eRep-A Digital Computer Program for the Subsonic Flow Past Turbomachine Blades Using a Matrix Method

7/30/2019 eRep-A Digital Computer Program for the Subsonic Flow Past Turbomachine Blades Using a Matrix Method

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6Z~3

R & M No.3838

PROCUREMENT EXECUTIVE, MINISTRYAeronauticaM Research Council

Reports and Memoranda

OF DEFENCE

A DIGITAL COMPUTER PROGRAMFOR THE SUBSONIC FLOW PAST

TURBOMACHHNE BLADES USINGA MATRIX METHOD

by'W.J. Calvert and D.J.L. Smith

National Gas Turbine Establishment

10 NETLondon" Her Majesty's Stationery Office


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UDC 518.5.07 : 533.6.011.31 : 621-135 -226.2

A D I G I T A L C O M P U T E R P R O G R A M F O R T H E S U B S O N I C F L O W P A S TT U R B O M AC H I N E B L A DE S U S IN G A M A T R I X M E T H O D

By W. J. Calver t and D. J. L. S mithN a t i o n a l G a s T u r b i n e E s t a b l i s h m e n t

R E P O R T S A N D M E M O R A N D A N o . 3 8 3 8 "November 1976

SUMMARYA detaile d des cript ion is given of a computer pr ogra m for analysing

the inviscid, subsonic compressi ble pressure d istribu tions around the bladeso f a x i al , r a d ia l a nd m i x e d - f l o w t u r b o m a c h i n es . T h e f l o w e q u a t i o n s a r ecombined to form a governing, Poiss on-typ e di fferenti al equati on for thestream functi on and the numeric al solution is obtain ed by a finitediffer ence matr ix method. Solutions for several turbomachine s, giving flowpatterns and velo city dist ributi ons are included.

* Replaces NGTE Repo rt 328 - ARC 37294


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C O N T E N T S

I. O Intro ductio n2.0 M a t h e m a t i c a l m o d e l

2.1 Stream function equati on2.2 Boundary conditions

2.2.1 Upper and lower boundarie s2.2.2 Up- and downs trea m boundaries

3.0 Numerical techniques

3.13.2 Finite difference gridF i n i t e d i f f e r e nc e a p p r o x i m a t i o n s3.2.1 Lapla cian operator3.2o 2 The de ri vat iv e a--x3.2.3 The derivative ~

3.3 Band matr ix4 . 0 S o l u t i o n p r o c e d u r e

4.14.24.3

Solution of matri x equat ionCalcul ation of gas stateCalcul ation of new right hand side

5 . 0 T h e c o m p u t e r p r o g r a m5.1 Data prepa ration5.2 Running instructi ons5.3 Output

5.3.1 Line printer output5.3.2 Graph plotter output5 . 3 . 3 D i a g n o s t i c o u t p u t

6.0 Numerical examples6.16.26.36.46.56.66.7

Impulse turbine cascade70- camber blade in cascadeConical flow past station ary and rotating blade rowsAxial compressor stator bladesRadial inflow turbineNASA turbine stator bladeAxial turbine stator blade

7.0 ConclusionsNotationReferences

Page

6

778

9iOiO1213131313

1414141415151516161717181922


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A p p e n d i x IA p p e n d i x I IAppendix IIIA p p e n d i x I V

Appendix V

Appendix VI

Appendix VllAppendix Vlll

Program tapeComputer program organisationData format for matrix prog ramData preparation program for blades specifiedby coordinatesData preparation progra m for double circulararc bladesMatri x program running instructions

Matri x program line printer resultsMatrix progra m diagnostic output

3Page232629

33

3740

4446

IllustrationsDetachable abstract cards

Figures I-2]


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41 . 0 I n t r o d u c t i o n

T h e d e s i g n e r o f t u r b o m a c h l n e r y b l a d i n g i s c o n t i n u a l l y f a c e d w i t hd e m a n d s f or h i g h e r d u t i e s a n d i m p r o v e d e f f i c i e n c i e s . I n o r d e r t o m e e t t h e s ed e m a n d s w i t h o u t e x t e n s i v e d e v e l o p m e n t t e s t i ng ~ s o p h i s t i c a t e d m e t h o d s m u s t b eused to predi ct the internal aerod ynami cs of a design.

In 1952 Wu I devel oped a general theory for the steady, three -dim ensio nali n v i s c id , s u b s o n i c f l o w t h r o u g h a x i a l, r a d i a l a n d m i x e d - f l o w t u r b o m a c h i n e s .T h e f l o w is c o n s i d e r e d o n t w o i n t e r s e c t i n g f a m i l i e s o f s t r e a m s u r f a c e s , t h eb l a d e - t o - b l a d e ( S i ) a n d h u b - t o - t i p ( $2 ), F i g u r e i. T h e g e o m e t r y of o n e f a m i l yi s g i v e n b y t h e f l o w p a t t e r n s w i t h i n t h e o t h e r , t h e c o m p l e t e f l o w s o l u t i o nb e i n g f o r m e d b y an i t e r a t i v e p r o c e d u r e b e t w e e n t h e tw o s u r f a c e s H o w e v e r , t h eequations are not generall y sol uble by analy tic means and at the time it wasi m p r a c t i c a l t o c a r r y o u t n u m e r i c a l s o l u t i o n s .

With the advent of the high spe ed digital co mputer, it has been pos-s i b l e t o u s e n u m e r i c a l t e c h n i q u e s t o s o l v e W u ~ s o r i g i n a l e q u a t i o n s . M a r s h 2p r o g r a m m e d a f i n i t e d i f f e r e n c e m a t r i x s o l u t i o n f o r t h e f l o w i n t h e h u b - t o - t i psurface, and Smith 3 adopted a similar appr oach to solve for the flow in theb l a d e - t o - b l a d e s u r f a c e . T o g e t h e r t h e t wo p r o g r a m s p r o v i d e a s o l u t i o n f o r t h eq u a s i t h r e e - d i m e n s i o n a l , s u b s o n i c , i n v i s c i d f l o w p a t t e r n t h r o u g h a b l a d e r o w.A l t e r n a t i v e l y t h e y c a n b e u s e d s e p a r a t e l y t o f i n d t h e f l o w p a t t e r n w i t h i n a n yg i v e n s t r e a m s u r f a c e .

The purpos e of this Report is to pres ent a full outli ne of the blad e-t o - b l a d e p r o g r a m e s t a b l i s h e d o n t he c o m p u t e r a t t h e C o m p u t e r A i d e d D e s i g nC e n t r e , C a mb r i d g e . S e c t i o n s 2 . 0, 3 . 0 a n d 4 0 d e s c r i b e t h e m a t h e m a t i c a l m o d e l ,t h e fi n i t e d i f f e r e n c e a p p r o x i m a t i o n s a n d t h e s o l u t i o n p r o c e d u r e r e s p e c t i v e l y .T h i s i n f o r m a t i o n i s n ot e s s e n t i a l f o r f o l l o w i n g t h e i n s t r u c t i o n s o n t h e u s eo f t h e p r o g r a m g i v e n i n S e c t i o n 5. 0. T h e e x a m p l e s i n S e c t i o n 6 0 h a v e b e e nchosen to illustra te the range of problem s that the pro gra m can solve.2 . 0 M a t h e m a t i c a l m o d e l

2 .1 S t r e a m f u n c t i o n e q u a t i o nThe equations of motion, continu ity, state and first and second laws

f o r t h e f lo w w i t h i n t h e 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 m a y b e c o m b i n e d t o g i v ea g o v e r n i ng , P o i s s o n - t y p e , d i f f e r e n t i a l e q u a t i o n i n te r m s o f a s t r e a m f u n c t i o n .In theory this equa tion can be solved for any stream surfac e geometry.However , the SI surfaces may be discontin uous at the blade trailing edge andt h i s w o u l d p r o d u c e v e r y c o m p l e x b o u n d a r y c o n d i t i o n s . S m i t h 3 ' 4 t h e r e f o r eintrod uced the assu mpti on that the $I surface is a surface of revolut ion.


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5T h i s i s c o r r e c t a t t h e h u b a n d t h e t i p o f t h e b l a d e , a n d i s c o n s i d e r e d t o b e ac l o s e a p p r o x i m a t i o n o v e r t h e r e s t o f t h e s p a n .

T h e c o o r d i n a t e s y s t e m s h o w n i n F i g u r e 2 i s u s e d i n s e t t i n g u p t h es t r e a m f u n c t i o n e q u a t io n . A n y s t r e a m s u r f a c e ( w h ic h d o e s n o t t u r n t h r o ug h a na n g l e o f 1 8 0 o r m o r e ) c a n b e m a d e s i n g l e v a l u e d i n y b y c h o o s i n g a s u i t a b l ev a l u e f o r 8, a n d h e n c e t h e s a m e e q u a t i o n m a y b e a p p l i e d t o a n y t y p e o ft u r b o m a c h i n e . 4T h e e q u a t i o n f o r t h e s t r e a m s u r f a c e t h e n b e c o m e s :

I ~ ~2 ~V 2 ~ : r-- + ~-~ -- : F (, x) . . . . ( i )

w h e r e1 ~ ~ ~ 0~F (,x) = ~ (in b' 0) ~ + ~ x (in b' 0) ~ x +

b' p (Wy)W x + (W sin % + Wy cos 0)

T h e d e r i v a t i v e s ~ a n d ~ a r e t h o s e w h i c h S m i t h r e f e r s t o a s s p e c i a ld e r i v a t i v e s , t a k e n o n t h e s t r e a m s u r f ac e . T h e s t r e a m f u n c t i o n ~ i s d e f i n e d by :

m

i ~ b' p W .... (2a)r x

~ b' We (2b)

F r o m E q u a t i o n ( 2a ) it c a n b e s e e n t h a t t h e i n t e g r a t i n g f a c t o r b' i s p r o p o r -t i o n a l t o t h e l o c a l t h i c k n e s s o f a t h i n s t r e a m s h e e t w h o s e m e a n s u r f a c e i st h e $I s u r f a c e c o n s i d e r e d .

T h e v e l o c i t y c o m p o n e n t s W x a n d W y a r e r e l a t e d b y t h e e q u a t i o n :

W = - W ta n ~' .... (3)y x

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


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62 . 2 B o u n d a r y c o n d i t i o n sT h e f l o w f i e l d i s s h o w n i n F i g u r e 3 . T h e b l a d e s u r f a c e s a n d l i n e s o f

c o n s t a n t ~ f o r m t h e u p p e r ( A BC D ) a n d l o w e r ( E FG H ) b o u n d a r i e s , a n d l i n e s o fc o n s t a n t x f o r m t h e u p s t r e a m ( AE ) a n d d o w n s t r e a m ( DH ) b o u n d a r i e s . T h e f o l l o w -i n g b o u n d a r y c o n d i t i o n s a r e im p o s e d :

2 . 2 . 1 U p p e r a n d l o w e r b o u n d a r i e sT h e b l a d e s u r f ac e s f o r m s t r e a ml i n e s a n d t h e r e f o r e t h e s t r ea m f u n c t i o n

a l o n g t h e m i s c o n s t a n t . F o r c o n v e n i e n c e t h e d a t u m v a l u e f o r t h e s t r e a mf u n c t i o n i s t a k e n a s z e r o o n th e l o w e r b l a d e s u r f a c e a n d t h i s g i v e s a v a l u e o nt h e u p p e r b l a d e s u r f a c e e q u a l t o Q , t h e m a s s f l o w w i t h i n t h e s t r e a m s h e e t .

O u t s i d e t h e b l a d e r o w th e f l o w p a t t e r n r e p e a t s i t s e l f a t i n t e r v a l s o fo n e b l a d e pi t c h . T h u s t h e f l ow c o n d i t i o n s o n t h e u p p e r a n d l o w e r b o u n d a r i e sa r e t h e s a m e f o r a g i v e n v a l u e o f x f o r t h e r e g i o n s A B F E a n d C D H G a n d a l s o :

( ~ ) u p p e r b o u n d a r y = ( ~ ) l o w e r b o u n d a r y + Q ( at c o n s t a n t x ) . .. . ( 4)

2 . 2 . 2 U p - a n d d o w n s t r e a m b o u n d a r i e sT h e f l o w o n t h e u p - a n d d o w n s t r e a m b o u n d a r i e s h a s b e e n a s s u m e d t o b e

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

( 0 ~ ) b - Q~ xx o u n d a r y - r A ~ t a n ~ b o u n d a r y . . .. ( 5)

3 . 0 N u m e r i c a l t e c h n i q u e sT h e s t r e a m f u n c t i o n E q u a t i o n ( I) is n o t g e n e r a l l y s o l u b l e b y a n a l y t i c

m e a n s a n d so a n u m e r i c a l m e t h o d h a s b e e n u s e d . T h e f l o w r e g i o n i s c o v e r e d b y af i n i t e d i f f e r e n c e g r i d a n d a t e a c h g r i d p o i n t t h e d e r i v a t i v e s a r e e x p r e s s e d i nt e r m s o f t h e v a l u e s o f t h e s t r e a m f u n c t i o n a t n e i g h b o u r i n g p o i n t s . T h e r e s u l t -i n g f i n i t e d i f f e r e n c e e q u a t i o n s f o r m a m a t r i x e q u a t i o n , a n d t hi s is t h e n s o l v e df o r t h e s t r e a m f u n c t i o n .

3 . 1 F i n i t e d i f f e r e n c e g r i dT h e i r r e g u l a r s h a p e o f t he f l o w r e g i o n d o e s n o t s u i t t h e u s e of a c o n -

v e n t i o n a l s q u a r e o r r e c t a n g u l a r f i n i t e d i f f e r e n c e g r i d , b e c a u s e t h e s t a r s o nt h e b o u n d a r i e s w o u l d h a v e s h o r t l im b s a n d c o n s e q u e n t l y l a r g e t r u n c a t i o n e r r o r s .I t i s o n l y t h e b o u n d a r y c o n d i t i o n s w h i c h m a k e a n y p r o b l e m u n i q u e a n d s o t h e s ee r r o r s a r e u n a c c e p t a b l e . T o o v e r c o m e t h is p r o b l e m t h e f l o w f i e l d is c o v e r e db y t h e d i s t o r t e d g r i d s h o w n in F i g u r e 3 . T h e g r i d c o n s i s t s o f s t r a i g h t l i n e s

a s t he b o u n d a r i e s a r e a t l e a s t o n e b l a d e p i t c h a w a y f r o m t h e b l a d e .f l o w a n g l e s a r e s p e c i f i e d , t h e n f r o m E q u a t i o n s ( 2a ) a n d ( 2 b ):


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7n o r m a l t o t h e x a x i s a n d e a c h l i n e h a s t h e s a m e n u m b e r o f p o i n t s e q u a l l y s p a c e db e t w e e n t h e u p p e r a n d l o w e r b o u n d a r i e s . T h e s p a c i n g of t he s t r a i g h t l i n e s m a yb e v a r i e d l o c a l l y t o p r o v i d e a d e t a i l e d f l o w p a t t e r n i n a n y r e g i o n ( e . g . n e a rt h e l e a d i n g e d g e ) .

3 . 2 F i n i t e d i f f e r e n c e a p p r o x i m a t i o n sT h e s t r e a m f u n c t i o n e q u a t i o n i s d e f i n e d i n te r m s o f s p e c i a l d e r i v a t i v e s

t a k e n o n t h e s t r e a m s u r fa c e . T h e f i n i t e d i f f e r e n c e g r i d a l s o l i es i n t h iss u r f a c e a n d h e n c e t h e s p e c i a l d e r i v a t i v e s m a y b e t r e a t e d a s n o r m a l p a r t i a ld e r i v a t i v e s .

3 . 2 . 1 L _ ~ l a c i a n o p e r a t o rW h e n t h e f u n c t i o n V 2 ~ i s e x p a n d e d a s a t w o - d i m e n s i o n a l T a y l o r s e r i es , i t

i s f ou n d t h a t t h e t e n p o i n t s t a r s h o w n i n F i g u r e 4 a is r e q u i r e d t o g i v e a na c c u r a c y o f 0 [ (r ~ ) 2 ] o r 0 [ ~ x 2 ]. H o w e v e r t h i s s t a r i n c lu d e s p o i n t s o n f o u rs t r a i g h t l i n e s a n d s o w o u l d g i v e a _ l a r g e b a n d w i d t h f o r th e m a t r i x . I f t h ec o n d i t i o n t h a t t h e c o e f f i c i e n t o f ~ x m u s t b e z e r o i s r e p l a c e d b y t h e c o n d i t i o nt h a t t h e c o e f f i c i e n t o f ~ y - ~ F m u s t b e z e r o, t h e n t h e s t a r s h o w n i n F i g u r e 4 bm a y b e u s e d. T h e s t r e a m f u n c t i o n e q u a t i o n t h e n b e c o m e s :

V ~ + E ~ x = F (~,x) + E . . . . ( 6 )

= F ' ( ~ , x )

w h e r eE = 2 ( I I Ixi+ I - x i x. - x.i i--]

3 . 2 . 2 T h e d e r i v a t i v e ~x xT h e f i r s t d e r i v a t i v e w i t h r e s p e c t t o x is r e q u ir e d :( i) t o e v a l u a t e t h e r i g h t h a n d s i d e o f t h e s t r e a m f u n c t i o n

e q u a t i o n ( E q u a t i o n ( 6 ) )( ii ) t o f i n d t he c o m p o n e n t o f v e l o c i t y i n t h e ~ d i r e c t i o n

( E q u a t i o n ( 2 b ) )a n d ( ii i) t o i n c o r p o r a t e t h e u p - a n d d o w n s t r e a m b o u n d a r y c o n d i t i o n s

( E q u a t i o n ( 5 ) ) .T e n p o i n t s t a r s o f t h e t y p e s h o w n i n F i g u r e 5 a r e u s e d i n t h e f i r s t t w o c a s e s ,g i v i n g a n a c c u r a c y o f O [ 6 x ~ ] . H o w e v e r i f t h e s e s t a r s w e r e u s e d t o

v 2 ~


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8i n c o r p o r a t e t h e b o u n d a r y c o n d i t i o n s , t h e n t he v a l u e o f t he s t r e a m f u n c t i o n o nt h e b o u n d a r y w o u l d d e p e n d o n t h e v a l u e s a t p o i n t s o n t h e t h r e e a d j a c e n t s t r a i g h tl i ne s . T h e r e s u l t w o u l d b e a n i nc r e a s e i n t h e b a n d w i d t h o f t h e m a t r i x o p e r a t o rfor ~oin ts on the stra ight lines i = 2 and i = m - i. To avoid this the valueo f ~ x i s a s s u m e d t o b e c o n s t a n t n e a r t h e u p s t r e a m b o u n d a r y a n d t o v a r y l l n -e a r l y w i t h x n e a r t h e d o w n s t r e a m b o u n d a r y . T h e e r r o r s i n t r o d u c e d b y t h i s a r en o r m a l l y s m a l l, b u t f o r f l ow s w i t h r a d i a l c o m p o n e n t s , i t s h o u l d b e n o t e d t h a ta n g u l a r m o m e n t u m w i l l n o t b e c o n s e r v e d b e t w e e n t h e f i r s t t w o a nd b e t w e e n t h elast two strai ght li nes.

3 . 2 . 3 T h e d e r i v a t i v e ~T h e f i r s t d e r i v a t i v e w i t h r e s p e c t t o ~ i s r e q u i r e d :(i) t o e v a l u a t e th e r i g h t h a n d si d e o f t h e s t r e a m f u n c t i o n e q u a t i o n

( E q u a t i o n ( 6 ) )( ii ) t o f i n d th e c o m p o n e n t o f v e l o c i t y i n t h e x d i r e c t i o n

(Equation (2a)).T h e g r i d s p a c i n g i s u n i f o r m i n t h e ~ d i r e c t i o n a n d s o s i m p l e a p p r o x i m a t i o n s m a yb e u s e d :G e n e r a l p o i n t s

? , j

P o i n t s o n b l a d e s u r f a c e s

fi,j+1 - fi,j-1 + 0 [ (r 6~)2 ]2r (6~)i, j . . . . ( 7 )

L o w e r

U p p e r

( ~ l i - 3 f. - f.~f 4fi,2 1,1 1,3r ,l 2( r 6~)i, I + 0 [ (r $ q ~ ) 2 ]

= l , n - 2 l,n-I l,n + O [ ( r ~ ) 2 ]r i , n 2 ( r 6 ~ ) i , n

. . . . ( 8 )

3 . 3 B a n d m a t r i xT h e f i n i t e d i f f e r e n c e e q u a t i o n s f o r t h e o p e r a t o r ( V + E ~ x ) a t e a c h

i n d e p e n d e n t ( i . e . ~ is n o t d e t e r m i n e d b y t h e b o u n d a r y c o n d i t i o n s ) g r i d p o i n tf o r m a m a t r i x o f o r d e r :

( m - 2) ( n - i) - ( t - ~ - I) 1


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E q u a t i o n (6) t h u s b e c o m e s :M ~ = F

T h e g r i d p o i n t s a r e a r r a n g e d i n th e o r de r :(i,j) = (2,1), (2,2) ..... ( 2 , n - i), (3,1)

.... (9)

(m - I, n - I)

s o t h a t t h e m a t r i x M i s b a n d e d . T h e m a x i m u m b a n d w i d t h o c c u r s f o r p o i n t so u t s i d e t h e b l a d e r o w o n th e u p p er a n d l o w e r b o u n d a r i e s o f t he c a l c u l a t i n gr e g i o n ( F i g u re 6) . U s i n g t h e r e p e a t b o u n d a r y c o n d i t i o n , t h e m a x i m u m d i f f e r -e n c e b e t w e e n t h e n u m b e r s o f t he c e n t r e a n d a n o t h e r p o i n t o n t h e s t a r is( 2 n - 3 ) , g i v i n g a b a n d w i d t h o f ( 4 n - 5 ). O n l y t h e e l e m e n t s w i t h i n t h i s b a n da r e s t o r e d ( F i g u r e 7) .4 . 0 S o l u t i o n p r o c e d u r e

T h e m a t r i x E q u a t i o n (9) i s n o n l i n e a r a n d t h e r e f o r e i t i s s o l v e d b y a ni t e r a t i v e p r o c e s s :

(i ) t h e r i g h t h a n d s i d e i s a s s u m e d c o n s t a n t a n d t h e l i n e a re q u a t i o n

M ~ = F = c o n s t a n ti s s o l v e d f o r

( ii ) t h e s o l u t i o n f o r t h e s t r e a m f u n c t i o n is u s e d t o c a l c u l a t et h e ga s s t a t e a t e a c h g r i d p o i n t

( i ii ) a n e w v a l u e f o r t h e r i g h t h a n d s i d e is c a l c u l a t e d f r o m t h eg a s s t a t e

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

Q i,j

. . . . ( l O )

w h e r e p i s t he c u r r e n t i t e r a t i o n , o r t h e s p e c i f i e d n u m b e r o f i t e r a t i o n s h a v eb e e n c o m p l e t e d .

T h e f i r s t r u n o f a n y g e o m e t r y r e q u i r e s a n i n i t i a l g u e s s t o s t a r t t h ei t e r a t i o n . F o r c a s c a d e o r c y l i n d r i c a l s t r e a m s u r f a c e s , t h e f l o w i s a s s u m e dt o b e u n i f o r m b e t w e e n t w o d i v i d i n g s t r e a m l i n e s w h i c h a r e s t r a i g h t u p - a n dd o w n s t r e a m o f t h e b l a d e r o w a n d f o l l o w t h e b l a d e s u r f a c e s w i t h i n t h e ro w.T h i s p r o v i d e s a n i n i t i a l g u e s s f o r t h e s t r e a m f u n c t i o n a n d s o t he p r o b l e me n t e r s t h e i t e r a t i v e l o o p a t p o i n t ( ii ) . F o r o t h e r f l o w s t h e r i g h t h a n d s i d ei s s e t t o z e r o a n d t h e l o o p i s e n t e r e d a t p o i n t ( i ) .

~ < T O L . . . . ( l l )


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I0The final soluti on for the stream function is stored on magne tic tape

(see App endi x I) so that it can be used as the start ing poi nt for su bseq uentruns of the same geom etry (i.eo enter loop at point (ii))~ Com pre ssi ble runsrequire a fairly accurate first guess if the problem is to converge, and sothey are always started from a previous solution.

4 ol S o l u t i o n o f m a t r i x e q u a t i o nT h e m a t r i x M i n E q u a t i o n ( i 0 ) i s o f o r d e r { ( m - 2) ( n - I) - ( t - ~ - i )}

and bandw idth (4n - 5): t ypical value s of m and n are 90 and 13 resp ectiv ely,givin g an order of ~i000 and ban dwi dth of 47. This is too large to be kep t inthe core store of present day computers and therefore use has been made of twomatri x solving routines whic h operate on a small section of the matr ix at atime (Figure 7) whi le the rest is kep t on bac kin g stores:

(i) Routine I splits the matr ix into the product of upper andlower (band) tri angu lar matr ices , so that

U L ~ = F .... (12)(i i) Rout ine 2 solves this eq uat ion for $._ usin g a forwa rd and

b a c k w a r d s u b s t i t u t i o n p r o c e s s.To ensure numerical stability, under r elaxat ion is used in calculati ng the newvalue of the stream function from this solution:

~p = ~p-1 + rf ( * - $p-I) .... (13)

where ~ is the solution to Equa tion (I0).The relax ation factor rf lies in the range 0.2 to 0.8, dep ending on the flowMach number.

4.2 Calc ulati on of sas stateGiven a soluti on for the stream func tion throug hout the flow field, theprodu cts 0Wx, 0Wy and 0W~ may be calculated_ from_Eq uati ons (2) and (3 ), usin g

athe finite differ ence appr oximation s for ~- and ~x ~ The prob lem is then tocalculate the static density 0.

For isentropic flow:I

0 _ _ _ _ . . . . ( 1 4 )0 o

This equa tion for O could be solved directly since the stagnation conditions atinlet are known and the local static ent halpy may be found in terms of p.


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II

H o w e v e r , S m i t h h a s f o u n d t h a t a m o r e s t a b l e a p p r o a c h i s t o l e t th e d e n s i t y l a gb y o n e i t e r a t i o n .

F o r t h e s u b j e c t b l a d e - t o - b l a d e f l o ws , i t c a n b e s h o w n 3 ' 4 t h a t t h er o t h a l p y i s c o n s e r v e d a l o n g s t r e a m li n e s :

2~o r V~I = H - = c o n s t a n tG . . . . ( 1 5 )

U s i n g t h e e n e r g y e q u a t i o n

h

E q u a t i o n ( 15 ) b e c o m e s :

V 2G

h = I + ~ 2 r 2 WG G . . . . ( 1 6 )

T h e r e f o r e , s u b s t i t u t i n g i n E q u a t i o n ( 14 ):I

P - (a - b) v-1Po

. . . . ( 1 7 )

w h e r e a =2 r 2I + - -G

H o

a n d b w =G H o

T h e t e r m W 2 is f o u n d f r o m t h e r e l a t i o n :

2 + 2

u 2 _- 2P P p - 1 . . . . ( 1 8 )

T h i s i s w h a t i s m e a n t b y t h e d e n s i t y l a g g i n g b y o n e i t e r a t i o n .o v e r a l l c a l c u l a t i o n c o n v e r g e s , Pp -1 a p p r o a c h e s p p.

T h e v e l o c i t y c o m p o n e n t s a r e t h e n c a l c u l a t e d u s i n g P p.

A s t h e


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12W u h a s s h o w n t h a t t h e c o n d i t i o n f o r t he s t r e a m f u n c t i o n e q u a t i o n t o

r e m a i n e l l i p t i c i s t h a t t he l o c a l r e l a t i v e M a c h n u m b e r i s l e s s t h a n u n i t y .T h e r e f o r e i f t h e l o c a l f l o w b e c o m e s s u p e r s o n i c , t h e d e n s i t y i s s e t t o t h ev a l u e c o r r e s p o n d i n g t o a M a c h n u m b e r o f u n i t y . T h i s a r t i f i c e e n a b l e s s o l u t i o n st o b e o b t a i n e d f o r t r a n s o n i c f l o w s : t h e s o l u t i o n s a r e n o t s t r i c t l y v a l i d , b u tt h e y h a v e b e e n f o u n d t o b e u s e f u l ( s e e S e c t i o n 6 . 7 ).

T h e v a l u e o f v e l o c i t y f o r u n i t y M a c h n u m b e r i s g i v e n b y :

W 2 = ( y - I) G h (i9)M = I - - . . . .2

S u b s t i t u t i n g f o r h f r o m E q u a t i o n ( 16 ) a n d r e a r r a n g i n g :

2 = y - I (G I + m2 r 2) (20)M=I y + I ....

H e n c e t h e c o m p l e t e s o l u t i o n f o r t h e s t a t i c d e n s i t y i s:

I

P = (a - b) y-1Po . . . . ( 21)

w h e r e G I + m2 rG H

o

a n d b _ W 2 f o r W 2 f ~ y - IG H G H ~ a ~ jo o

(y---$--~) W2 > a y - i= a - i f o r G H O ( y - - ~ )

4 . 3 C a l c u l a t i o n o f n e w r i g h t h a n d s i d eG i v e n t h e v a l u e s o f t h e s t r e a m f u n c t i o n , d e n s i t y a n d c o m p o n e n t s o f

v e l o c i t y a t e a c h g ri d p o i n t , t h e n e w r i g h t h a n d s i d e m a y b e c a l c u l a t e d u s i n gt h e f i n i t e d i f f e r e n c e s t a r s f or ~ x a n d ~ .


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135 . 0 T h e c o m p u t e r p r o s r a m

T h e m a t r i x b l a d e - t o - b l a d e p r o g r a m h a s b e e n w r i t t e n i n s e g m e n t e d f o r m(Appendix II) to economi se on core store. For a max imu m grid size of m = 90and n = 13, each segment require s 2OK words of core or less. This grid sizehas proved ample for all types of axial blade rows and for simple radial flowmachines. If necessa ry the segments can be re-dim ension ed to allow a finergrid for mixed flow problems, though this will i ncrease the cost

5. I D a t a p r e p a r a t i o nThe data required by the prog ram falls into three sections: titles,

blade and stream surface geometry, and flow data. The specifications of eachare given in App endi x IIIo Partic ular care should be taken to ensure thatthe blade data are smooth, since this is a potentia l flow solu tion - example64 bel ow i llustrates the effect of sudden changes in the blade surface curva-ture on the solution. Limita tions on the grid geometry and suggest ed spa cingsfor the straight lines are given in Figures 8 and 9.

Two data prepa ratio n programs are available to prepare the geomet ricdata for cascade flow problems The first (Appendix IV) is suitable forblades with blunt leading edges (Figure I0) which are specifie d by a tableof coordinates. The second (App endix V) produces data for double circu lar arccompressor blades

Each new blade geom etry and each solutio n are automa ticall y stored on amaster magnet ic tape (henceforth called the progr am tape) (Appendix I).Furthe r runs using the same geometry will then read this section of the datafrom the prog ram tape (i.eo only the title and flow data sections arerequired) and they may also read a previous soluti on to use as a first guessto start the solution procedure. In order to safeguar d this informa tion it isadvisable to keep at least one copy of this tape, and to update it regula rlyusing the 'COPY' command.

5 2 R u n n i n g i n s t r u c t i o n sThe job has been split into three parts, to suit the CADC Atlas, but

the user sets up only the first; the other two are run automaticallyT y p i c a l r u n n i n g i n s t r u c t i on s a r e g i v e n i n A p p e n d i x V l .5 . 3 O u t p u tThe progra m produces output on both the line printer and the graph

plotter The titles used are defined in the Notation.


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145 . 3 . 1 L i n e p r i n t e r o u t p u t

(i ) I n p u t d a t a. T h e p r o b l e m s p e c i f i c a t i o n a n d i n p u t d a t a a r e p r i n t e do u t b y p r o g r a m 4 F I R S T .

( i i ) C o n v e r g e n c e r e c o r d . T h e m a x i m u m d i f f e r e n c e ( E q u a t i o n ( ii ))b e t w e e n t w o s u c c e s s i v e s o l u t i o n s f o r t h e s t r e a m f u n c t i o n i sp r i n t e d o u t b y p r o g r a m S O L U T I O N .

( ii i) R e s u l t s . T h e r e s u l t s a r e p r i n t e d o u t b y p r o g r a m 6 R E S . I f t h ep r o b l e m h a s c o n v e r g e d t o t h e s p e c i f i e d t o l e r a n c e , t h e y a r ep r o d u c e d f o r t h e l a s t i t e r a t i o n o n l y : f o r a p r o b l e m w h i c h h a sn o t c o n v e r g e d t o t h e s p e c i f i e d t o l e r a n c e t h e y a r e p r o d u c e d f o rt h e l a s t t w o i t e r a t i o n s . I f f u l l o u t p u t i s s p e c i f i e d t h e n t h ef l o w c o n d i t i o n s a r e c a l c u l a t e d a t e a c h g r i d p o i n t ( A p p e n d i x V I I ) .O t h e r w i s e o n l y t h e b l a d e s u r f a c e v e l o c i t y d i s t r i b u t i o n i sp r i n t e d o u t .

( i v ) S u m m a r y o f p r o g r a m t a p e. T h e p r o g r a m w r i t e s o u t a l i s t o f t h et i t l e s o f t h e a r r a y s s t o r e d o n t h e p r o g r a m t a p e ( A p p e n d i x I ).5 . 3 . 2 G r a p h p l o t t e r o u t p u t

T h e g r a p h p l o t t e r i s u s e d t o p l o t o u t t h e b l a d e s u r f a c e v e l o c i t y a n dM a c h n u m b e r ( c o m p r e s s i b l e r u n s o n ly ) d i s t r i b u t i o n s T h e s t r e a m l i n e p a t t e r n i sc a l c u l a t e d u s i n g l i n e a r i n t e r p o l a t i o n f o r t h e s t r e a m f u n c t i o n b e t w e e n t h e g r i dp o i n t s o n e a c h s t r a i g h t l i n e, a n d t h e s t r e a m l i n e s a r e p l o t t e d o u t i n t h eM - P H I p l a n e .

5 . 3 . 3 D i a g n o s t i c o u t p u tT h e r u n n i n g i n s t r u c t i o n s a r e w r i t t e n t o t h e f i l e s p e c i f i e d a s t h e y a r e

e x e c u t e d , a n d p r o g r a m w a r n i n g a n d e r r o r m e s s a g e s w i l l a l s o b e w r i t t e n t o t h i sf i l e ( se e A p p e n d i x V I I I ) .6 . 0 N u m e r i c a l e x a m p l e s

T h e m a t r i x p r o g r a m m a y b e u s e d t o a n a l y s e t h e i n v i s c i d , s u b s o n i c c o m -p r e s s i b l e f l o w a r o u n d t h e b l a d e s o f a n y c a s c a d e o r t u r b o m a c h i n e . T h e f o l l o w i n ge x a m p l e s h a v e b e e n c h o s e n t o i l l u s t r a t e s o m e of t he p r o b l e m s w h i c h c a n b es o l v e d . I n t h e f i r s t f o u r c a s e s m a t r i x s o l u t i o n s a r e c o m p a r e d w i t h s o l u t i o n sf r o m o t h e r t h e o r e t i c a l m e t h o d s w h i c h m a k e t h e s a m e b a s i c a s s u m p t i o n s a b o u t t h ef l ow . T h i s s h o w s t h at t h e m a t r i x p r o g r a m w o r k s c o r r e c t l y a n d d e m o n s t r a t e s t h eo v e r a l l l e v e l o f a c c u r a c y w h i c h i s a c h i e v e d . T h e f if t h e x a m p l e w a s c h o s e n t os h o w t ha t t h e p r o g r a m c a n a l s o d ea l w i t h r e l a t i v e e d d y m o t i o n . F i n a l l y t h es i x t h a n d s e v e n t h c a s e s g i v e c o m p a r i s o n s w i t h e x p e r i m e n t a l r e s u l t s t o d e m o n -s t r a t e t h e a g r e e m e n t a c h i e v e d w i t h r e a l fl o w s


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156.1 Impulse turbine cascadeThe first example is for simple, i ncompr essibl e flow through a two-

dimen sional cascade The profil e is typical of an impulse type, turbineblade, w ith 112 camber and the cascade has a pitch /chor d ratio of 059 andI01 flow deflection. Goste low 5 has obta ined an exact solutio n for this case,a n d t h e m a t r i x p r o g r a m w a s r u n u s i n g t he s a m e f l o w a n g le s . T h e tw o b l a d esurface velocit y dist ributi ons are compared in Figur e ii: they show excellen tagreement.

6.2 70 camber blade in cascadeThe second example is also for the incomp ressibl e flow through a two-

dimens ional cascad e, but at very high negat ive incidence (-700) The profil eis a turbine bl ade wit h 70 camber, and the cascade has a pitc h/cho rd ratio of0.9.

T h e m a t r i x s o l u t i o n i s c o mp a r e d w i t h a n e x a c t s o l u t i o n b y G o s t e l o w 5 i nFigure 12: the only discrep ancy occurs in the region of the blade trailingedge. For this region the exact profi le co ordinates are a long way apart andit is proba ble that errors in interp olatin g for the coordinates for the mat rixprog ram have caused the difference. There seems no reason able doubt thatcomplete agreeme nt would have been obtained if the exact aerofoil shape hadb e e n m o r e f u l l y d e f i n ed .

This examp le is far more severe t han could occur in practice, but itshows that there is no prob lem in analy sing high incidence flows. The stream-l i n e p a t t e r n c a l c u l a t e d b y t h e m a t r i x p r o g r a m i s s h ow n i n F i g u r e 1 3 a n d i t m a ybe seen that the leading edge s tagna tion poi nt is well round on the suctionsurface.

6.3 Conical flow past stationa ry and rotatin g blade rowsThe above compariso ns are for plane cascade flows. Three -dime nsion al

effects have two main forms; cha nge of stream surface thickness, and change ofstream surface radius wit h the result ing Coriolis forces for rotor rows. Tocheck that the prog ram deals correct ly with these aspects it was used to ana-lyse the incom press ible flow on a conical stream surface through a turbine rowfor both station ary and rotating cases. The cone half an gle was 45 and thes t r e a m s u r f a c e t h i c k n e s s v a r i e d l i n e a r l y w i t h a x i al d i s t a n c e , g i v i n g a ni n c r e a s e o f 4 4 .6 p e r c e n t b e t e e n t h e u p - a n d d o w n s t r e a m b o u n d a r i e s .

6T h e m a t r i x s o l u t i o n s a r e c o m p a r e d w i t h t h e ex a c t W i l k i n s o n / M a r t e n s e nsingular ities solutions in Figures 14 and 15. The absciss a ~ is defined as


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16

/ ! dm , an d g i ve s a c o n f o r m a l t r a n s f o r m a t i o n o f th e f l o w i n t h e m e r i d i o n a lj rs t r e a m s u r f a c e ( w h i c h g e n e r a l l y h a s d o u b l e c u r v a t u r e ) o n t o a f l a t p l a n e i nw h i c h a l l a n g l e s a r e p r e s e r v e d . I n b o t h t h e s t a t i o n a r y a n d r o t a t i n g c a s e s t h eo n l y s i g n i f i c a n t d i f f e r e n c e b e t w e e n t h e t w o s o l u t i o n s i s a t t h e l e a d i n g e d g e ,a n d t h i s is p r o b a b l y d u e t o t h e c o a r s e g r i d u s e d f o r t h e m a t r i x s o l u t i o n .(The spacing at the lea ding edge was set to 4.2 per cent of an axial chor d toa v o i d t h e n e e d f o r i n t e r p o l a t i o n b e t w e e n t h e d a t a p o i nt s . )

6 .4 A x i a l c o m p r e s s o r s t a t o r b l a d e sT h e s e c o m p a r i s o n s a r e f o r th e c o m p r e s s i b l e f l o w p a s t t w o c o m p r e s s o r

b l a d e pr o f i l e s w h i c h w e r e d e s i g n e d u s i n g a p r e s c r i b e d v e l o c i t y d i s t r i b u t i o n7(PVD) pro gra m . In both cases the duty is:I n l e t M a c h n u m b e rI n l e t f l o w a n g l eO u t l e t f l o w a n g l eP i t c h / c h o r d r a t i o

= O . 8 5= 5 0 = 0 0

= 0.45T h e P V D p r o g r a m c a l c u l a t e s t h e e f f e c t i v e s h a p e r e q u i r e d t o fu l f i l t h i s

d u t y a n d p r o d u c e t h e s p e c i f i e d v e l o c i t i e s . T h e b l a d e s u r f a c e b o u n d a r y l a y e rsa r e t h e n e s t i m a t e d a n d s u b t r a c t e d f r o m t h e e f f e c t i v e s h a p e t o g i v e t h e b l a d ep r o f i l e .

T h e b l a d e s u r f a c e M a c h n u m b e r d i s t r i b u t i o n s c a l c u l a t e d b y t h e m a t r i xp r o g r a m a r e c o m p a r e d w i t h t h o s e s p e c i f i e d a s in p u t t o t h e P V D p r o g r a m i nF i g u r e s 1 6 a n d 1 7. T h e r e i s g e n e r a l l y g o o d a g r e e m e n t , e x c e p t t h a t th e m a t r i xs o l u t i o n i s n o t q u i t e s m o o t h a n d i t p r e d i c t s l o w e r M a c h n u m b e r s o v e r t h e b a c kh a l f of t h e b l a d e s . T h e s e d i f f e r e n c e s o c c u r b e c a u s e t h e m a t r i x p r o g r a m i su s i n g t h e b l a d e p r o f i l e a n d n ot t h e e f f e c t i v e s h a p e f o u n d b y t h e P V D p r o g r a m :t h e m a t r i x s o l u t i o n i s t h e r e f o r e f o r a t h i n n e r p r o f i l e w h i c h i s n o t p e r f e c t l ysmooth.

T h e s e e x a m p l e s i l l u s t r a t e t h e a b i l i t y o f t h e m a t r i x p r o g r a m t o p r o d u c ea n s w e r s f o r f l o w s w i t h s u p e r s o n i c p a t c h e s .

6 . 5 R a d i a l i n f l o w t u r b i n eT h e f i f t h e x a m p l e w a s c h o s e n t o i l l u s t r a t e t h e a b i l i t y o f t he m a t r i x

p r o g r a m t o d e a l w i t h r e l a t i v e e d d y m o t i o n . I t i s a s i m p l i f i e d c a s e o f ar a d i a l f l o w m a c h i n e w i t h r a d i a l b l a d e s o f z er o t h i c k n e s s 8 T h e f l o w i s i n c o m -p r e s s i b l e a n d h a s n o a x i al c o m p o n e n t .

T h e b l a d e r e l a t i v e v e l o c i t y d i s t r i b u t i o n s a n d t h e s t r e a m l i n e p a t t e r n(plotted in the r-r~ plane) are shown in Figu res 18 and 19. Both clearly sh owt h e r e g i o n o f r e v e r s e f l o w.


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176 . 6 N A S A t u r b i n e s t a t o r b l a d eT h i s e x a m p l e c o m p a r e s a m a t r i x s o l u t i o n w i t h t h e e x p e r i m e n t a l r e s u l t s

for the compre ssible flow through an annular cascade. The profil e is the midsection of a NASA turbine stator b lade 9 and it is typical of a high tempera-ture section in havin g blunt leading and tra iling edges to allow for coolingpassages.

The blade was anal ysed for the design value of inlet Mac h number (0.212)and inlet and exit gas angles (0 and -67o). The flow was assumed to be two-d i m e n s i o n a l. T h e c o m p u t e d d i s t r i b u t i o n s o f b l a d e s u r f a c e M a c h nu m b e r ar ec o m p a r e d w i t h t h e v a l u e s c a l c u l a t e d f r o m th e m e a s u r e d s t a t i c p r e s s u r e d i s t r i -buti on in Figure 20 There is generall y good agreement, the main discrepan ciesoccurr ing over the last 50 per cent of the blade suct ion surface. This regionis boun ded by the trailing edge stagnat ion streamli ne and it is conside red thatthe errors a re due to the fact that the blade surface bound ary layers aren e g l e c t e d i n t h e m a t r i x a n a ly s i s . T h e o s c i l l a t i o n o f t h e c o m p u t e d d i s t r i b u t i o nin this area is caused by the blade surface being slightly uneven. It is notappare nt in the experime ntal results becau se of the smoothi ng effect of theb o u n d a r y l a y e r .

6.7 Axial turbine stator bladeThe last example is for compressi ble flow through a cascade of turbine

b l a d e s w i t h f l o w c o n t r a c t i o n d u e to w a l l b o u n d a r y l a y e r g ro w th . T h e b l a d eprofi le was the mid- span sectio n of the second stage stator of a two stageturbine IO. The experimental cascade resu lts for blade surface Mach numbera r e c o m p a r e d w i t h t w o m a t r i x s o l u t i o n s i n F i g u r e 2 1.

T h e f i r s t m a t r i x s o l u t i o n u s e d t h e o u t l e t a n g l e p r e d i c t e d b y t h eA i n l e y / M a t h i e s o n m e t h o d ( - 4 9. 9 5 c o m p a r e d w i t h t h e e x p e ri m e n t a l v a l u e o f-48.56 ) and assume d that the strea m surface thickness was constant. There isreaso nable agree ment over the first third of the blade, but the matr ix s olutio nt h e n d i v e r g e s f r o m t he e x p e r i m e n t a l r e s u l t s a s t h e b o u n d a r y l a y e r b l o c k a g eincreases.

For the second solution, the stream surface thicknes s was varied inorder to give the meas ured value of axial veloc ity ratio (this required acontr actio n of 8 per cent), and the experim ental value for the outlet an glewas used. Since there was no soluti on available for the $2 (hub-to-tip) plane,the strea m surface thickness wi thin the blade ro w was assumed to vary line arlywit h axial distance Comp ariso n of this solutio n with the experime ntalresults shows that the predic ted Ma ch number over the first half of the


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18

s u c t i o n s u r f a c e is t oo h i g h . A p a r t f r o m t h i s t h e a g r e e m e n t i s g o o d a n d t h ep e a k M a c h n u m b e r i s w e l l p r e d i c t e d , d e s p i t e t h e s u p e r s o n i c p a t c h c o v e r i n gn e a r l y ha l f o f t he s u c t i o n s u r f ac e . T h i s s u g g e s t s t h a t m o s t o f t h e s t r e a ms u r f a c e c o n t r a c t i o n s h o u l d i n f a c t b e c o n c e n t r a t e d o v e r t h e s e c o n d h a l f o f t heb l a d e i n th e r e g i o n o f s u c t i o n s u r f a c e d i f f u s i o n T h e c h a n g e s i n s lo p e o f t h em a t r i x s o l u t i o n s a t M = I a r e d u e t o t h e a s s u m p t i o n t h a t t h e d e n s i t y i s c o n -stant for M > I (Sec tion 4.2).

T h i s c o m p a r i s o n w i t h e x p e r i m e n t a l r e s u l t s i l l u s t r a t e s t h a t s o l u t i o n sw i t h t r a n s o n i c f l o w s c a n b e p h y s i c a l l y m e a n i n g f u l .7 . 0 C o n c l u s i o n s

A c o m p u te r p r o g r a m f o r a n a l y s i n g t h e b l a d e - t o - b l a d e f l o w p a t t e r n s i nt u r b o m a c h i n e s h a s b e e n d e s c r i b e d . T h e t h e o r y i s b a s e d o n t h e e a r l i e r w o r k o fW u I a n d t h e n u m e r i c a l s o l u t i o n i s o b t a i n e d b y f i n i t e d i f f e r e n c e a p p r o x i m a t i o n sto the gove rnin g equations .

T h e p r o g r a m h a s b e e n d e m o n s t r a t e d t o p r o d u c e g o o d e s t i m a t e s f o r a w i d er a n g e o f f l o w s , i n c l u d i n g t h e e f f e c t s o f c o m p r e s s i b i l i t y , s t r e a m t u b e c o n t r a c -t io n, c h a n g e o f s t r e a m t u b e r ad i u s , C o r i o l i s f o r c e s a n d r e l a t i v e e d d y m o t i o n

A l t h o u g h s t r i c t l y o n l y v a l i d f o r s u b s o n i c f l o ~ s , t h e p r o g r a m c a n p r o -d u c e s o l u t i o n s f o r f lo w s w i t h s u p e r s o n i c p a t c h e s , a n d t h e r e s u l t s a g r e e w e l lw i t h o t h e r t h e o r e t i c a l s o l u t i o n s a n d e x p e r i m e n t a l d a t a


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.Coordinate s ystem s (Figur e 2)N O T A T I O N

r , ~ , z r a d i a l , c i r c u m f e r e n t i a l a n d a x i a l c o o r d i n a t e s ( ~ i s m e a s u r e di n d i r e c t i o n o f r o t a t i o n)x , y C a r t e s i a n c o o r d i n a t e s o b t a i n e d b y r o t a t i n g t h e r , z a x e s

t h r o u g h a n a n g l e eF i n i t e d i f f e r e n c e g r i d ( F i g u r e 3)m t o t al n u m b e r o f s t r a i g h t l i n e sn n u m b e r o f g r i d p o i n t s p e r s t r a i g ht l i n e

n u m b e r o f s t r a i g h t l i n e s u p s t r e a m o f b l a d e r o wt n u m b e r o f s t r a i g h t l i n e s u p s t r e a m o f b l a d e r o w, p l u s n u m b e r

o f s t r a i g h t l i n e s w i t h i n b l a d e r o w , p l u s o n en u m b e r o f a g e n e r a l s t r a i g h t l i n e ( s t a r ti n g f r o m u p s t r e a mb o u n d a r y )n u m b e r o f a g r i d p o i n t a l o n g a s t r a i g h t l i n e ( s t a r t i n g a tl o w e r b o u n d a r y )

~ l o c a l g r i d s p a c i n g i n d i r e c t i o nd x l o c a l g r i d s p a c i n g i n x d i r e c t i o ng ~ b l a d e p i t c h i n r a d i a n s ( = 2 ~ / n u m b e r o f b l a d e s )S.tream. su rfa ce (Fi gure 2)

u n i t v e c t o r n o r m a l t o s t r e a m s u r f a c ea n g l e o f s t r e a m s u r f a c e ( i n m e r i d i o n a l p l a n e ) t o z a x i s

b i n t e g r a t i n g f a c t o r u s e d i n d e f i n i n g s t r e a m f u n c ti o n ;p r o p o r t i o n a l t o t h e l o c a l r a d i a l t h i c k n e s s o f t h e s t r e a ms u r f a c e

F l u i d p r o p e r t i e s

W

VL0

M

PPoH

Ho

h

s t r e a m f u n c t i o nr e l a t i v e v e l o c i t ya b s o l u t e v e l o c i t ya n g u l a r v e l o c i t yr e l a t i v e M a c h n u m b e rs t a t i c d e n s i t ys t a g n a t i o n d e n s i t y ( a b s o l u t e) a t i n l e ts t a g n a t i o n e n t h a l p y ( a b s o l u t e )s t a g n a t i o n e n t h a l p y a t i n l ets t a t i c e n t h a l p y

19


21/69

20IQ

G

O t h e r

X COORDYM COORD

P HIL A M B D AS T H I C KB T H I C KVXVYW UVUW

VABS MREL MS T R F NSTAG HSTAT HSTAG PR

STAT PRDENSITYABS ANG

rothalpymass flow with in stream surfaceflow angle (relative)ratio of specific heatsconstant relating velo city and enthalpy

H---h

rf relaxatio n factorp number of current iteration' prime d values are rela tive to the xy axes,C0mputer output

x coordinate of calculating planey coordinate of calculating planedistance of calculati ng plane fr om upstrea m boundary, mea suredalong a meridi onal streamline

coord inate of a grid point (in radians)tan ~' (ieo meas ured from x axis)stream surface thickne ss/upst ream thicknessblade thickness in radianscomponent of velocity in x direct ioncomponent of velocity in y directi oncomponent of relative velocity in ~ directioncomponent of absolute velo city in ~ directionr e l a t i v e v e l o c i t ya b s o l u t e v e l o c i t yabsolute Mach numberr e l a t i v e M a c h n u m b e rstream function, non dimensionali sed by the mass flow Qa b s o l u t e s t a g n a t i o n e n t h a l p ys t a t i c e n t h a l p ya b s o l u t e s t a g n a t i o n p r e s s u r e / a b s o l ut e s t a g n a t i o n p r e s s ur e a tinletstatic press ure/a bsolu te stagnati on pressure at inletstatic densityabsolute flow angle = tan I ~ U~\vxj


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R E L A N GV/VD

W/WD

C P

SL

r e l a t i v e f l o w a n g l e = t a n -I ( ~ X )a b s o l u t e bl a d e s u r f ac e v e l o c i t y / a b s o l u t e d o w n s t r e a mv e l o c i t yr e l a t i v e bl a d e s u r f ac e v e l oc i t y / r e l a t i v e d o w n s t r e a mv e l o c i t y

P - Pds t a t i c p r e s s u r e c o e f f i c i e n t - P d - P dw h e r e p = l o c a l s t a t i c p r e s s u r e

P d = d o w n s t r e a m s t a t i c p r e s s u r eP d = a b s o l u t e d o w n s t r e a m t o t a l p r e s s u r e

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

2]


23/69

22

NO.

I

2

4

I0

A u t h o r ( s )

C. H. Wu

H . M a r s h

Do J. L. Smith

D. J. L. SmithD. H. Frost

J . P . G o s t e l o w

D . H . W i l k i n s o n

N. J. SeybC, B o s m a nW. J. WhitneyM. E. SzanceP. T. MoffitD. E. MonroeIo H. JohnstonD. E. Smart

R E F E R E N C E STitle, etc.

A g e n e r a l t h e o r y o f t h e t h r e e - d i m e n s i o n a lf l o w i n s u b s o n i c a n d s u p e r s o n i c t u r b o -m a c h i n e s o f a x i al , r a d i a l a n d m i x e d f l o wtypes.N A S A T e c h N o t e 2 6 0 4 , 1 9 5 2A d i g i t a l c o m p u t e r p r o g r a m f o r t h e t h r o u g h -f l o w f l u i d m e c h a n i c s i n a n a r b i t r a r y t u r b o -m a c h i n e , u s i n g a m a t r i x m e t h o d .A R C R & M 3 5 0 9 , 1 9 6 8F l o w p a s t t u r b o - m a c h i n e b l ad e s .P h D T h es i s , U n i v e r s i t y o f L o n d o n , 1 9 7 0C a l c u l a t i o n of t h e f l o w p a s t t u r b o - m a c h i n eb l a d e s .T h e r m o d y n a m i c s a n d F l u i d M e c h a n i c sC o n v e n t i o n .I n s t M e c h E n g G l a s g o w , 1 9 7 0T h e a c c u r a t e p r e d i c t i o n o f c a s c a d ep e r f o r m a n c e .P h D T h e s i s , U n i v e r s i t y o f L i v e r p o o l , 1 9 6 5C a l c u l a t i o n o f b l a d e - t o - b l a d e f l o w i n at u r b o m a c h i n e b y s t r e a m l i n e c u r v a t u re A R C R & M 3 7 0 4 , 1 9 7 2P r i v a t e c o m m u n i c a t i o n .P r i v a t e c o m m u n i c a t i o n .C o l d a i r i n v e s t i g a t i o n o f a t u r b i n e f o rh i g h t e m p e r a t u r e e n g i n e a p p l i c a t i o n .N A S A T N D - 3 7 5 1 , 1 9 6 7

A n e x p e r i m e n t i n b l a d e p r o f i l e d e s i g n .ARC CP 941, 1966


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A P P E N D I X I23

P r o g r a m t a p ea

T h e g r i d d at a , b a n d m a t r i x , i n f l u e n c e c o e f f i c i e n t s f o r ~ x a n dp r e v i o u s s o l u t i o n s f o r e a c h g e o m e t r y a r e s t o r ed o n a m a s t e r m a g n e t i c t ap e,k n o w n a s t h e p r o g r a m t ap e. T h e i n f o r m a t i o n i s w r i t t e n i n b i n a r y f o r m a n deach array is identified by a title of up to eight alp hanume ric characters.Most of the titles are set with in the program, but dif ferent ones must bespecifi ed (in the input data) to identify eac h set of geomet ric data (i.e.grid, band matrix , and influenc e coef ficients) and each solution, so thatthey can be accessed by fu ture runs.,Example

S t a r t i n g w i t h a n e w p r o g r a m t a p e, t h r e e p r o b l e m s a r e r u n:(i) First run of geome try A, the geometr ic data being labelled as

( i i )

(iii)

'GEOA' and the result as 'SF@~I'.First run of geomet ry B, the geome tric da ta being labelled as'BGEOMETY' and the resu lt as 'SF~II'.Second run of geometry A (NB blade secti on of input data isnot required) prod ucin g result 'SF@@2' (this run may usesol uti on 'SF@@I' as a firs t guess).

The arrays stored on the tape will then be:

I B a n d m a t r i x s t o r e d i n { ( m - 2)

G E O AIGVYXD E L XD E L P H IEP H IB A N D D A T Ao

I

B A N D D A T ANI

N2

|( n - I) - (t - - I) + i~ s e c t i o n s2n - 3 !

(see Figure 7)

N3N4


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24N 5N 6N N 6

o

1I CsP1BGEONETYIGVYXDELXD E L P H IE

P H IB A N D D A T Ao

B A N D D A T AN IN N IN 2N 3N 4N 5N 6N N 6I Co

o

I CS F I 1S F @ @ 2E N D F I L E

( m - 2 ) se ct io ns _ o f i n f l u e n c e c o e f f i c i e n t s f o r f i n i t e d i f f e r e n c easta rs fo r a--x


26/69

2 5

A s u m m a r y o f t h i s f o r m i s p r o d u c e d a f t e r e a c h r u n ( t h e t i t l e sB A N D D A T A a n d I C a r e w r i t t e n o u t o n l y o n c e f o r e a c h s e t o f g e o m e t r i c d a t a) .W i t h t h e p r e s e n t p r o g r a m d i m e n s i o n s ( m < 9 0; n < 1 3 ) , e a c h s e t of g e o m e t r i cd a t a o c c u p i e s a p p r o x i m a t e l y 1 3 0 b l o c k s a n d e a c h n e w s o l u t i o n a p p r o x i m a t e l y2 ~ b l o c k s . T h e r e a r e 4 0 0 0 b l o c k s o n e a c h m a g n e t i c t a pe .


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2 6A P P E N D I X I I

C o m p u t e r p r o g r a m o r g a n i s a t i o n _ nT h e m a t r i x p r o g r a m i s w r i t t e n i n s e g m e n t e d f o r m t o m i n i m i s e t h e a m o u n t

o f c o r e s t o r e r e q u i r e d. D i s c a n d m a g n e t i c t a p e b a c k i n g s t o r e s a r e u s e d t os t o r e d a t a a n d t r a n s f e r i t f r o m o n e s e g m e n t t o t h e n e x t ( s e e b e l o w ) .

I n o r d e r t o s u i t t h e C A D C s y s t e m , t h e s e g m e n t s ~ r e r u n i n t h r e ed i f f e r e n t j o b s, t h e s e c o n d a n d t h i r d b e i n g s e t u p a u t o m a t i c a l l y a t t h e e n do f t he p r e v i o u s j o b ( s e e A p p e n d i x V l ) .J o b I

P r o g r a m F i l eN A T G A S ~ I / S I / I I N D

N A T G A S ~ I / S I / 2 B A N D

N A T G A S ~ I / S I / T R A N S F E R

J o b 2N A T G A S ~ I / S I / 3 R E D

N A T G A S ~ I / S l / 4 F I R S T

o

D e s q r i p t i o nR e a d s i n a l l d a t a , a n d s e t s u p t h ef i n i t e d i f f e r e n c e g r i d o n t h e p r o g r a mm a g n e t i c t a p e ( f i rs t r u n o n l y ) .C a l c u l a t e s t h e w e i g h t i n g c o e f f i c i e n t sf o r t h e f i n i t e d i f f e r e n c e a p p r o x i m a t i o n sa 0f o r ~ xx a n d ( V + E ~ x ). T h e c o e f f i c i e n t sa r e g e n e r a t e d i n b l o c k s a n d s t o r e d o nt h e p r o g r a m m a g n e t i c t a p e ( if th i s i s a nu p d a t e r u n of a g e o m e t r y , t h is i n f o r m -a t i o n w i l l a l r e a d y b e s t o r e d o n t h e t a p ea n d t h i s s e g m e n t d o e s n o t h i n g ) .T r a n s f e r s t h e b a n d m a t r i x f r o m t h ep r o g r a m t a p e o n to t h e i n p ut m a g n e t i ct a p e f o r th e m a t r i x r o u t i n e s a n d t h eo t h e r d a t a o n t he p r o g r a m t a p e o n t o ad i s c b a c k i n g s t o r e.

U s e s t h e C A D C r o u t i n e R E D U C E t o s p l i tt h e b a n d m a t r i x i n t o t h e p r o d u c t o f t w ot r i a n g u l a r m a t r i c e s w h i c h a r e s t o r e d o nt w o t e m p o r a r y m a g n e t i c t a p e s (7 a n d 8) .I t o p e n s t h e d i s c f i l e / M A T R I X / I N T E R t oc a r r y d e t a i l s o f c h an g e s i n t h e m a t r i x( f o r m a x i m u m a c c u r a c y ) i n t o S O L VE .( i) W r i t e s o u t t h e g r i d d a t a o n t h el i n e p r i n t e r( ii ) G e n e r a t e s t h e f i r s t g u e s s t o s t a r tt h e i t e r a t i v e s o l u t i o n


28/69

27P r o g r a m F i l e

L

N A T G A S ~ I / S I / S O L U T I O N

N A T G A S @ I / S / V E L O C I T Y

NATGAS~I/SI/RHS

m

I t e ra t i v

S o l ut i o n

D e s c r i p t i o nS o l v e s t h e b a n d m a t r i x u s i n g C A D Cr o u t i n e S O L V E .C a l c u l a t e s t h e v e l o c i t y c o m p o n e n t s a n dd e n s i t y f r o m t h e s t r e a m f u n c t i o n.G e n e r a t e s t h e n e x t r i g h t h a n d s i d e .

NOTE T h e s e g m e n t s S O L U T I O N, V E L O C I T Y a n d R H S a r e s t o r e d i n c o m p i l e d f o r mo n f i l e s / D U M P / S O L U T I O N , / D U M P / V E L O C I T Y , a n d / D U M P /R H S , s o t h at t he ym a y b e o v e r l a i d .

J o b 3N A T G A S ~ I / S I / 6 R E S

N A T G A S ~ I / S I / 7 G R A P H

( i) C a l c u l a t e s a n d p r i n t s o u t t h e l i n ep r i n t e r r e s u l t s ( s ee S e c t i o n 5. 3 ).( i i ) W r i t e s t h e f i n a l s o l u t i o n f o r t h es t r e a m f u n c t i o n o n t o t he p r o g r a mm a g n e t i c t a p e .P l o t s o u t g r a p h s o f t h e b l a d e s u r f a c ev e l o c i t y a n d M a c h n u m b e r d i s t r i b u t i o n s ,a n d a p i c t u r e o f t h e s t r e a m l i n e p a t t e r n .

T h e f o l l o w i n g b a c k i n g s t o r es a r e u s e d d u r i n g a r u n:L o g i c a l N u m b e r T y p e

2 M a g n e t i c t a p e

2 T e m p o r a r y d i s c

3 T e m p o r a r y d i s c

D e s c r i p t i o nI n p u t t a p e f o r m a t r i x s o l v i n gr o u t i n e s : s e t u p i n t r a n s f e r(Job I) for use in 3RED (Job 2)a n d t h e n c l o s e d .H o l d s s o l u t i o n t o s t r e a m f u n c t i o ne q u a t i o n : s e t u p i n S O L U T I O N ( J o b 2 ).( A d i s c B S D i s u s e d t o a v o i d o v e r -w r i t i n g t h e b a n d m a t r i x s t o r e d o nm a g n e t i c t a p e 2. )T e m p o r a r y s t o r a ge f o r t h e w e i g h t i n gc o e f f i c i e n t s o f t h e f i n i t e d i f f e r e n c estars in 2BA ND (Job i).


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28

L o g i c a l N u m b e r

4 ))

5 )

7 ))

8 )

9 ))I 0 )

)ii )12

13

1415

T y p eT e m p o r a r y d i s c

T e m p o r a r y d i s c

N a m e d d i s c

M a g n e t i c t a p e s(temporary)

N a m e d d i s c f i l e s

N a m e d d i s c

T e m p o r a r y d i s c

M a g n e t i c t a p eN a m e d d i s c

D e s c r i p t i o nH o l d s n e x t a p p r o x i m a t i o n f o r r ig h th a n d s i d e of s t r e a m f u n c t i o n e q u a t i o n :set up in RHS for use in S OLUTION(Job 2).T e m p o r a r y s t o r ag e u s e d b y m a t r i xs o l v i n g r o u t i n e s R E D U C E A N D S O L V E(Job 2).D i s c f i l e c a l l e d / M A T R I X / I N T E R(size = 5 blocks). Set up in 3REDand used by S OLUTI ON (Job 2).S t o r a g e f o r u p p e r a n d l o w e rt r i a n g u l a r m a t r i c e s : s et u p i n3RED for use in SOLUTION (Job 2).S t o r a g e f o r t h e c o m p i l e d v e r s i o n so f s e g m e n t s S O L U T I O N , V E L O C I T Y a n dRHS.

H o l d s d a t a f r o m t h e p r o g r a m t a p ewhi ch is need ed in the second job.It is set up in TRA NSF ER and dele tedafter the third job (ma ximum size= 60 blocks).C a r r i es i n f o r m a t i o n b e t w e e nS O L U T I O N , V E L O C I T Y a n d R H S d u r i n gi t e r a t i v e s o l u t i o n f o r s t r e a mfunc tion (Job 2).P r o g r a m t a p e .C a r r i e s f l o w d a t a b e t w e e n a l lsegments: set up in lIND(maxi mum size = I0 blocks).


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A P P E N D I X I I IR a t a f o r m a t f o r m a t r i x p r o g r a m

29

( i) T i t l e s e c t i o n o f d a t a f o r m a t r i x b l a d e - t o - b l a d e p r o g r a m

F o r t r a n N a m e D e s c r i p t i o n

J o b t i t l e - u p t o 4 0 a l p h a n u m e r i c c h a r a c t e r sN A ~ ( 1 ) , I = I , I

T I T L E (I) , I=I, 2

I S P E C

F o r m a t

I A 4

2 A 4 T i t l e i d e n t i f y i n g g e o m e t r i c d a t a o n t h ep r o g r a m m a g n e t i c t a p e - u p t o 8 a l p h a n u m e r i cc h a r a c t e r s ( se e A p p e n d i x I )

C o n t r o l c h a r a c t e r=

=i

f o r t h e f i r s t r u n o f a p r o b l e m i . e.b l a d e g e o m e t r y s e c t i o n i n c l u d e d o n d a t at a p ef o r a n u p d a t e r u n , i . e t h e f i n i t ed i f f e r e n c e g r i d , b a n d m a t r i x a n di n f l u e n c e c o e f f i c i e n t s h a v e b e en s et u pb y a p r e v i o u s r u n a n d a r e s t o r e d o n t h ep r o g r a m t a pe u n d e r ' T I T L E ( I ) , I= I , 2 '

( ii ) B ! a d e g e o m e t r y s e c t i o n o f d a t a f o r m a t r i x p r o g r a mN B T h i s s e c t i o n

(i) i s o n l y r e q u i r e d f o r I S P E C = O( ii ) c a n b e p r o d u c e d b y t h e d a t a p r e p a r a t i o n p r o g r a m s -

A p p e n d i c e s I V an d V

F o r t r a n N a m e

L

IT

M

F o r m a t

4 X

D e s c r i p t i o n

N u m b e r o f s t r a ig h t l i n es u p s t r e a m o f b l a d er o w =N u m b e r o f f i r s t s t r a i g h t l i n e d o w n s t r e a m o fb l a d e r o w = tT o t a l n u m b e r o f s t r a i g h t l i n e s = m ( m < 9 ~ )


31/69

3 0


T H E T A

l O R D

N B LI E S

I C T

I C R

S P E E D

X ( I )

I=I,MY ( I , I )

Y ( 2 , 1 )

F o r m a t

2 F

5 1

I F

6 F


T y p i c a l b l a d e p i t c h ( u se d t o n o n - d i m e n s i o n a l -i s e l e n g t h s )A n g l e i n r a d i a n s t h r o u g h w h i c h x a n d y a xe sa r e t u r n e d f r o m r a n d z a x e s ( s e F i g u r e 2f o r s i g n c o n v e n t i o n )

S t r e a m s u r f a c e a n g l e c o n tr o l c h a r a c t e r= ~ i f % i s m e a s u r e d f r o m t h e x a x i s= I i f % i s m e a s u r e d f r o m t h e z a x i sN u m b e r o f b l a d e s i n r o w ( = ~ f o r c a s c a d e f l o w )G r i d s p a c i ng c o n t r o l c h a r a c t e r= @ f or n o n - u n i f o r m s p a c i n g= i fo r u n i f o r m s p a c i n gS t r e a m s u r f a c e t h i c k n e s s c o n t r o l c h a r a c t e r

= @ f or v a r y i n g t h i c k n e s s= I f o r u n i f o r m t h i c k n e s sS t r e a m s u r f a c e r a d i u s c o n t r o l c h a r a c t e r= 9 f o r v a r y i n g r a d i u s= i f o r c a s c a d e o r c y l i n d r i c a l f l

R o t a t i o n a l s p e ed p a r a m e t e r= ~ f o r s t a t i o n a r y b l a d e s= I f o r r o t a t i n g b l a d e s

x c o o r d i n a t e o f s t r a i g h t l i n eo A n y p o s i t i o n sm a y b e c h o s e n s o l o n g a s s k e w g r i d s 9 h i g ha s p e c t r a t i o s a n d s u d d e n c h a n g e s i n s p a c i n ga r e a v o i d e d ( s e e F i g u r e 8 ) F i g u r e 9 g i v e st y p i c a l p o s i t i o n s .t a n % (s e e l O R D a b o v e f o r d a t u m a n d F i g u r e 2f o r s i g n c o n v e n t i o n )S t r e a m s u r f a c e t h i c k n e s s b ' ( m e a s u r e dp a r a l l e l t o y a x i s )


32/69

31


P H I ( I , I ) I

A l

F o r m a t D e s c r i p t i o n

c o o r d i n a t e o f l ow e r b o u n d a r y i n r a d i a n s( m e a s u r e d i n t h e d i r e c t i o n o f r o t a t i o n )N B L i n e s E F a n d G H i n F i g u r e 2 a re a tc o n s t a n t ~ .y c o o r d i n a t e o f s t r a i g h t l i n eB l a d e t h i c k n e s s i n r a d i a n s

( ii i ) F l o w da t a s e c t i o n of d a t a f o r m a t r i x p r o g r a m

P G A

O G A

G A M M A

2 F

6F

T a n g e n t o f i n l e t r e l a t i v e g as a n g l e =

T a n g e n t o f o u t l e t r e l a t i v e g a s a n g l e =

C o n s t a n t r e l a t i n g v e l o c i t y a n d e n t h a l p y

R a t i o o f s p e c i f i c h e a t sG A M M A > I 0 - i n c o m p r e s s i b l e r u n , p r o g r a m

c a l c u l a t e s i t s o w n f i r s tg u e s s

O < G A M M A < I O - c o m p r e s s i b l e r u n - t h e a c t u a lv a l u e f o r t he f l u i d s h o u l db e u s e d

G A M M A < 0 - i n c o m p r e s s i b l e r u n , s t a r t i n gf r o m t h e p r e v i o u s r e s u l td e f i n e d b y th e t i t l e' S F I ( 1 ) , I = I , 2 '

N B F o r G A M M A < I 0 t h is m u s t b e a n u p d a t e r u ni . e . I S P E C = I


33/69

F o r t r a n N a m e F o r m a t

S P E E D

S T HS T DV E L I

R F

T 0 L

M A X I T

S F I ( I ) , I = I , 2

T R I G G E R

S F 2 ( I ) , I = I , 2


2 F , I

2 A 4

2 A 4

32

A n g u l a r v e l o c i t y i n r a d i a n s p e r s e c o nd . F o ra s t a t o r b l a d e a t y p i c a l v a l u e s h o u l d b e g i v e ns i n c e S. S P E E D i s u s e d t o n o n - d i m e n s i o n a l i s ev e l o c i t i e s w i t h i n t h e p r o g ra m A b s o l u t e s t a g n a t i o n e n t h a l p y a t u p s t r e a m p l a n eA b s o l u t e s t a g n a t i o n d e n s i t y a t u p s t r e a m p l a n eI n l e t v e l o c i t y r e l a t i v e t o b l a d e r o w

R e l a x a t i o n f a c t o r r f ( s ee E q u a t i o n ( 1 4) ). Au s e f u l g u i d e i s R F = 1 - M f o r 0 . 2 < M < 0 . 8( M = 0 . 5 ( M i n l e t + M e x i t ) )C o n v e r g e n c e c r i t e r i o n f o r s t r e a m f u n ct i o n ,u s u a l l y b e t w e e n 0 o 0 0 1 a n d 0. 0 0 0 1 M a x i m u m n u m b e r o f i t e r a t i o n s , u s u a l l y 2 0 .M A X I T m u s t b e c h o s e n s u c h t h a t t h e s e c o n d j o bd o e s n o t r u n o u t o f c o m p u t a t i o n t i m e ( s e eA p p e n d i x V I )

T i t l e i d e n t i f y i n g p r e v i o u s r e s u l t t o b e u s e da s a f i r s t g u e s s - up t o 8 c h a r a c t e r s ( s e eA p p e n d i x I )F o r G A M M A > I 0, a d u m m y s h o u l d b e u s e d

L i n e p r i n t e r o u t p u t c o n t r o l c h a r a c t e r= @ f o r f ul l o u t p u t= I f o r l i m i t e d o u t p u t

T i t l e t o b e u s e d t o i d e n t i f y t h e r e s u l t s o ft h i s r u n - u p t o 8 c h a r a c t e r s ( s ee A p p e n d i x I )


34/69

33A P P E N D I X I V

D a t a p r e p a r a t i o n p r o g r a m f o r b l a d e s s p e c i f i e d.bY coordinates

T h i s p r o g r a m p r o d u c e s t h e b l a d e d a t a f or t h e m a t r i x p r o g r a m f o rc a s c a d e o r c y l i n d r i c a l f l o ws p a s t b l a d e s e c t i o n s s p e c i f i e d b y a t a b l e ofc o o r d i n a t e s . I t u s e s a s p l i n e f it t o i n t e r p o l a t e b e t w e e n t h e p o i n t s , a n dt h e d a t a i s p r o d u c e d i n t he f o r m a t r e q u i r e d b y t h e m a t r i x p r o g r a m .i. B l a d e s p e c i f i c a t i o n

T h e b l a d e c o o r d i n a t e s m a y b e s p e c i f i e d w i t h r e s p e c t t o a ny c o n v e n i e n taxes (Figure I0). The spline fit canno t deal wit h regions of highc u r v a t u r e , a n d s o t he p r o g r a m c a n o n l y b e us e d f o r b l a d e s w i t h b l u n t l e a d i n ge d g e s. T h e l i m i t i n g c o n d i t i o n i s g i v e n i n F i g u r e l O a . A t t he t r a i l i n g e d g e,t h e b l a d e m u s t b e d e f i n e d b y t h e r a d i u s a n d c e n t r e o f t he t r a i l i n g e d g ec i r c l e a n d t h e t wo t a n g e n t p o i n t s a t w h i c h t h e p r o f i l e m e r g e s i n t o t h ecircl e (Fi gure lOc).2. Posi tion s of straig ht lines

T h e p o s i t i o n s o f t h e s t r a i g h t l i n e s m a y e i t h e r b e s p e c i f i e d i n th ei n p u t d a t a, o r t h e s t a n d a r d p o s i t i o n s s t o r e d i n t h e p r o g r a m m a y b e u s e d( F i g u r e 9) . I n b o t h c a s e s t he p r o g r a m c h e c k s t h e d i s t o r t i o n o f t he g r i dw i t h i n t h e b l a d e r o w a nd i n s e r t s a n e x t r a s t r a i g h t l i n e i f th e s k e w i sgrea ter than 70 per cent (Figure 8).3. S t r e a m s u r f a c e t h i c k n e s s d i s t r i b u t i o n

T h e s t r e a m s u r f a c e t h i c kn e s s d i s t r i b u t i o n m a y b e e i t h e r u n i f o r mt h r o u g h o u t , o r t h r e e l i n e a r v a r i a t i o n s w i t h x b e t w e e n v a l u e s s p e c i f i e d a tt h e u p s t r e a m p l a n e, t h e b l a d e l e a d i n g e d g e, t h e b l a d e t r a i l i n g e d g e a n dt h e d o w n s t r e a m p l a n e.4. Dat a input

The data requir ed is as follows:


35/69

34


T I T L E ( I ) , I = I , I O

P I T C HMPM SS T A G G E R

L E T R I G

X P ( 1 ) , Y P ( 1 ) ,I = I,MP

X S ( 1 ) , Y S ( 1 ) ,I = I,MS

X C E NY C E NT E R A D

I f L E T R I G = 1X L EY L E

T I , T 2 , T 3 , T 4

N

N U S

F o r m a t

I O A 4

F , 2 1 ,F , I

2 F

2 F

3 F

2 F

4F


J o b t i t l e - u p to 4 0 a l p h a n u m e r i c c h a r a c t e r sB l a d e p i t c hN u m b e r o f p o i n t s g i v e n o n p r e s s u r e s u r f a c eN u m b e r o f p o i n t s g i v e n o n s u c t i o n s u r f a c eA n g l e b e t w e e n x - y a n d x ' - y ' a x e s ( s e e F i g u r el O b ) i n d e g r e e sC o n t r o l c h a r a c t e r= 0 if l e a d i n g e d g e t a n g e n t p o i n t i s n o t g i v e n= 1 i f l e a d i n g e d g e t a n g e n t i s s p e c i f i e d

C o o r d i n a t e s o f p o i n t s o n p r e s s u r e s u r f a c e i nx ' - y ' s y s t e m . L a s t p o i n t i s a s s p e c i f i e d i nF i g u r e l O c

C o o r d i n a t e s o f p o i n t s o n s u c t i o n s u r f a c e i nx ' - y ' s y s t e m . L a s t p o i n t i s a s s p e c i f i e d i nF i g u r e I O c

C o o r d i n a t e s o f c e n t r e o f t r a i l i n g e d g e c i r c lei n x ' - y ' s y s t e mR a d i u s o f t r a i l i n g e d g e c i r c l e

C o o r d i n a t e s o f l e a d i n g e d g e t a n g e n t p o i n t i nx - y s y s t e m

S t r e a m s u r f a c e t h i c k n e s s e s a t: u p s t r e a mp l a n e , b l a d e l e a d i n g e d g e , b l a d e t r a i l i n ge d g e a nd d o w n s t r e a m p l a n e

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

N u m b e r o f u p s t r e a m c a l c u l a t i n g p l a n e s .s e t to z e ro t h e u p s t r e a m p l a n e s w i l l b ep o s i t i o n e d a s i n F i g u r e 9

If


36/69

35


If i~S > OU S P L ( 1 ) , I = I , N U S

NP

I f N P > 0P G ( 1 ) , I = I , N P

ND S

If NDS > 0D S P L ( 1 ) , I = I , N D S

F o r m a t

IOF

IOF

10F


P o s i t i o n s o f u p s t r e a m p l a n e s a s a p e r c e n t a g eo f a x i a l c h o r d , m e a s u r e d f r o m t h e l e a d i n g e d g ein t h g \ u p s t r e a m d i r e c t i o n

, P\

N u m b e r o f c a l c u l a t i n g p l a n e s w i t h i n t h e b l a d erow (incl leading and trailing ed ge points).I f s e t t o Z e r o t h e p l a n es w i l l b e p o s i t i o n e das in Fig ure 9.

P o s i t i o n s o f p l a n e s w i t h i n b l a d e r o w a s ap e r c e n t a g e o f a x i a l c h o r d, m e a s u r e d f r o m t h el e a d i n g e d g e i n t h e d o w n s t r e a m d i r e c t i o n .The first wil l be 0 and the last I00.

N u m b e r o f c a l c ul a t i n g p l a n e s d o w n s t r e a m o fblad e row. If set to zero the plane s willbe pos iti one d as in Figure 9.

P o s i t i o n s o f d o w n s t r e a m p l a n e s a s a p e r c e n t a g eo f a x i a l c h o r d , m e a s u r e d f r o m t he t r a i l i n ge d g e i n t h e d o w n s t r e a m d i r e c t i o n .

5 . R u n n i n g i n s t r u c t i o n sT h e p r o g r a m i s s t o r e d o n f i l e N A T G A S / T K H / P R O G , t y p i c a l r u n n i n g

i n s t r u c t i o n s w o u l d b e :C O M M A N D ( 7 1 1 7 N A T G A S ~ 3 / T K H D A T A P R E P A R A T I O N)COMP 2 MINSE X E C I M I N SL I M S T O R E 3 2 Ke I M O U T 3 ~O U T P U T ~ T O F l e e ( N A T G A S / T K H / D I A G S )N O T E SS W I T C H L A R G E


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36.ASA(S 16K) NATGAS/TKH/PROG. L I B R A R Y C A D / G I N O / G R A P H C A D / G I N O S A L *o E N T E R ( 1 3 N A T G A S / T K H / D A T A 0 4 N A T G A S / B L A D E / D A T A08 PRINTER) S 32K FR 2254CLOSEL I S T N A T G A S / B L A D E / D A T AF I N I S H

where NATG AS/T KH/D ATA is the input filea n d N A T G A S / B L A D E / D A T A i s t h e o u t p u t f i l e

6. OutputThe line printer output gives an echo of the input data, the amount of

grid skew and the gradients and curvatures of the surfaces for planes withi nthe blade row, and a listing of the output fileo The progr am also produc esa picture of the blade profil e and a graph of the blade su rface gradien tsagainst Xo If this graph is smooth then the blade surface veloc ityd i s t r i b u t i o n s p r o d u c e d b y t h e m a t r i x p r o g r a m s h o u l d a l s o b e s m oo t h.

T h e o u t p u t f i l e N A T G A S / B L A D E / D A T A c o n t a i n s t h e b l a d e g e o m e t r y d a t afor the matr ix pro gra m (see Appen dix III~2)o This file is then edited toinsert the title and flow data sections (see App endi x III.I and 3)


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37

A P P E N D I X VD a t a p r e p a r a t i o n p r o g r a m f o r d o u b l e c i r c u l a r a r c b l a d e s

T h i s p r o g r a m p r o d u c e s t h e b l a d e d a t a f o r t h e m a t r i x p r o g r a m f o rc a s c a d e o r c y l i n d r i c a l f l o w p a s t d o u b l e c i r c u l a r a r c c o m p r e s s o r b l a d e s .i. B l a d e s p e c i f i c a t i 0 n

T h e b l a d e p r o f i l e i s s p e c i f i e d b y t h e u s u a l p a r a m e t e r s . T h e p r o g r a mt h e n s e t s u p t h e e q u a t i o n s o f t h e b l a d e s u r f a c e s a n d h e n c e c a n f i n d t hee x a c t c o o r d i n a t e s o f a n y p o i n t o n t h e b l a d e .2. P o s i t i o n s o f c a l c u l a t i n g p l a n e s

T h e p o s i t i o n s o f th e c a l c u l a t i n g p l a n e s m a y e i t h e r b e s p e c i f i e d i nt h e in p u t d a t a, o r th e s t a n d a r d p o s i t i o n s s t o r e d i n t h e p r o g r a m m a y b e u s e d( F i g u r e 9) . I n b o t h c a s e s t h e d i s t o r t i o n o f t h e g r i d w i t h i n t h e b l a d e r o wi s c h e c k e d , a n d a n e x t r a p l a n e i n s e r t e d i f th e s k e w i s g r e a t e r t h a n 7 0 p e rcent (Figure 8).3. S t r e a m s u r f a c e t h i c k n e s s d i s t r i b u t i o n

T h e s t r e a m s u r f a c e t h i c k n es s d i s t r i b u t i o n m a y b e ei t h e r u n i f o r mt h r o u g h o u t , o r t h r e e l i n e a r v a r i a t i o n s w i t h x b e t w e e n v a l u e s s p e c i f i e d a tt h e u p s t r e a m p l a n e , t h e b l a d e l e a d i n g e d g e , t h e b l a d e t r a i l i n g e d g e a n dt h e d o w n s t r e a m p l a n e .4. Data inpu t

The data input is as follows:


T I T L E ( 1 ) , I = I , I O

S T A G G E R

C A M B E RC H O R DSCT MRLE

F o r m a t

IOA4

6F


J o b t i t l e - u p to 40 a l p h a n u m e r i c c h a r a c t e r s

S t a g g e r i n d e g r e e s ( p o s i t i v e f o r a c o m p r e s s o rblade)C a m b e r i n d e g r e e sC h o r d l e n g t hP i t c h / c h o r d r a t i oT h i c k n e s s / c h o r d r a t i oR a d i u s o f l e a d i n g a n d t r a i l i n g e d g e s


39/69

38


T I , T 2 , T 3 , T 4

NUS

If NUS > 0USP L (I) , =I ,NUS

NP

If NP > 0PG(1) ,I=I,NP

NDS

If NDS > ODSP L (I) ,I=I ,NDS

F o r m a t

4F

I

I(~F

1OF

IOF


S t r e a m s u r f a c e t h i c k n e s s e s a t t h e u p s t r e a mp l a n e , b l a d e l e a d i n g e dg e , b l a d e t r a i l i n ge d g e a n d d o w n s t r e a m p l a n e

N u m b e r o f p o i n t s o n e a c h s t r a i gh t l i n e

N u m b e r o f u p s t r e a m c a l c u l a t i n g p l a n e s . I fset to zero the planes will be posi tion edas in Figur e 9o

P o s i t i o n s o f u p s t r e a m p l a n e s a s a p e r c e n t a g eo f ax i a l c h o r d , m e a s u r e d f r o m t h e l e ad i n ge d g e i n t h e u p s t r e a m d i r e c t i o n .

N u m b e r o f c a l c u l a t i n g p l a n e s w i t h i n b l a d erow (incl leadi ng and trailing edge points)If set to zero the planes will be posit ione das in Figu re 9.

P o s i t i o n s o f p l a n e s w i t h i n t h e b l a d e r o w asa p e r c e n t a g e o f a x i a l c h o r d, m e a s u r e d f r o mt h e l e a d i n g e d g e i n t h e d o w n s t r e a m d i r e c ti o n .The first will be 0 and the last I00.

N u m b e r o f c a l c u l a t i n g p l a n e s d o w n s t r e a m o fblade row If set to zero the plane s willbe pos itio ned as in Figure 9.

P o s i t i o n s o f d o w n s t r e a m p l a n e s a s a p e r c e n t a g eo f a x i a l c h o r d , m e a s u r e d f r o m t h e t r a i l i n ge d g e i n t he d o w n s t r e a m d i r e c t i on

5. . R u n n i n g n s t r u c t io n sT h e p r o g r a m i s s t o r e d o n f i l e N A T G A S / D C A / P R O G .

i n s t r u c t i o n s w o u l d b e :T y p i c a l r u n n i n g


40/69

C O M M A N D ( 7 1 1 7 N A T G A S ~ 3 / D C A D A T A P RE P A R A T I O N )C O M P 2 M I N SE X E C 8 M I N SL I M S T O R E 1 2 KO U T P U T ~ T O F I LE ( N A T G A S / D C A / D I A G S )N O T E SS W I T C H M E D I U M, A SA N A T G A S / D C A / P R O G. L I B RA R Y C A D / G I N O / G R A P H C A D / G I N O S A L / *. E N T E R ( Ii N A T G A S / D C A / D A T A 0 2 N A T G A S / B L A D E / D A T A0 8 P R I N T E R ) S 1 2 K F R 1 2 ~CLOSEL I S T N A T G A S / B L A D E / D A T AF I N I S H

w h e r e N A T G A S / D C A / D A T A is t h e i n p ut f i l ea n d N A T G A S / B L A D E / D A T A h o l d s t h e b l a d e d a t a f o r t he

m a t r i x p r o g r a m6. Output

The line pri nter output giv es an echo of the input data, the amountof skew of the grid and a listing of the output file. A pict ure of theblade pr ofil e is plo tted out, but there is no need to check the smoothn essof the surface gr adie nts since the poin ts wil l be exact.

T h e o u t p u t f i l e N A T G A S / B L A D E / D A T A c o n t a i n s t h e b l a d e g e o m e t r y d a t af o r t h e m a t r i x p r o g r a m ( s ee A p p e n d i x 1 1 1. 2 ) . T h i s f i le is t h e n e d i t e d t oinsert the title and flow data sections (see App end ix II I.I and 3).

39


41/69

4 0A P P E N D I X V l

M a t r i x p r o g r a m - r u n ni n g i ns t r u ct i o nsL

T h e m a t r i x p r o g r a m i s r u n i n t h r e e s e p a r a t e j o b s, t o s u i t t h e C A D Cs y s t em . N o r m a l l y t h e s e c o n d a n d t h i r d j ob s a re a u t o m a t i c a l l y s et u p b y t h ep r e v i o u s j o b, b u t t h e y c a n a l s o b e r e s t a r t e d b y h a n d i f t he j o b ha s f a i l e dd u e to a s y s t e m f a u l t . T h e r u n n i n g c o m m a n d s f o r e a c h j o b ar e k e p t o n m a s t e rf i l e s, w h i c h a r e e d i t e d b e f o r e e a c h r u n: t h e e d i t e d v e r s i o n s a r e n o t k e p t.T y p i c a l e x a m p l e s o f t h e e d i t i n g i n s t r u c t i o n s a n d t h e f i l e s p r o d u c e d a r eg i v e n b e l o w .J o b i

E d i t C o m m a n d sE D I T N A T G A S ~ I / S I / R U N J O BG ; P R O J E C T ; 7 1 1 7 N A T G A S ~ 3G ; D A T A ; U S E R / I N P U T / D A T AG ; / B S D I 2 ; U S E R / B A C K I N G / S T O R E I

G ; / B S D I 5 ; U S E R / B A C K I N G / S T O R E 2

G ; P R O G ; T A P IG ; M A T ; T A P 2

WR U N J O B


C o s t c e n t r e f o r j o bI n p u t d a t a f i l eN a m e o f a d i s c B S D ( m a x s i z e = 6 0b l o c k s )N a m e o f a d i s c B S D ( m a x s i z e = I 0b l o c k s )N a m e o f a p r o g r a m t a p eN a m e o f in p u t t a p e f o r m a t r i xr o u t i n e s

T h e s e i n s t r u c t i o n s w o u l d p r o d u c e t h e r u n n i n g f i le :C O M M A N D ( 7 1 1 7 N A T G A S ~ 3 / S E T U P P R O B L E M )L I M S T O R E 1 6 KL I M O U T 3 ~C O M P 5 M I N SE X E C 2 ~ M I N ST A P E 2 ( T A P 2 )T A P E 1 4 (T A P I )O U T P U T ~ T O F I L E ( N A T G A S / S I / D I A G S )N O T E SS E T N A T G A S ~ I. E D I T / S I / M A S T E R W I T H H *G ; X X; U S E R / I N P U T / D A T A ; G ; I T A P E ;


42/69

G ; B S D I 2 ; S E R / B A C K I N G / S T O R E I ; ; S D I 5 ; S E R / B A C K I N G / S T O R E 2C O M M A N D. E D I T / S I / R U N J O B 2 W I T H H *G ; 1 2 B S D; U S E R / B A C K I N G / S T O R E I ; ; 15 B S D ; U S E R / B A C K I N G / S T O R E 2G ; C O S T C E N T R E ; 1 1 7 N A T G A S ~ 3 ; G ; P T A P ; T A P I ; ; M T A P ; T A P 2

Q U E U E J O B @F I N I S H

N o t eT h e e x p r e s s i o n ' G ; I T A P E ; I ' o n l in e 1 2 t e l l s t h e p r o g r a m t h a t i t i s

u s i n g a n e x i s

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