The effect of a nonuniform flow field on the Alford force calculation is investigated. The ideas used here are based on those developed by Horlock and Greitzer. It is shown that the nonuniformity of the flow field does contribute to the Alford force calculation. An attempt is also made to include the effect of whirl speed. The values predicted by the model are compared with those obtained experimentally by Urlicks and Wohlrab. The possibility of using existing turbine tip loss correlations to predict beta is also exploited. The nonuniform flow field induced by the tip clearnance variation tends to increase the resultant destabilizing force over and above what would be predicted on the basis of the local variation of efficiency. On the one hand, the pressure force due to the nonuniform inlet and exit pressure also plays a part even for unshrouded blades, and this counteracts the flow field effects, so that the simple Alford prediction remains a reasonable approximation. Once the efficiency variation with clearance is known, the presented model gives a slightly overpredicted, but reasonably accurate destabilizing force. In the absence of efficiency vs. clearance data, an empirical tip loss coefficient can be used to give a reasonable prediction of destabilizing force. To a first approximation, the whirl does have a damping effect, but only of small magnitude, and thus it can be ignored for some purposes

Martinezsanchez, M.

Qiu, Y. J.

English

NASA Technical Reports Server

PREDICTION OF DESTABILIZING BLADE TIP FORCES 
FOR SHROUDED AND UNSHROUDED TURBINES* 
Yuan J .  Qiu and M. Martinez-Sanchez 
Massachusetts Institute o'f Technology 
Cambridge, Massachusetts 02139 
Highly loaded turbomachinery tends t o  e x h i b i t  no t i ceab le  subsynchronous w h i r l  
a t  a frequency close t o  one of the  r o t o r  c r i t i ca l  f requencies  when t h e  r o t a t i n g  
speed is w e l l  i n  excess o f  t h a t  cri t ical .  The subsynchxonous wh i r l  is  n o t  a 
"forced" v i b r a t i o n  phenomenon where t h e  v i b r a t i o n  frequency i s  the  same as t h e  
f o r c i n g  frequency; e.g., r o t o r  unbalancing. I t  is  b e t t e r  c l a s s i f i e d  a s  a pos i -  
t i v e  feedback phenonenon tending t o  amplify small. d i s tu rbances  a t  the system 
n a t u r a l  f requencies .  The cause of wh i r l  can be t r aced  back to a number of forc- 
i n g  mechanisms. One of these is the  v a r i a t i o n  of  t h e  blade t i p  forces due t o  
nonuniform blade t i p  c learance.  Alford ( 1 )  w a s  t h e  f i r s t  to model t h i s  aerody- 
namic f o r c e  on the  blade. By assuming uniform upstream and downstream flow 
f i e l d s ,  he w a s  a b l e  t o  argue t h a t  t he  v a r i a t i c n  of t h e  b lade  t i p  force  would be 
p r o p o r t i o n a l  t o  the  e f f i c i e n c y  va r i a t ion .  Therefore ,  i f  a tu rb ine  r o t o r  has  some 
small e c c e n t r i c i t y  (Fig.  I ? ,  then due t o  t h e  e f f i c i e n c y  v a r i a t i o n ,  t he  b lade  
f o r c e  on the  narrow-gap s i d e  must be g r e a t e r  than  t h e  b lade  f o r c e  on t h e  oppos i t e  
side. The r e s u l t a n t  forc-e Fy on the  r o t o r ,  a c t i n g  perpendicular  t o  the  e c c e n t r i -  
c i t y  is then no zero.  -This force can cause t h e  r o t o r  t o  w h i r l  f o m a r d ,  i . e . ,  i n  
t h e  same d i r e c t i o n  as  the  ro t a t ion .  Q u a l i t a t i v e l y :  
i f  
then 
T m  a 'o F y =  -- H 
where 6 t  is the  local t i p  c learance a t  azimuth e ,  eo is the  e c c e n t r i c i t y  and H 
is the  blade he igh t ,  Dm is the  aean d iameter  of t h e  rotor and T, is torque on 
t h e  r o t o r ,  9 is a numerical f a c t o r  of orde r  unity depending on the  blading. 
Contrary t o  Alford 's  assumptions, t he  presence of t he  blade t i p  clearance 
va r i a t ion  i s  bound t o  influence the  whole flow f i e l d .  The purpose of t h i s  paper 
i s  t o  inves t iga te  the  e f f e c t  of t h i s  non-uniform flow f i e l d  on t h e  Alford force  
ca lcu la t ion .  The ideas  used here are based on those developed by Horlock and 
Th i s  work was supported by NASA c o n t r a c t  NXS8-350L8. 
287 
https://ntrs.nasa.gov/search.jsp?R=19860020711 2020-03-20T14:39:36+00:00Z
Greitzer ( 2 ) .  It will' be shown that the non-uniformity of the flow field does 
contribute to the Alford force calculation. An attempt is also made to include 
the effect of whirl speed. The values predicted by the model are comnpared with 
those obtained experimentally by Urlicks(3) and Wohlrab ( 4 ) .  The possibility of 
using existing turbine tip loss correlations to predict $ is also exploited. 
The Model 
The method of approach is to first find the flow field due to 
variation, and then, using the velocity triangles, to calculate 
Alford force. The detail of the analysis is given in Ref. 5. It 
the tip clearance 
the resultant 
is to be stressed 
that a complete-three-dimensioal investigation of the flow field with the 
non-uniform tip clearance is not attempted here. 
Consider a single-stage turbine with its rotor whirling at a constant eccen- 
tricity e, (Fig. 2 ) .  
approximated to be two dimensional. Fig.3 shows an actuator disc representing a 
two dimensional unwrapped blade row of circumference ~ T R ,  where w is the relative 
flow velocity and c is the absolute flow velocity. If the Mach numbers upstream 
and downstream are small, the flow can be considered incompressible, as well as 
inviscid, outside of the blade row. 
If the hub/tip ratio is high, the €low field can be 
The variation of the tip clearance around the annulus can be represented by 
where 51 is the whirling speed, y is the distance in the tangential direction, R 
is the mean blade radius, H the blade height, and the En are constants. 
Since only the first harmonic is of interest, as will be explained later, 
higher harmonics are not included in the analysis and E1 = eo/H. 
c 4 
The perturbation velocities c and c are related to the stream 
function J, by 
2 Upstream of the stagy, the flow is irrotational, V 
flow is rotational V $3 = r;. The flow is steady in a reference frame 
rotating at the whirl speed 51, therefore the stream functions can be written as 
$1 = 0 and downstream the 
1 
= A exp [ fi (x + i y) - 51 ti] upstream of the stage ( 3 )  
288 
1 
$ 3  = B exp [E ( - x + i y )  - Sa t i ]  
1 
+ C exp [ ( - x t a n  a i + y i - Sa t i ) ]  downstream of t h e  s t a g e  ( 4 )  3 
where Z3 i s  t h e  mean flow angle  downstream of  t h e  rotor. Notice t h e  
i r r o t a t i o n a l  term d i e s  o u t  f a r  upstream and downstream. 
i s  convected by t h e  mean flow a t  an angle  E 3 .  
are considered.’ 
The v o r t i c i t y  downstream 
Again only  f i r s t  harmonics 
There are t h r e e  matching condi t ions  across t h e  a c t u a t o r  disc sur face :  
1) The axial mass f l u x  i s  cons tan t  
. 
2 )  The r e l a t i v e  flow leaving angle  i s  c o n s t a n t  
r r c c 
t a n $  c = w = c + U  
a t x = O  
3 3x 3x 3Y 
3 )  
using t h e  b.own experimental  information. 
The t h i r d  boundary condi t ion  involves  matching p e r t u r b a t i o n  q u a n t i t i e s  by 
The t o t a l - t o - s t a t i c  t u r b i n e  e f f i c i e n c y  can be w r i t t e n  as 
- c  ) u  t u r b i n e  work - (cy3 y2  
Y - l / Y  n =  
1 ideal work p3 
Po 1 
c T (1- (-) 
P 01 
( 7 )  
where cy2 and cy3 are t h e  y-components of t h e  absolute flow v e l o c i t y  
b e f o r e  and a f t e r  t h e  rotor (def ined  p o s i t i v e  i n  t h e  y d i r e c t i o n ) ,  u i s  t h e  b lade  
speed, p3 is t h e  s ta t ic  p r e s s u r e  downstream of t h e  disc, and 
t o t a l  p r e s s u r e  upstream. 
poi  i s  t h e  
Due to  
as t h e  sum 
r l =  
where t o  a 
c 
r l =  
t h e  local blade t i p  c learance  v a r i a t i o n ,  t h e  e f f i c i e n c y  can be w r i t t e n  
of t h e  mean e f f i c i e n c y  and i t s  p e r t u r b a t i o n :  
ii + n  
f i r s t  approximation 
(9) 
289 
It is known that the second term in the above expression is small compared 
with the first term, if f3 is.not too far from the designed value. Hence the 
second term in Eq.(9) will be ignored in the analysis. 
We now perturb eq.(7) and use the Bdditional condition that the stator leaving 
angle is also constant to eliminate c y2, the third boundary condition is 
obtained. 
downstream pressure perturbation. In terms of the perturbation streamfunctions, 
the.three matching conditions can then be written as 
We also use the downstream y-momentum equation to eliminate the 
4 - tan 8, = - - + u  a Y  ax 
2 2 2 
2 2 
c 
au 1 - Y2 a Y  I + (cy3- c 
a $1 
axay aY2 
+ tan a2 -= - u t -  a *3 
where 
1 cu-ryu 
Substituting the forms given by E q s .  ( 3 )  and ( 4 )  for $1 and $3, these 
conditions become three linear algebraic equations for the coefficients A, B and 
C. To obtain the blade force variation, the momentum equation is applied across 
the rotor blades. The tangential component of the force is given by 
fy = p c (c x3 y3- cy2) 
Since only the perturbation of force 
force 
If we consider a reference frame 
together the forces perpendicular to 
net cross-force 
- a t  F = &2nR Real(f e ) cos (y/R) 
Y Y 
contributes to the resultant destabilizing 
0 0 
(C - C  ) H  
Y 3  Y2 
rotating at the whirl speed a ,  and add 
the instantaneous eccentricity, we find the 
d Y  
290 
This expression justifies the inclusion of only the first harmonic in the 
stream function expansion, since the higher harmonics contribute nothing to the 
Alford force. A non-dimensional excitation coefficient kxy is defined as 
Fy eo k = y j - / ~  
xy s 
where Us is the ideal total force in the circumferential direction on the 
blade, and eo/H is the dimensionless eccentricity. 
Results 
Cases with stationary eccentricity are considered first. A typical plot of 
the axial velocity perturbation and the blade force variation vs. angular posi- 
tion are shown in Fig. 4. 
counterpbase with the tip clearance variation, which indicates that the nonuni- 
form flow field will tend to increase the Alford forces. 
As was to be expected, these quantities are almost in 
Although Alford published his paper in 1965, very few experiments have been 
done to verify the proposed formula. 
comparison between the Alford model, the present analysis, and experimental 
results. Two tests were done by Urlich (3) and Wolhrab ( 4 ) ,  and more recently 
Vance and Laudadio (6) did verify the linearity between the Alford forces and 
eccentricity for a fan. Accurate experimental data seems very difficult to 
obtain due to the small magnitude of the forces to be measured. 
Hence it is difficult to make any good 
For the present analysis the data of Urlich seems to be the more suitable, 
since he also determined the efficiency vs. average clearance for an unshrouded 
turbine stage. From this information the values of X = aq/a(t/H) are found by a 
second order polynomial curve fit to the efficiency vs. clearance data (Table 1). 
Values of % are found by using A ,  together with appropriate flow angles (Ref. 
3 )  in the analytical model. 
values of hY are plotted in Figs. 5 and 6, together with the values of kxy 
obtained from the Alford formula. It can be seen that the analysis gives a value 
of % higher than the Alford formula, indicating that the non-uniformity does 
increase the destabilizing forces. 
The predicted values and experimentally obtained 
In addition, it is also clear from Figs. 5 and 6 that the new theory over- 
predicts the side forces, when compared to the experimental data. However, we 
should at this point also consider the effect of the nonuniform pattern of pres- 
sure acting on the turbine wheel rim. Fig. 7 shows that, although the pressure 
field just upstream of the rotor is in phase with the clearance variation, it is 
shifted by 90' just downstream, and that those nonuniform pressures act in such a 
way as to reduce the forward cross-force. 
A precise accounting of the effect of these pressure forces is not possible 
in the context of our actuator-disc theory. 
rotation of the pressure pattern must occur in the rotor passages, and so it is 
logical to calculate the induced side force by letting the rotated pressure wave 
act on an axial rim width equal to half a bladde chord. 
ents, including this pressure force, are also shown in Fig. 6. Although the 
precise axial length over which the pressure forces act is difficult to deter- 
mine, it can be seen that they do contribute significantly to the net force, and 
that their effect is to essentially remove the overprediction. 
However, we can see that the 90° 
New excitation coeffici- 
291 
The Use of Tip Clearance Correlations 
In many situations, curves of efficiency vs. clearance are not available. 
The possibility of using some of the existing tip clearance loss correlations is 
investigated here. The formula used is given by Dunham and Came (7), in the form 
of a tip loss coefficient which depends on flow angles and other stage variables, 
as well as on the clearance to chord ratio . This allows determination of q, and 
hence h for use in our model. Fig. 8 shows the predicted Alford forces based on 
the tip loss clearance correlation(Ref.(7)), together with the experimental 
results. It can be seen that the prediction is very reasonable, as it compares 
well with the data for small axial (stator-rotor) gap, although not as well for 
large axial gap. The fact that + varies with the gap between the rotor and 
stator is not well understood, and it may be highly dependent on the detailed 
geometrical arrangement of stator and rotor. 
Wohlrab (Ref. 4) tested a shrouded turbine. Since he did not supply data on 
efficiency vs. tip clearance, X has to be estimated, in order to compare the . 
present analysis with the experimental values obtained. Encouraged by the re- 
sults for the unshrouded turbine, an estimated tip loss coefficient is used 
again, but this time the effective tip clearance is given by (Ref.(7)). 
-0.42 t = (geometric tip clearance) x (number of seals) 
Since there is also a shroud, we would expect some seal forces acting on the 
rotor shroud band. 
( 8 ) .  The results (Fig. 9) again show reasonable agreement, falling between the 
data obtained by Wohlrab for the two axial gaps tested. 
These are found by applying a model developed by Lee, et al. 
Effect of Whirl 
All the cases examined so far have been for an assumed static offset. The 
effect of whirl for a typical case is shown in Fig. 10. For a whirl speed equal 
to 50% of the rotating speed, the effect on k is small ( 2  10%). This is consis- 
tent with Wohlrab's experiment, in which he found that within experimental accu- 
racy there is no difference between the values of k, obtained from kinetic and 
static tests. It is noted, however, that the predicfed effect of whirl (which 
could be expressed as an Alford damping coefficient), could not have been obtain- 
ed from the original Alford model, since it depends crucially on the delays 
between wheelmotion and disc passage of the induced flow non-uniformities, and 
these were ignored in the Alford model. 
Conclusion 
The non-uniform flow field induced by the tip clearance variation tends to 
increase the resultant destabilizing force over and above what would be predicted 
on the basis of the local variation of efficiency. On the one hand, the pressure 
force due to the non-uniform inlet and exit pressure also plays a part even for 
unshrouded blades, and this counteracts the flow field effects, so that the 
simple Alford prediction remains a reasonable approximation. Once the efficiency 
variation with clearance is known, the present model gives a slightly overpre- 
dicted, but reasonably accurate destabilizing force. 
ency vs. clearance data, an empirical tip loss coefficient can be used to give a 
reasonable prediction of destabilizing force. To a first approximation, the 
whirl does have a damping effect, but only of small magnitude, and thus it can be 
ignored for some purposes. To gain more insight and understanding, more accurate 
In the absence of effici- 
29 2 
experimental determinations of the tip force must be made. 
out that the destabilizing force is also highly dependent on the geometry of the 
rotor and stator, as shown by the experimental dependence of the magnitude of 
destabilizing force on the axial gap between the stator and rotor. 
of axial gap on the Alford force points to the need for more detailed study of 
the flow field around tip regions. 
It must be pointed 
This effect 
References 
1. Alford, E.F., "Protecting Turbomachinery from Self-Excited Rotor Whirl." 
Journal of Engineering for Power, Oct. 1965, pp338-344. 
2. Horlock, J . H .  and Greitzer, E.M., "Non-uniform Flows in Axial Compressors 
Due to Tip Clearance Variation,'' Proc. Instn. Mechanical Engrs., 
V o l .  m, pp173 
3. Urlichs, K., Clearance Flow Generated Transverse Force at the Rotors of 
Thermo Turbomachines," Dissertation, Technical University of Munich, 
(1975). English translation in NASA TM-77 292, Oct. 1983. 
4. Wohlrab, R., "Experimental Determination of Gap Flow-Conditioned Force at 
Turbine Stages and Their Effect on the Running Stability of Simple Rotors," 
Dissertation, Munich Institute of Technology (1975). English translation 
in NASA TM-77 293, Oct. 1983. 
5. Martinez-Sanchez, M, Greitzer, E. and Qiu, Y., "Turbine Blade-Tip Clearance 
Excitation Forces," Report on NASA Contract Number NAS8-35018, June, 1985. 
6. Vance and Laudadio, "Experimental Measurement of Alford's Force in Axial 
Flow Turbomachinery," Journal of Engineering for Gas Turbines and Power, 
1984, pp585. 
7. Dunham and Came, "Improvements to the Ainley-Mathieson Method of Turbine 
Performance Prediction," ASME Transactions , July 1970. 
8. Lee, D.W.K., Martinez-Sanchez, M. and Czajkowski, E., "The Prediction of 
Force Coefficients for Labyrinth Seals," 3rd Rotordynamic Instability 
Workshop, Texas A&M University, May 1984. 
List of Symbols 
C absolute velocity 
cP 
Dm 
e0 eccentricity 
fY 
FY 
S Y  
specific heat 
mean rotor diameter 
blade force per unit circumferential length 
tangential force 
blade span 
excitation coefficient 
pressure 
H 
P 
293 
R 
t 
T 
Tm 
US 
W 
U 
U 
@1,2,3 
B 
blade radius 
time; tip clearance 
temperature 
mean torque on rotor 
relative velocity 
blade speed 
ideal blade force=(ideal turbine work)/(blade speed) 
absolute flow angle 
relative flow angle 
Alford factor 
perturbation vorticity 
circumferential angle 
stream function 
whirl speed 
denity 
efficiency 
Superscript 
I perturbation 
- mean value 
subscript 
Y tangential direction 
X axial direction 
1 upstream of stator 
2 downstream of stator 
3 downstream of rotor 
0 stagnation 
Toblc.1 Efficiency for tip clearances 
294 
Figure 1. - Tip clearance effect for turbine rotor. 
Figure 2. - Rotor under whirl. 
295 
13 
FLOW 
Figure 3. - Two-dimensional flow model. 
- tip clearance vuriatim 
Figure 4. - Axial velocity and local blade force distribution. 
296 
20 4.0 6.0 0.0 
Figure 5. - Variation of  excitation coefficient with t i p  clearance. 
% 
30 
2 8  
1.0 
OC 
'+ 
axial cylod =21% - t -predicted - 0 - experimcntial 
-A -Alford model 
-o---prediitd with 
pressure correcticn 
2.0 4.0 6.0 81) 
Figure 6. - Variation of excitation coefficient with t i p  clearance. 
297 
_.f 
Figure 8. - Variation of excitation coefficient with t ip  clearance. 
298 
@ \  a. 
CALCULATED 
blade force variation only 
seal force +blade force 
EXPERIMENTAL 
ai SAX = 0.6 mm 
{+ SAX =2.8 mm 
31 = 2Ahs/U2 =2isentropic enthalpy drop/U2 
Figure 9. - Variation of clearance excitation coefficient with $. 
% 
whirl sped % 
Figure 10. - Effect of whirl. 
299 


Prediction of destabilizing blade tip forces for shrouded and unshrouded turbines

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