To describe the compressible, turbulent flow in a labyrinth seal, a three volume bulk flow model is presented. The conservation equations for mass, momentum, and energy are established in every control volume. A perturbation analysis is performed, yielding zeroth order equations for centric rotor position and first order equations describing the flow field for small rotor motions around the seal center. The equations are integrated numerically. From perturbation pressure, the forces on the shaft and the dynamic coefficients are calculated

Nordmann, R.

Weiser, P.

English

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EVALUATION OF ROTORDYNAMIC COEFFICIENTS OF LOOK-THROUGH LABYRINTHS 
BY MEANS OF A THREE VOLUME BULK FLOW MODEL 
R. Nordmann and P. Weiser 
Department o f  Mechanical Engineering 
Univers i ty  o f  Kaiserslautern 
Kaiserslautern, Federal Republic o f  Germany 
To describe the compressible, turbulent flow in a labyrinth seal, a three volume 
bulk flow model is presented. The conservation equations for  mass, momentum 
and energy are  established in every control volume. A perturbation analysis is 
performed, yielding zeroth order equations for  centric rotor  position and f i r s t  
order equations describing the flow field for  small rotor  motions around the 
seal center. The equations are  integrated numerically. From perturbation pressure, 
the forces on the shaf t  and the dynamic coefficients are  calculated. 
NOMENCLATURE 
Forces on the  shaft  in x,y direction 
shaf t  displacements 
direct and cross-coupled stiffness 
direct and cross-coupled damping 
pressure 
axial, radial and tangential velocity 
temperature 
density 
axial, radial and tangential coordinate 
time 
length of labyrinth chamber 
depth of labyrinth chamber 
shaf t  radius 
https://ntrs.nasa.gov/search.jsp?R=19920005138 2020-03-24T07:05:20+00:00Z
SUBSCRIPTS 
nominal clearance 
seal fin thickness 
to ta l  seal length 
eccentricity ratio, perturbation parameter 
radius of circular shaf t  orbit  
seal clearance 
rotational frequency of t h e  shaf t  
precession frequency of t he  shaf t  
pressure before labyrinth 
pressure behind labyrinth 
swirl  
perfect gas  constant  
specific heat ratio 
shear  s t r e s s  
friction factor  
Reynolds number 
hydraulic diameter 
Mach number 
relative surface roughness 
entrance loss  factor  
loss  factor  a t  seal fin entrance 
laminar viscosity 
imaginary unit 
zeroth order  
f i r s t  order 
s t a to r  
rotor 
axial 
circumferential 
fluid- 
chamber, upper wall 
chamber, side wall 
Sine 
Cosine 
seal  entrance 
relative 
Indices f o r  the  different control volumes 
INTRODUCTION 
Labyrinth seals, commonly used in turbomachinery t o  reduce the leakage loss, 
can have a strong influence on the dynamic behavior of a turbine or a compressor, 
especially for high performance machines. In some cases, the fluid forces gene- 
rated in labyrinths caused severe rotor instabilities. Therefore, the design engi- 
neer needs information concerning the possible influences that  a labyrinth can 
have on the dynamics of a rotating system. 
For mathematical modeling, the seal forces can be described in terms of stiffness 
and damping parameters, the so-called rotordynamic seal coefficients (see Eq.1). 
These coefficients are input parameters for rotordynamic investigations and have 
t o  be determined by fluid mechanics methods. In contrast t o  WEISER and 
NORDMANN (1987,1988,19891, where we used Finite-Differencing methods (FDM) 
to  solve the conservation equations in conjunction with a turbulence model, 
Florjancic developed a more conventional three volume bulk flow method which 
takes advantage from FDM calculations and reduces the computational effort 
dramatically. In 1987, KLAUK has already developed a three volume model for 
the compressible flow in labyrinths, which is extended in WEISER (1989). In 
this approach, the energy equation and the equation of state for a perfect gas 
is used in addition t o  the previously existing model. 
MATHEMATICAL MODELING 
Many &tempts have been made for the calculation of labyrinth coefficients, 
e.g. see the publications of KOSTYUK(197S), CHILDS(1984) or  WYSSMANN(1984). 
Flow visualization experiments show, that  for look-through labyrinths, the flow 
field can be divided in two characteristic regions: a vortex flow in the seal 
chamber and a jet flow region beneath the seal strip and the groove. 
Therefore, the calculation domain is divided into three control volumes (CV). 
see Fig.1: one in the seal chamber accounting for the vortex flow and two 
volumes representing the jet flow beneath chamber and seal fain. Fig.1 also gives 
an impression of the bulk velocities in the different. volumes. 
Coordinate System 
T-l! I1 ' L.: J \', \ \ \.\"\\'\ ', \\.\' 
Rotor 
Control Volumes 0 th  Order Bulk Velocities 
--- 
Fig. 1: 3-volume bulk flow model 
For mathematical description, we use the momentum, continuity and energy 
equation in every control volume. Paying attention t o  the  following assumptions: 
- compressible medium (Q # const.) 
- constant  laminar viscosity p 
- modelling of turbulence by wall shear s t resses  
- no shear s t resses  within the fluid (exception: jet friction shear s t r e s s  
in the  contact  region of C V  I1 and CV 111) 
- no  radial shear s t resses  
- f i r s t  order radial velocities only in CV I1 and I11 
we obtain the following set  o f  equations to  describe the turbulent, compressible 
seal flow (equations shown here for teeth-on-stator seal): 
1 3  u2 v2 
-- { p w h  ( + 2 + 2 r 3cp + F~) ]  + 
z ( w  - R o )  
r v  
r ( w  - R o )  
r v  
151 
As already mentioned, the  modeling of flow turbulence is performed using semi- 
empirical wall shear s t ress  formulations which i s  the usual way in bulk flow 
analysis, see  for  example CHILDS(19841, SCHARRER(1987). The fluid shear s t r e s s  
in the  contact region of CV I1 and CV I11 is described in the same manner a s  
WYSSMANN(1984) suggested. A1 1 shear s t ress  components can be wsi t ten in 
a generalized form: 
1 
T.. = - Q Ai Vj Vrel 
'J 2 
with 
and 
D,Vre, 
Re = 
P/Q 
Vrel is  t he  relative velocity, Vi the component of the  resulting velocity vector 
which i s  in t he  same direction i s  the  shear s t ress  component t o  be calculated. 
Index i indicates the  appropiate shear s t ress  (s: stator-, r: rotor  etc.) and index 
j gives the  direction (z: axial, cp: tangential). With these definitions, we obtain 
the following relations: 
cv I: 
-
Rotor: Vrel = ( (w1-  R W ) ~  + U:)O.~ 2 )0.5 Stator:  vreI = ( W: + u1 
V,= wI- R o  V,= wI 
vz= UI vz= UI 
cv 11: 
--
Rotor: Vrel = ( ( wI1- R oI2 + U F ~ ) ~ ' ~  Stator:  Vrel = ( wf1 + u:~)"" 
Jet: 're, = ( ( WII- WIII) + (uI1- u ' ~ ~ ~ ) ~  )O" with uiII = uII(  1 - f ir)  
cv  111: 
-- 
Jet: vreI = ( (w1r  w ~ ~ ~ ) ~  ' (U 11- u ' ~ ~ ) ~  )'.' 
vz = u~~ - ui11 
Chamber side wall: 
2 )0.5 B 
're1 = ( WIII + U'~II  with uiII = u' - I1 L 
Upper wall in t he  seal groove: 
r e  = ( Wrl l  + 2 )O.' ~'111 
The parameters introduced in t he  shear  s t ress  formulations according t o  MOODY 
(1944) are  
For t he  fluid shear  s t ress ,  m, C1 and C2 are zero; n is s e t  t o  3.762 (according 
t o  WYSSMANN). To account fo r  the  real flow situation in the labyrinth, n is  
multiplied with t he  fac tor  6 ,  which depends on  geometry and performance data. 
BOUNDARY CONDITIONS 
For t he  solution of t he  differential equations, boundary conditions have t o  be 
specified. To calculate the  pressure, density and axial velocity a t  t he  seal en- 
trance, we  use t he  same relations a s  NELSON(1989, calculating t he  entrance 
values depending on the  Mach number: 
At the  entrance of every seal fin, there  occurs a pressure drop due  to the  
acceleration of t h e  fluid. Therefore, t he  equations 7 are  used in a modified form 
with a constant  loss  factor  rI: 
The model parameters B,  By and TI are determined with the help of the Finite- 
Difference programs of NORDMANN and WEISER(1987,1988,1989). 
PERTURBATION ANALYSIS 
In order t o  determine the seal forces, we assume tha t  the  rotor  performs only 
small motions around the  seal center. Then we can describe the dependent 
variables with a perturbations series, truncated af ter  the linear term: 
As perturbation parameter, we choose the relative eccentricity e, which i s  the 
ratio of the  shaft  eccentricity and the  nominal clearance: 
Inserting these expressions into the governing equations, separating terms without 
e and with ei yields two s e t s  of equations: the zeroth order relations describe 
the flow field for  a centric rotor  position and the  first  order terms s tand  f o r  
the flow situation a t  small eccentric shaft  motion. 
ELIMINATION OF TEMPORAL AND CIRCUMFERENTIAL DERIVATIVES 
While t h e  zeroth order  equations depend only on the  axial coordinate z,  the  
f i rs t  order  set contains a l so  derivatives with respect t o  time t ,  tangential and 
radial coordinates. Assuming tha t  the shaf t  moves on a small  circular orbit  
(see Fig. 2), the  seal clearance can be  written a s  ' 
h = ho + e hl = C, - X(t)coscp - Y(t) sincp (11 ) 
I 
Fig. 2: Circular shaf t  orbit  
For t he  dependent variables, we prescribe a corresponding solution: 
- coscp + ul sincp pl = pl coscp + pl sincp ul - UIC 
S C S 
- coscp + vl  sincp Tl = Tl coscp + T1 sincp v1 - vlc 
S C S 
(12) 
wl = w, coscp + wl sincp pl = Q lc coscp + Q sincp 
C S S 
Now, t h e  circumferential derivatives can be  calculated analytically. Separating 
the  equations with respect t o  sine and cosine terms and introducing complex 
variables: 
we obtain a now complex system of f i rs t  order equations. 
As already mentioned, we  assume a circular shaft  orbit, which gives fo r  the 
time dependence of the  clearance 
Again, similar expressions for  the  variables are established: 
Thereby, the derivatives with respect t o  time can be expressed analytically. Fi- 
nally, the  radial derivatives in C V  I1 and I11 are modeled with 
After having performed these s teps,  the  f i rs t  order equations do  only depend 
on z. The solution s t a r t s  with a numerical integration of zeroth order equations 
with respect  t o  t h e  boundary conditions. The f i rs t  order solution using t h e  per- 
turbed boundary conditions yields t he  pressure distribution in t he  seal for  small 
ro tor  motions. The forces are obtained by a pressure integration: 
Because we have to determine 4 parameters (K, k, D, d), t he  calculation of f i r s t  
order equations has t o  be  done twice for  two different ro tor  precession fre- 
quencies (f2=0 and Q=w). Then the  rotordynamic coefficients can b e  calculated: 
COMPARISON TO EXPERIMENTS AND OTHER BULK-FLOW-THEORIES 
1. Example 
BENCKERT(1980) investigated experimentally a three chamber look-through laby- 
rinth. Due t o  the  design of his t e s t  rig, he could only measure stiffnesses.  The 
data  of t he  labyrinth are  
Ta: 300 K 
working fluid: air 
number of chambers : 3 
No shaft  rotation ! 
Fig.3 and 4 show the  comparison t o  the experimental data for  different preswirl 
conditions. While the  3-volume approach predicts a positive direct stiffness 
according t o  the measurements, the theory of CHILDS(1984) gives the opposite 
sign fo r  K. The agreement between experiment and 3-volume model is good, 
especially showing tha t  the  model presented is capable t o  ensure sufficient 
accuracy in the  calculation of dynamic seal parameters even for  shor t  labyrinth 
seals where the  approaches of CHILDS o r  WYSSMANN have some problems. 
Comparison of measurement, bulk theory of CHILDS 
and 3-volume model 
-
- Experiments 
0 
BENCKERT 
- - 
+ Theory of 
CHILDS 
-.- 
:-:: 3-volume 
model 
0 100 200 
preswirl [m/s] 
Fig. 3: Direct stiffness K 
Comparison of measurement, bulk theory of CHILDS 
and 3-volume model 
400000 
- Experiments 
0 
BENCKERT 
k CN/m] -+-Theory of 
CHILDS 
- .- 
::-:: 3-Volume 
model 
0 
0 100 200 
preswirl [rn/s] 
Fig. 4: Cross-coupled stiffness k 
2. Example 
In a paper published 1986, WYSSMANN shows a comparison of measured data  
for  a 15.-chamber look-through labyrinth. The experiments were carried o u t  by 
CHILDS(1984). WYSSMANN compares the  measurements t o  his 2-volume approach. 
For a 16-strip seal with fins on  s tator ,  t he  resul ts  of t he  experiments, of WYSS- 
MANN and the  3-volume model are  shown in Fig. 5 - 7. 
comparison of measurement, bulk theory of WYSSMANN 
and 3-volume model 
-
Experiments 
- CHILDS 
0 
--Theory of 
t WYSSMANN 
model 
preswirl [m/s] 
Fig. 5: Direct stiffness 
Comparison of measurements, bulk theory of WYSSMANN 
and 3-volume model 
100000 
- Experiments 
CHILDS 
- ' Theory of 
k [N/m] 0 + WYSSMANN 
- . -  
:,: 3-volume 
model 
preswirl [m/s] 
Fig. 6: Cross-coupled stiffness 
Comparison of measurement, bulk theory of WYSSMANN 
and 3-volume model 
0 
preswirl [m/s] 
- measurement 
0 
CHILDS 
- - 
+ theory of 
WYSSMANN 
-. - 
x 3-volume 
model 
Fig. 7: Direct damping 
In contrast  t o  t he  2-volume theory, the model presented here predicts the  direct 
s t i f fness  more accurate. 
CONCLUSION 
A new bulk f low model has been presented for  the  calculation of t he  rotor- 
dynamic seal parameters. The theory uses a three volume approach, and conser- 
vation equations fo r  mass,  momentum and energy a re  s e t  up for  every control 
volume. A perturbation analysis yields zeroth and f i rs t  order equations. The 
integration of f i r s t  order  pressure perturbation resul ts  in t h e  desired dynamic 
coefficients. 
The comparison t o  experiments shows good agreement; compared to o ther  bulk 
flow models, t he  present theory allows a much bet ter  calculation of the direct 
st iffness.  This is  mostly due to the  determination of the  model parameters with 
t he  help of t he  FDM programs, a s  a comparison t o  an earlier 3-volume model 
of KLAUK, which was developed in 1987 and did no t  have the  benefit of using 
FDM results,  shows. 
REFERENCES 
BENCKERT, H.: 
"Stromungsbedingte Federkennwerte in Labyrinthdichtungen" 
Dissertation TU Stu t tgar t ,  1980 
CHILDS, D. W., SCHARRER, J. K.: 
"An Iwatsubo Solution for  Labyrinth Seals Comparison of 
Experimental Results" 
NASA 2338, May 1984 
CHILDS, D.W.; SCHARRER, J.K.: 
"Experimental Rotordynamic Coefficients Results for  Teeth-on-Rotor- 
and Teeth-on-Stator Labyrinth Gas Seals" 
NASA CP 2443, 1986, p.259-276 
FLORJANCIC, S.: 
"Annular Seals of High Energy Centrifugal Pumps" 
Dissertation, Swiss Federal Insti tute of Zurich, 1990 
KLAUK, E.: 
"Berechnung de r  rotordynamischen Koeffizienten von mit kompressiblen 
Medien durchstromten Labyrinthdichtungen unter Anwendung von 
Mehr-Volumen-Modellen" 
Diplomarbeit, University of Kaiserslautern, 1987; AG Machine Dynamics 
KOSTYUK, A. G.: 
"Circulation Forces over the  Shrouding and their Influence on the  
Threshold Capacity of Large Turbine Units" 
Teploenergetika, 197.5, Vo1.22,No.3, p. 41-46 
MOODY, L. F.: 
"Friction Factors for  Pipe Flow" 
Trans. ASME 66 (November 1944), p. 671-,684 
NELSON, C. C.: 
"Rotordynamic Coefficients for  Compressible Flow in Tapered Annular 
Seals" 
Journal of Tribology, Vol 107, July 1985 
NORDMANN, R., DIETZEN. F. J., WEISER, H. P.: 
"Calculation of Rotordynamic Coefficients and Leakage for Annular 
Gas Seals by Means of Finite-Difference Techniques" 
llth Biennial Conference on Mechanical Vibration and Noise, 
Boston, 27-30 Sept. 1987 
NORDMANN, R., WEISER, H.P.: 
"Rotordynamic Coefficients for Labyrinth Seals Calculated by 
Means of a Finite-Difference Technique" 
NASA, CP 3026, 1988 
SCHARRER, J. K.: 
"Theory versus Experiment for  the Rotordynamic Coefficients 
of Labyrinth Gas Seals: Part I - A Two Control Volume Model" 
The l l th Biennial Conference on Mechanical Vibrations and Noise, 
Boston, 27-30 Sept. 1987 
WEISER, H. P., NORDMANN, R.: 
"Calculation of Rotordynamic Labyrinth Seal Coefficients by Means 
of a Threedimensional Finite-Difference Method" 
Accepted for presentation for: 12th Biennial Conference on 
Mechanical Vibration and Noise, Montreal, Sept. 1989 
WEISER, H. P.: 
"Ein Beitrag zur Berechnung der dynamischen Koeffizienten von 
Labyrinthdichtungssystemen bei turbulenter Durchstromung mit 
kompressiblen Medien" 
Ph. D. Thesis, University of Kaiserslautern 1989 
WYSSMANN, H. R., PHAM, T. C., JENNY, R. J.: 
"Prediction of Stiffness and Damping Coefficients for 
Centrifugal Compressor Labyrinth Seals" 
ASME Journal of Engineering for Gas Turbines and Power, 
Oct. 1984, B. 106, p. 920-,926 
WYSSMANN, H.R.: 
"Theory and Measurement of Labyrinth Gas Seal Coefficients for Rotor 
Stability of Turbocompressors" 
NASA CP 2443, 1986, p.237-258 


Evaluation of Rotordynamic Coefficients of Look-through Labyrinths by Means of a Three Volume Bulk Flow Model

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