The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively

Nordmann, R.

Weiser, P.

English

NASA Technical Reports Server

ROTORDYNAMIC COEFFICIENTS FOR LABYRINTH SEALS CALCULATED BY 
MEANS OF A F I N I T E  DIFFERENCE TECHNIQUE 
R. Nordmann and P. Weiser 
Department o f  Mechanical Engineering 
University o f  Kaiserslautern 
Kaiserslautern, Federal Republic o f  Germany 
The compressible. turbulent, time dependent and three dimensional flow in a 
labyrinth seal can be described by the Navier-Stokes equations in conjunction with 
a turbulence model. Additionally, equations for masi and energy conservation and an 
equation of state are required. To solve these equations, a perturbation analysis 
is performed yielding zeroth order equations for centric shaft position and first 
order equations describing the flow field for small motions around the seal center. 
For numerical solution a finite difference method is applied to the zeroth and 
first order equations resulting in leakage and dynamic seal coefficients respec- 
tively. 
F1' F2 
ul* u2 
K. k 
c. c 
u.v.w 
p.k.e 
T 
PI *Pt *Pe 
P 
C 
P 
t 
z,r,B 
rl 
Pr 
u a  
P 
G 
Df ss + 
k' e 
c * C1' 
% 
cO 
6 
r 
0 
c2 
Forces on the shaft in z, y direction 
shaft displacements 
direct and cross-coupled stiffness 
direct and cross-coupled damping 
axial, radial and circumferential velocity 
pressure, turbulence energy, turbulent energy dissipation 
temperature 
laminar, turbulent and effective viscosity 
(we = Pl+VJ 
dens i ty 
specific heat for constant pressure 
time 
axial, radial and circumferential coordinate 
radial coordinate after transformation 
turbulent Prandtl number 
constants of the k-6 model 
constants of the k-E model 
production term in turbulence model 
laminar dissipation 
general variable 
general source term 
seal clearance for centric shaft position 
seal clearance for eccentric shaft position 
radius of the precession motion of the shaft 
161 
PRECEDING PAGE BLANK MOT FLMED 
https://ntrs.nasa.gov/search.jsp?R=19890013529 2020-03-20T03:08:29+00:00Z
e = ro/Co 
w 
R 
r 
r 
i 
a 
D1-10 
0 
1 
R 
S 
C 
S 
perturbation parameter 
rotational frequency of the shaft 
precession frequency of the shaft 
radius from seal center to end of seal fin 
radius from seal center to stator 
Transformation constants 
zeroth order variables 
first order variables 
Rotor 
Stator 
cosine 
sine 
The problem of subsynchronous vibrations in high performance turbomachinery has 
been investigated for many years. However, for some destabilizing effects there Is 
no satisfactory solution from the qualitative as well as quantitative points of 
view. In machines like turbocompressors and turbines, g a s  seals and especially 
labyrinths have an important influence on the dynamic behavior. These elements with 
turbulent flow conditions have the potential to develop significant forces 
which may lead to self-excited vibrations of the shaft. 
To predict the stability behavior of a high performance rotating machinery. the 
rotordynamic coefficients (eq. 1) of the turbulent seals have to be known. Many 
theoretical studies, numerical calculations and measurements have been carried out 
to determine the stiffness and damping Characteristics. Most of the recent theore- 
tical developments are based on a bulk flow theory, considering the shear stresses 
only at the wall but not within the fluid. Nelson (ref. 1) has derived a procedure 
with a 'one volume' model for straight and tapered seals. Iwatsubo (ref. 2) as well 
as Childs (ref. 3) have also used the one volume model for labyrinth seals. How- 
ever, in case of such complicated seal geometries difficulties m y  occur, when 
neglecting the local turbulence characteristics within the fluid. particularly at 
locations with strong velocity changes. 
Wyssmann et al. and Wyssmann (ref. 4 and 5) have extended the bulk flow theory for 
labyrinth seals. working with a two volume model. The latter takes into account the 
shear stresses between the core flow and the jet flow. Wyssmann's two volume model 
is derived from results of a finite difference calculation for a rotationally sym- 
metric single cavity turbulent flow (centered position of the shaft), based on the 
time averaged Navier-Stokes equations with a k-E model. A first attempt to apply 
finite difference methods to labyrinth seals was made by Rhode et.al. (ref. 6). who 
I 1 6 2  
calculated only one chamber seals for centric shaft position. However, bulk flow 
theories show a substantial lack in the prediction of labyrinth seal coefficients. 
As already shown by Nordmnn, Dietzen and Weiser (ref. 7), the solution of the 
time-averaged conservation equations for momentum. mass and energy in conjunction 
with a k-e turbulence model by means of a finite difference technique is a physi- 
cally more realistic way for calculating rotordynamic coefficients of gas seals. 
This paper is concerned with the extension of the already mentioned calculation 
procedure for annular seals to straight-through labyrinths. 
To describe the flow in a labyrinth seal, we use the time-averaged conservation 
equations for momentum, mdss and energy and the equation of state for a perfect 
gas .  The correlation terms of the turbulent fluctuation quantities are modelled via 
the k-e turbulence model of Launder and Spalding (ref. 8). 
All these differential equations can be arranged in the following generalized form: 
where 9 stands for any of the dependent variables, and the corresponding values 
of r+ and S+ are indicated in Table 1 (see Appendix A ) .  
PERTURBATION ANALYSIS 
Obviously, this set of equations is not solvable due to the limited performance 
capabilities of today's computers. Therefore, a stepwise procedure is performed In 
order to reduce the three dimensional and time dependent flow problem to a two 
dimensional time independent one. 
First. the governing equations are transformed to another coordinate system by 
introducing a new radial coordinate whereby the eccentrically moving shaft is 
converted to a shaft rotating in the center of the seal (Fig. 1). 
Fig. 1: Coordinate Transformation 
163 
The transform functions are given in Fig. 2. 
Strips on stator: 
Fig. 2: Transform functions 
Assuming small shaft motions on a circular orbit around the centric position we 
perform a perturbation analysis by introducing the following expressions for the 
dependent variables into the governing equations: 
r = n  + e rl 1 0 v = v  + e v  0 
0 
u = u  + e u1 
0 
T = T  + e T 1  P = Po + e P1 P = Po + e PI (4) 
This procedure yields a set of zeroth order equations governing the centric flow 
field and a set of first order equations describing the flow field for small 
eccentric shaft motions. 
The first order equations depend also on the circumferential coordinate 0 and on 
time t. To eliminate the circumferential derivations. we insert circumferentially 
periodic solutions for the first order variables in the corresponding set of equa- 
t ions : 
cos0 + v sine 
cos0 + w sin0 p1 = plc cos0 + p sine 
cose + p sine cos6 + Tls sin0 
IC 1s cos0 + ulS sine v1 = v 
wl = IC 1s 1s 
P1 = Plc 1s T1 = 
IC 
u = u  1 
( 5 )  
After separating the equations into sine and cosine terms and rearranging them by 
introducing the following complex variables, 
164 
- - - 
= IC + IUls 1 IC IC + lWlS v = v + iVlS 
T, = TIC + ITls P1 = PIc + iPls 
w = w  1 - - 
p1 - Plc + IPlS (6) 
a circular shaft precession orbit (Fig. 3) with frequency R and corresponding 
solutions for the first order variables are assumed whereby the temporal deri- 
vations can be eliminated: 
int El = roe 
(change in clearance) 
Fig. 3: Circular shaft orb1 t 
- int w = W  e 1 1 
int - int 
A iRt - v l = v  e int ul = u1 e 
- iRt 
- 
1 
TI = TI e P1 = P1 e P1 = PI e (7) 
Finally the zeroth and first order equations are arranged to a generalized form: 
Table 2 (see Appendix B) indicates the values for the zeroth and first order 
equations. 
The constants D -D (given in the Appendix C) arise from the coordinate trans- 
formation and depend only on zeroth order variables. These terms are zero for the 
labyrinth chambers. 
1 10 
FINITE D1-E WEIIDD 
For the numerical solution of the governing equations we apply a finite difference 
technique in the same m e r  as suggested in nrany publications (see e.g. ref. 9). 
First the calculation domain is covered by a finite difference grid (Fig. 4). Dis- 
cretizing the equations results in algebraic expressions linking every node point 
to his four neighbours. Solution of zeroth order equations results in flow field. 
pressure distribution, and leakage. For first order solution. the equations for the 
turbulence quantities k, e can be dropped due to assumed small rotor motions. 
Because E q .  (1) contains four unknown parameters (K.k,C.c). first order solution is 
performed for two different shaft precession frequencies 0. This results in two 
sets of dynamic forces F1, Fz obtained from the integration of the pressure pertur- 
bations (see Eq. 9). I 
165 
r 
0 
r 
0 
Finally the dyrramic coefficients are calculated from E q .  10. 
(9) 
Fig. 4: Nonuniform finite difference grid for labyrinths 
- Zeroth order equations 
For the velocities the near wall regions are represented by the logarithmic wall 
law. The entrance conditions are iteratively calculated depending on entrance 
Mach number and loss coefficient. The entrance swirl is assumed to be known. 
- Firs t  order equations 
Due to shaft precession and coordinate transformation. one has to pay attention 
to the following boundary conditions for the radial and circumferential velo- 
cities: 
A 
v = (0.. 0.) WIS = (0. .O.)  1s 
A A 
VIR = (0.0 (n-w)co) WIR = (nco.o.) 
166 
For the pressure, the exit perturbation is zero in case of exit Mach number less 
than one ("undercritical" flow) which occurred in the application examples. 
APPLICATION 
For testing our program. we d e  comparative calculations to measurements performed 
by Benckert (ref. 10). 
seal mta: 
Shaft diameter: 0.150 rn 
Strip height : 5.5 m 
Strip pitch : 8 mm 
Clearance 0.5 mm 
Number of chambers : 3 
Entrance pressure p : 1.425 bar 
: 0.950 bar Exit pressure 
Reservoir temperature: 300 K 
Laminar viscosity : 1 .8*10--5 Pa s 
Perfect gas constant : 287.06 J/kg.K 
Specific heat ratio : 1.4 
Rotor surface velocity: 0 m/s 
0 
pa 
The calculation grid. the flow field and the pressure distribution are shown in 
Fig. 5-7. 
In Fig. 8-10 mass flow, direct and crosscoupled stiffness are plotted versus the 
swirl parameter. as defined by Benckert (E 2 = 0.5*p0*w / (po-pa)). 0 0 
F i g .  5: Finite difference grid 
167  
........ ........ ........ .. . . e * + *  ......... .. 
1-1 0 * . *.,,** ......... ......... .. . . r e * ( +  
4 * * * * .  . a .... r t r t t  
............ ............ ............ 
............. ........... . . a a * * * f ? f  ............ ............. ................ ............... ............... ............... ................ ............... 
. . e - . , ~ ? ?  
+- -- 4 -4 
---..- 
Fig. 6: Flow f i e l d  
Fig. 7: Pressure distribution 
168 
1. I1 
. 1.18 
Q) 
\ 
LD 
1.15 
0 
E 
1.11 
1.11 
Fig. 8 :  Mass flow 
S E R L Z O  
'"w"T" 
mjr77mm 
CALCULATION O f  
L A B Y R I N T H  S E A L  
COEFFICIENTS 
* I E X P .  - I C A L C .  
211111. 11 
151000.11 
n 
L 
\ z 
Y 
U 111111.11 
51111. 11 
1.11 1.25 1 . 5 1  1. 75 1.11 
1. e e  
1. 11 8.25 1 . 5 1  1.75 1.00 
Fig. 9: Direct Stiffness e 
Ea C 3 
SEAL20 
C A L C U L A T I O H  OF 
C O E F F I C I E N T S  
L A B V R I H T H  S E A L  
* : E X P .  - : C A L C .  
169 
200000.00 
H0000. 1 0  
100001.00 
s0010.0c 
0.00 
0.00 0.25 0.50 0.75 1 .01  
SEFILZD 
CRLCULRTION OF 
COEFFICIENTS 
L R e y R I m i u  sEaL 
* . :  EXP. 
b 
Fig. 10: Crosscoupled Stiffness Ee c 1 
We have shown that a finite difference method based on the Navier-Stokes equations 
in conjunction with a turbulence model is more suitable to describe the dynamic 
coefficients of gas seals than the simpler models based on bulk flow theories. Of 
course the calculation time is also much greater. But great progress is still 
possible. For example, advanced solution algorithms like Mu1 ti-Grid. Newton-Sche- 
mes. SIP (strongly implicit procedure) can be implemented to speed up the computer 
program. 
Further on. a three dimensional calculation procedure is currently under develop- 
ment to determine the coefficients of more complex geometries like stepped and 
interlocking labyrinths (Fig. 11.12). 
... ...... * 
Fig. 11: Stepped seal 
170 
F i g .  12: Interlocking seal 
T 
k 
APPMDIX A 
Table 1: Governing equations of turbulent seal flow 
pe/Pr 5% + u 2 + v 2 + 2) + & (Diss + p e )  
P P 
pe/uk G - p e  
2 a w  
Equation of State  for a perfect gas: 
p = p R T  
Cons t a t s  of the k-e model 
C = 0.09 C1 = 1.45 Cz = 1 . 9 1  uk = 1.0 uE = 1.22 Pr = 0.9 
Cr 
171 
Production term in the equations for the turbulence quantities: 
Laminar Dissipation in the energy equation: 
APPEM)M B 
Table a: Zeroth order equations 
0 
2 
c1 ; c - c  2 p o i i  
1 b o  *o 1 c (uo + vo 8r)) + c (D~DD + pot) 
P P 
PO Equation of State: po - -   ReTo 
Table 2b: First order equations 
1 7 2  
2 
a l a  * w  0- vowo - -(u v p )- - -+rlV0VOP1)+ T1+i--- rl ax 0 0 1  T i *  
173 
n 
only the first order continuity-equation to determine p1 shows a slightly modified 
form. 
APPENDIX c 
First order transforPration constants 
Note: r = r for strips on stator 
X a 
r = r1 for strips on rotor 
X 
V 
2 
( rx+ 
cO" 
-- *o + p w w - p  v v - *eo $) 2 ( % O F  0 0 0  0 0 0  
174 
REFERENCES 
/1/ 
/2Y 
A/ 
/4/ 
/5/ 
/6/ 
/7/ 
/8 /  
/9/ 
Nelson, C.C.; 1985: Rotordynamic coefficients for compressible flow in tapered 
annular seals. Journal of Tribology, Vol. 107. July 1985 
Iwatsubo. T.; 1980: Evaluation of instability forces of labyrinth seals in 
turbines or compressors. NASA conference publication 2133. May 1980 
Childs. D.W.: Scharrer. J.K.: 1984: An Iwatsubo-based solution for labyrinth 
seals-comparison with experimental results. NASA conference publication 2338 
Rotordynamic instability problems in high performance turbomachinery 1984 
Wyssmann. H.R.; Pham, T.C.; Jenny, R.J.; 1984: Prediction of stiffness and 
damping coefficients for centrifugal compressor labyrinth seals. ASME Journal 
of Engineering for Gas Turbines and Power. Oct. 1984. Vol. 106, pp. 920-926 
Wyssmann. H.R.: 1986: Theory and measurements of turbocompressors. The fourth 
workshop on rotordynamics instability problems In high performance turbo- 
machinery. June 2-4. 1986 Texas A&M University, College Station, Texas 
Rhode. D.L.; Sobolik. S.R.; 1986: Simulation of Subsonic Flow Through a 
Generic Labyrinth Seal. Journal of Engineering for Gas Turbines and Power, 
V O ~ .  108. Oct. 1986. pp. 674-680 
Nordmann. R.; Dietzen. F.J.: Weiser. H.P.: Calculation of Rotordynamic 
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Techniques. The 11th Biennial Conference on Mechanical Vibration and Noise. 
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Launder. B.E.; !3palding. D.B.; 1974: The numerical computation of turbulent 
flows. Computer methods In applied mechanics and engineering. 3 (1974) 269-289 
Gosm, A.D.: Pun. W.: 1974: Calculation of recirculation flows. Imperial 
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1101 Benckert. H.: Stromungsbedlngte Federkennwerte In Labyrlnthdichtungen. Dlsser- 
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175 


Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique

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