc L ubrqc Engrs, Vol 7, N o 4 (1964)pp 322-3325 Taylor, G. I. Stability of a viscous liquid contained betweentwo rotating cylinder , Trans R Soc, Lond on, Ser A, Voi 223(1923) p 2896 Taylor, G. I. Flui d frictio n between rotating cylinder ,Proc R Soc, Lon don, Set A, Vol 157 (1936) pp 546-5647 Schultz-Grunow and Hein, H. Beitrag zur Coue ttestrom ungZ/w Flugwiss, Vol 4 (1956) p 288 Pal, S. I. T urbulent flow between rotating cylinders ,NACA TN 892 (1943)9 Wilcock, D. F. Turbulen ce in high speed journal bearings ,TransAm Soc mech Engrs, Vol 72 (195 0) pp 825-833

10 Abram ovitz, S. Turbulence in a tilting pad thrust bearing ,TransAm Soc mech Engrs, Vol 78 (1956) p p 7-1111 Smith, M. I. and Fuller, D. D. J ou rna l bearing oper ation atsuper laminar speeds , TransAm Soc mech Engrs, Vol 73,(1956) p46912 Kaye, J. and Elgar, E. C. M odes of adiabatic and diabaticfluid flow in an annulus with an inner rotating cylinder ,TransAm Soc mech Engrs, Vol 80 (1958) p p 753-76513 Yamada, Y. R esi~ance of flow throug h an annulus with aninner rotating cylinder , Bull Jap Soc mech Engrs, Vol 5 ,No 18 (1962) pp 302-31014 Yam ada, Y. To rqu e resistance of flow between rotating co-axial cylinders having axial flow , Bull .lap Soc mech Engrs,Vol 5, No 20 (19 62) pp 634454115 Yamada, Y. On pressure loss of flow between rotating co-axial cylinders with rectangular grooves , Bull Jap Soc mechEngrs, Vol 5, No 20 (1962 ) pp 642455116 Burton, R. A. and Carper, H. J . An experimental s tudy ofannular flow s wi th applications in turbu lent film lubrication ,

Trans Am Soc mech Engrs, J Lubric Technol, Vol 89, Set F,No 3 (Jul 1967) pp 381-39117 Traugott, S. C. Inf luence on solid-body rotat ion of screen-produced turbulence , N A C A T e ch N o t e No 4135 (1958)

TRIBOLOGY August 972 59


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1 8 T a o , L . N . A t h e o r y o f l u b r i c a t io n in sh o r t j o u r n a l b ea r in g sw i t h t u r b u l e n t f l o w , T ra n s A m S o c m e c h E n g rs V o l 7 8( 1 9 5 8 ) p p 1 7 3 4 - 1 7 4 01 9 T a o , L . N . A t h e o r y o f l u b r i c a t io n w i t h t u r b u l e n t f lo w a n di t s a ppl ic a t ion to s l ide r be a r ings , T r a n s A m S o c m e c h E n g rsJ a p p M e c h Vol 27 , Se t E , No 1 (Ma r 1960) pp 1-1520 Cho w, V . T . a nd Sa ibe l , E . The e f f e c t of turbule n c e on s lide rbe a r ing lubr ic a t ion , Trans A m Soc m e c h Engrs J appl Me c hVol 26 , Se t E , No 1 (Ma r 1 959) pp 122-1262 1 C o n s t a n t i n e sc u , V . N . O n t u r b u l e n t l u b r i c a t i o n , P r o c l n s t

me c h Engrs V o l 1 7 3 , N o 3 8 ( 1 9 5 9 ) p p 8 8 1 - 9 0 022 Con s ta nt ine sc u , V . N . Ana lys i s of be a r ings ope ra t ing int u r b u l e n t r e g i m e , Trans A m Soc m e c h Engrs J bas EngngVol 84 , Se r D , No 1 (Ma r 1962) p p 139-15123 Arwa s , E . B . , S te rn l i c h t , G . a nd We rnic k , R . J . A na lys i s ofp la in c y l indr ic a l jour na l be a r ings in turbule nt r e g im e , TransA m S o c m e c h E n gr s J b as E n g n g V o 1 8 6 , S e r D , N o 2( J u n 1 9 6 4 ) p p 3 8 7 - 3 9 524 Sc hl ic h t ing , H . Boundary lay e r the ory Mc Gra w Hi l l , 4 thE d i t i o n ( 1 9 6 0 )25 Re yno lds , A . J . A na lys i s of turbu le nt be a r ing f i lm s , J m e c hE n g n g S c i V o l 5 , N o 3 ( 1 9 6 3 ) p p 2 5 8 - 2 7 22 6 N g , C . W . F l u i d d y n a m i c f o u n d a t i o n o f t u r b u l e n t lu b r i c a ti o nt h e o r y , Trans A m Soc L ubr ic Engrs V o l 7 , N o 4 ( 1 9 6 4 )

pp 311-32127 Ng, C . W. , a nd Pa n , C . H . T . A l ine a r iz e d turbu le nt lubr ic a t iont h e o r y , Trans A m Soc m e c h Engrs J bas Engng V o l 8 7 ,S e r D , N o 3 ( 1 9 6 5 ) p p 6 7 5 - 6 8 828 E k o d , H . G . J r a n d N g , C . W . A t h e o r y o f t u r b u l e n t f l u i df i lm s a nd i t s a ppl ic a t ion to be a r ings , T r a n s A m S o c m e c hE n g r s J L u b r i c T e c h n o l V o l 8 9 , S e r F , N o 3 ( J u l ( 1 9 6 7 )p p 3 4 6 - 3 6 2

2 9 C h o w , C . Y . a n d V o h r , J . H . H e f i c al g r o o v e d j o u r n a l b e ar i n go p e r a t e d i n t u r b u l e n t r e g i m e , Trans A m S oc me c h Engrs JL u b r i c T e c h n o l V o l 9 2 , S e t F , N o 2 ( A p r 1 9 7 0 ) p p 3 4 6 - 3 5 830 Voh r , J . H . a nd Cho w, C . Y . The ore t i c a l a na lys i s of sp ir al -groove d sc rew se al for turbule nt ope ra t io n , T ra n s A m S o em e c h E n gr s J L u b r i c T e c h n o l Vol 91 , Se r F , No 4 (Oc t 1969)p p 6 7 5 - 6 8 631 Pa pe , J . G . a nd Vra kking , W. J . V isc ose a l -pre s sure ge ne ra t iona n d f r i c t i o n l o ss u n d e r t u r b u l e n t c o n d i t i o n s , Trans A m So cLubr ic Engrs V o l 1 1 , N o 4 (1 9 6 8 ) p p 3 1 0 - 3 2 03 2 K e t o i a , H . N . a n d M c G r e w , J . M . T u r b u l e n t o p e r a t i o n o f t h evi sc ose a l , Trans Am Soc Lubr ic Engrs Vol 10 , No 3 (1967)p p 2 5 6 - 2 7 233 Kul inski , E . S . a nd Os t r ac h , S . A c r i t i c a l e va lua t ion of thes ta tus a nd t r e nd s in h igh spe e d f lu id f ilm lubr ic a t ion ,N A S A C r 1 0 5 8 ( 1 9 6 8 )34 Zuk, J . e t a l . Con ve c t ive ine r t i a a nd ga s inge s t ion e f f e c t s onf low re gim e s of the v i sc ose a l the o ry a nd e xpe r im e n t , TransA m S o c L u b r i c E n gr s V o l 1 0 , N o 3 (1 9 6 7 ) p p 2 7 3 - 2 9 335 Cons ta n t ine sc u , V . N . O n the inf lue nc e on ine r t i a forc e s inturbule n t a nd l a m ina r s e l f - a ct ing f i lm s , T ra n s A m S o c m e c hE n gr s J L u b r i c T e c h n o l Vol 92 , Se r F , No 3 ( Ju l 1970)p p 4 7 3 - 4 8 136 P inkus , O . P . a nd S te rnf ic h t , B . T h e o r y o f h y d r o d y n a m i c

lubrication Mc Gra w Hi l l ( 1961)37 Os te d e , F . a nd Sa ibe l , E . O n the e f f e c t of lubr ic a nt ine r t i ai n h y d r o d y n a m i c l u b r i c a t i o n , Z A M P V o l 6 ( 1 9 5 5 ) p 3 3 438 Ka hle r t , W. De r e inf lur s de r t r a ghe i t skra f te be i de r hydro -d y n a m i s c h e n s c h m i e r m i t e l t h e o r i e , Ing-Arc h V o l 1 6 ( 1 9 4 7 )p 32139 Ke t t l e bo roug h, C . F . Tu rbule nt a nd ine r t i a f low in s l ide rbe a r ings , T r a n s A m S o c L u b r i c E n g r s V o l 8 , N o 3 ( 1 9 6 5 )p p 2 8 6 - 2 9 5

cr iealr e v i e w s i nt lb ogy1 9 7 Th e f i r s t issue o f a n e w a n n u a l p u b l i ca t io nc r i t i ca l re v ie w s in t r i b o l o g yi s now ava i lab le .

Th e L e e d s I n d u s t r ia l U n i t o f T r ib o lo g y h ave sca n ne d a l l t h e w o r ld ' s l i t e ra tu re in t r i b o lo g yfo r 1 9 7 0 a n d c la ss i fi e d i t i n t o s i x ca te g o rie s : d e ve lo p m e n ts - co n ta c t , f r i c t i o n a n d w e a rlu b r i c a t io n a n d lu b r i ca n t s - ma te r ia l s - me ch a n isms - i n d u s t r ia l a p p l i ca t io n s . Th isl i t e ra tu re h as b e en c r i t i ca l l y re v ie w e d b y le a d in g a u th o r i t i e s in th e f i e ld , u n d e r t h eGe n e ra l Ed i t o rsh ip o f P ro fe sso r D . D o w so n . I n a d d i t i o n t o t h e re v ie w s , t h e p u b l i c a t io nco n ta in s a co mp le te b ib l i o g ra p h y , a n d a su b je c t i n d e x . 1 0 0 p a ge s A4 s ize.c r i t i ca l re v ie w s in t r i b o l o g y 1 9 7 0 c o st s 6 a c o p y , o r 4 f o r s u bs c ri be rs to T R I B O L O G Y .A i r m a il p o s ta g e 1 . 5 0 e x t r a . O r d e r y o u r c o p y n o w f r o mSa le s Ma n a g e r , IPC Sc ie n ce a n d Te c h n o lo g y Pre ss L td ,3 2 H i g h S t r e e t , G u i l d f o r d , S u r r e y , E n g l a n d

160 TRIB OLO GY August 1972

turbulence and inertia effects in bearings

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