navier stokes equations for 2d cascades

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J . o f T h e r m a l S c i e n c e Vo l .7 , N o . 1

J l ~ ~'~j

M u l t i g r i d N a v i e r - S t o k e s C a l c u l a t i o n fo r T w o

D i m e n s i o n a l C a s c a d e s

Yang Ce Lao Da zhon g J i ang Z ikangT h e r m a l E n g i n e e r i n g D e p a r t m e n t , T s i n g h u a U n i v e r s i t y , B e i j in g , 1 00 0 84 , C h i n a

A f a s t and a c c u r a t e numer i c a l m e thod fo r so lv ing t he t w o d imens iona l Reyno ld s ave r aged N av i e r-Stokes i s appl ied to ca lcu la te the in te rn a l f lu id of turb in es and c omp ressors . Th e code is based onan expl ic i t , t im e-m arch ing , f in i te volum e technique . In order to acce lera te convergence , loca l t im es t e p p in g , m u l t i g ri d m e t h o d i s e m p l o y e d. F o u r s ta g e R u n g e - K u t t a m e t h o d i s i m p l e m e n t e d t o e x t e n dt he s t ab i l i t y d o m a in . Tes t c a se s o f H o b son ' s i mpu l se ca sc a d e, NA SA R o to r 37 and San z ' s s upe rc r i t ic a Icomp res so r c a scade a r e p re s e n t ed . R esu l t s o f Mach n u mb er d i s t r i bu t i on on b l a de su r f ace s an d M achnumber co n tou r p l o t s i nd i c a t e good ag reemen t w i th expe r i men t a l da t a . Compared w i t h fu l l t h r ee 3DNav ie r-S tokes (N-S ) c odes , t he two d ime n s iona l code on ly t ake s a sho r t t ime t o ob t a i n p r e d i c t e d r e -su l t s . This code can be used wide ly in prac t ica l engineer ing des ign .

K e y w o r d s : n u m e r i c a l m e t h o d , two dimension, Navier-Stokes equation,t u r b o m a c h i n e r y .

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

O v e r t h e p a s t 2 0 y e a r s s t e a d y p r o g r e s s h a s b e e nm a d e i n t h e d e v e l o p m e n t o f C F D c o d es f or t u r b o m a -c h i n e r y b l a d e ro w s . T h e n u m e r i c a l m e t h o d s t h a t h a v eb e e n t h e m o s t h i g h ly d e v e l o p e d a n d h a v e p r o v i d e dt h e g r e a t e s t a d v a n c e m e n t s i n t u r b o m a c h i n e r y f i el d a r et h e t i m e m a r c h i n g s o l v e r s b a s e d o n a f u l l y c o n s e r v a -t i v e f o r m o f g o v e r n i n g e q u a t i o n . T h e y p r o v i d e a s in -g l e a p p r o a c h f o r s u b s o n i c , t r a n s o n i c , a n d s u p e r s o n i cf lo w s, a n d t h e y i n h e r e n t l y p r o v i d e n a t u r a l s h o c k c a p -

t u r i n g c a p a b i l i t y. I s s u e s o f s o l u t i o n a c c u r a c y, g e o m e t -r i c c o m p l e x i t y, s p e e d ( c o s t ) , u n s t e a d i n e s s , t u r b u l e n c e( in p a r t i c u l a r n e a r - w a l l m o d e l l i n g , s e p a r a t e d r e g i o nm o d e l l i n g ) a n d o v e r a l l t i m e t o a n a l y z e a d e s i g n s t i l le x i st . T h e e v e n t u a l g o a l o f C F D c o d e s is t i m e a c c u -r a t e m o d e l o f t h e t h r e e - d i m e n s i o n a l f l ow t h r o u g h t h eb l a d e r o w s .

T h e c o n v e n t i o n a l t w o a n d q u a s i - t h r e e d i m e n s i o n a lm e t h o d s h a s b e e n e x t e n s i v e ly u s e d in t h e d e v e l o p m e n to f t u r b o m a c h i n e r y b l ad i n g. T h e d e si g n o f m o d e r n t u r -b o m a c h i n e s w i t h h i g h e f f ic i en c y a n d p o w e r m a k e i t i n -c r e a s in g l y i m p o r t a n t t h a t a t h r e e d i m e n s i o n a l m e t h o d

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

Received October, 1997.

B e c a u s e s o m e p r o b l e m s o f t h r e e d i m e n s i o n a l v i s co u sc o d e s c a n n ' t b e s o lv e d , w h i c h i n c l ud e t u r b u l e n c e m o d -e l i n g a n d t h e l a rg e a m o u n t o f t i m e s t i ll n e c e s s a ry,t w o a n d q u a s i - t h r e e d i m e n s i o n a l m e t h o d i s s ti ll u s e dw i d e l y b y t u r b o m a c h i n e r y d e s i g n e r .

T h i s p a p e r d e s c r i b e d a n t w o d i m e n s i o n a l v i s c o u sc o d e . I t s ol v es th e g o v e r n i n g e q u a t i o n s f o r c o m p r e s s -i b l e f lo w o n a b l a d e t o b l a d e s t r e a m s u r f a c e . T h eg e n e r a l b l a d e - t o - b l a d e s u r f a c e g e o m e t r y i s s h o w n i nF i g .1 . T h e s t r e a m t u b e t h i c k n es s , th e d i s ta n c e b e -t w e e n s t r e a m s u r f a c e a n d a x i a l d e s c r i b e d b y F i g . 1

a r e a l lo w e d t o c h a n g e a l o n g w i t h a x i s d i r e c t i o n . T h ec o d e o f s o l u t io n c h o s e n f o r t u r b i n e a n d c o m p r e s s o rf l u i d f i e l d p r e s e n t e d h e r e i s b a s e d o n a n e x p l i c i t ,t i m e - m a r c h i n g , f i n it e v o l u m e m e t h o d . T h e f u ll y c o n -s e r v a t iv e f o r m o f g o v e r n i n g e q u a t i o n s u s e s a c e l l -c o r n e r f i n i t e - f l u m e d i s c r e t i z a t i o n c o u p l e d w i t h a f l e x -i b l e m u l t i - s t e p p i n g s c h e m e . Vi s c o u s ef f e c ts i s s i m -u l a t e d b y a v e r y s i m p l e t w o - l a y e r a l g e b r a ic t u r b u -l e nc e m o d e l (B a l d w i n a n d L o m a x ) . To e x t e n d t h es t a b i l i t y l i m i t a n d a c c e l e r a t e c o n v e rg e n c e , l o c a l t i m es t e p p i n g , m u l t i g r i d a n d r e s i d u a l a v e r a g i n g a re a p p l i ed .T h e m u l t i g r i d m e t h o d w h i c h c o n t r i b u t e s m u c h t o t h e

e f f ic i e n c y o f o u r c a l c u l a t i o n w i ll b e d e s c r i b e d i n t h ef o l l o w i n g p a r a g r a p h . To v e r i f y t h e c a p a b i l i t i e s o f t h e

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Yang Ce et al . Multigrid Navier-Stokes Calculation for Two Dimensional Cascades 17

numer i ca l me thod used in t h i s pape r, Hobson ' s im-pu l se ca scade , NASA ro to r 37 tr anson ic compres so rro to r and Sanz ' s supe rc r i t i ca l compres so r ca scade a reca l cu l a t ed . In add i t ion , a compar i son wi ll be madebe tween expe r imen ta l da t a and s imu la t ion p red ic t ion .

II 1

. Z

Fig.1 Blade-to-blade surface revolution

G O V E R N I N G E Q U AT I O N S

The in t eg ra l fo rm o f gove rn ing equa t ions i n a ro -t a t i ng cy l ind r i ca l coo rd ina t e sys t em can be wr i t t en a s

0O - - ~ i C d v + ~ . ~ s = f q d d v + / ~ . ~ s s (1 )

conse rva t ion var i ab le s ¢ , f lux vec to r ~ , sou rce t e rm

kV, viscou s for ce te rm ;J~ are given b y

¢ = (p, pw¢ , pWm) r,

U = Fi, + Gim,

F = ( p W, , p + p w L p w m w , ) T,

c = ( p w m , p w ¢ w m , p + p w ~ ) ~ ,

¢ = (0 , -PWmsinO[W¢ + 2wr],

p de p "~-P sin O[W¢ + wr] 2 + -h d--mm+ - s i n e ]r r

T = T~i~ + Tmim,

T m = ( 0 , - r m ¢ , _ ~ ) r ,

T¢ = (0 , - -T¢,¢ , - -Tern) T,

2 [ IOW ¢ OWmj],T ¢¢ = ~ # i 2 r 0 ¢ Om

T

2 r OWm IOW¢]~mm=~.[2~---~ r ~ J '

F°W¢ l OWm ]TmC = VC~m= # L Om + r 0¢ J'

where v i sce l l volume , s i s a rea of cont ro l volum e face ,We , t angen t i a l d i r ec t ion (¢ ) and Wm mer id iona l (rn)veloci ty, c i s th ickness of adjac ent s t rea m surfac e , 0i s ang le be tween s t r eaml ine o f m and ro t a t i ng ax i a lof Z, ~- is she ar stress , # is tot al ed dy viscosity, an d

# = ]'~L "q- # T "

The gove rn ing equa t ions u sed in t h i s s tudy a re t hetwo-d imens iona l N-S equa t ions coup led wi th co r io l i sand cen t r i fuga l fo rce t e rms in mom en t um equa t ions .Re la t ive ve loc ity comp onen t s a r e r e t a ined a s depen-den t va r i ab l e s i n a sys t em a t t ached to a ro t a t i ng o rs t a t i ona ry b l ade row. T he equa t ions so lved a re : con -

t inu i ty equa t ion , momen tum equa t ions i n t he mer id -iona l and t ange n t i a l d i r ec t ion . To dec rease ex t r a com-pu ta t ion a l cos t , t he ene rgy equa t ion i s r ep l aced by theas sumpt ion o f cons t an t s t agna t ion en tha lpy in s ide s t a -to r b l ade and cons t an t r e l a t i ve s t agna t ion en tha lpyins ide ro tor b lade . For a perfec t gas , we can wri te

where h i s ro thalpy.Turbu lence i s mode led by a two- l aye r a lgeb ra i c

mix ing l eng th eddy v i scos ity mode l (Ba ldwin and Lo-max) . Th i s t ype o f t u rbu lence mod e l has t he advan-t age o f i nc reas ing a l i t t le more comp u ta t iona l cos tthan tha t o f an inv i scid codes , bu t i t c an s imu la t e a l llevel of turb ulen ce v iscos i ty appro pr ia te ly . I t has be-come advan tageous fo r so lving the Re yno lds -a ve rage dN-S equa t ions for turbu lent shea r f lows. I t is s t i ll be-ing used in bo th ex t e rna l a nd in t e rna l f lows. The to t a ledd y viscos i ty consis ts of a laminar par t #L a nd a tur-bu len t pa r t #T" The l amina r v i scos i ty i s mode led bySuth er lan d 's law in which #L is a funct io n of the lo-ca l s t a t i c t emp era tu re . The tu rbu lenc e v iscos ity # r i s

o b t a i n e d b y

{ ( # r ) i . . . . Y<-Ycrossove~, T = (3 )

(#r)oute* Y > Y . . . . . . . .

where y i s t he no rma l d i s t ance f rom the nea re s t wa l la nd ycrossove, i s the smal les t va lue of y a t which val -ues f rom inne r and ou te r fo rmulas a r e equa l. ( # r ) i . . . .and (#r)oute~ a re t he i nne r l aye r and the ou te r l aye reddy viscos i ty, respect ive ly. The inner turbulent v is -cos i ty i s def ined as

( ' T ) ' n n e r = P t 2 I ~1 ( 4 )

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18 Jou rna l of The rm M Science , Vol . 7 , No . l , 1998

w h e r e [ f~ [ i s t h e m a g n i t u d e o f t h e v o r t i c i t y a n d l =

kv( - + =

T h e o u t e r t u r b u l e n t v i s c o si t y is d ef i n e d a s

(#T)O**t~ = O.O168C~ppF,~ak~Fk (5 )

w h e r e

F~,~ke= Y m a x F m a x (6 )

F m a x i n E q .( 6 ) i s t h e m a x i m u m v a l u e o f F = y [ f ~ ] ( t -e - y + / A + ) ,a n d y . .. . i s t h e v a l u e o f y c o r r e s p o n d i n g t oF, . . . . T h e K l e b a n o f f i n t e r m i t t e n c e f a c t o r i s g i v e n b y

• ~" Ym ax ~ J

T h e v a r i o u s c o n s t a n t s a r e t a k e n a s k = 0 . 4 1 , A + =

26, C~p = 1.6, Ck = 0 .3.

N U M E R I C A L A L G O R I T H M

T h e a l g o r i t h m u s e d t o so l v e t h e g o v e r n i n g e q u a t i o n sis a n e x p l i ci t , t i m e - m a r c h i n g , f i n i te v o l u m e m e t h o d .T h e p r i n c i p le o f a t i m e m a r c h i n g m e t h o d is t o c o n -s i d er t h e s o l u t i o n o f a s t a t i o n a r y p r o b l e m a s t h e s o-l u t i o n a f t e r a s u f fi c ie n t l o n g c a l c u l a t i o n t i m e o f t h ei n s t a t i o n a r y e q u a t i o n s d e s c r ib i n g th i s p r o b l e m . T h ec o m p u t a t i o n s t a r ts f r o m a r o u g h a p p r o x i m a t i o n o f t h e

f i na l s o l u t i o n , c o n s i d e r e d a s a la rg e p e r t u r b a t i o n o f t h es t e a d y f l o w f ie ld , a n d d e v e l o p s u n d e r c e r t a i n b o u n d -a r y c o n d i t i o n s u n t i l c o n v e rg e n c e . I n o u r s tu d y, w e u s eH m e s h t o d i s c r e ti z e e q u a t i o n s. T h i s t y p e o f m e s h h a st h e a d v a n t a g e o f e a s y g e n e r a t io n , a n d i t c an b e u s e dt o a h n o s t a l l k i n d s o f g e o m e t r i c a l s h a p e . I n th i s p a -p e r , H m e s h i s m a d e u p b y t w o f a m i l y p la n e s , o n e i st h e r e v o l u t i o n s u r f a c e s ( b l a d e - t o - b l a d e ) a n d t h e o t h e ri s q u a s i o r t h o g r a p h i c s u r f a c e s . O n e d i s a d v a n t a g e o fH m e s h i s t h a t f o r t h i c k l e a d i n g e d g e a n d t r a i l i n ge d g e t h e e r r o r s m a y b e i n c u r d u e t o u s i n g a h i g h l ys h e a r e d H m e s h . I t c a n b e o v e r c o m e b y u s in g su f fi -c i e n t m e s h p o i n t s o r p l a c i n g c u s p s a t t h e l e a d i n g a n dt r a i l i n g e d g e . T h e n o d e s c a n b e c h o s e n a t a t y p i c a ll o c a t i o n s o f t h e e l e m e n t s , s u c h a s c e l l - c e n t r e s , c e l l-v e r t i c e s o r m i d - s i d e s . I n o u r in v e s t i g a t i o n , n o d e s a r el o c a t e d a t e a c h c e ll o f t h e 4 c o r n e r s . T h e f l u x e s o fm a s s a n d m o m e n t u m t h r o u g h e a c h fa ce a r e f o u nd u s -i n g a v e r a g e s o f t h e i n d e p e n d e n t v a r i a b l e s o f e q u a t i o n ss t o r e d a t t h e c o r n e r s o f t h a t f a c e . D e n t o n ( 1 9 8 2 ) fi r stu s ed th i s m e t h o d . C o m p a r e d w i t h t h e ol d m e t h o d o fJ i a n g ( 1 9 8 5 ) , H u a n g a n d J i a n g ( 1 9 9 5 ) , C h e n ( 1 9 9 5 ) ,w h i c h s t o r e s v a r i a b l e s a t c e ll c e n t e r , t h e c e ll s o f c o v -e r in g w h o l e d o m a i n d o n ' t o v e r l a p in n e w m e t h o d s ot h a t t h e n e w m e t h o d s h o r t e n s c o m p u t a t i o n a l t i m e . I na d d i t i o n , t h e n e w m e t h o d c a n g u a r a n t e e t h e f o r m a l

a c c u r a c y n o t t o d e c r e a s e f o r r a p i d c h a n g e s o f m e s hs p a c in g . A s y s t e m o f o r d i n a r y d i f f e re n t i al e q u a t i o n sc a n b e o b t a i n e d b y a p p l y i n g E q . ( 1 ) t o e a c h c e ll a n da p p r o x i m a t i n g t h e s u r f a c e i n t e g r a l w i t h a f i n i t e v o l -u m e s c h e m e ,

d p _ 1 4v Z (8)

i = 1

d(pWm) 1 4d t - v{ Z 3 ) ,

/ = 14

+ E ~ t A S m t A S , , a( 9 )

p dh P s in ¢ ]+ 7 }

d(pl/V¢) 1 4

l = 14

4 (10)

/ = 1

- V . - P w m s in O ( W ¢ + 2 w r ) ]r

( 8 ) , ( 9 ) and (10 ) h a ve t h e fo rm

d ( V. ~ ) + Q = 0 ( l l )d t

I n o r d e r t o s u p p r e s s t h e t e n d e n c y f o r s p u r i o u s o d da n d e v e n p o i n t o s c i l l a ti o n s , a n d t o p r e v e n t u n s i g h t l yo v e r s h o o t s n e a r s h o c k w a v e , t h e s c h e m e i s a u g m e n t e db y a d i s s ip a t i v e t e r m s o th a t E q . ( l l ) b e c o m e s

d ( V .¢ )d ~ + Q - D = 0 (12 )

w h e r e V i s t h e v o l u m e o f e l e m e n t v o l u m e a n d A S

i s t he a r e a o f c e l l f a c e . Q i s t he n e t f l ux ou t o f t i l ece ll . D i s t he d i s s i p a t i on . E q s . (8 ) , ( 9 ) , ( 10 ) a r e in t e -g r a t e d i n t im e b y a f o u r s t a g e R u n g e - K u t t a e x p l ic i ts c he m e . T h e a i m o f u s in g m u l t i - s t a g e R u n g e - K u t t am e t h o d is t h a t i t c a n e x t e n d c o m p u t a t i o n a l s t a b i l i t yl i m i t . J a m e s o n e t a l. ( 1 9 81 ) d e v e l o p e d a n d i m p l e -m e n t e d t h i s te c h n i q u e f o r t h e E u l e r eq u a t i o n s . T h i st e c h n i q u e h a s r e c ei v e d w i d e s p r e a d a c c e p t a n c e f o r b o t he x t e r n a l a n d i n t e r n a l f l o w s , i n c l u d i n g t u r b o m a c h i n e r yf l O W S .

M U L T I G R I D M E T H O D

B e s i d e s l o c al t im e s t e p p i n g m e t h o d , m u l t i g r i d t e c h -

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Yang Ce et al. Multigrid Navier-Stokes Calculation for Two Dimensional Cascades 19

nique is also utilized for accelerating the convergenceto steady state, multigrid method is the most effectivetechnique of accelerating convergence. T he multig ridtechnique can have several levels, one kind of sim-plified multi grid tech nique and two levels of grid areused in our codes. Define residua l R = (Q - D) /V .For a coarse grid, its residual can be expressed by

f t

Re = X Rf , where Ry is the flux change of fine gridi = 1

and R,, is the c hange of coarse grid, n is the n umb er o ffine grid inside one coarse grid. The final chang e of afine grid is obtai ned by R} = R I + k- Re, where k is acoefficient, some test cases indicate that this methodis able to reduce about one third of computation timecomparing with single-level grid method.

B O U N D A R Y C O N D I T I O N S

For cascade calculations, four types of boundariesare usually encountered: wall, periodic, inlet and out-let. All solid surfaces are model ed as rigid, nonslip,and impermeable. For periodic boundaries, equiva-lent flow variables are imposed at corresponding cells.At the inlet boundary, the relative stagnation pressureand relative stagnation temperature are specified, inaddition, the relative flow direction or relative swirlvelocity is also specified. Stat ic pressu re is extr ap-olate d from the points inside the flow field. At thedownstream boundary the static pressure is fixed andheld constant at the hub surface and simple radialequilibrimn is supplemented. Other variables are ob-tained by extrapolation fi'om the points inside the flowfield.

RESULTS

tion /3 = 438 °, out let stat ic pressur e P2 -= 1.01325MPa. Detailed cascade geom etry dat a is given byHobson (1972). Both Euler method and N-S methodare utilized to calcula te the flow field. Mesh pointschosen in our calc ulat ion is 13 x 58. 798 and 776 it-erative steps are needed for convergence to engineer-ing accu racy of two metho ds, respectively. Bot h ofthe methods only took less then 1 rain to produce re-sults. C omp ute d results and accurat e analyti cal solu-tions are showed in Fig.2. The results of relative Machnumber distribution on cascade surface in Fig.2 indi-cates that the flow predictions are reasonably good.

'.e

1 . 2 -

1.0 ̧

0.8 ̧

0 . 6 -

0.4-

0.2 i

0.0 02

z~ Designo Euler

n ~ n N-S

i i i i

0.4 0.6 0.8 1.0Relative chord .V C

Fig.2 Relative Mach number distribution

near the cascade surface

S a n z ' s s u p e r c r i t i c a l c o m p r e s s o r c a s c a de .The second test case is the computation of the flow inthe supercritical compressor cascade of Sanz (1984).Inlet is ambient condition. The main characteris-tics of the flow are the following: inlet M ach nu mbe rM1 = 0.711, inlet flow angle fl = 30.81 °, outl et Ma thnumbe r M2 = 0.544. In bl ade -to -bl ade direction, 16

Three test cases are calculated in order to verify

that the metho d described in this paper can be appliedto predict internal fluid in stator s and rotors. Thefirst case presented here is Hobson's impulse cascade.The next case examine Sanz's supercritical compressorcascade. Lastly, we demons trat es an isolated rotor(NASA Rotor 37).

Hob son 's impu lse cascade . Hobson's impulsecascade is usually used by many papers as a test casebecause it offers accura te analyt ical solutions. Calcu-lation is performed on a pentium personal computerof main frequency 100 MHz and 32 Mk memory space.The values of some major parameters used in our cal-cula tion are: inlet total pr essur e P~v = 1.26853 MP,inlet total temperature T~v = 640 K, inlet flow direc-

1.2

1.0

0.8

0 , 6 -

0 .4

0.2 i

0.0 0.2

z~ z~ ]', z~ ~ ~ ComPutation

0:4 0:6 08 1.0Relative chord .r / C

Fig.3 Relative Mach number distributionnear the cascade surface

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20 Journ al of The rma l Science, Vol. 7, No.l , 1998

g r i d p o i n t s we re u s ed. Up s t r e am o f t he l e ad ing edge ,t h e r e a r e 30 g ri d po i n t s ; t h e r e ax e 61 g ri d po in t s i n -s i d e p a s sage , and t he r e a r e 30 g r i d p o in t s dow n s t r eamo f t h e t r a i l i ng edg e . A t o t a l o f 121 g r i d po in t s i ns t r e am w ise d i r ec t i o n we re emp l oyed . Th i s i s a t o t a l o f1936 gr id poin ts ; T he ma in d i ff icu l ty of th i s te s t i s toc o m p u t e t h e d e c e l e r a t i o n o n t h e s u c t i o n s i d e w i t h o u tn u m er i c a l l y gene r a t i n g a s hock . F ig .3 shows t h e ve ryg o o d a g r ee m e n t b e t w e e n t h e c o m p u t e d M a c h n u m b e rd i s t r i b u t i on on ca sc a de su r f ace a n d t h e exac t so lu t i on .No s ho ck appea r s on t h e s u c t i o n s i de d u r ing t he d e -ce lera t ion .

N A S A R o t o r 3 7 . R o t o r 3 7 w a s d e s ig n e d a n dt e s t e d a t N A S A L e w i s , w h ic h is a l o w - a s p e c t - r a t i oro tor. I t has bee n as a tes t case for a carefu l assess -m e n t o f t h e a b i l i ty o f t h e v a r i o u s t u r b o m a c h i n e r y C F Dcodes by AS ME i n 1994. The ro t o r ha s 3 6 b l adeswi th a de s ign p r e s s u re r a t i o o f 2 . 10 6 and ma ss f l owo f 20 .19 kg / s a t 100% ro to r speed , co r r e s p o n d ing t o1 7 ,188 rpm wi th a t i p r e l a t i ve Mach num b e r 1 .48 . I ti s t h e t h i rd ro to r o f a comp re s so r w i th a p r e s s u re r a t i oo f a b o u t 2 0. O t h e r m a i n d e s ig n d a t a a r e : t o t a l t e m -pe ra t u r e r a t i o 1 . 2 7 , hub t o t i p r a t i o 0 . 7 , a s p e c t r a t i o1 .19, pea k e f f ic i en c y 0 .876 . The ex t ens iv e t e s t da t a

fl ~ V t"

{ . / ; 7 / ~ I _

/

~ ~' 30% Span !J l / . I l,,,,,x"0 ,98

( b ) Tcs,t

F ig .4 Rela t ive Mach number contour p lo ts for NASA ro tor 37at 30 percent span at the design point

\\, \ " / J2 4 / .. :;

k ug// /,..kL\ .;;i,

Fig .5 Rela t ive Mach number contour p lo ts for NASA ro tor 37at 70 percent span at the design point

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