7/28/2019 A Study of Boundary Layer Characteristics of
Turbomachines Blade Rows And Their Relation To Over-All Blade
Loss
1/5
W A R N E R L . S T E W A R TW A R R E N J . W M T N E Y
R O B E R T Y . W O N GLew is Research Cente r , NASA,
C l e v e l a n d , O h i o
Th i s pape r p resen t s the resu l t s o f a num be r o f i
nves t i ga t i ons conce rned w i t h the bound -a r y - l a ye r
cha rac te r i s t i c s o f t u r bomach i ne b lad e rows and the
i r re l a t i on to the over -a l lb l ad e loss. I t i s demons t
r a t ed how the over -a l l b l ad e loss can be ob ta i ned f r
om themom en tum bound a ry - l a ye r t h i c kn ess . Th e mom en
tum bounda ry - l a ye r t h i c kness i sin u r n shown to be co r
re l a t ed by f l ow Reyno lds num be r and to t a l b l ad e su r
f ace d i f f u s i on .By assum in g Zw e i f el ' s f o rm o f b
l ad e -l oad i n g d iag r am the to t a l b l ad e su r f ace d i
f f u s i onpa ra me te r can be de term ined as a func t i on o f
b l ad e sol i d i t y and reac t i on ac ross theb lad e row . Th
us , t h i s t ype o f l oss ana l ys i s enab les an app r ox ima
t e p rede term ina t i ono f t he over -a l l b l ad e row loss as
de r i ved f r om fund am en ta l bound a ry - l a ye r concep t
s.I n add i t i on , i t shows the e f f ec t on over -a l l b l ad
e loss o f va r y i ng such desi gn fea tu resas so l i d i t y and
reac t i on .
I n t r o d u c t i o nTHE BASIC BOUNDARY-LAYER charac te r i s
t i c s o f
t u r bomach in e b lade row s have cons ide ra b le im por tan
ce becausethe v iscous losses are genera ted in th e bou nd ar y
layer s of theb lade sur face s and end wa l l s . T hu s , whe n
cons ide r i ng a sub-son ic t u rb ine ope rat ing at i t s de s
ign po in t , o r po in t o f h ighe s te f f ic iency, the major
source of loss that l imits this e f f ic iency ist h e b o u n d a
r y - l a y e r l o ss . T h e N A S A o v e r t h e p a s t f e w
y e a r shas condu c te d a nu mbe r o f i nve s t igat i ons , bot
h ana ly t i ca l ande xpe r ime nta l , i n o rde r t o be t t e r
und e rs tand th e bound ar y - l aye rph e nome n a occur r i ng
in b lad e rows . T h i s i n for mat i on i s o f useto t he de s
igne r i n o rde r t o p rope r l y a l l ow for t he d i sp lace
me ntth i ckne ss in s i z ing f l ow passage s and in p re de te
rm in ing the e f f i -c iency of a turbine .
T h i s pape r p re sent s t he re su l t s of t he se inve s t
i gat ions andshows how the bas i c boundary - l aye r quant i t i
e s can be re la te d t othe over-a l l b lad e loss . T hi s is
don e by f i rst cons id eri ng th emom e nt um l oss fo r an i so
la te d a i r fo i l and t r ans for m in g th i s l os sin t o a t
o ta l -mome n tum - loss para me te r fo r a cascade inc lud i
ng
Cont r i buted by the H ydrau l i c D iv is i on and presented
at theAnn ua l M e e t ing , A t l ant i c C i t y , N . J . , N
ove mbe r 29-De ce mbe r 4 ,1 9 5 9 , o f T H E A M E R I C A N SO
C I E T Y O F M E C H A N I C A L E N G I N E E R S .NOTE : Sta
tements and opin ions advan ced in papers are to beun derstood as
in divi dual expressions of their auth ors and not those ofthe
Society . M anu scrip t received at A S M E Headquar ters , Apri l
S,1959. Paper N o. 59 A -23.
the e f fect of the end wal ls and the e f fect of mix ing to
uni form f lowcond i t i ons down s t r e am of t he b lade row . T
h e mome ntu m- l osspar ameter can then be used to obta in a k i
net i c -ener gy loss forthe blade row, whi ch is an indi cat i on
of the ef f i c iency of th eb lade row . F ina l l y , t he ove r
-a l l b lade l oss i s de ve lope d f rom th ebas i c boun dar y -
l aye r mom e nt um loss as a fun c t ion o f b lade sur -face d i
f fu s ion , re ac t ion , and so l i d i t y . T he ef fe c t o f
R e yno l dsnu mb er is also discussed.
A i r f o i l B o u n d a r y - L a y e r C h a r a c t e r i s
t i c sT he de f in i t i ve e quat i ons o f t he var ious bounda
ry - l a j ' e r t h i ck -
nesses and the re lat ionships among them are deve loped in re
fer-ence [1 ] ' for compr ess ibl e f low. T h e quan t i t ies
that wi l l be usedhe re in are t he d i sp lace me nt t h i ckne
ss 5 , mom e nt um t h i ckn e ss 0,f o r m f a c t o r I I , and e
ne rgy fac tor E . T h e for m fac tor i s t he ra t i oo f t o ta
l d i sp la ce ment t h i ckn e ss t o t o ta l mome n tu m th i
ckne ss ,and th e energy fact or is the rat io of tota l ener gy
thi ckness t ototal mom en tu m thi ckness . F ig . 1 show s a
typical ai r foi l andi t s boun dar y - l aye r g r owth as wel l
as an a lge bra i c desc r ip t i on o ft h e b o u n d a r y - l a
y e r q u a n t i t i e s. T h e f a c t o r s H a n d E we re de
-r i ve d in re fere nce [ 1 ] as fun c t ion s o f M ac h nu mbe r
and thes imple powe r veloc i t y p ro f i l e e xpone n t n . T h
i s t yp e o f ve loc i t ypro f i l e i s comm onl y assume d for
t u rbu le nt bound ar y l aye r , andfu l l y de ve lope d tu r bu
le nt boun dar y l aye r i s ge nera l l y assu me d
1 N um ber s in bra ckets designate R eferences at end of
paper.- N o m e n c l a t u r e -
.4 = ar ea, sq ft( J = as pect rat i o, h / cc = chord , f t
D = di f fus i on factorE = k ine t i c -e ne rgy fac tore =
kinet ic -energjr - I oss coef f ic ient
H = for m fact orh = bla de he ight , f tI = mean cam ber lengt
h, f t
M = M a c h n u m b e rm = mom ent um -l oss coef f ic ientn = e
xpone nt use d for t u r bu le nt - f l ow
s imple powe r ve loc i t y p ro f i l ep = absolute pressure ,
lb/sq f t .
R re ac t ionR e = R e m o l d s n u m b e r
s = spa cin g, ftt = t ra i l i ng -e dgc th i ckne ss , f tV =
ve loc i ty , f t /sec
a = flow an gle meas ur ed fr om axial d i-rect ion, deg
a s t = stagger angl e measu red f r om ax ialdi rect ion,
deg
5 = disp la cement thi ckness , f t6 = mom e nt um th i ckne ss
, f tcr = sol id it y, c/sip k ine t i c -e ne rgy th i ckne ss , f
t
Subscr ip ts :b = base R e yno l ds nu mbe r
eff = effecti vefs = f ree-st reamp = pressur e sur faces = suct
i on surfa cet = tot al
0 = bl ade entr an ce1 = blade t ra i l ing edge2 = after mix i
ng
2d = two-d ime ns i ona l3d = th ree-di mensi onalSuperscr ip t
:' = absolu te tota l state
5 8 8 / S E P T E M B E R 1 9 6 0 T r a n s a c t i o n s of t h
e A S i V i E
Copyright 1960 by ASME
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7/28/2019 A Study of Boundary Layer Characteristics of
Turbomachines Blade Rows And Their Relation To Over-All Blade
Loss
2/5
t " tF i g . 1 A i r f o i l a n d b o u n d a r y - l a y e r p
a r a m e t e r s
t o p re va i l i n t u rb omach i ne b la de rows . T he var i
a t i ons o f H an dE are show n in F igs , 2 and 3. T h e poi nt s
on the cur ve are experi -ment al dat a f ro m re fer ence [2 ]
.
T h e fo r m fac tor i n F ig . 2 i nc re ases w i th M ach n um
be r and hasa s l i ght var ia t i on w i th t he ve loc i t y p ro
f i l e e xpone n t n . I n F i g . 3the ener gy factor is even
less af fected by th e ve loc i ty pr of i l e ex-p o n e n t a n d
v i r t u a l l y u n a f f e c t ed b y M a c h n u m b e r . T h
e r a n g eof 7i used in the f igures inc ludes that general ly
encountered int u r b u l e n t b ou n d a r y l a y er . E x c el
l e n t a g r e e m en t w a s o b t a i n e dbetween the
experimental resul ts of re ference [2] and those cal -cu la te d
fo r a s imp le powe r ve loc i t y p ro f i l e hav i ng an e xpon
e ntn o f 'A - T hu s , by assuming a ve loc i t y p ro f i l e e
xpone nt of 1 / t , thef a c t o r s H a n d E can be de te rm ine
d w i th re asonab le accuracy i ft he b lad e -out le t M ach nu
mbe r i s kn own . T h e s ign i f i cance o ft h i s w i l l be
come appar e nt i n t he fo l l ow in g se c t ions .
B o u n d a r y - L a y e r C h a r a c t e r i s t i c s R e l
a i e d t o B l a d e R o w L o s s e sI n t u r bomach i ne r y t
he a i r fo i l s a re space d arou nd an annu l us
to for m an ann ul ar cascade. I t is of in terest to deter mi
ne theloss as a f r act i on of the mo me n tu m or energy of the
ideal or f r ee-st r eam f low . T h e f r act i on of the mo men
tu m loss (see F ig. 4) canbe expressed as
TOl (s cos t 8, 8P )
m = I c cos Ctit + 8 ,S CO S1 -
F i g . 2 V a r i a t i o n i n b o u n d a r y - l a y e r f o
r m f ac t o r w i t h M a c h n u m b e r a n dv e l o c i t y p r
o f i l e e x p o n e n t
2 . 6
2 . 2
1.0
C A L C U L A T E D F OR S I M P L E P O W E RV E L O C I T Y P
R O F I L EE X P E R I M E N T A L P O I N T S O B T A I NE D
A T B L A D E O U T L E T ( R E F 2 )
T h e nu me ra tor 6, re pre sent s a l e ngth w h ich , whe n m
ul t ip l i e d bythe f re e -s t re am mome ntum, y ie lds t he l
os s or de f i c i e ncy o f mo-me nt um in t he boun dar y l aye r
. T h e de nomi nat or (s cos a i t 8S 8P) i s a l e ngth re pre se
nt ing t h e ide a l mom e nt um of t h eactua l we i ght f low. T
hu s, the ter m vh repr esents th e f r act i on ofmo men tu m loss
based on th e actua l we i ght f low. T h e express ionfor m i may
a l so be w r i t t e n in t e rms o f d j l , wh i ch i s t he
comm onw a y of n o n d i m e n s i o n a l i z i n g t h e m o m e
n t u m b o u n d a r y - l a y e rthickness of ai r foi ls ,
as
0 . I a
0 .4 .8 1.2F R E E - S T R E A M M A C H N O .. M f $
F i g . 3 V a r i a t i o n i n b o u n d a r y - l a y e r e n
e r g y f a c to r w i t h M a c h n u m b e ra n d v e l o c i t y
p r o f i l e e x p o n e n t
bla de r ows wh ere in the bla des ha ve inf in i te length . I
n a cascade,howe ve r , t he re are e nd wa l l s t hat a l so have
boundary - l aye rlosses th at are app rec i abl e . T h e loss on
th e end wal ls can be as-sumed to be equal to the average loss
occurr ing on the blade sur-faces. T her efor e , th e fact or to
corr ect for the end wal ls can beexpressed as the area rat io:
^.bladc + A walli4blade
B y re ferr in g to F ig. 5 the wett ed surfa ce area of th e
bla des perpassage can be approx imate d as
2 c 2 T h e wet ted sur face of th e wal l area can be app r ox
i mat ed b y
2es cos a a tT h u s ,
2c 2 (3. + 2cs cos a s t cos a s tA m / A U ^ TT TT ; = L +T he
k in e t i c -e ne rgy l os s can be obta in e d by app ly i ng t
he fac torE , o r s imp l y by
ei = m i Es ince E me re ly re la te s e ne rgy th i ckne ss t o
mome n tu m th i ck ne ssan d the deri vat ion of ej wou ld be ana
logou s to that of i i . T her e-fore , by us i ng F i g. 3, ei can
be obta in ed as
ei = 1. 8 m-iT h e for egoin g discuss ion has, in e f fect ,
cons id ered th e loss of
T h e r e f o r e ,2c 2 e
TI.3D _ E Y D _ ^ COS GATn h , i d ei,2d c
(7
T he e f f i cacy o f t he l os s assumpt ions and e nd wa l l a
rea appr ox im a-t i ons i s show n in F ig . 6 , wh ich i s t ake
n f r om re fe re nce [3 ] . T h etwo po in t s re pre se nt t he t
o ta l -p re ssure l os s compute d f rom c i r -cumfe re nt ia l
su rve ys t ake n at ma ny r ad i i spaced ac ross t he b l adespan
for t he tw o in d i cate d b lade -out le t M ach numbe rs . T he
sol i dl i ne re pre se nt s t he t o ta l -p r e ssure l oss as a
fun c t ion o f M ac h
C A L C U L A T E D F OR S I M P L E P O W E RV E L O C I T Y P
R O F I L E
E X P E R I M E N T A L P O I N T S O B T A I N E DA T B L A D E
O U T L E T ( R E F 2)
.4 .6 .8 1.0F R E E - S T R E A M M A C H N O ., M ,
J o u r n a l of B a s i c E n g i n e e r i n g S E P T E M B E
R 1 9 6 0 / 5 8 9
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