The asymmetric flow within a volute exerts a radial force on a centrifugal impeller.  The present paper presents experimental measurements of the radial forces on the impeller in the presence of cavitation

Acosta, A. J.

Brennen, C. E.

Caughey, T. K.

Franz, R. J.

English

Caltech Authors - Main

ON THE EFFECT OF CAVITATION ON THE RADIAL FORCES AND-HYDRODYNAMIC 
STIFFNESS OF A CENTRIFUGAL PUMP* 
R.J. Franz, C.E.  Brennen, A.J. Acosta, and T.K. Caughey 
Caljfornqa Instltuta o f  Technology 
Pasadena, California 91125 
The asymmetric flow within a volute exerts a radial foroe on a centrifugal 
impeller. The present paper presents experimental measurements of the radial farces 
on the impeller i n  the  prassnoe of' oavi ta t iun .  
NOMEN CL Am RE 
2 2 hydrodynamic force matrix, non-dimensionalized by pnw rZbZ 
2 
impeller inlet  area (nrl), outlet area (2nr2b2) 
impeller disoharge width ( 0 . 6 2  in1 
6-co~uponent generalized force vector 
components of the instantaneous l a t e r a l  force on the impeller in the 
rotating dynamometer reference frame 
components of the instantaneous lateral force on the impeller in the fixed 
2 3 laboratory referenae frame (X.Y), non-dimensionalized by pnw rZbZ 
values of Fx and Py i f  the impeller was located a t  the the orig in  of the  
2 3 (X ,Y )  coordinate aystem (vo lu te  center) ,  non-dirnensionalieed by pnw rZbZ 
* The authors are indebted to  the NASA George Marshall Space Flight Center, 
Alabama for continued sponsorship of t h i s  research under contract NAS8-33108. 
We are also grateful for the help  given by D. Brennen. 
F~ J F~ components of the  l a t e r a l  fo rce  on the  impel ler  normal t o  and t angen t i a l  
t o  the  whirl  o r b i t ,  averaged over the  o r b i t ,  non-dimensionalized by 
2 2 pnw r b E 2 2 
FN = (A,, + Ayy)/2 FT = (-Axy * Ayx)/2 
P 1  ' P t l  upstream s t a t i c ,  t o t a l  pressure  
P2 'Pt2 downstream s t a t i c ,  t o t a l  pressure  
% s t a t i c  ppessure a t  impeller  i n l e t ,  ptl - p(91 2/2 A1 
Pv vapor pressure of water 
Q flow r a t e  
"1 ~~2 impeller  i n l e t ,  discharge rad ius  (1.594 i n . ,  3.188 in . )  
t time 
( X J Y )  f ixed  labora tory  reference  frame 
X, Y instantaneous coordinates of the  impeller  cen te r  i n  t h e  f ixed  labora tory  
reference  frame ( X , Y ) ,  non-dimensionalized by r2 
a r ad ius  of t h e  c i r c u l a r  whir l  o r b i t  
8 angle of the impeller  on t h e  eccen t r i c  o i r c l e ,  measured from the  volute  
tongue i n  the  d i r e c t i o n  of impel ler  r o t a t i o n  
P densi ty  of' water 
I I c a v i t a t i o n  number, 
,2r ; /2  
9 f l a  ooef f lo ien t  based on the  impel ler  discharge a rea  and t i p  apes& 
Pt2-Pu 
t o t a l  head c o f f i o i e n t ,  
r ad ian  frequency of t h e  impeller  ( s h a f t )  r o t a t i o n  
INTRODUCTION 
Earlier papers (refs.1-8) have described measurements of the  r a d i a l  fo rces  and 
hydrodynermioally induoed rotordynamic c o e f f i c i e n t s  of oen t r i fuga l  pumps with var ious  
impellera and volutes .  A l l  of these  e a r l i e r  measurements were made i n  the absence 
of any c a v i t a t i o n  wi th in  t h e  pump. Yet the re  is some evidence t h a t  the  preaence of 
c a v i t a t i o n  may have B s i g n i f i c a n t  e f f e c t  on these  f o r c e s  and coef f i c i en t s .  Indeed 
aame t e a t s  of t h e  high speed pumps i n  t h e  Space S h u t t l e  Main Engine have suggested a 
change i n  t h e  rotordynamics when c a v i t a t i o n  occurs ( r e f .  9 ) .  The present  paper i s  a 
supplement t o  our e a r l i e r  measurements of fo roes  and c o e f f i c i e n t s  and c o n s t i t u t e s  a n  
exp lo ra t ion  of t h e  inf luence  of cav i t a t ion .  
The referenoes  6-8 provide a complete desc r ip t ion  of the  f a c i l i t y .  Br ief ly ,  
the  dynamometer, composed of two p a r a l l e l  p l a t e s  connected by four s t r a i n  gaged 
pos ts ,  is mounted between the  impel ler  and the  d r ive  s h a f t .  It measures t h e  six 
components of a generaLized hydrodynamic f o r c e  vector  EFI a c t i n g  on the  impeller, 
The impeller  can be sub jec t  t o  whir l ing  motion i n  an  o r b i t  eccen t r i c  t o  t h e  vo lu te  
center ,  i n  a d d i t i o n  t o  the  normal impeller  r o t a t i o n .  Since the  eccen t r i c  motion i s  
i n  the l a t e r a l  plane, perpendicular  t o  t h e  impel ler  cen te r l ine ,  only the  two 
components of the fo rce  vector  EF) i n  t h i s  l a t e r a l  plane w i l l  be discussed. 
These fo roes  can be represented by 
Refering t o  f i g u r e  1, Fx and F a r e  i n  the  volute  frame of reference,  and x and y 
represent  t h e  coordinates oF the  impeller cen te r  measured from the  volute  center. 
Dimensionless q u a n t i t i e s  a r e  used throughout (see  Nomenclature f o r  d e f i n i t i o n s ) .  
The preaent  results are only f o r  the  oase of no w h i r l :  x and y are f i x e d  i n  time. 
When t h e  impel ler  is looated  a t  an angular pos i t ion ,  8 ,  on t h e  c i r c u l a r  whirl  o r b i t  
of r ad ius  e ,  equation (1) is w r i t t e n  as 
The s teady l a t e r a l  foroes,  represented by F and F can be considered as t h e  
Y' sum of two forces :  a f ixed fo rce ,  represante8  by Fox 
and F 8 ~ s  
which the impeller  
would experience if located  a t  the  volute  cen te r ,  and a fo rce  e t o  t h e  e c c e n t r i c  
displacement of the  impel ler ,  represented by a " s t i f fness r r  matr ix [A1 . By taking 
d a t a  a t  four f ixed  e c c e n t r i c  p o s i t i o n s  of the  impel ler ,  90 degrees a p a r t ,  t h e  matrix 
[A1 and t h e  vector  IFo} can be ext rac ted .  The g r a v i t a t i o n a l  and buoyancy fo rces  on 
t h e  r o t o r  are subtracted out.  
EXPERIMENTS 
I n  references  6-8 r e s u l t s  were presented f o r  a typ ica l  i m p e l l e r  ( impel ler  X I ,  a 
f i v e  bladed Byron-Jackson cen t r i fuga l  impeller  with a s p e c i f i c  speed of 0.57, 
operat ing i n  a s p i r a l  volute  (volute  A) a t  var ious  impel ler  speeds (I 2000 rpm) and 
flow r a t e s .  I n  order  t o  t e s t  under c a v i t a t i n g  condit ions,  t h e  impel ler  speed was 
increased t o  3000 rpm and t h e  water was s u b s t a n t i a l l y  de-aerated. 
I n  the  following pages t h e  r e s u l t s  f o r  t h r e e  d i f f e r e n t  flow r a t e s  a r e  
presented: t h e  flow r a t e s  chosen a r e  below design (d=0.060), design (d=0.092) and 
above design (d=0.104). Cavi ta t ion  performance curves f o r  each of these  flow rates 
a r e  presented i n  f i g u r e  2. We observe t h a t  the  breakdown c a v i t a t i o n  numbers f o r  the  
t h r e e  flow r a t e s  t e s t e d  are 0.17, 0.26 and 0.30 respect ive ly .  
Results  f o r  the  r a d i a l  forces ,  F and F , and f o r  the  s t i f f n e s s  c o e f f i c i e n t s  
a r e  presented i n  f i g u r e s  3 through 8: A qufEk glance w i l l  show t h a t  l a r g e  changes 
occur i n  both c h a r a c t e r i s t i c s  a s  t h e  pump approaches and passes through breakdown. 
Notice t h a t  the  steady fo rce  shown i n  f i g u r e s  3 and 4 changes i n  both magnitude and 
d i rec t ion .  
Figures 3 and 4 show t h a t  f o r  d=0.060 the  magnitude of {Fol decreases  with 
performance l o s s  with a small  change i n  d i rec t ion .  For both d=0 .O92 and d=0 .lO4, 
t h e  magnitude of {F I has  a minimum with decreasing head coeff icent .  For each flow 
c o e f f i c i e n t ,  t he  d f r e c t i o n  of {Fol r o t a t e s  away from the  tongue. For d-0.092, {Fol 
r o t a t e s  through more than 180° as t h e  pump progresses through breakdown. 
Figures 5-8 show the  hydrodynamic f o r c e  matrix CAI from equation (2)  . These 
no-whirl r e s u l t s  corrrespond t o  -[K1 of the  quadra t i c  f i t  of reference  2 t o  whir l  
d a t a  f o r  mass, damping and s t i f f n e s s  matrices.  The t h r e e  flow c o e f f i c i e n t s  e x h i b i t  
t h e  same trends.  The diagonal elements decrease with performance l o s s .  The off- 
diagonal elements first decrease s l i g h t l y  i n  magnitude with lower c a v i t a t i o n  number 
then increase  with performance l o s s .  Figure 8 i n d i c a t e s  t h a t  t h e  off-diagonal ele-  
ments do not change monotonically with head coef f i c i en t .  
I n  summary, c a v i t a t i o n  a f f e c t s  t h e  steady fo rces ,  both the  impeller-centered 
f o r c e  {Fol and the  hydrodynamic f o r c e  matrix CAI, exerted upon an  e c c e n t r i c  
impeller .  It is useful  t o  i n t e r p r e t e  the  matrix CAI i n  terms of t h e  average normal 
force ,  FU, and the  t angen t i a l  fo rce ,  F , ac t ing  on a whir l ing  impel ler  i n  t h e  limit 
a s  the  whwl speed approaches zero.  ~ i f h  c a v i t a t i o n  breakdown, t h e  normal fo rce  is 
reduced while the  t angen t i a l  fo rce  i s  increased.  Fur ther  t e s t s  a r e  planned t o  
examine the  unsteady flow e f f e c t s  which occur a t  non-zero whir l  speed. More specif-  
i c a l l y ,  the  frequency dependence i n  t h e  matrix [A1 ( t h e  damping and added mass 
components) w i l l  s h o r t l y  be examined as a func t ion  of c a v i t a t i o n  number. 
REFERENCES 
1 Agoatinelli, A .  , Nobles, D ,  , and Mockridge, C +  R .  , "An Experimental Investige- 
t i o n  of Radial Thrust i n  Centrifugal Pumps", ASME Journal of Engineering for 
Power, Vole 82, pg. 120-126, Apri l  1960. 
2 .  D m ,  H., and Hergt, P, , tlRadial Forces on Impeller of Volute Caaing PumpeM, 
Flow Reaearch on Blading (L. 8. Dzung, ed.1, Elsevier Pub. Co., The Nether- 
lands, pp 305-321, 1 9 7 0 ,  
3 .  Ohashi, H .  and Shoji ,  H . ,  "Lateral Fluid Forces Aoting on a Whirling Centrifu- 
gal Impeller i n  Vanelese and Vaned Dif fuser ,"  Third Hclrkahop an Rotordynamic 
h a t a b i l i t y  Problems in High-Perfarmanee Turbomachinwy, Texas AW University, 
May 28-30, 1984. 
4 .  Chamieh, D. S . ,  ''Forces on a Whirling Centrifugal Pump-Impeller", Ph. D. 
Thesis, Division of Engineering and Appl ied  Soienoe, California Institute of 
Teahnology, February 1983. 
5 ,  Chamieh, D.  S,, Acosta, A.  J . ,  Brennen, C. E. and Caughey, T .  K,, "Experimental 
Measurements of Hydrodynamic Radial Forces and S t i f f n e s s  Matrioes for a 
Centrifugal Pump-Impeller,'l ASME Journal of F l u i d s  Engineering, Vol, 107,  Sept, 
1985, pp. 307-315. 
6 ,  Jery, B,, Aoosta, A.J., Brennen, C . E . ,  and Caughey, T.K., "Hydrodynamic 
Impeller S t i f f n e s s ,  Damping and Inertia i n  the Rotordynamias of Centrifugal 
Pumps," "Third Workshop on Rotordynamic Ins tab i l i ty  Problems i n  High 
Performance Turbomaohinery", Texas A&! University, May 28-30,1984. 
7 .  Jery, B., Brennen, C.E., Caughey, T . K . ,  and Acosta, A.J., "Forces on Centrifu- 
gal Pump rmpsllers, " "Second International Pump Symposium", Houston, Texas, 
A p r i l  2 9-May 2,1985. 
8 .  Jery, B e D  "Experimental Study of Unsteady Hydrodynamic Force Matrices on Whir- 
l i n g  Centrifugal Pump-Impellers", Ph. D .  Thesis. Division of Engineering and 
Applied Science, California Institute of Teohnology. October 1985. 
9 .  Rocketdyne Division, Rodwel l ,  personal oommunfcatlon 
/. CUTWATER 
'- WHIRL CIRCULAR 
O R B I T  
Figure 1 Schematic representation of the posi t ion of the impel ler  within the 
volute,  The measured forces are indicated i n  the rotating dynamometer 
frame (as FI. F2 and i n  the stationary volute frame (as F, FYI . 
-- 
I I I I I I 
HERD C O E F  0 PHI=.060 
PHI2.092 
A PHI=,104 
L ,  1 . --- 
0  
I 
0.5 1 . 0  1 .5  
s igma 
Figure 2 Cavitation performance curves for  the three flow coef f i c i ents :  6-0.060 
(below design) , d=O.O92 (design) and d=0 .I04 (above design) , for  volute 
A/impeller X a t  3000 rpm. 
I 1 
I 
I 
I 
I 
I 
0 0 , s  1, B 1  a 5 
sigma 
. 1 0  
Figure 3 The dependence of the magnitude and direct ion of the volute force, CF I, 
on cavitat ion number for  the three flaw coef f i c i ents :  d=0.060, d=0,0%, 
d=0 .lO4, for  volute A/irnpeller X a t  3000 rpm. 
r I I I I I I - Fomag 0 PHI= .068  
P H I = . 0 9 2  - A P H I = . l B 4  - 
8 . 1  0.2 0,f 6 , 4  0 . 5  0.6 
HEFtD COEF 
n u a n =  B .e n 0 I 
Figure 4 The dependence of the magnitude and direction of t h e  volute force, {F I, 
on head coef f icbent for the three flow coef f i c i ents  : d-0.060, d=O.082, 
Q4.104, for volute A/impeller X a t  3000 rpm. 
Figure 5 
I I I I 
0 0.5 1 .0  1 .5  
sigma 
The dependence of the elements of the hydrodynamic force matrix, [ A ] ,  on 
cavitat ion number for  d=0.060 (below des ign) ,  for  volute A/impeller X at 
3000 rpm. 
Figure 6 The dependence of the elements of the hydrodynamic force matrix, I A l ,  on 
cavitat ion number for  Q=0,092 (at design) ,  for  volute A/impeller X a t  
3000 rpm. 
I I I I 
u Axx 0 Axy PHI=,092 
A Ayx v Ayy 
B 
D 
v 
2 - v :  - 
1 - 
i 
# A A A ;  A A  
A  - 
0 I I _. . I I .  1 I 
r 0 0 4 C E 
-1 - - 
L.1 I I 1 
0 0.5 1.0 1 , 5  
s i gea 
Figure 7 The dependence of the elements of the hydrodynamic force matrix, C A I ,  on 
cavitat ion number for  Q=0.104 (above design), for volute A/impeller X at 
3000 rpm. 
I 1 
R x x  6 Rxy I 
r I I * 
PHI= ,  104 
A Ayx v A Y V  
$9 B s Q B u 2 - - 
1 - 
i 
A4 a. A A A A 
Y 
- 
0 I I I I I I I I 
P" O0 ' 
0 O 0 0 
0.1 0n2 0 , 3  0 . 4  . 085  0,6 
HEAD COEF 
-1  
Figure 8 The dependence of the elements of the hydrodynamic force matrfx, [ A ] ,  on 
head c o e f f i c i e n t  for d=0,104 (above des ign) ,  for  volute Alimpeller X at 
3000 rpm. 
- - 
I I I I 
0 0 . 5  1.0 1,5 
sigma 


On the Effect of Cavitation on the Radial Forces and Hydrodynamic Stiffness of a Centrifugal Pump

ON THE EFFECT OF CAVITATION ON THE RADIAL FORCES AND-HYDRODYNAMIC 
STIFFNESS OF A CENTRIFUGAL PUMP* 
R.J. Franz, C.E.  Brennen, A.J. Acosta, and T.K. Caughey 
Caljfornqa Instltuta o f  Technology 
Pasadena, California 91125 
The asymmetric flow within a volute exerts a radial foroe on a centrifugal 
impeller. The present paper presents experimental measurements of the radial farces 
on the impeller i n  the  prassnoe of' oavi ta t iun .  
NOMEN CL Am RE 
2 2 hydrodynamic force matrix, non-dimensionalized by pnw rZbZ 
2 impeller inlet  area (nrl), outlet area (2nr2b2) 
impeller disoharge width ( 0 . 6 2  in1 
6-co~uponent generalized force vector 
components of the instantaneous l a t e r a l  force on the impeller in the 
rotating dynamometer reference frame 
components of the instantaneous lateral force on the impeller in the fixed 
2 3 laboratory referenae frame (X.Y), non-dimensionalized by pnw rZbZ 
values of Fx and Py i f  the impeller was located a t  the the orig in  of the  
2 3 (X ,Y )  coordinate aystem (vo lu te  center) ,  non-dirnensionalieed by pnw rZbZ 
* The authors are indebted to  the NASA George Marshall Space Flight Center, 
Alabama for continued sponsorship of t h i s  research under contract NAS8-33108. 
We are also grateful for the help  given by D. Brennen. 
F~ J F~ components of the  l a t e r a l  fo rce  on the  impel ler  normal t o  and t angen t i a l  
t o  the  whirl  o r b i t ,  averaged over the  o r b i t ,  non-dimensionalized by 
2 2 pnw r b E 2 2 
FN = (A,, + Ayy)/2 FT = (-Axy * Ayx)/2 
P 1  ' P t l  upstream s t a t i c ,  t o t a l  pressure  
P2 'Pt2 downstream s t a t i c ,  t o t a l  pressure  
% s t a t i c  ppessure a t  impeller  i n l e t ,  ptl - p(91 2/2 A1 
Pv vapor pressure of water 
Q flow r a t e  
"1 ~~2 impeller  i n l e t ,  discharge rad ius  (1.594 i n . ,  3.188 in . )  
t time 
( X J Y )  f ixed  labora tory  reference  frame 
X, Y instantaneous coordinates of the  impeller  cen te r  i n  t h e  f ixed  labora tory  
reference  frame ( X , Y ) ,  non-dimensionalized by r2 
a r ad ius  of t h e  c i r c u l a r  whir l  o r b i t  
8 angle of the impeller  on t h e  eccen t r i c  o i r c l e ,  measured from the  volute  
tongue i n  the  d i r e c t i o n  of impel ler  r o t a t i o n  
P densi ty  of' water 
I I c a v i t a t i o n  number, 
,2r ; /2  
9 f l a  ooef f lo ien t  based on the  impel ler  discharge a rea  and t i p  apes& 
Pt2-Pu 
t o t a l  head c o f f i o i e n t ,  
r ad ian  frequency of t h e  impeller  ( s h a f t )  r o t a t i o n  
INTRODUCTION 
Earlier papers (refs.1-8) have described measurements of the  r a d i a l  fo rces  and 
hydrodynermioally induoed rotordynamic c o e f f i c i e n t s  of oen t r i fuga l  pumps with var ious  
impellera and volutes .  A l l  of these  e a r l i e r  measurements were made i n  the absence 
of any c a v i t a t i o n  wi th in  t h e  pump. Yet the re  is some evidence t h a t  the  preaence of 
c a v i t a t i o n  may have B s i g n i f i c a n t  e f f e c t  on these  f o r c e s  and coef f i c i en t s .  Indeed 
aame t e a t s  of t h e  high speed pumps i n  t h e  Space S h u t t l e  Main Engine have suggested a 
change i n  t h e  rotordynamics when c a v i t a t i o n  occurs ( r e f .  9 ) .  The present  paper i s  a 
supplement t o  our e a r l i e r  measurements of fo roes  and c o e f f i c i e n t s  and c o n s t i t u t e s  a n  
exp lo ra t ion  of t h e  inf luence  of cav i t a t ion .  
The referenoes  6-8 provide a complete desc r ip t ion  of the  f a c i l i t y .  Br ief ly ,  
the  dynamometer, composed of two p a r a l l e l  p l a t e s  connected by four s t r a i n  gaged 
pos ts ,  is mounted between the  impel ler  and the  d r ive  s h a f t .  It measures t h e  six 
components of a generaLized hydrodynamic f o r c e  vector  EFI a c t i n g  on the  impeller, 
The impeller  can be sub jec t  t o  whir l ing  motion i n  an  o r b i t  eccen t r i c  t o  t h e  vo lu te  
center ,  i n  a d d i t i o n  t o  the  normal impeller  r o t a t i o n .  Since the  eccen t r i c  motion i s  
i n  the l a t e r a l  plane, perpendicular  t o  t h e  impel ler  cen te r l ine ,  only the  two 
components of the fo rce  vector  EF) i n  t h i s  l a t e r a l  plane w i l l  be discussed. 
These fo roes  can be represented by 
Refering t o  f i g u r e  1, Fx and F a r e  i n  the  volute  frame of reference,  and x and y 
represent  t h e  coordinates oF the  impeller cen te r  measured from the  volute  center. 
Dimensionless q u a n t i t i e s  a r e  used throughout (see  Nomenclature f o r  d e f i n i t i o n s ) .  
The preaent  results are only f o r  the  oase of no w h i r l :  x and y are f i x e d  i n  time. 
When t h e  impel ler  is looated  a t  an angular pos i t ion ,  8 ,  on t h e  c i r c u l a r  whirl  o r b i t  
of r ad ius  e ,  equation (1) is w r i t t e n  as 
The s teady l a t e r a l  foroes,  represented by F and F can be considered as t h e  
Y' sum of two forces :  a f ixed fo rce ,  represante8  by Fox 
and F 8 ~ s  
which the impeller  
would experience if located  a t  the  volute  cen te r ,  and a fo rce  e t o  t h e  e c c e n t r i c  
displacement of the  impel ler ,  represented by a " s t i f fness r r  matr ix [A1 . By taking 
d a t a  a t  four f ixed  e c c e n t r i c  p o s i t i o n s  of the  impel ler ,  90 degrees a p a r t ,  t h e  matrix 
[A1 and t h e  vector  IFo} can be ext rac ted .  The g r a v i t a t i o n a l  and buoyancy fo rces  on 
t h e  r o t o r  are subtracted out.  
EXPERIMENTS 
I n  references  6-8 r e s u l t s  were presented f o r  a typ ica l  i m p e l l e r  ( impel ler  X I ,  a 
f i v e  bladed Byron-Jackson cen t r i fuga l  impeller  with a s p e c i f i c  speed of 0.57, 
operat ing i n  a s p i r a l  volute  (volute  A) a t  var ious  impel ler  speeds (I 2000 rpm) and 
flow r a t e s .  I n  order  t o  t e s t  under c a v i t a t i n g  condit ions,  t h e  impel ler  speed was 
increased t o  3000 rpm and t h e  water was s u b s t a n t i a l l y  de-aerated. 
I n  the  following pages t h e  r e s u l t s  f o r  t h r e e  d i f f e r e n t  flow r a t e s  a r e  
presented: t h e  flow r a t e s  chosen a r e  below design (d=0.060), design (d=0.092) and 
above design (d=0.104). Cavi ta t ion  performance curves f o r  each of these  flow rates 
a r e  presented i n  f i g u r e  2. We observe t h a t  the  breakdown c a v i t a t i o n  numbers f o r  the  
t h r e e  flow r a t e s  t e s t e d  are 0.17, 0.26 and 0.30 respect ive ly .  
Results  f o r  the  r a d i a l  forces ,  F and F , and f o r  the  s t i f f n e s s  c o e f f i c i e n t s  
a r e  presented i n  f i g u r e s  3 through 8: A qufEk glance w i l l  show t h a t  l a r g e  changes 
occur i n  both c h a r a c t e r i s t i c s  a s  t h e  pump approaches and passes through breakdown. 
Notice t h a t  the  steady fo rce  shown i n  f i g u r e s  3 and 4 changes i n  both magnitude and 
d i rec t ion .  
Figures 3 and 4 show t h a t  f o r  d=0.060 the  magnitude of {Fol decreases  with 
performance l o s s  with a small  change i n  d i rec t ion .  For both d=0 .O92 and d=0 .lO4, 
t h e  magnitude of {F I has  a minimum with decreasing head coeff icent .  For each flow 
c o e f f i c i e n t ,  t he  d f r e c t i o n  of {Fol r o t a t e s  away from the  tongue. For d-0.092, {Fol 
r o t a t e s  through more than 180° as t h e  pump progresses through breakdown. 
Figures 5-8 show the  hydrodynamic f o r c e  matrix CAI from equation (2)  . These 
no-whirl r e s u l t s  corrrespond t o  -[K1 of the  quadra t i c  f i t  of reference  2 t o  whir l  
d a t a  f o r  mass, damping and s t i f f n e s s  matrices.  The t h r e e  flow c o e f f i c i e n t s  e x h i b i t  
t h e  same trends.  The diagonal elements decrease with performance l o s s .  The off- 
diagonal elements first decrease s l i g h t l y  i n  magnitude with lower c a v i t a t i o n  number 
then increase  with performance l o s s .  Figure 8 i n d i c a t e s  t h a t  t h e  off-diagonal ele-  
ments do not change monotonically with head coef f i c i en t .  
I n  summary, c a v i t a t i o n  a f f e c t s  t h e  steady fo rces ,  both the  impeller-centered 
f o r c e  {Fol and the  hydrodynamic f o r c e  matrix CAI, exerted upon an  e c c e n t r i c  
impeller .  It is useful  t o  i n t e r p r e t e  the  matrix CAI i n  terms of t h e  average normal 
force ,  FU, and the  t angen t i a l  fo rce ,  F , ac t ing  on a whir l ing  impel ler  i n  t h e  limit 
a s  the  whwl speed approaches zero.  ~ i f h  c a v i t a t i o n  breakdown, t h e  normal fo rce  is 
reduced while the  t angen t i a l  fo rce  i s  increased.  Fur ther  t e s t s  a r e  planned t o  
examine the  unsteady flow e f f e c t s  which occur a t  non-zero whir l  speed. More specif-  
i c a l l y ,  the  frequency dependence i n  t h e  matrix [A1 ( t h e  damping and added mass 
components) w i l l  s h o r t l y  be examined as a func t ion  of c a v i t a t i o n  number. 
REFERENCES 
1 Agoatinelli, A .  , Nobles, D ,  , and Mockridge, C +  R .  , "An Experimental Investige- 
t i o n  of Radial Thrust i n  Centrifugal Pumps", ASME Journal of Engineering for 
Power, Vole 82, pg. 120-126, Apri l  1960. 
2 .  D m ,  H., and Hergt, P, , tlRadial Forces on Impeller of Volute Caaing PumpeM, 
Flow Reaearch on Blading (L. 8. Dzung, ed.1, Elsevier Pub. Co., The Nether- 
lands, pp 305-321, 1 9 7 0 ,  
3 .  Ohashi, H .  and Shoji ,  H . ,  "Lateral Fluid Forces Aoting on a Whirling Centrifu- 
gal Impeller i n  Vanelese and Vaned Dif fuser ,"  Third Hclrkahop an Rotordynamic 
h a t a b i l i t y  Problems in High-Perfarmanee Turbomachinwy, Texas AW University, 
May 28-30, 1984. 
4 .  Chamieh, D. S . ,  ''Forces on a Whirling Centrifugal Pump-Impeller", Ph. D. 
Thesis, Division of Engineering and Appl ied  Soienoe, California Institute of 
Teahnology, February 1983. 
5 ,  Chamieh, D.  S,, Acosta, A.  J . ,  Brennen, C. E. and Caughey, T .  K,, "Experimental 
Measurements of Hydrodynamic Radial Forces and S t i f f n e s s  Matrioes for a 
Centrifugal Pump-Impeller,'l ASME Journal of F l u i d s  Engineering, Vol, 107,  Sept, 
1985, pp. 307-315. 
6 ,  Jery, B,, Aoosta, A.J., Brennen, C . E . ,  and Caughey, T.K., "Hydrodynamic 
Impeller S t i f f n e s s ,  Damping and Inertia i n  the Rotordynamias of Centrifugal 
Pumps," "Third Workshop on Rotordynamic Ins tab i l i ty  Problems i n  High 
Performance Turbomaohinery", Texas A&! University, May 28-30,1984. 
7 .  Jery, B., Brennen, C.E., Caughey, T . K . ,  and Acosta, A.J., "Forces on Centrifu- 
gal Pump rmpsllers, " "Second International Pump Symposium", Houston, Texas, 
A p r i l  2 9-May 2,1985. 
8 .  Jery, B e D  "Experimental Study of Unsteady Hydrodynamic Force Matrices on Whir- 
l i n g  Centrifugal Pump-Impellers", Ph. D .  Thesis. Division of Engineering and 
Applied Science, California Institute of Teohnology. October 1985. 
9 .  Rocketdyne Division, Rodwel l ,  personal oommunfcatlon 
/. CUTWATER 
'- WHIRL CIRCULAR 
O R B I T  
Figure 1 Schematic representation of the posi t ion of the impel ler  within the 
volute,  The measured forces are indicated i n  the rotating dynamometer 
frame (as FI. F2 and i n  the stationary volute frame (as F, FYI . 
-- 
I I I I I I 
HERD C O E F  0 PHI=.060 
PHI2.092 
A PHI=,104 
L ,  1 . --- 
0  
I 
0.5 1 . 0  1 .5  
s igma 
Figure 2 Cavitation performance curves for  the three flow coef f i c i ents :  6-0.060 
(below design) , d=O.O92 (design) and d=0 .I04 (above design) , for  volute 
A/impeller X a t  3000 rpm. 
I 1 
I 
I 
I 
I 
I 
0 0 , s  1, B 1  a 5 
sigma 
. 1 0  
Figure 3 The dependence of the magnitude and direct ion of the volute force, CF I, 
on cavitat ion number for  the three flaw coef f i c i ents :  d=0.060, d=0,0%, 
d=0 .lO4, for  volute A/irnpeller X a t  3000 rpm. 
r I I I I I I 
- Fomag 0 PHI= .068  
P H I = . 0 9 2  
- A P H I = . l B 4  - 
8 . 1  0.2 0,f 6 , 4  0 . 5  0.6 
HEFtD COEF 
n u a n =  B .e n 0 I 
Figure 4 The dependence of the magnitude and direction of t h e  volute force, {F I, 
on head coef f icbent for the three flow coef f i c i ents  : d-0.060, d=O.082, 
Q4.104, for volute A/impeller X a t  3000 rpm. 
Figure 5 
I I I I 
0 0.5 1 .0  1 .5  
sigma 
The dependence of the elements of the hydrodynamic force matrix, [ A ] ,  on 
cavitat ion number for  d=0.060 (below des ign) ,  for  volute A/impeller X at 
3000 rpm. 
Figure 6 The dependence of the elements of the hydrodynamic force matrix, I A l ,  on 
cavitat ion number for  Q=0,092 (at design) ,  for  volute A/impeller X a t  
3000 rpm. 
I I I I 
u Axx 0 Axy PHI=,092 
A Ayx v Ayy 
B 
D 
v 
2 - v :  - 
1 - 
i 
# A A A ;  A A  
A  - 
0 I I _. . I I .  1 I 
r 0 0 4 C E 
-1 - - 
L.1 I I 1 
0 0.5 1.0 1 , 5  
s i gea 
Figure 7 The dependence of the elements of the hydrodynamic force matrix, C A I ,  on 
cavitat ion number for  Q=0.104 (above design), for volute A/impeller X at 
3000 rpm. 
I 1 
R x x  6 Rxy I r I I * PHI= ,  104 
A Ayx v A Y V  
$9 B s Q B u 2 - - 
1 - 
i 
A4 a. A A A A 
Y 
- 
0 I I I I I I I I 
P" O0 ' 0 O 0 0 
0.1 0n2 0 , 3  0 . 4  . 085  0,6 
HEAD COEF 
-1  
Figure 8 The dependence of the elements of the hydrodynamic force matrfx, [ A ] ,  on 
head c o e f f i c i e n t  for d=0,104 (above des ign) ,  for  volute Alimpeller X at 
3000 rpm. 
- - 
I I I I 
0 0 . 5  1.0 1,5 
sigma 


The asymmetric flow within a volute exerts a radial force on a centrifugal impeller. Experimental measurements of the radial forces on the impeller in the presence of cavitation are presented

NASA Technical Reports Server

N87-22225
ON THE EFFECT OF CAVITATION ON THE RADIAL FORCES AND HYDRODYNAMIC
STIFFNESS OF A CENTRIFUGAL PUMP*
R.S. Franz, C.£. Brennen, A.S. Acosta, and T.K. Caughey
California Institute of Technology
Pasadena, California 91125
The asymmetric flow within a volute exerts a radial force on a centrifugal
impeller. The present paper presents experimental measurements of the radial forces
on the impeller in the presence of cavitation.
NOMEN CLATU RE
[A]
A1 ,A2
22
hydrodynamic force matrix, non-dimensionalized by prim r2b 2
impeller inlet area (_r12), outlet area (2_r2b 2)
b 2 impeller discharge width (0.62 in)
{F} 6-component generalized force vector
F1,F 2 components of the instantaneous lateral force on the impeller in the
rotating dynamometer reference frame
Fx,Fy components of the instantaneous lateral force on the impeller in the fixed
23
laboratory reference frame (X,Y), non-dimensionalized by p_ r2b 2
Fox,Foy values of Fx and Fy if the impeller was located at the the origin of the
23
(X,Y) coordinate system (volute center), non-dimensionalized by p_ r2b 2
* The authors are indebted to the NASA George Marshall Space Flight Center,
Alabama for continued sponsorship of this research under contract NAS8-33108.
We are also grateful for the help given by D. Brennen.
493
 ' CZD a PAGE:  .ZO
https://ntrs.nasa.gov/search.jsp?R=19870012792 2020-03-20T11:51:01+00:00Z
FN,F T components of the lateral force on the impeller normal to and tangential
to the whirl orbit, averaged over the orbit, non-dimensionalized by
22
Pn_ r2b2e
F N = (Axx + Ayy)/2 F T = (-Axy + Ayx)/2
pl,Pt I upstream static, total pressure
P2,Pt2
PI
downstream static, total pressure
static pressure at impeller inlet, Ptl - P(A_I )2/2
Pv vapor pressure of water
Q flow rate
rl,r 2 impeller inlet, discharge radius (1.594 in., 3.188 in.)
t time
(X,Y) fixed laboratory reference frame
x,y instantaneous coordinates of the impeller center in the fixed laboratory
reference frame (X,Y), non-dimensionalized by r2
radius of the circular whirl orbit
e angle of the impeller on the eccentric circle, measured from the volute
tongue in the direction of impeller rotation
density of water
cavitation number,
494
?flow coefficient based on the impeller discharge area and tip speed, --_
_r 2 A2
Pt2-Ptl
total head cofficient, p2r^
W radian frequency of the impeller (shaft) rotation
INTRODUCTION
Earlier papers (refs.l-8) have described measurements of the radial forces and
hydrodynamically induced rotordynamic coefficients of centrifugal pumps with various
impellers and volutes. All of these earlier measurements were made in the absence
of any cavitation within the pump. Yet there is some evidence that the presence of
cavitation may have a significant effect on these forces and coefficients. Indeed
some tests of the high speed pumps in the Space Shuttle Main Engine have suggested a
change in the rotordynamics when cavitation occurs (ref. 9). The present paper is a
supplement to our earlier measurements of forces and coefficients and constitutes an
exploration of the influence of cavitation.
The references 6-8 provide a complete description of the facility. Briefly,
the dynamometer, composed of two parallel plates connected by four strain gaged
posts, is mounted between the impeller and the drive shaft. It measures the six
components of a generalized hydrodynamic force vector {F] acting on the impeller.
The impeller can be subject to whirling motion in an orbit eccentric to the volute
center, in addition to the normal impeller rotation. Since the eccentric motion is
in the lateral plane, perpendicular to the impeller centerline, only the two
components of the force vector {F] in this lateral plane will be discussed.
These forces can be represented by
Refering to figure I, F and F are in the volute frame of reference, and x and y
represent the coordinates oF the impeller center measured from the volute center.
Dimensionless quantities are used throughout (see Nomenclature for definitions).
The present results are only for the case of no whirl: x and y are fixed in time.
When the impeller is located at an angular position, e, on the circular whirl orbit
of radius e, equation (1) is written as
IFxl I I + [ A ] I 1 (2)
Fox e/r 2 cos8
Fy = Foy ' {e/r 2 sine
The steady lateral forces, represented by F and F, can be considered as the
sum of two forces: a fixed force, represente_ by F^Y and F., which the impeller
would experience if located at the volute center, and _force _e to the eccentric
495
displacement of the impeller, represented by a "stiffness" matrix [A]. By taking
data at four fixed eccentric positions of the impeller, 90 degrees apart, the matrix
[A] and the vector {F } can be extracted. The gravitational and buoyancy forces on
O
the rotor are subtracted out.
EXPERIMENTS
In references 6-8 results were presented for a typical impeller (impeller X), a
five bladed Byron-Jackson centrifugal impeller with a specific speed of 0.5V,
operating in a spiral volute (volute A) at various impeller speeds (_ 2000 rpm) and
flow rates. In order to test under cavitating conditions, the impeller speed was
increased to 3000 rpm and the water was substantially de-aerated.
In the following pages the results for three different flow rates are
presented: the flow rates chosen are below design (_=0.060), design (_=0.092) and
above design (_=0.i04). Cavitation performance curves for each of these flow rates
are presented in figure 2. We observe that the breakdown cavitation numbers for the
three flow rates tested are 0.17, 0.26 and 0.30 respectively.
RESULTS
Results for the radial forces, F and FO , and for the stiffness coefficientsare presented in figures 3 through A qu1_k glance will show that large changes
occur in both characteristics as the pump approaches and passes through breakdown.
Notice that the steady force shown in figures S and 4 changes in both magnitude and
direction.
Figures 3 and 4 show that for g=0.060 the magnitude of {F } decreases with
performance loss with a small change in direction. For both g_0.092 and _=0.I04,
the magnitude of {F} has a minimum with decreasing head coefficent. For each flow
coefficient, the d_rection of {F } rotates away from the tongue. For _=0.092, {Fo}
Q
rotates through more than 180 ° as the pump progresses through breakdown.
Figures 5-8 show the hydrodynamic force matrix [A] from equation (2). These
no-whirl results corrrespond to -[K] of the quadratic fit of reference 2 to whirl
data for mass, damping and stiffness matrices. The three flow coefficients exhibit
the same trends. The diagonal elements decrease with performance loss. The off-
diagonal elements first decrease slightly in magnitude with lower cavitation number
then increase with performance loss. Figure 8 indicates that the off-diagonal ele-
ments do not change monotonically with head coefficient.
In summary, cavitation affects the steady forces, both the impeller-centered
force {F } and the hydrodynamic force matrix [A], exerted upon an eccentric
impeller. U It is useful to interprete the matrix [A] in terms of the average normal
force, FN, and the tangential force, F-, acting on a whirling impeller in the limit
as the whlrl speed approaches zero. Wi_h cavitation breakdown, the normal force is
reduced while the tangential force is increased. Further tests are planned to
examine the unsteady flow effects which occur at non-zero whirl speed. More specif-
ically, the frequency dependence in the matrix [A] (the damping and added mass
components) will shortly be examined as a function of cavitation number.
496
REFERENCES
•
.
•
o
.
o
•
.
Agostinelli, A., Nobles, D., and Mockridge, C. R., "An Experimental Investiga-
tion of Radial Thrust in Centrifugal Pumps", ASME Journal of Engineering for
Power, Vol. 82, pp. 120-126, April 1960.
Domm, H., and Hergt, P., "Radial Forces on Impeller of Volute
Flow Research on Blading (L. S. Dzung, ed.), Elsevier Pub.
lands, pp 305-321, 1970.
Casing Pumps",
Co., The Nether-
Ohashi, H. and ShoJi, H., "Lateral Fluid Forces Acting on a Whirling Centrifu-
gal Impeller in Vaneless and Vaned Diffuser," Third Workshop on Rotordynamic
Instability Problems in High-Performance Turbomachinery, Texas A6M University,
May 28-30, 1984.
Chamieh, D. S., "Forces on a Whirling Centrifugal Pump-Impeller", Ph. D.
Thesis, Division of Engineering and Applied Science, California Institute of
Technology, February 1983.
Chamieh, D. S., Acosta, A. J., Brennen, C. E. and Caughey, T. K., "Experimental
Measurements of Hydrodynamic Radial Forces and Stiffness Matrices for a
Centrifugal Pump-lmpeller," ASME Journal of Fluids Engineering, Vol. 107, Sept.
1985, pp. 307-315.
Jery, B., Aoosta, A.J., Brennen, C.E., and Caughey, T.K., "Hydrodynamic
Impeller Stiffness, Damping and Inertia in the Rotordynamics of Centrifugal
Pumps," "Third Workshop on Rotordynamic Instability Problems in High
Performance Turbomachinery", Texas A_4University, May 28-30,1984.
Jery, B., Brennen, C.E., Caughey, T.K., and Acosta, A.J., "Forces on Centrifu-
gal Pump Impellers," "Second International Pump Symposium", Houston, Texas,
April 29-May 2,1985.
Jery, B., "Experimental Study of Unsteady Hydrodynamic Force Matrices on Whir-
ling Centrifugal Pump-Impellers", Ph.D. Thesis, Division of Engineering and
Applied Science, California Institute of Technology, October 1985.
9. Rocketdyne Division, Rockwell, personal communication
497
Y/
/
/
/
I
I
\
\
\
\
\
Fy(l )
VOLUTE
CENTER
--x(_)-- J
/
/
/
/ 7MP_ 0_,
CENTER #' \\
E y(_') \
F,(O) = F_(O) y'\
VOLUTE _ X
CUTWATER
/
WHIRL CIRCULAR
ORBIT
Figure I Schematic representation of the position of the impeller within the
volute. The measured forces are indicated in the rotating dynamometer
frame (as F 1, F 2 ) and in the stationary volute frame (as F x, Fy).
Figure 2
8.6
8.5
8.4
8,3
8,2
8.I
' ' I l
'HERb COEF ° PHI=.@60
o PHI=,@92
,. PHI=,IB4
lid II O Q
o
o ¢,o@ o 0
Q ¢,
a • A _
¢.
_ C" A
A
&
0
A
' ' ' I ' ' ' ' I
m r = 0 Q []
' ' ' I ' ' ' ' I ' ' ' ' I '
e 0,5 1,8 1.5
sigma
Cavitation performance curves for the three flow coefficients: @=0.060
(below design), @=0.092 (design) and @=0.104 (above design), for volute
A/impeller X at 3000 rpm.
498
,10
,95
360
270
180
9e
0
'N 'PHi e68 ' ' ' ' ' ' ' '' Fo a_ = =.
* PHI:.992
,, PHI=.104
{: Qua • = a = ,,,, u a a
L_, a& a &* & ,Ih, 4,
I0_ ° 0 0 ° ° 0 C
, ' l I b i ' ' I ' ' I , I i ,
i Foang ' , I ' ' ' ' I
_, ,,,
°
° @
DQ a m D m 0 C 0 m m
i , , , I _ ' ' ' I , , , , I
I
8 9,5 1,8 1.5
sigma
Q
I
Figure 3 The dependence of the magnitude and direction of the volute force, {F },
on cavitation number for the three flow coefficients: d=O.060, _=0.0_2,
$=O.104, for volute Alimpeller X at 3000 rpm.
Figure 4
,19
,95
368
27B
188
9@
@
''''''''H'''F,P1o. • -.o6o.... '....'....'....'
° PHI=.092
,*PHI=.184
=q
Q
Q • Q C
[] Q
m
&, & && &
a a a A J,_I_,
i° © {' o ©
I " " I .... I .... I .... I .... I
I ' ' ' ' I ' ' ' ' I ' ' ' ' i ' ' ' I ' ' ' ' I ' ' ' ' I '
Foan_ _ %_' ,. • ,. ,, ,,. ,,
C, @ & &_
,D
,,n 0% _= % _' _ o "d
I ' _ " I " _ " , I ,. ,. _ , I ' '. '. I I ,. , ,. ,. I ,. , ,. ,. I ,. I I I
• I I ' I I I
0,1 e,2 8,3 6.4 8,5 0,6
HERD COEF
The dependence of the magnitude and direction of the volute force, {F },
on head coefficient for the three flow coefficients: d=O.060, $=0.0_2,
_=0.104, for volute A/impeller X at 3000 rpm.
499
-i
Axx o Axy PH .
Ayx • Ayy
I I I I I I I I i I I I I I i I
o,::,_o o o o ,:, ,:, _, ,:, o _,
0 8.5 1.8 1,5
sigma
Figure 5 The dependence of the elements of the hydrodynamic force matrix, [A], on
cavitation number for $=0.060 (below design), for volute A/impeller X at
3000 rpm.
Figure 6
-i
' ' ' A'x ' ' ' ' I-' 09i '= Axx _ y PH .
Ayx _ Ayy
[]
T
I I I I i I I I I I , i i i I i i
I , i i i I , i , , I , i , i I
0 0,5 1.8 1,5
sisma
i I
The dependence of the elements of the hydrodynamic force matrix, [A], on
cavitation number for $=0.092 (at design), for volute A/impeller X at
3000 rpm.
500
Figure 7
-i
p
I
0
' =' Bxx '<_i_xy 'PHI=,184
Ayx • Ayy
, 9 g
' I
I I I I I i i i i l I I I I I I
@ @@ @ @ @ @ @
0.5 1.0 1.5
sigma
The dependence of the elements of the hydrodynamic force matrix, [A], on
cavitation number for _=0.104 (above design), for volute A/impeller X at
SO00 rpm.
-i
, .... ,._..,°_ ,_y'_'
Ayx • Ayy
'.... 'P_i;.ie_'.... ''''
' I I ' ' I: : : : I : ; I I ! ! ! _ ! ! _ I I ? : : ) I ! : : : I : : :
@
l , , , a I I I i i i I ¢ i i i , _ , , I , , , , I , , I I I , i i i
0 0,1 0,2 0,3 9,4 6.5 6,6
HEBDCOEF
Figure 8 The dependence of the elements of the hydrodynamic force matrix, [A], on
head coefficient for _=0.i04 (above design), for volute A/impeller X at
3000 rpm.
501


On the effect of cavitation on the radial forces and hydrodynamic stiffness of a centrifugal pump

Agoatinelli, A . , Nobles, D , , and Mockridge, C + R . , &quot;An Experimental Investiget i o n of Radial Thrust i n Centrifugal Pumps&quot;, ASME Journal of Engineering for Power, Vole 82, pg. 120-126, Apri l 1960.

An Experimental Investiget i o n of Radial Thrust i n Centrifugal Pumps&quot;,

B e D &quot;Experimental Study of Unsteady Hydrodynamic Force Matrices on Whirl i n g Centrifugal Pump-Impellers&quot;,

Chamieh, D. S . , ''Forces on a Whirling Centrifugal Pump-Impeller&quot;, Ph. D. Thesis, Division of Engineering and Appl ied Soienoe, California Institute of Teahnology, February 1983.

Chamieh, D. S,, Acosta, A. J . , Brennen, C. E. and Caughey, T . K,, &quot;Experimental Measurements of Hydrodynamic Radial Forces and S t i f f n e s s Matrioes for a Centrifugal Pump-Impeller,'l ASME Journal of F l u i d s Engineering, Vol, 107, Sept,

D m , H., and Hergt, P, , tlRadial Forces on Impeller of Volute Caaing PumpeM, Flow Reaearch on Blading (L. 8. Dzung, ed.1, Elsevier Pub. Co., The Netherlands, pp 305-321, 1 9 7 0 ,

Division, Rodwel l , personal oommunfcatlon /. CUTWATER '-

Experimental Measurements of Hydrodynamic Radial Forces and S t i f f n e s s Matrioes for a Centrifugal Pump-Impeller,'l

Forces on a Whirling Centrifugal Pump-Impeller&quot;,

Forces on Centrifugal Pump rmpsllers, &quot; &quot;Second International Pump Symposium&quot;, Houston, Texas, A p r i l 2 9-May 2,1985.

Hydrodynamic Impeller S t i f f n e s s , Damping and Inertia i n the Rotordynamias of Centrifugal Pumps,&quot; &quot;Third Workshop on Rotordynamic Ins tab i l i ty Problems i n High Performance Turbomaohinery&quot;, Texas A&! University,

Jery, B e D &quot;Experimental Study of Unsteady Hydrodynamic Force Matrices on Whirl i n g Centrifugal Pump-Impellers&quot;, Ph. D . Thesis. Division of Engineering and Applied Science, California Institute of Teohnology. October 1985.

Jery, B., Brennen, C.E., Caughey, T . K . , and Acosta, A.J., &quot;Forces on Centrifugal Pump rmpsllers, &quot; &quot;Second International Pump Symposium&quot;, Houston, Texas, A p r i l 2 9-May 2,1985.

Lateral Fluid Forces Aoting on a Whirling Centrifugal Impeller i n Vanelese and Vaned Dif fuser ,&quot; Third Hclrkahop an Rotordynamic h a t a b i l i t y Problems in High-Perfarmanee Turbomachinwy,

pp. 307-315.

Rocketdyne Division, Rodwel l , personal oommunfcatlon /. CUTWATER '- WHIRL CIRCULAR O R B I T

http://authors.library.caltech.edu/208/1/FRA073.pdf

On the Effect of Cavitation on the Radial Forces and Hydrodynamic Stiffness of a Centrifugal Pump

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Caltech Authors - Main

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