Destabilizing fluid forces on a whirling centrifugal impeller rotating in a volute were observed. A quasisteady analysis neglecting shed vorticity or an unsteady analysis without a volute does not predict the existence of such destabilizing fluid forces on a whirling impeller. The effects of a volute and the shed vorticity are considered. We treat cases when an impeller with an infinite number of vanes rotates with a constant velocity omega and its center whirls with a constant eccentric radius epsilon and a constant whirling velocity psi. It is assumed that: (1) the number of the vanes is so large that the impeller can be treated as an actuator impeller in which the flow is perfectly guided; (2) flow is inviscid, incompressible and two dimensional; (3) the eccentricity epsilon is so small that unsteady components can be linearized; (4) vorticity is transported on a prescribed mean flow, the operating point is near design flow rate; and (5) the volute can be represented by a curved plate

Acosta, A. J.

Brennen, C. E.

Tsujimoto, Y.

English

NASA Technical Reports Server

TWO-DIMEHSIOI?AL UNSTEADY ABUiLYSIS OF FLUID FORCES OH 
A WHIRLING CEIYTRIFUGAL IMPELLER II? A VOLUTE* 
Y. T s u j h t o +  
Osaka University 
560 Toyonaka, Osaka, Japan 
A. J. Acosta and C. E. Brennen 
California Ins t i t u t e  of Technology 
Pasadena, California 91125 
Destabilizing f l u i d  forces on a whirling centrifugal impeller ro ta t ing  i n  a 
volute have been observed (Ref.1). A quasisteady analysis neglecting shed v o r t i c i t y  
(Ref.2) o r  an unsteady analysis without a volute (Ref.3) does not predict the 
existence of such destabil izing f l u i d  forces on a whirling impeller, 
The present report  i s  intended t o  take i n t o  account the e f f ec t s  of a volute and 
the shed vor t ic i ty .  We t r e a t  cases when an impeller with an i n f i n i t e  number of 
vanes r o t a t e s  with a constant ve loc i ty  B and i t s  center whirls with a constant 
eccentric radius e and a constant whirling velocity a. 
Major assumptions a re  a s  follows: 
The number of the vanes i s  so large tha t  the impeller can be t rea ted  a s  
an actuator impeller i n  which the flow i s  perfectly guided. 
Flow is inviscid, incompressible and two-dimensional. 
The eccent r ic i ty  e i s  so small t ha t  unsteady components can be linear- 
iz  ed. 
Vorticity i s  transported on a prescribed mean flow, i.e., the operating 
point i s  near design flow ra te .  
The volute can be represented by a curved p l a t e  (see Fig.1). 
* Par t i a l  support by the Yamada Science Foundation, Japan and the National 
Aeronautics and Space Administration under Contract NAS8-33108. 
+ Visiting Associate, California Institute of Technology (1983-84). 
161 
https://ntrs.nasa.gov/search.jsp?R=19850005817 2020-03-20T19:43:02+00:00Z
SPMBOLS 
2 = Xt;Y : stationary frame with i t s  or ig in  0 fixed t o  the  center of the w h i t  
l ing  motion 
#=X'+iT : t rans la t ing  frame with i t s  o r ig in  0'  fixed t o  the  center of the 
=2-ze;Ut impeller and i t s  axes pa ra l l e l  t o  those of Z. 
c , r ,  : inner and outer radius of the impeller 
gCr? 
E : eccent r ic i ty  
: vane angle measured from circumferential d i rec t ion  
w : angular velocity of whirling motion 
J2 : angular ve loc i ty  of the impeller 
E : prerotation, Q: flow r a t e  
= U +LU : absolute ve loc i ty  
( Vr, Uel 
r=u'+iv' : velocity r e l a t ive  t o  x'y' frame 
cu;, V i )  
(Wr, We) 
w : veloc i ty  r e l a t ive  t o  the  impeller 
t : time, t-0 when 00' is  i n  the  d i rec t ion  of I. 
n : circumferential mode number 
Subscripts 
1 , 2  : quan t i t i e s  a t  the inner and outer radius respectively 
Super sc r ip t  s 
- 
: steady component - ; unsteady component 
B A S I C  EQUATIONS 
Euler's equation i n  the  ro ta t ing  and t rans la t ing  frame f i r e d  t o  the ro tor  is, 
* 
where a/at  i s  the time derivative i n  the ro ta t ing  frame and g=ioeeiot and 
cy Q Xz'=irOsie are the t rans la t iona l  velocity due t o  whirling and the rotational 
velocity of the impeller. In an actuator impeller with an i n f i n i t e  number of vanes, 
162 
the e f fec ts  of the v o r t i c i t y  d is t r ibu t ion  on the vanes and the forces exerted by the 
vanes a re  represented by the vor t ic i ty  V X x  and the external force 5 
respectively in  the above equation. For inviscid flow through actuator impellers, 
f and J X ( V X x )  are  normal t o  the vane surface and the component of Quat ion  (1) 
para l le l  t o  the vane surface i s  
el 
Integrating Equation (2) along the vane surface we get the following t o t a l  pressure 
increase, 
where 
PLEMENTARY FLOW COMPONENTS 
The flow tangency condition a t  the impeller ou t l e t  is, 
The f i r s t  term on the r i g h t  hand si& of the above equation i s  cancelled by the flow 
compone nt 
and other disturbance components should s a t i s f y  the following equation 
For the region outside the impeller we consider two types of elementary velocity 
disturbances satisfying Equation (8) : 
F i r s t ,  consider the following velocity f i e ld :  
This velocity f i e l d  has a vortex a t  z '=z'  and s a t i s f i e s  Equation (8) with no 
circulation/net flow arorurd/from the impeller. Consider a vortex d i s t r ibu t ion  
0 
nS) = r S w  4 e c ( s ) s i n  w t  + s q ( s ) c o s  w t  on the volute srrrface zo(s)=z'o+tte i w t  . 
Assuming E << r2, we get the following steady and unsteady velocity components 
C/O) 
where z '  i s  fixed t o  x'y'-plane. A t  z=zs=z'+eeiwt, which i s  fixed t o  the volute, 
we get the following expressions: 
164 
In the same way the ve loc i ty  component (7) has the  following steady and unsteady 
components a t  the volute sllrface 
ie 
zs=rse s=z'+sei"t. 
Next consider the ve loc i ty  f i e l d  due t o  shed vo r t i c i ty .  I f  we assume tha t  the 
v o r t i c i t y  is transported on a streamline 
Equation (17) includes the  e f f e c t s  of the  v o r t i c i t y  i n  r <r(B. Since G ( r )  i s  
convergent only  for n23 as  R+, we should a r tZ f i c i a l ly  prescribe some 
appropriate f i n i t e  value f o r  R. Equation (17) does not s a t i s f y  Equation (8) and we 
should add a potential  component 
165 
such tha t  
s a t i s f i e s  the boundary condition of Equation ( 8 ) .  We may now note tha t  there a re  
steady and unsteady components a s  follows: 
(i) Steady component; i n  Equations (16)-(19) - we put w=O and represent the  
re la ted  quangities with superscript and suf f ix  n (n=1,2,. .) for  each 
mode, gm etc. 
represent the re la ted  quant i t ies  such tha t  cs etc. (ii) Unsteady component; correspondingly t o  the sign on w and the  mode n, we 
z=z =z@+eeiwt. the steady and unsteady velocity f i e l d s  a re  On the volute 
as  follows: 
expressed 
S 
In  the region upstream of the impeller (rlr 1 ,  we consider a source Q and 
prero ta t ion  of strength ri a t  the center of t i e  volute. Then the velocity can be 
expressed i n  the se r i e s  
- 
where =-h+iAIn, AS=$ +iA~n.A~=f$n+i<n a r e  complex constants. n 
Now we have given a l l  of the elementary flow components necessary fo r  the 
construction of the e n t i r e  flow f ie ld .  Each of them contains several unknowns tha t  
a r e  determined i n  the following sections. 
166 
BouMlABY CONDITIONS ON THE VOLUTE 
The flow tangency condition on the  vola te  surface is, 
where a i s  the angle between volute  and x-axis. I f  we put 
h 
b/ = i- (ye/oswt f @shut 
v'= U'?- xcoswt 3- & h W t  - fb 
we ge t  the following conditions,  
p'md - = 0 ,  (24) 
%@Sol - 2 L & o c =  - E Q W O ( ,  d2!3 
$@so( -Eg&C(= -Eu&a.  ( 26) 
The steady ve loc i ty  a',~' is  given a s  a sum of the ve loc i ty  components i n  Equations 
(121, (14) and ( 2 0 ) .  The unsteady components u' 7 and u * ~ ,  uts  a re  the cosine 
and s ine  components of the ve loc i ty  of Equations (131, (15) and (21). Equations 
(241426) cons t i tu te  in tegra l  equations f o r  P s ( s ) , c ( s )  and c ( s ) .  
- -  
tu & N  
C' 0 
CONTINUITY EQUATION 
The cont inui ty  equation across  the  impeller i s  
where Po is the  angle between corresponding leading and t r a i l i n g  edges of a vane. 
A t  the  outer radius  the t o t a l  of the steady veloci ty  components given by Equations 
( 71, (10) and (19) can be expressed a s  a Fourier se r ies ;  namely, 
167 
In the same way, the unsteady components of Equations (111, and (19) can be 
expressed a s  
A t  the inner radius 
then, 
t=rl we expand Equation (22) i n  9iel-$P0 ra ther  than i n  
From Equation (27) we get the following r e l a t ions  
STRENGTH OF SHED VORTICITY 
If we use the expressions (28-31) i n  Equation (51, the steady vo r t i c i ty  
components can be expressed a s  
Q where we have used W~=~71s;, because of the assumption on the transport of the 
vor t ic i ty .  In the same way, i f  we express the v o r t i c i t y  by 
168 
4 
and use the expressions (28-31) i n  Equation (51, we can express 1; i n  terms of the 
Fourier coef f ic ien ts  i n  Equations (281431) .  Comparing Equation (40) and Equation 
(161, we get the following r e l a t ions ,  
METEQD OF SOLUTIQN 
We have used the  following unknowns f o r  the expression of the flaw f i e l d  
steadv comvonent unsteady component 
These Mknowns a r e  determined by the following re la t ions :  
Steady Comvonent Unsteady Component 
B.C. on t he  volute:  Eq. (24) Eqs. (251, (26) 
Continuity : Eqs. (32) ,(33) EqS. (34) ,(35) ,(36) s(37) 
Vor t ic i ty  : Eqs. (381, (39) Eq s ( 41 1 p ( 42 ( 43 ) n ( 44) 
These equations include in t eg ra l s  r e l a t ed  t o  the vortex d i s t r i b u t i o n  on the  volute 
surface,  which should be evaluated by some appropriate method. Equations (241426) 
a re  in tegra l  equations f o r  the vortex d i s t r ibu t ions  on the  volute surface and could 
be reduced t o  simultaneous l i n e a r  equations by a s ingular i ty  method. In the  solu- 
t i o n  of the vortex d i s t r i b u t i o n s  the “gutta condition” a t  the t r a i l i n g  edge should 
be applied. S t r i c t l y  speaking the c i r cu la t ion  around the volute f luc tua te s  and a 
f r e e  vortex sheet i s  shed from the t r a i l i n g  edge of the  volute.  Since we a re  mainly 
in te res ted  i n  the forces  on the  impeller, we w i l l  neglect the e f f ec t  of the f r ee  
vortex sheet but apply the following conditions at. the  t r a i l i n g  edge. 
Steady par t :  
m a >  = 0 
Unsteady par ts :  
C4f> 
Now w e  can express a l l  the r e l a t ions  a s  a s e t  of simultaneous l i n e a r  equations which 
can be solved numerically. The steady component may be solved independently of 
169 
unsteady component, and the r e s u l t  used i n  the  ana lys i s  of the unsteady components. 
UNSTEADY FORCES ON THE IMPELLER 
By considering the  balance of the momentum of the  f l u i d  i n  the  impeller, we can 
express the  forces  on t he  impeller a s  follows; 
Steady component 
(4%) 
and unsteady component 
The t o t a l  pressure i s  given by Eqaation (3 )  and the  in t eg ra l s  can be evaluated 
ana ly t i ca l ly  by using the  expressions (28-31). 
CDNCLUDING REMARKS 
The unsteady forces  can eventually be expressed i n  the form of s t i f f n e s s  
matrix, 
The t imzaverage of the force component i n  the  d i r ec t ion  of whirl ing motion i s  given 
by Ih (Yc-rs) and the  sign of t h i s  quant i ty  determines whether or n o t z h e  f l u i d  
forces  have des tab i l iz ing  e f f e c t s  on the  whirl ing motion. gives 
the  time average of the force component i n  the r ad ia l  d i r ec t ion  and thus the 
hydrodynamic s t i f fnes s .  The ult imate goal of the present study i s  t o  examine these 
f ac to r s  f o r  r e a l  i s t i c  impeller-volute combinations. 
The sum v. gc+Ys) 
170 
REFERENCES 
1. Chamieh, D. S., Acosta, A. J., Brennen, C. E., Caughey, T. IC. and Franz, R., 
"Experimental Measurements of Hydrodynamic Stiffness Matrices for a Centrifu- 
gal Pump Impeller", Proceedings of NASA/ARO Workshop on Rotordynamic 
Instability Problems in High Performance Turbomachinery, NASA CP-2250, pp. 
382-398. 
2. Cbamieh, D. S. "Forces on a Whirling Centrifugal Pump Impeller'f, Report 
E249.2, Div. of Eng. 8 Appl. Sci., Calif .  Ins t .  of Tech., 1983: see a l s o  
Chamieh, D., Acosta, A. J., Brennen, C. E. and Caughey, T. K. 1980. "A Brief 
Note on the Interaction of an Actuator Cascade with a Singularity. Proceedings 
of NASAIARO Workshop on Rotordynamic Instability Problems in High Performance 
Turbomachinery", NASA CP-2133,1980, pp. 237-248. Which formed the stimulus for 
the present work. 
3. Shoji ,  E. and Ohashi, E. "Fluid forces  on ro t a t ing  centr i fugal  impeller with 
whirl ing motion", 1st Workshop on Rotordynamic I n s t a b i l i t y  Problems in High 
Performance Turbomachinery, Texas Am Univ., NASA Conf. Pub. 2133, (1980). 
Shoji ,  E. and Ohashi, E. "Fluid forces  on ro t a t ing  centr i fugal  impeller with 
whirl ing motion", Japan SOC. Mech. Engrs. ( i n  Japanese),  Vo1.47, No.1, B(1981- 
71, p. 1187. 
171 
0"- ------ r----.. ---- .I----- - - L A - -  - - 0 - - -  - -  
172 


Two-dimensional unsteady analysis of fluid forces on a whirling centrifugal impeller in a volute

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