We have recently become concerned with making estimates of steady forces that may be exerted between moving blade rows and stationary blade rows or volutes.  Our present interest is with time averaged forces for estimation of shaft loads and flow asymmetry forces rather than with transient processes. For this purpose we have adopted the well-known "actuator" model for the blade row in which the flow leaving the row or cascade is assumed to have a constant leaving angle.  The disturbances external to this row such as a volute may be represented by distributions of vortex elements as was done for example by Domm and Hergt [1]

Acosta, A. J.

Brennen, C. E.

Caughey, T. K.

Chamieh, D.

English

Caltech Authors - Main

A BRIEF NOTE ON THE INTERACTION OF AK 
ACTUATOR CASCADE WITH A SINGULARITY' 
D i m i t r i  Chamieh, A. J .  Acos'ta, C.E .  Rrennen, T.K. Caughey 
Cal i fornia  I n s t i t u t e  of Technology 
Pasadena, Ca l i fo rn ia  91125 
Int roduct ion 
W e  have recen t ly  become concerned with making es t imates  of s teady 
fo rces  t h a t  may b e  exer ted  between moving blade  rows and s t a t i o n a r y  b lade  
rows o r  vo lu tes .  Our present  i n t e r e s t  is  with rime averaged fo rces  for 
es t imat ion of s h a f t  loads and flow asymmetry fo rces  r a t h e r  than wi th  
t r a n s i e n t  processes.  For t h i s  purpose w e  have adopted the well-known 
t I a c tua to r t '  model f o r  t h e  b lade  row i n  which the f low leaving the r o w  o r  
cascade i s  assumed t o  have a constant  leaving angle. The  d is turbances  
e x t e r n a l  t o  t h i s  row such as a vo lu te  may be represented by d i s t r i b u t i o n s  
of v o r t e x  elements as was done f o r  example by Domm and Hergt [ l ] .  
I n  the present  case t h i s  s i n g u l a r i t y  causes per turbat ions  of t h e  b a s i c  
one-dimensional flow through t h e  a c t u a t o r  cascade which l e a d  t o  o v e r a l l  
r o t o r  f o r c e s  and flow per tu rba t ions  which are t h e  sub jec t  of i n t e r e s t  here. 
The problem then is one of const ruct ing a v e l o c i t y  field t h a t  inc ludes  the 
dis turbance  (but  adds no mare) and s a t i s f i e s  t h e  flow tangency condi t ion  
leaving t h e  b lade  row. With reference  t o  F ig .  1 t h i s  r e q u i r e s  
v 
- - 
- cos i3 
u (1) 
a t  the  r o w  exit, y = 0, where v includes the  disturbance ve loc i ty  as 
" ~ o t  presented a t  workshop. 
well. as added pe r tu rba t ions  needed t o  s a t i s f y  Eq. (1) .  This is a p a r t i c -  
u l a r l y  simple problem when t h e  flow f i e l d  leaving t h e  a c t u a t o r  row i s  
i r r o t a t i o n a l .  In  t h e  next paragraph we consider two such cases  where t h i s  
assumption is  va l id .  
The Actuator Cascade 
Here w e  consider s teady flow of constant  t o t a l  pressure  through an 
a c t u a t o r  cascade. The flow leaving t h i s  cascade has  a given d i r e c t i o n  Bv 
(see Fig. 1) as t h i s  is equivalent  t o  t h e  Kutta condit ion.  W e  now consider  
two s i t u a t i o n s :  i n  t h e  f i r s t ,  w e  may imagine t h a t  t h e r e  a r e  d is turbances  
downstream of t h e  cascade. These d is turbances  may be due t o  t h e  e f f e c t  of  
downstream d i f f u s e r  vanes o r  a vo lu te  s t r u c t u r e  f o r  example. In t h e  second, 
a s  a p a r t i c u l a r l y  sim~le example, w e  consider t h e  e f f e c t  of pe r iod ic  changes 
i n  t h e  blade leaving angle  f3v on t h e  leaving flow without any downstream 
disturbances.  
(i) Downstream disturbances.  
In the nota t ion  of Fig. 1, t h e  t r a i l i n g  edge is  s i t u a t e d  on the  real 
z axis. The flow i s  assumed t o  be i r r o t a t i o n a l  so t h a t  complex v a r i a b l e  
methods may be used. L e t  us consider t h e  problem of the i n t e r a c t i o n  of the 
cascade with a single dis turbance  loca ted  i n  t h e  upper h a l f  p lane  at z = z 0 ' 
This is denoted by 
where (u,v) a r e  t h e  v e l o c i t y  components i n  t h e  (x,y)  d i r e c t i o n s  respect ive-  
l y .  The e f f e c t  of  t h e  disturbance gives rise t o  a d d i t i o n a l  co r rec t ion  terms 
wi(z) which cannot have any s i n g u l a r i t i e s  i n  t h e  upper ha l f  plane. The 
DISTURBANCES . :... 
. . .. 
,. . .. 
. I '. 
4 A. '4 
- * -  - - - -0- - - - - 
DOWNSTREAM 
UP STREAM 
FIGURE 1. SKETCH OF CASCADE ACTUATOR ROW 
( z  = x + iy) 
sum 
w - w *  T =  d i 
must sa t i s fy  the f l o w  angle leaving condi t ion  
vd+vi+v 
cat Bv 
udfUi+U 
where U,V are mean flow velocity components in the aBsence of any d i s tu rb -  
ance. Then on y = 0 
v hr -(u * . ) c o t  Py = d i  d l  
since V = U cot 8,. Here (vd,ud) are known. 
w must resul t  in the t o t a l  velocity compunrnts i 
au, f Bv = 0 f T 
(where here a = I, b = -tan fi ). Thus w has 
v T 
The induced distuzbance 
satisfying (4a ) ,  i.e., 
to s a t i s f y  a mixed bound- 
ary condition on y = 0. This tu rns  out  to be  neat ly  handled by the 
methuds described by Cheng and R o t t  [2] .  The i.nduced disturbauce w is i 
a-ib - w = - -  
i a+ib w d (2-G) 
which is seen merely to be an "image" of 
wd in the cascade exit plane. 
This is easy to show; l e t  
H(z) = (a-ib)w T 
Then Re{H(z)] = au -k bv i s  requi red  to be  zero on the real axis. Now 
set 
s o  that on y = O , H ( x , O )  is  the d i f f e r e n c e  of two complex conjugate func- 
tions and is therefore purely imaginary. A s  an example, cons ider  t h e  row 
o f  vortices of strength r and p e r i o d  d seen in Fig. 1. Then 
i I' 00 W = - -  Z 
d 2~ z-z,,-IL~ (7) 
n=-03 
Then 
and finally w has the well-known sum T 
We should point out here tha t  this image system is almost th* same 
as that  used by D v m  and Hergt t o  set  up the interference problem f o r  a velure 
i n  the presence of a point source-vortex. The p r i n c i p a l  differrrlce i s  tIlar 
here the leaving angle of the cascade is fully modelled instead of being 
approximated in the mean. 
(ti) Leaving cngle  var ia t ion.  
In the above ic was assumed tha t  the leaving angle  @v w a s  constant . 
This is not essential as the following example shows. Suppose B, = B,(x) 
and t h a t  w = 0. Then Eq. (3)  becumes d 
L e t  ' s assume that 
- 
Bv(x) = Bv+hB(x) 
where Af3 is a small change. Then w is proportional to A B  and we have i 
approximately 
v 
u - tan B v = -,- i v i ABv(x) , 
cos 'BV 
which is of t he  form 
Disturbance flows of t h i s  type may again be tackled by the methods of [Z ] ;  
more complete formulations are given in the book by Carrier, Krook and 
Pearson [ 3 ] .  Again we note t ha t  Eq. (9b) is equivalent to requiring that 
on y = 0. A solution of this equation is 
provided c(z) is chosen to have no singularities in the upper half plane. 
As a  practical example we  may imagine c(x) is of the form 
C(X) = consr. cos x 
and then it is easily seen that 
cons t iz w =-  e i a-k5.b 
I s  the required disturbance flow since w vanishes f o r  Y -+ a. i 
Moving Cascades 
The above examples were for irrotational, constant-errergy f lows .  Idhen 
the cascade moves i n  a t angent ia l  d i r e c t i o n  (parallel t o  t he  x axis)  work 
is done on the  f l u i d  i n  accordance with the Euler  formula. In what fof~.ows 
the absolute f l o w  is assurrcrd tu be steady and the cascade inoves at speed Ut 
parallel to the x axis. Assume also that the Bernoulli constant upstream 
of the cascade, i s  constant everywhere. Then it follows that the 
Leaving Bernou l l i  constant BZ is given by 
where w is the relative v e l o c i t y  parallsl to tlre blades and ds is an 
increment of blade arc. The relarive flow -Ls tinsteady s ince  t he  absolute 
f l o w  is steady. Thus 
With th i s  and continuity t g e  f i n d  
and here u2, ul are t a n g e n t i a l  
upstream of corresponding po in t s  
velocFtL&s immediately duwnsrream of arld 
of the  moving blade r o w .  In  general, B2 
is  n o t  constant  a t  every point along the exit from the row and we there- 
f o r e  expect the leaving flow to be rotational through the r e l a t i on  
from which we f i n d  t h e  only component of o t o  be  & or  simply 
From Fig. 1 the  blade e x i t  i s  a t  y = 0 ,  the i n l e t  i s  a t  y = -a and L 
r e f e r s  t o  t he  "length" 
We see  that the downstream fiow i s  then ro t a t i ona l .  Progress i s  
read i ly  made now only i f  w e  assume the  disturbances t o  the flow f i e l d  are 
s m a l l  compared t o  t h e  mean ve loc i ty  components (U,V). I n  t h a t  case  it can 
be  assumed t h a t  w and B a r e  constant* on mean flow streamlines given by 
dy - V 
- t a n u  , dx U 
thus 
and 
w(x,y) = u(x-y cot  a ) 
We now separate  the unknown downstream and upstream flow f i e l d  i n t o  
components a s  follows 
(i) downstream ( u . v ) ~  = (U,V) + (ud,vd) + (ui,vi) + (ur,vr) 
(Ti) upstream ( U , V ) ~  = (U1,V) + (ul,vl) 
where (U,V), (UIV) a r e  mean components, (ud,vd) is the downstream 
A. l.e., w e  l i n e a r i z e  t h e  v o r t i c i t y  equations. 
p o t e n t i a l  d is turbance ,  (ui,vi) i s  a downstream i r r o t a t i o n a l  flow 
and (ur,v,) i s  a r o t a t i o n a l  (shear)  flow which accounrs f o r  t h e  v o r t i c i t y  
avr 
au 
r u = - - -  
ax ay 
It follows t h a t  u , v  a r e  constant  along l i n e s  of x-y c o t  CY = constant  
r r 
and t h a t  vr = t a n  u u . The upstream dis turbance  (ul,vl) i s  i r r o t a t i o n a l .  
r 
I n  t h i s  decomposition (ud,vd) are given dis turbances .  The problem 
then is  t o  f ind  the t h r e e  sets of components , u i  and (ulYvl). 
One r e l a t i o n  between these is given by Eq. ( l l a ) .  Two more r e l a t i o n s  a r e  
needed. One of t h e s e  is given by con t inu i ty  ac ross  t h e  cascade, i,e., 
(here (xc,-a) and (x,O) a r e  po in t s  corresponding t o  t h e  same vane t r a c e ) .  
The o t h e r  is  by t h e  flow tangency condi t ion  a t  y = 0, i.e., 
ui+"df"r = (vi+Vdh',) t a n  k . 
This  i s  apparent ly  a complicated system of r e l a t i o n s  t o  solve .  To 
s u m  up w e  have the i n i t i a l l y  unknown s i x  v e l o c i t y  components (urYvr), (ui,vi) 
and ( u 1 v )  U and v a r e  r e l a t e d  t o  each other  though the require-  
r r 
ment t h a t  far downstream the mean flow angle i s  undisturbed. Both sets 
(uI,,vi) downstream and (ul,vl) upstream arc conjugate ( p o t e n t i a l )  func- 
t i o n s  so  t h a t  u and v a r e  re la ted .  There are then only t h r e e  unknown 
funct ions  l e f t  and w e  have t h e  Eqns. ( l l a )  (with the previous d e f i n i t i o n  
of w ) ,  (12) and (13) t o  r e l a t e  them. Thus a closed system is obtained 
from which s o l u t i o n s  analogous t o  Eq. (8) can b e  found. Then i n  p r i n c i p l e ,  
245 
complete vo lu te  ac tua to r  impel ler  i n t e r a c t i o n s  can be worked ou t .  
W e  should mention t h a t  the type of problem addressed i n  t h i s  sec t ion  
is  not  new except i n  its app l ica t ion  t o  s ingu la r  d is turbances .  E a r l i e r  
Ehrich [4] s tudied t h e  e f f e c t  of i n l e t  wakes passing through r o t o r  and 
s t a t o r  b lade  rows with a  set of equations e s s e n t i a l l y  i d e n t i c a l  t o  t he  
present  ones. Subsequently Katz [5] c a r r i e d  out  a s i m i l a r  computation 
using t h e  acce le ra t ion  p o t e n t i a l  ins tead  o f  t h e  v e l o c i t y  components but  
wi th  t h e  inc lus ion of losses through t h e  b lade  row. Again t h e  matching 
problem ac ross  t h e  b lade  row i s  e s s e n t i a l l y  the same as t h e  present  one. 
The i n t e r a c t i o n  flow f i e l d s  i n  these  works w e r e  determined by Four ier  s e r i e s  
expansion which is a s u i t a b l e  procedure when only a  f e w  terms are needed 
t o  represent  t h e  disturbance.  
Discussion 
We have used t h e  s i n g u l a r i t i e s  of Eq. (8) t o  study t h e  i n t e r a c t i o n  
between a r o t a t i n g  a c t u a t o r  impel ler  and a  vo lu te  (assuming, i r r o t a t i o n a l  
volute f l o w ) .  The unknown v o r t i c i t y  d i s t r i b u t i o n  on t h e  vo lu te  i s  expres- 
sed i n  a  Glauert s e r i e s  t h e  c o e f f i c i e n t s  of which a r e  then determined i n  
t h e  usual  way t o  make the vo lu te  su r face  a streamline.  With t h e s e ~ f u r c e s  
can be found. This t a s k  i s  near ly  complete; w e  intend subsequently t o  i n -  
c lude  t h e  r o t a t i o n a l  e f f e c t s  described i n  the  previous sec t ion .  
Acknowledgment 
This work was supported i n  p a r t  under NASA Contract NAS8-33108 and 
the Byron Jackson Company. This support  is gratefully acknowledged. 
References 
1. D m ,  H .  and Hergt, P. 1970. "Radial Forces on Impeller of Valute 
Casing Pumps". Flow Kesearch on Blading, Dzung, L.S. (Ed . ) ,  E l sev ie r ,  
pp. 365-321. 
2. Chung, W.K., Rott ,  N. "Generalizations of the Inversion Formula of 
Thin A i r f o i l  Theory". J. of Rational  Mech. and Analysis, Vol. 3 ,  No. 3, 
May 1954. 
3. Carr ie r ,  G., Krook, PI., Pearson, C . ,  Functions of a Comp1,ex Variable, 
McGraw-Hill, 1966, Ch. 7. 
4. Ehrich, F., "Circumferential Inlet Dis to r t ions  i n  Axial Flow Turbo- 
macl-rinery", J. hero. Sc i . ,  - 24, N o ,  6,  p.  413, 1957. 
5. Katz, R.,  "Performance of Axial Compressors with Asymmetric I n l e t  FLOWS". 
Calif .  Inst. of Tech., Div. of Eng. & Appl. Sci . ,  J e t  Propulsion Center, 
1453. 


A Brief Note on the Interaction of an Actuator Cascade with a Singularity

Carr ie r , G., Krook, PI., Pearson, C . , Functions of a Comp1,ex Variable, McGraw-Hill, 1966, Ch. 7.

Chung, W.K., Rott , N. &quot;Generalizations of the Inversion Formula of Thin A i r f o i l Theory&quot;. J. of Rational Mech. and Analysis, Vol. 3 , No. 3, May 1954.

Circumferential Inlet Dis to r t ions i n Axial Flow Turbomacl-rinery&quot;,

D m , H . and Hergt, P. 1970. &quot;Radial Forces on Impeller of Valute Casing Pumps&quot;. Flow Kesearch on Blading, Dzung, L.S. (Ed . ) , E l sev ie r , pp. 365-321.

Ehrich, F., &quot;Circumferential Inlet Dis to r t ions i n Axial Flow Turbomacl-rinery&quot;, J. hero. Sc i . , - 24, N o , 6, p. 413, 1957.

Functions of a Comp1,ex Variable, McGraw-Hill,

Generalizations of the Inversion Formula of Thin A i r f o i l Theory&quot;.

Katz, R., &quot;Performance of Axial Compressors with Asymmetric I n l e t FLOWS&quot;. Calif . Inst. of Tech., Div. of Eng. & Appl. Sci . , J e t Propulsion Center,

Performance of Axial Compressors with Asymmetric I n l e t FLOWS&quot;. Calif .

Radial Forces on Impeller of Valute Casing Pumps&quot;. Flow Kesearch on

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A Brief Note on the Interaction of an Actuator Cascade with a Singularity

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