rAD—A0h5 122 PEMISYLVANIA STATE UNIV UNIVERSITY PARK APPLIED RESE—ETC F/a SO,;A STREAItIIE CLQVATURE P~ TKOD OF ANA LYZ IN AXISYI CRIC AXIAL, —ETC(U)~ A. 77 N W MCBRIDE N00017—73—C—t41$

MCLASSIFIED TN..77.’219

END____________ 0

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7A STREANLINE CURVATURE METBO]) OF AN ALYZI N G A XTSY MME T Ri C

AXIAL4 , MIXED AND RADIAL FLOW TURB ONA CHINE RY

M. W. McBride

Technical MemorandumFile No. TM 77—21921 July 1977Contract No. N00017—73—C—1418

Copy No.

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~~~ ‘,

The Pennsylv3nia State University ~~~~~

.

APPLIED RESEARCH LABORATORY 1/

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Approved f o r Public Rele~ scU n l im i t e d D i s t r i b u ti o n

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1;- “ rr t~~ ~_~J ~~~~~~~A ~~TREA •1LINE~~ URV~Vr i~E I1~T11OD OF /~N \L\’Z ii~C 9 Technicci Mern~~ nj f l m

I AXI S?MNETRIC AXIAL , ~~I1~~~D AND RADIAL FLOW _______

J ~11~URBo~.1~\c }1INERY , ~~. ~~~~~~~~

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II. CONT~~OLL I~~ .’ O F E I C(~ NA ’ c. .F~O A D D R L - $ S .~~~~

Naval Sea Systems Command / 21 Jui~~ &77 -

Washington , DC 20362 .~~F I ~~~ F P ~~ç C

21~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~: I T ~~~C L d ( ~~! t~~~~~4 j ~~. r~)

Ia.. ~~S i I & UT l Q i~~ TA ’1 CML~ I (Qi (.1: is(’~ c.?t,)

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turbo : ~c h incry r~~ici i al ~ lowflow anal y s i sS ro~ rn I I r~ L U rv i~ ur ’ r: hod

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A Str ani i no Curvature ~~~ hod of t h r o u g h f1o~’ o n oly s i s for ax I ~~y , o ot l I ax i~.1,m l x t t d mid i - r id 1 f i n~’ t i oonch I n t ~ v has bco :~ i mp lc i r : nt : tH . ‘flic r:o hod con hou scd f o r o i t ho r t htt’ d i r t e t . 01~ “~~~Hm ct prol~~~i . and can f . O I V O open f1c~’ , ‘rob ] oxrisuch f 1c’ ~ tlir:i 1 lt a Torn :-. 1r’~ ,‘i~ a horl’’ of vc’vo] :’ l ç,n ::o ~~‘ t 1 1

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F i l e No. 7 7 — 2 1 921 July 1977MWM :jepTotal P a g e s : 21Total Figure s: 6

APPL iED RE SEAI~CH LABORATORY

In te rna l Memorandum

To: B. R. Park in , R. H. 1-henderson , W. S. Gearha rt , M. J. Picrzga ,A. L. Treaster , F. H . So it h , J . P.. Ross, N. L. Billet,ARL Files (4), ~ater Tunnel Files

Information : J. C. Johnson , N. A. Ahourezk , F. P . Pinion , C. L . Key ,(Abstract N. T. Pigoi t , R. F. Cus ta rd , F. S. A r c h i ba l d , H . P. Bruce ,

Only) N. F. Davis , J. J. H i s e nh i u t h , C. 13. Gurney, W. N. hail ,C. H. Hoffman, J. W. Holl, C. C. Laurh ie , 3. L. Lumley ,B. H. Robb ins , C. L. Sanders , D. N. Stinebring , D. E. Thompson,A. Yocum

Froum: M. W. McBride

Subject : A Strean-i i iv e Curva tij r e ethod of A n n l v z f ~vor~: e t r f c A’ :ia l ,Mix ed ai~d IUid ia l Fko~ ‘IOu i o ma c iL i i j v

References: See page 15.

Abstract: A Streamline Curvature Method of through flo~; analys i s fo rax i symmet r i c axial , mixed and radial . f low S t u r b om a c h in e r yhas been implemented . The method can be used for c i ti : erthe direct or in d i r e c t prob lem and can solve open flowproblems such as f low th rough a propel le r on a body ofr evo lu t ion os vei l as the more f a m i l i a r duc t f lo~.’ probicus .Computa t iona l r e f i n e m e n t s allow m o d el i n g of blade rowchordwise and spanwise loading and blockage d i s t r i b u t i o nsand se lec t ion of st rcm: ;i ine l o c a t i o n s . Several examp lesof the ab i l i t i e s of t i m e analysis are p r e s e n t e d .

Acknowledgment : The work r e p o r t e d in t h i f ; paper vas sponsored by the NavalSea Systems Command , Code ~LA 0351. Special t hanks isg iver: to H. 3. Pi : r7ga who assist ed in deve lop ing t h e d i rec tf low anal y s is . ~ ~~~~~~~ -

—

~

- - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~ -

~~~~~‘--——“

~—— :~~~~:~~~~~

- -~~

‘~~~~

- ~~~~ -_

. ~~~~~ I—2— 21 Jul y 1977

MWM:jep

Table of Contents

C

Abstract 1

Acknowledgment I

Nomenclature . 3

List of Fi gures 5

Introduction 6

Dev.elopment of til e Equat ions of Motion 6

Application of the Cont inui ty and Energy Equations 10

Computational Procedure 12

E f f e c t s of Rotors and S ta tors 12

Losses 13

Examp les 13

Summary 14

Refe rences 15

F igmu es 16

~

— - --

~

- _- ----— -~~~~~~~~~- -~~~~~--- ~~~~~~~~~_ ----- ..-

~~~~~ ,~~~

_. .- - - _ _ -. --

~,‘— -n’--- - -~~~ ~~~~~~~~~~~~~~~~~~ -‘ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - , _ _~~~~~~~~~~~~~~~~~~~

_~~~~~~~~~~~~~~~~~ ‘5

—3— 21 Jul y 1977MWN :jep

Nomenclature

a angle between a streamline and a reference line

dummy variable of in tegra tion in i~ direc t ion

r~ ordina te parallel to reference line

k streamline curvature (radius of curvature Rk) 1

n ordina te normal to a streamline

P s ta t ic pressure

angle between streamline and axis of rotation

2 2 1/2Vm meridional velocity (u + v )

p f l uid density

r ordina te along a radial l ine

s ordinate parallel to a streamline

U rotor rotational velocity

u velocity parallel to axis of rotation

v velocity normal to axis of ro ta t ion

R radius

V0 tangential velocity

V~ total velocity at reference conditions

X ordinate parallel to axis of rotation

Y ordina te normal to axis of rotation

coordina te loca tion on Ti axis

Subs c rip~~

x deno tes partial differentiation W.N.T. X

y denotes partia l differentiation W .R.T. Y

I denotes inner boundary

o denotes outot i ’ u h I ’ I I l I V

-~~~~~~~~~~~~~~~~~~~~~~~.-- _ -~~~~~~~~~~-~~~~~~~ - -~~~~~~~~~~~~~~~

—4 — 21 J ul y 1977MWN:3ep

denotes a f u n c t i o n of r~

denotes a property of a particular streamline corresponding toa value of ~ on fl

1 blade row inle t

2 blade row ex it

~

—5— 21 July 1977MWM : j ~ p

List of Figures

Figure No. Title Pace

1 General Reference Station Parameters (MeridionalPlane) 16

2 Differen tial Streamline Element 17

3 Overall View of the Streamlines Through aCoun terro tating Open Propeller Set 18

4 Detail View of tile Streamlines Through aCoun terro tating Open Propeller Se t 19

5 Streamlines in a Francis Type Turbine 20

6 Comparison of Theoret ical and Exper imenta lVelocities Behind an Open Propeller as Solvedby the Direc t (Spec i f i ed Angu la r i t y ) Method . . . 21

—6— 21 July 1977M1.’ -~: j c-p

l nt r o d t t c t ion

The S t r e a m l i n e Curva tu re M cthod (SC~~) has been developed over manyyears to solve va r ious classes of axisynmc ’ t r i c and q u a s i — t h r e e d imensional

• turh omachino th rough f l o w problems . T u e method relies on the ab i l i ty tode f ine a c c u r a t e ly t i m e s t r eam l i n e s on a m e r i d i o n a l p lane and to determineradial and convect ive a c c e l e r a t i o n s based on the i r geome t ry . In ti le past ,the SCM has been succes s fu l ly app l i e d to axial flow pumps and compressor!turbines , th is being p e r f o rm e d w i t h t i m e e q uat i o n s w r i t t e n in o r t i m c : g o na l o~intrinsic coordinate sys tems . Except fo r the cumbersome q u a s i — t h r e e d i m e o : ; i o m t a lanalysis ( the d i rec t problem) , ap p r o x i mat i on s are o f t e n made which l imi t t i m eaccuracy of the anal ysis , p a r t i c u l a r ly w i t h regard to t ime b l a d e rows . BladeTows arc usual ly t r ea ted as t h i c k a c t u a t o r disks w i t h chordw ise load ing ,blockage , and rad ia l bod y forces ignored .

Computa t iona lly , many of these program s are i n e f f i c i e n t and l imitedin the types of cases they can h a n d l e . T h i s paper d o c u m e n t s a computeranalysis which addresses the problem s d e s c r i b e d . The program is capableof modeling axial , mixed and r ad ial f l o w s w i t h mu l t i p le blade rows andcan sol ve an approx imate l y d i r e c t as well as the i n d i r e c t probicia . Bladeto blade e f f e c t s are i nco rpo rat e d as a c i r c umfer e n t ia l l y averaged rad ia lbody for ce . P a r t i c u l a r a t t en t : ion to s t a b i li t y of convergence i-eihes t h i sanalysis sui table for problem s which arc usual ly very d i f f i c u l t to solve.

In order to improve tile u sefu lness of tile SCM , equa t ions were w r i t t e nsuch tha t r e fc rcn rc s tat ions I t a y take an a r b i t r a ry p a L l : i i m r o u g h the j a c h i m m e .This allows the comp lete n i o d e l i r m g of tile leading and t r a i l i n g edges of t i l eblade row;; . The op t ion of i nt r ab l ade compu t ing s t a t i o n s models t he bladeloading and blockage d i s t r i b u t i o n, and r a d i a l bod y fo rces . The USC ofspec ia lized curve f i t t i n g r o u t i n e s al lows rad ia l as wel l as ln )x cdI f lowand ay i a l flows to be anal y z e d . Time s t a b i l i t y of the program allo’ ’s v e ryhigh s ta t i o n aspect r a t i o s to be used. Pseudo— st reaa lj n es and t h e a b i l i tyto d e fin e s t r e a m lin e loca t ions i U p r o v e s the accuracy and speed of the pro~~r a:uand allows concen t ra t ion of da ta in reg ions of i n t e r e s t .

To da t e , the program has been used to analyze axial f low and r a d i a lflow pu alo : and has p e r f o rm e d t i m e d i rec t ana lys i s of an open p rope l l e r inan i n f i n i t e mediun , a l l w i t h good success. Examples of these cases w i l ]be p r e s e nt e d in t h i s 1-oport . D o c u m en t at i o n of tile a c tua l p rogr am w i l lappear imi a la ter r epo r t .

Developmen t of t he I~~u a t i o n s of M o t i o n

In tim e axisyi:i- . ’iric inviac ~d an a ly s i s , Euler ’s m omentum e q u a t i o n [srecast into a r adi~~i c q u i l : i b m - i i r ; e q u a t i o n . C o n s e r v a t i o n of mas s, totalc u e op t i m i d angul ar no: :cim t i l i en:- 1 etc t i t C analys i s . ‘i l i e equa t i n t :- : r 0 -be so lved by site et s;;jvo appro .: h u n t i o n s , w t h OCCC ;IU ; 1i \ ’ d at a and :1eeiv; :t ~~v e ;t ak en f r o m t i m e i t e r a t i v e l y approx i m :m a t e d v , ’l o c l t v I i c - i d and st r c ;t : : l i n egeome t ry .

L et x and v bc ’ r r i - t : : n ~~t m l ; . i c c ) r.~ I n a t ’ ; i i , fl 1 : 1 i : i i C t l l , l l p 1 ; : anj 1 l t : t m ~~t , ~~ n l ’ i y t i 1. J u i ~ - m ’ ; ; ( t 1 1 : . - L ,: l - -. h i m t i . , - : : : :- i ; l i ; L l m l ; t r

t c s , 1cm a L::u - d ; . . ::- , i , : ; a ~ i i ~~~~~~~~~~~~ ii:; a ’ t :

~~~~~~~~ . .~~ -~~~~~~. , -~~~~

- —~ -—— - ~~ -----—— _--_-w_~~~~ ~~~~~~~~~~~~ - - -- ~~ -~~ fl ~~~~~~~~~ r

— 7— 21 Jul y 1977NWM: j op

I ~hP~lx 2~y p~~ y

(1)

• ~u ~u 1 ~

pU -— + V •-—- - —

p~~x

Fi gures (1) and (2) r epre semlt the meridional p lane of the solut ion Sand geotietric quan t i ti e s used in the fo l low ing equations are showi m onthese f i gur e s .

The pressure g r a d i e n t between points a and b , Figure 2 , is represen tedby:

~~ k d sI+~~~dnI . (2)

To ti m e pressure grad ien t al .ong ri re quires the p a r t i a l deriva-tives e determined .

1 e - ;- Lr camwise d i r e c t i o n one has

- (3 )

Combining (1) and (3) we have -

— -~~

-~~~~

-~~~~~= [u + v } sin~~ + ~~~~~~~~~ cos ~ . (4)

Note t ha t

v V sin~~Taand (5)

u Vm con

The f o ll owl np do rivat I ye:-; ni-c de term ned by t i ;c c ha i n ru 1 ~‘:

- -

~

- - .- -— — - - - -- -

~ -

~— -

~ -.--- ,-—,--

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~

—8— 21 Jul y 1977NW~f: j ci p

= V~ cos ~

+ s iii q

= ~~~ ~~~~~~~~

~ ~x + COS

~(6)

V cos ~) + sin

~~~~= _ V

mSff l

~~~~~ y+ COS

~~) V ni y

.

. .

The quant i t i es determined in (5) and (6) arc subs t i tu t ed intoEquatiorm (4) and a f t e r r e d uc t i o n the r e su l t is:

— -~~ = V COS ~ V~1 + V sin ~ V~1 . (7)

Equation (7) reduces to

I -~~~

-- _.-~~m. (8)p~~s m

which is seen to be tile d i ff e r e n t i a l form of t he s tead y Bernou l l i ’se q u a t i o n f o r an incompress ib le f low.

Ti m e st r e am wis e normal component of Eq uat i o n (2) is developed in asimilar manner by no t ing tha t

~P ~iP 211’= 7~~ cos q) —

~~~~ Sin q) , (9)

and

— p-’-- [u ~~~~~ + v ~~ cos ~ —

~u + v -~-~~ ]

sin ~ . (10)p 21n 21x dy cx dy

A f t e r comb ining (5) , (6) and (10) , we obt a in t i m e f o i l Os- I up r e : ,u l t.

~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~~~ ~~~~~~~~j-~~~ ~~~~~~- - - --~~~~~~~~ --~~~~-~~~~~. - - - - ~~~~---~~~ . -- -~~~~ - -

~ - ~~-w_ -~~~~~~~~

- —~~~~~~~~~~~

- .- - -- - -~

21 Jul y 1977j op

— ~~ ~~~~ = v 2 con ~ + v 2 s i m m C’)

q~ (11)p 2111 11) X f T y

which is e q u i va l e n t to

_ i~~~~ = V 2{;~~~)>: +~~~~~~

}

. (1-2)

The quan t i t y in the hr a ch c -t s in (12) i s r e c o g n i ze d to be e q u i v a l e n t to21q!~~s , t ime c u r v a t u r e (k) of t h e st r o j : - l .i no.

Finally , Equation (12) becomes

k v 2 (13)p 2 1 n am

Combin ing E q u a t i o n S (2) , (8) and (13) , we do t c r m - : i tie the r ad i al cqui l i b; Ieq uu l_ 10m m f or no t ’ n;. i i h u g f i u~ to be

— 3P 21s

+ 21P 21n

21~~~~~~2 1s 21ri 21n 21r i

and f i n a l l y ,

— I ~~~ = k V 2 sin o~ + V —~~~ c o s a . (14)

am am

Time pressure gradient dime to swirl is de te rmined in a sir ;i ia r r ; ;mme r andleads to the term ,

— ~~ ~~~~~ = 1 2 (15)p d R R 0

- -~~~ ~~~~~ -~~~~~~-- ~~ - —-~~~~~~~~~~~~~~ ——-~~ -—

—10— 21 July 1977Mh’M : j ep

where t h e r e ar e no circumfc-ic-ntial d e r i v a t i v e s , i .e. , t ime f low is an: t n . ci tobe u n i f o r m Iii t l i c ci rc u in i~ :ent a a l di r e uL i o n . Tim e c o m p u t a t ion a l forum of ther a d i a l e q u i l i b r i u m e q u a t i o n is

— k V 2 sin a ± 4 v02 slfl( - + A ) + v ~~~ cos a . (16)

The three t er m s on the r i g h t hand side of E q uat i o n ( 16) opresca tt i m e mer id ional cu rva tu re , the r adial and the convec tive a cce l e r at i onsr e s pe ct i v e l y . l rmt egra t i on of the equat ion will y io ld the static pressured i f f e r e n c e be tween any two points in the f low f i e l d .

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~

From Equa t ion (16) , t h e s t a t i c p ressure d i f f e r e n c e r e la t i v e to . on- c:point , say , in the f low , to another poin t , 1 , a long the r efe renc l i n emay he found . To sa t i s f y conserva t ion of mass and energy acrcm ss t1 .~r e f e ren c e line , an a b s o i t m i e value of s t a ti c p ressure must be found at t i .t

reference point , q 1. The fo l lowing d i r e c t s o l u t i o n fo r thi s value is animprovement over o t h e r procedures which ar e i t e r a t i ve in n a t u r e .

A continuity eqmmat ion may be wr i t t e n f o r every s t a t i o n irm the p r ob i c a :

fl

p V (r i) r ( q ) sin(cc ri ) dm1 coilst . (17)

1l~

In words , the mass f low across a reference line regardless of i t s p a thbetween two stream ] i nes , is cons tan t , assuming an i in oupre ; s I b ] . e f l u i d .The energy equa t ion may be w r i t t e n for a p a r t i cu l a r po in t , I~, on timereference h u e (along a streamline) as

{P~ + 4 P V 2 - 4 P V

62

- - 1[- ~~

- c- ~~~~ . (18)

The f i r s t two term s on the l e f t hand si de rc !pre sent the t o t a l en e r gy(sta tic plus d v m s i n t i c ) a v a i l :ii,l a t i e p o in t , The t ci rm a ~~~ con ia I t in a] 1 h e adloss and rotc: r ( ‘flc ig’~’ c i ta mi : 5 ho tvec:n the i-c cr ence c o n d i t i o n and t heS ta t l o m i of i n t e r e s t . The t hj r d 1 : - i . is derived f t - i::; t i m e ~~t t t t ; , t t ) ~~~ t ho: ; n ; ngu i ; i rmnon ; ci urn amid i L i t h e r ot a t i o n :11 k i ii I i t c u r pv . TI mc- qua n i i t ‘. 1 sa;.t:ui- .~ d cO1i:-Lj i f l t in t h e ; mb , ~~ :~~~e of i a - -es o r ns- : ;i t on; c h : ;mh n - s d u e to aroN~m- or stat or . The stint of I h f o u r i I ; and f i t I i i 1 e m : - .s i t ; Lb ; s t a t j e

at . ~lin ’ t €~~ ; t l i e i a t - ;~rn 1. 1 n Li ‘ ‘ m t t I’: pro - - - t i r e d 11 c - i - - nc ’

I ru n E q u it i n t i ( i i )

—- - - - - ~~~ ---- - - __ _ _ _

_ _ _ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -_----.—-- ~~~-- - - -~~~~~-~~

—11-- 2]. July 1977MWM:jep

Equa tions (17) and (18) mtmay be combined and i n teg r at e d :

Ti 0 1/2

J {P

+ 4 p v~~~ - 4 p Vo~2

- -

~{_ 1 ~~Jd~

}

r sin(a0) dii =

4 p V~1 r~~ Sin(a q ) d~ . (1~ )

Because-Pa is a constant , Equation (1.9) umay be rearranged to give

1’flj :2

dr ~ =

~: {~

+ 4 4 p - {~

~~~ fi}d~

}

-

-~ 4 p Vm 2}

W 2d~ (20)

where,

W = r~ sin (an

) .

and f inally; -

Ii

=

J { ~~~+4 ~ (V 2V

2: - P

_

Vm 2 J ( 1

_

Q~J d ~f}

S4 2d

. (21)

The static ;)rcssimrc: any~ .mere along a r e f e r e n c e l i n e is then :

-- —.~~ - —_ .— —~--.----- —_ —. -~~~~~~~~~.- - — — —-— - --—- — -. ---—----.----~~ —— . - - ———-‘ -,-=~-.- - - -

- --- ---~~~-—-

— 12— 2~ J u ly 1977Mi~M :j e p

= Pfl1

~ {— 4 -~3-J dii . (22)

The velocity p ro f i l e wimich s a t i s f i e s angu lar momentum , con t inu i ty andtotal energy is given by rear ranging (18):

1/2

Vm~ = ~ {p

+ 4 p(V 2 - v0~

2 ) - P~~. - f

{- 4 ~~J d~

}

. - (23)

The flow f ield is solved by marching douns t ream to each s ta t ion in t u rn andin tegra t ing Equa t ion (16) . The s t a t i c pressure is then obtained fAo : Lqt m ; ’t i o r( 2 2 ) . The improved veloc i ty prof i les are genera ted by Eq u a t i o n (23) u n t i lthe changes in time p rof i l es are small between two successive passes.

Com~ u tat iona l Pro cc dure

Certain data are necessary to start the iterative computation cycle.Data specif y ing the des ign , i.e., the geometry, blade loca ti on , loadingand thickness , and referenc e f lu id cond itions are input . lni t ial one—dimensional approximat ions of all the velocity profi] es and the st r en i - l incp a t t e r n are made. From the in it i a l guess for the ve l o c ity p r o f i l e s andstreamline geometry, deriv atives nece ssary for the terms in Equat ion (16)can be determined . This equation is in tegra ted and the pressure d i f fe r en c eas a function of r~ is saved. T h i s in fo rmat ion is used to solve E q u a t i o n(21) f or P1k. Once solved , Equa t ion (23) is used to generate i.mprovcdvelocity p r o f i l e s . The improved veloci ty p r o f i l e s are in t eg ra ted to g ivethe amas s f low as a f unction of n and by specif y i n g percentages of the t o t - almass flow , new streamline locations are determined. This data is fed backinto the program and t i m e cycle is repeated un t i l convergence is m e t .

All of the m a j o r program var iables such as s t reaml ine location , r a d i u sof curva ture a t every poin t , velocities and other derivatives are severlvdamped agains t thei r previous values to prevent i n s t a b i l i t ie s from gro~- - i n gduring the computations . As a result of t h i s t rea tment , problems w i t i m t I m er a t io of radial distance to axial spacing (station aspect ratio) tha t al it

large can be so lved . This f ea t u r e is impor tant when prob lerat such aspropel lers in an a p p r o x i : n m t - l y i n f i n i te m e d ium are to be analyzed . Thecomputa t ional a sp e c t s of the SCM anal y s i s wi l l he fu l ly r epor ted in a l a t e rdocument , along ~ i th a co mi mp let e problem solut :ion .

— E f f e c t s of R o t o r s and . St a t o r s

In an i nvisc id s o l u t i o n , t h e e f fe c t of a rot - or or u t - i t m : r i t ; to r lm : t ; , ’c-the angu ] ar rnomc n turn of t im e f 1 ui d p ans i n t i m r o m i p li a h i ad - a,:s’ and in t i tocasc- of a ro to r t o e b m ; m : 1 c h a t o t a l p 1- - a - t i r e , i b m ’ t o m e (P

- + [ / b - I’ ~

) i tL 9 I I C t i o n ( 2 1 ) . In i n : c a - - - t i m e t : ~ 1/ 2 ” -

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—13-- 21 July 1977MWM : j op

A rotor chaimg es the t o t a l iierm d i i i p ropor t ion to the change in angularmomen tum of the f l ui d passing through the rotor. The total pressure isincreased (or in the case of a tu rb ine , decreased) in accordance wi th ti m eequa t ion ,

= 4 p (U ~ V 0 2 - U1 V01

) , ( 2 4 )

where V0 is positive in time direction of rotation.

Fr ic t iona l and secon dary f low lo~~~es p lay an importan t role i.n deter-mining the per formance of any fluid imandling machine . Our a b i l i t y to predictthese l- sses t heo re t i c a l l y f o r a gene ra l i zed tu rbomach ine in nonex i . st an t ,theref ~ rc cor re lat iona l da t a must i c used i f the e f f e c t s are to be includedin the ana lyn is . To ob ta in these cP~ta , a mach ine m u s t he a n a l y z e d forinviscid flow . Compar ison wi th e x p e r i m en t a l r esu l t s then ind icates Li memagni tude and d i s t r i b u t i o n of the losses encountered . These data can beincorportated into the analysis as a distribut :ed total pressure loss andthe pu-ogram ‘tuned’ for a particular machine. It must then 1)0 assumedthat the losses are similar for other machines of time same general t y p e .Da ta exist for a varie ty of pumpjet comi figurati am ;- ; arid have been usedsuccessf ull y in predi ct ing the p e r f o r m a n c e of new m ach i n e s .

Exam

In this section we presen t several examples of the uses of the StreamlineCurva ture Method as described in th is r e p o r t . The f irs t examp le . is of acounterrotating set of open propellers on a bod y of revolution. The outerbo u ndary streaml ine is def ined by a potential flow around tile body shag-.- .The e f f e c t of t he rotors on th is s t reaml ine are small enough to be negle cted.

The p lo t t ed s t reaml ines fo r th is c o n f i g u r a t i o n are simown in Fi gmi r m-s (3) and(4) demonstrate t h e more ; i gnificant results. First , there is a strea ;11r;econ trac t ion throug h the rotors as the flow is accelerated near the body.The velocity profiles exhibit the char ac tesi s t i . c bul ge or j e t behind therotor and in th is case t i m e a ve rage jet velocity is about 1.6 times the f i ccs t ream v e l o c i t y . The p lot also d em o n s t r a t e s the a b i l i t y of the p m o g r t ; mto use curved reference stations and to solve prob lems wit!1 h i g h s t a t i — naspect r at i o s ( in th i s case , A~=2 5) . The anal ysi s is able 10 deterrni mm cei timer the ti p r ad i us of I 1 m m. propeil c-rn for a

~‘, i \ e m ~ i-ni ct ; f l (1W rate , or

g i v e n t h e t i p r ad i us as h i m the (1].recl solution , to determine t h e mass f l owand powering r o q i m i r e m - - e m i t s fo r t ime corm f i p m m u a t i o n .

The second examp le is of a F r a n c i s— t y p e I ur l inc .- wi iii t-~i sk i - I i~a te;; a n da s i ; m m l a t c d m u - I \ O l i m t O . i i l L s ; ’ :m r t i c u l . i r p : u l - i r - t ; m ! ; n r n t r : : t ; - :: t i m e : ;b i l i vof t b - ; m : : . : 1 ” ; ;j : ; t e - I- en - r ;g . a s i~~l , ; - . d i d i y . ; d i ; i l f h m ; ~: . AS t m c - : : - I h o c i - l o t i S pr- -: - m-mmi & d in I ~~, m mr ; -

_ _ _ _ _ _ _ _ _ _ _

—14— 21 July 1977N l - M : j ep

Ti m e t h i rd exnr p ie is time direct solution of time flow through an oi~enp ropell e r . In t h i s case onl y the ang ular i ty d istr i b u t ion in time rotorexit p l a n e is sp c - c i f i e c i , r a t her than thic usual tangential velocity distribution .A tangential velocity distribution is it e r a t i v e ly d e t e r m i n e d tha t s a t i s fi e sthe an gu l a r i t y while at the 5mm t ime all other equa tions of mo t ion ares a t i s f i e d . Figure (6) shows t i m e theore t ical veloci t- y p r o f i l e obta ined f romthis procedure and correspond-hug oxper i im ental da ta for one typ ica l case.

Sununary - - .

A method of th rough—flow anal ysis fo r turhorn achinery has been developedwhich h a s proven successful for a va r i e ty of types , including ax ia l , r a d ia land mixed flow machin es . The eq ua tions of motion are solved by a d i f f eren tapproach than is commonl y used , in t ha t a d i f f e r e n t i a l equa t ion for thepres~ ure g rad ien t r a t i m e r than the ve loc i ty g rad ien t is used. A d d i t i o n a l l y ,the energy and continuity equations are satisfied by a direct integrationrather than by i t e ra t ive appr o :daat ions .

Re f inemen t s to the numerica l method r em it more accu ra t e model ing ofthe s t reaml ines , a l l o w i n g mixed and r ad i a l f lows to be ana lyzed . Nim i t lp lc-blade rows wi t h in t r ab l a d e c oi- tg1m t : i. n~ s t a t i o ns al lows m o d e l i n g of t i m e b J ;md ~-spanwise and chord~~iae loading d i st r ib u t i o m ; s and the b la d e t h i c k ne s sd i s t r i b u t i o n. Se lec t ion of s t r e a m lin e l o c a tio n and dens i ty i i : g r O V m t s accui a syin regions of i n tere s t .

P a r t i c u l a r a t t e n ti o n to t ime s t a b i l i t y of t ime n um e r i c a l p rocedure n i l e - -a-very h igh s t a t i o n aspect rat ios to be used and allows problems ;,u ii aspropel lers in an unb ounded medium to be ana l yzed , as ~ceil as the st :andardin ternal flow probi emns.

Frict ion losses and secondary flow e f f e c ts may be included in anemp irical fashion when a p a r t i c u l a r m a c h i n e type is p resc r ibed .

_ __ _ _ _ _

_ _ _ _ _ _ _ ______________________________ —

—15 -- 21 Jul y 1977MWM :jep

References

1. Smith , L. I I . , “The Radial. Equi lib~ iuin Equa t ion of Turbomachinery ,”J. Eng . I’ower , J a n u a r y 1966.

2. Novak , R. A . , “S t reaml ine Curva tu re Computing Procedures for FluidFlow Problems , ” ASNE Paper No. 66—l -~’A/GT3 .

3. Treast e r , A. L . , “C omp u t c -r i z e d App l i ca t ion of the St reamlineC u r v a t u re Method to the Ind i rec t , Ax i symametr ie TurbomachineProblem , ” ORL TM 514.249 1—16 , October 31, 1969.

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- -.-- - -— —--.- s--.~~~ —----s~~~~~~~ — s - — — - --~~~~~~~ ------s-~~~~~~~~~~~~~~~~~~~~~ -

DISTR I BUTION LIST 1-OR UNCLASSIFIED TM 77— 219 , by 1-1. W . McBride , da ted21 July 1977

Commar id ir Naval Shi p Eng inee r ing CenterNaval Sea Systems Command A t tn : R . J . CauleyDepar tment of the Navy Code NgEC—6l40BWashington , DC 20362 (Copy No. 11)A t t n : l ib ra ry

Code NSEA-~O9G32 Naval. Ship Eng ine er ing Cen te r(Copy No. 1 and 2) At tn : F . W e l l i n g

Code Nsr :C—61 44Naval Sea Systems Command (Copy No. 12)A t t m : C. G. McCui gan

Code NS EA— 03l33 commanding O f f i c e r(Copy No. 3) Naval Underwater System s Center

Newport , RI. 02840Naval Sea Systems Command A t tn : R. N a d o l inkAt tn : L. Benen Code SB323

Code NSEA—0 322 (Copy No. 13)(Copy No. 4)

Naval Underwater Systems CenterNaval Sea Systems Command Attn: R. TrainorAttn : E . C. Liszka Code SB323

Code NSEA—0 342 ) . (Copy No. 14)(Copy No. 5)

Naval U n d e rw a t e r Systems Cen te rNaval Sea Systems Command A t tn : L ibra ryAt tn : T. B . Peircc- Code LA15

Code NSEA— 0351 (Copy No. 15)(Copy No. 6)

Corm snanding O f f i c e rNaval Sea Systems Command Naval Ocean s- st ems CenterA t t n : J . C. Juergens San Diego Labora tory

Code N SEA—037 San Diego , CA 92132(Copy No. 7) At tn : J . W . 1ioyt

- Code 2501Naval Sea Systems Command (Copy No. 16)At ~n : A. R . Pa ladi no

Code NSEA— 0 3 72 Naval Ocean Systems Cent:er(Copy No. 8) At tn : D . Nel son

Code 2 5/42U. S. N a v a l Post Graduate School (Copy No. 17)Mom iti.-rc- ’- , CAAttn: Library Naval Ocean Systems Center(Copy N o . 9) At t m m : A . C. l a l - t i l a

Code ~Omb’Commander (Copy No. 18)Nava l Sii . i p I - . i m g i m n - e m - i ng C e n t e rWash I mig to n , DC 20360 N a v ; i I (u - i - an b~- a t i -m s (~en t&.-rA t t n : t-? . L. l o u i s At t n : ~~~. I - i a - 1 a ~an

(~~de N d ( : — u l a . i ; ((‘~~~-c :.e. 1~ )( ( z i p ’ . ~~a . 10)

_

- - -—--~~~~--~~~~~- --—---- --

D i S i m J t — P l 1 f t ~ F i ~ 1- : c . i . \ s s l I i i-:u l I- h 77 — 2 19 , by M. ~- . M c B r i d e , d a t ed

21 J uly 1977

Co : . z i a J I a; i ~- n r & Di n i - e t c h David W . 5 .n y l or N a v a l Sh i p R&I ) C e n t e rDay d U . i : m y l o m ~-~\al di ip i~~I) Cent e r A t t n : U . Blake

- m l ! i f t I n - - v . Code 1942Bet hIL-5d . ~, :t ; j -~~~~ (Copy No. 29)Attn : U . I- . Ci: - - . j i i~

Code 15 David W. Tay lor N a v a l Shi p R&D C- a: cv

(Copy N o . 20) A t t n : Tech. I n f o . Lib .Code 522.1

David W. Taylor U:ival Shi p UdU Ccnt -r (Copy No. 30)A t tn : B. Cox

Code 3544 Comm anding O f f i c e r & D i r e c t o r(Copy No. 21) David U . Ta~ 1or Naval U f l i p US IJ C t z - er

D e p a r t m e n t of the NavyDavid W . Taylor i- aval Shi p R&D Center Annapol is Labora to ryAttu : R. Werm t ci r A n n a p o l is , NI ) 21402

Code 152 A t t n : J. C . S t r i ck er(Copy No. 22) Code 2521 .

(Copy No. 31)David U . Taylor U ava l Shi p R&D CenterA t m : W. B. Morgan David W. Tay lor Naval Shi p R&I) Cc- m ; t e r

Code 154 At t n : M. C. Brop hy(Copy No. 23) Code 2721

(Copy No. 32)David U . Taylor Naval Shi p R&D Cente rA t t n : R. Cunning Commander

Code 1544 Naval Su r f ace Weapon Cm n t er(Copy No. 24) Silver S p r i n g , I-iD 20910

At Ln: V. C. D. DawsonDavid U . Tay lor Naval Ship R&D Center Code WU — 02A t t n : J . McCar th y (Copy No. 33)

Code 1552(Copy N o . 25) O f f i c e of N ava l Research

- Depar tmen t of the NavyDavid N . Taylor Naval Ship R&D Center 800 N. Quincy S t r e e tAt tn : T . Brocket t Ar l ing ton , VA 222 17

Code 1544 (Copy No. 34)(Copy No. 26)

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( ( U - p ~1 ::~ . -‘; 1)

- .-

- - .-~~~~~~~~~~-,--~~~~~—-~~~~~~~-—--- - - -~~ .--

~~~

D I S T R i B U T I O N LiST FOR U N C I . A S P I F] ED TN 77— 2 19 , by N. U . M c b r i d e , d a t e d21 J u l y 1977

Professor It. ~-ba r sIm Dr. P. van Oossan enDurham Univers i t y N ethe r l a mi th ; Ship Model basinDurham llaagsteeg 2ENGLAND P . 0. Box 28(Copy N o. 48) Wagen ingen

- TIlE N EII IER I AND SDr. L. H. Smi th (Copy No. 57)Compressor & Farm Design Tech. Oper.DTO , Main Drop 11—43 Dr. E. GreitzerCinc inna t i , Oh i~o 45215 M S— J. 6 Un i t ed Technolog ies Resea rch Cen te r(Copy No. 49) Silver Lane

F . H a r t f o r d , Conn .Whi t t ie Turhomach inery Laboratory (Copy No. 58)Mading ley RoadCambrid ge Von Karman Tn s t . for Fluid 1)ynamicsEN G l AND Turbomachinery Labora toryAtt:n: Dr. D. S. Whitehead Rhode—Saint—Cemiese(Copy No. 50) BELGU 1M

At t n : L i by a y rWhit t le Turbomac-hinery L a b o r a t o r y (Copy No. 59)Attmm : Sir Wil liama h awthorne:(Copy No. 51) Mr. N. Whipp en

All i s— C lm a l n:e rsW h i t t l e Turbom na ch in c - ry Labora tory box 712At tn : Library York , PA 17405(Copy No. 52) (Copy No. 60)

J. h lor l ock Dr . }Isuan YehVice Chancellor Direc torU n i v e r s i t y of Sa lford Towne School. of Civil & (tech . Eng.Salford , MS 4WT University of PennsylvaniaENGLAND Room 113 Towne(Copy No. 53) 220 S. 33rd S t ree t

Phi ladelp h ia , PA 19104Ily d ron au t i c s , In c . (Copy No. 61)Prindeli School I !oadLaure l , I-il ) 20810 Dr . C. K. SerovyAttn: Libra ry Professor(Copy No. 54) Mechanical E ng i n e e r i n g D e p a r t m en t

Iowa S t a t e U n i v e r s i t yA d m i r a l t y Research Labora tory Sames , Iowa 50010Tedd inc ;t on , Mi ddle sc x (Copy No. 62)l-P-: aI -~~~~~ )

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D i v . of Eng . & App i led Sc i c-necA dm r a l l y R e - s e a r ch L a b o r a t o r y Ca l i f o r n i a I n s ti t u t e of l i -hn o l ogvAt In : Dr . A . U -ore P n a ; i d c - m ~:m , C \ ~1 .10( -~m y : . . ~~ ) (( - I ’ - Un. 6~ )

_ _ _ _-

~~~~~~~~~

j

DIST RI E U ] J O N LI UT FOR UN cLASE J1- ’lED ‘I’M 77 — 219 , b y N . U. M cbr i d e , dated21 J uly 1977

NASA I n~~- is Rosca rc i m Center I-h- . M. U . Mcbride21000 llrook p zmrk Roach The Pennsy lvania State Un ive r s i t yCleveland , Ohi o 44135 Al ’PLlED RESEARC h LABORATORYAttn : Mr. N . .1. h lartmnaun Post Office Box 30(Copy No. 64) State College , PA 16801.

(Copy No. 72)NASA Ta ’ais Research CenterA t t n : D. Sandercock GTWT L ibra ry(Copy No. 65) The P e n n s y l v a n ia S ta te Unive r s i ty

APPLJ. El) RESEARCH LABORATORYNASA Lea-is Research Center Post O f f i c e Box 30A t t n : N . Sanger State College , PA 16801(Copy N~~. 66) (Copy No. 73)

Dr. P. LeeheyD epar t m mm en t of Naval A r c h i t e c t u r eMassachuset ts I n s t i t u t e of Technology77 Massachuse t t s Av enueCambridge , M assachuse t t s 02139(Copy N o. 67)

Stevens Ins titute of Technology711 Hudson StreetCastle Point StationHoboken , New Jersey 07030A t t n : S. Isakormas(Copy N o. 68)

Northern Research & Eng . Corpora t ion219 Vassar StreetCambridge , Ma ssachuse tts 02139Attn: Library(Copy No. 69)

Mr. Lin~-:ood C. WrightChief of AerodynamicsAiflesearch M f g. Co.2525 N. 190th S t r ee tTorrance , CA 90509(Copy No. 70)

Dr. R. F. HendersonThe P en nsy l v a nia S t a t e U n i v e r s i t yA1’PLi GD R1D ;EAR CII LA BORATORYPont O f f ic - c Pox 30Sta t.e ( o . l ic-ge , PA 16801(Copy N o . 71)

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