First experimental investigations performed on a new test rig are presented. For a staggered labyrinth seal with fourteen cavities the stiffness coefficient and the leakage flow are measured. The experimental results are compared to calculated results which are obtained by a one-volume bulk-flow theory. A perturbation analysis is made for seven terms. It is found out that the friction factors have great impact on the dynamic coefficients. They are obtained by turbulent flow computation by a finite-volume model with the Reynolds equations used as basic equations

Kwanka, K.

Ortinger, W.

Steckel, J.

English

NASA Technical Reports Server

N94- 34187 y
CALCULATION AND MEASUREMENT OF THE INFLUENCE OF FLOW PARAMETERS ON
ROTORDYNAMICCOEFFICIENTS IN LABYRINTH SEALS:
K. Kwanka, W. Ortinger, and J. Steckel
Lehrstuhl Thermische Kraftanlagen
Technische Universitat MiJnchen
MiJnchen, Germany
<,/--37
Abstract
First experimental investigations performed on a new test rig are presented. For a staggered
labyrinth seal with fourteen cavities the stiffness coefficient and the leakage flow are measured.
The experimental results are compared to calculated results which are obtained by a one-
volume bulk-flow theory. A perturbation analysis is made for seven terms. It is found out that
the friction factors have great impact on the dynamic coefficients. They are obtained by
turbulent flow computation by a finite-volume model with the Reynolds-equations used as basic
equations.
=
E
Nomenclature
C,E,K
F
H
U
Czt,, COX
e
f
l
n
P
RR
5
(p
K
_t
"C
W,_,A,_-,n,;,ii
O)
P
Damping, inertia, stiffness coefficients
Force
Labyrinth height
Part of cavity perimeter
Friction factor
Circumferential velocity, axial velocity
Eccentricity
Cross-sectional area ofeavity
Mass flow related to circumference
Length of cavity
Rotating speed
Pressure
Radius of rotor
Radial clearance
Peripheral angle
lsentropic coefficient
Friction factor
Flow coefficient, dynamic viscosity
Shear stress
Perturbation terms
Angular velocity
Density
PIlBGEOiN6 PAGE BLANK I_T FILML_D 209
PAGE INTEBTIONALLYBI.A_
https://ntrs.nasa.gov/search.jsp?R=19940029681 2020-06-16T12:51:37+00:00Z
Introduction
Despite the numerous efforts in the past, there still remain some open questions in
understanding tip-clearance excitation and, as a consequence, a lack in modelling and
predicting forces and coefficients induced by labyrinth flow. The efforts to increase the
efficiency and to minimize the specific costs or the specific weight of a rotor-easing
configuration ot_en lead to parameters which are in some cases very close to the stability
margin. Beyond the stability limit the rotational energy which is transferred to the rotor bending
exceeds the damping that dissipates this energy. As a consequence, self-excited vibration
mechanisms impede the scheduled operation regime of the turbomachine. In addition to the
labyrinth seal forces several other mechanisms (e.g. oil whip) act on the rotor and an
amplification of the excitation may occur [5].
The labyrinth seal forces are usually described in a linearized formulation for small
deflections out of the centered position by using dynamic coefficients. These are stiffness,
damping and inertia coefficients.
Many investigations were performed to estimate the dynamic coefficients on several seal
geometries. In the experimental field two procedures are frequently used. First, static
measurements were made [4, 2,12] which allow the determination of the direct and cross-
coupled stiffness coefficients of the labyrinth seal. Second, dynamic measurements [2, 7],
which were carried out to identify the other coefficients in eq. (1).
The experimental results can be used for verification of the various theoretical approaches.
The computational effort is low when a bulk-flow theory - either a one-, two- or three-volume
model [1, 3, 8]- is used. An increased amount of computational time is needed if the seal
geometry is discretized by a grid, and if the local variables are calculated by the finite-
difference or the finite-volume method [9]. The forces acting on the rotor are obtained by a final
integration.
In this paper a staggered labyrinth seal is investigated. Only few experimental results are
published for this geometry in comparison to others. With the help of the static identification
method a maximum accuracy for the measured stiffness coefficients is expected. In most cases
these coefficients are of outstanding importance for the dynamic behaviour of the rotor.
The calculated results are obtained by a one-volume bulk-flow theory. It is found out that
the bulk-flow theory in connection with the perturbation method is highly sensitive to friction
factors. In this case they are calculated by means of turbulent flow computation. The Reynolds
equations in cylindrical coordinates including the standard k-e model give the velocity
components and the turbulent values. A boundary-layer theory is applied and the friction
factors are calculated depending on the axial and circumferential status of the flow.
Test Rig
The test rig is designed with the purpose to measure labyrinth flow induced radial forces on
the eccentric rotor. In fig. la. a schematic representation of the test rig can be seen. Two
identical grooved labyrinth seals with fourteen cavities each are located symmetrically to the
inflow region. The inflow region is outlined in fig. lb. The inlet swirl can be varied by the
combination of radial and circumferential flow components.
210
///
/
Fig. la. Schematic of the test fig
The geometric data of the staggered labyrinth are indicated in fig. 2. At the center position
the gap between the rotor and the seal tips attached to the casing is 0.5 nun. The eccentricity
can be adjusted in horizontal direction by moving the casing relative to the rotor.
One of the seals is used to measure the pressure distribution in circumferential direction and, as
a consequence, the acting force in each cavity and on the whole rotor. Ten pressure lines are
located in each cavity. This gives a total of some 160 data points per labyrinth seal unit. The
pressure lines arc connected to a multifunction pressure measurement rig. The flow field in the
cavities and the inlet swirl is determined by using a probe which is located in the second
labyrinth seal.
05 ! •
• I p-,,I
_//
_3: /
24O
Fig. lb. Inflow configuration of the test rig Fig. 2. Geometric data of the labyrinth
(dimensions in mm)
211
Some test rig dimensions and experimental conditions are listed in table 1. The maximum
preswiri velocity (Cu_22 m/s) is reached whcn there is an inflow only via the tangential feeders.
Due to the fact that the direction of the rotational speed can be changed the preswirl can be
negative and positive. With high prcssurc differences on the labyrinth seal the critical pressure
ratio prevailes on the last seal tip.
Table 1. Test rig dimensions and experimental conditions
Diameter (stator)
Length (seal)
Clearance (e=0)
Eccentricity
Pressure difference
Rotating' speed
.D [mini
L [mm]
d [mm]
e, [mm]
/_o [kPa]
n [.rpm]
240
150
0,5
max. 0,4
max. 450
*8000
First measurements at the center position show that the pressure is declining in axial
direction, alternately in big and small steps. Together with the geometry dependent flow area
this causes a zigzag course of the flow coefficient (fig. 3), which is almost entirely independent
on the rotational speed, inlet swirl and eccentricity. The mean value of the flow coefficient in
the neutral position of the rotor is 0.69 and 0.675 when the eccentricity is 0.4 ram.
1
0,g5
0,g
'u2 0,85
0,8
_0,75
u
o_ 0,7
,'i" 0,65
0,6
0,55
0,5 i i i I I I I [ • i i i i ii p i i * • i i 1 i i , • •
N ¢_ _i" U') (D I_" CO O'J o _- N ¢_ _1" u13
Seal tip
= e=O.Omm
o e=0.4mm
Fig. 3. Flow coefficient for centric and eccentric position of the rotor ( Ap = 160000 Pa;
n = 4000 rpm )
Considering the direction of displacement, the force acting on the rotor can be splitted in a
restoring and a cross force. Despite the fact that thc restoring force is conservative it has an
influence on the critical speeds and the vibrational modes and thus on the stability. When the
restoring force has a decentralized effect the bending modes are in general more deflected and
more excited by cross forces, and the stability limit declines.
In center position of the rotor the circumferential pressure distribution in each cavity has
shown that almost no resulting force is acting on the rotor and which means that the rotor is
well manufactured and exactly alined. At the off-center position of the rotor the restoring force
has a zigzag course like the flow coefficient (fig. 4). The resulting force for the whole labyrinth
212
is decenterizing for all parameters investigated. This behavior of the restoring force can be
explained by the change of state of the throttled flow in the alternating large flow areas in the
cavaty and the small flow areas between the strips and rotor. Only in the gap behind the last
seal tip the direction of the force is directed against the deflection as is normally expected forplain seals.
"--'-*----- 0,000 RPM
-- - n- -. 2,000 RPM
- " _- "' 4,000 RPM
6,000 RPM
Fig. 4. Restoring force versus cavity in dependence of rotating speed ( Ap = 300000 Pa;
e = 0.4 ram; cu _. 22 m/s)
From a maximum in the first cavities the cross force declines towards the end of the
labyrinth seal (fig. 5). The influence of the rotational speed on the restoring and the cross force
is small. For all investigated parameters the restoring force was higher than the cross force.
Fig. 5. Cross force versus cavity in dependence of rotating speed ( Ap = 300000 Pa;
e = 0.4 ram; cu _ 22 m/s)
213
Therotationalspeedandthe entry swirl act together and influence the magnitude of the
destabilizing force (fig. 6). When the direction of rotation and inlet swirl coincide then the cross
force is smaller than in case of counteracting swirl and drag exerted by the rotating surface of
the rotor to the flow. The influence of swirl exceeds the influence of rotation. Decisive for the
magnitude of the cross force is the change of circumferential velocity and not the absolute value
of this velocity. In case of conspiring swift and drag the circumferential velocity has the highest
values but not the cross force.
Further details about the experimental investigations effected under consideration of
numerous parameters are contained in [1 l].The circumferential velocity cu in front of the seal
is about 20 m/s for all investigations in this paper which are performed with swirl.
15
10
5
0¢1,1
1/I
tn -10
.o
¢o -15
-20
-25
[]
[]
D
I _' I I I
_I o_ o,a 0,4
• rotation, no swirl
D rotation, positive swirl
• rotation, negative swirl
Eccentricity [mm]
n
Fig. 6. Influence of entry swirl and rotating speed on the cross force ( Ap = 460000 Pa;
n = 4000 rpm )
Theoretical Treatment
With the help of a bulk-flow theory the circumferential pressure distribution due to the
eccentric position of the rotor relative to the casing is obtained.
The governing equations are the following:
)+ ,)+/h,+,-/h,=o
)2 c_it79C,,, Cui - C,,i I ,:Z /(Ri ( RRo) - C"i =C?Cui+ C=__ +/hi + - -
-_ R Rc_p "p--]'_ - tcs, c ., p iRR &
(2) - (5)
U s,
":s,: )
:zj, _,,, = ._,,,_ sg.( R,,(o- cu,)
214
This is thewell-knownsetof partial differential equations. In order to find an analytical
solution the differential equation system is linearized by splitting the main variables into an
average value (for the centric position) and an additional value for the eccentric position. This
perturbation method is applicated for eight terms.
6=
p= p,(l+_)
- XR.(I+A )
f =f.(l+E)
=
(6)(a-h)
The most important ones are g for the pressure distribution and AS, AR for the friction
factor.
Eqs. (6) are put in eqs. (2) - (5) and an exponential fommlation for the perturbation term
leeds to the cross force acting on the rotor of the labyrinth seal.
2£
F = -RRI p. ] _sin cpd_p
o
(7)
With the help of this perturbation method it is possible to calculate the forces in a first step
and, in a second step, to obtain the dynamic coefficients of the seal. Details of the calculation
method can be seen in [1].
The value of the cross force is influenced by two factors. First, by the difference of the
circumferential velocity from chamber to chamber. This leads to methods to control the swirl,
for example swirl-brakes or helically grooved seals, a well-known and often discussed
phenomenon.
Second, a greet influence of the cross forces on the friction factor is found. This can be seen
in figure 7. For the given labyrinth geometry in the test rig a great variety of results for the
cross forces is obtained by the application of several friction factors for labyrinth seals.
1,5 1-
o
o
t,,,
o
o 0,5-
o
1+
Fig.7.
0
O
O
o
O
• O
',- t'N 03 _ t/_ ¢D I_ 00 (_ O _ 04 03 "d"
Cavity
II
Cross force in dependence on different methods to calculate the friction factor
( Ap = 80000 Pa; n = 4000 _m )
[] Serkov
Childs
* Lee
• Serkov (mod.)
• Rajakumar
o Ortinger
215
For the following calculations the friction factor is obtained by computing the turbulent flow
field in the labyrinth seal with the Reynolds equations in connection with a k-_ turbulence
model. The circumferential velocity near the wall is determined and a theory for the
computation of the friction factor is applied.
r* 2 = 4c£ (8)
c/ = O. 5pc
For the shear stress on the labyrinth wall the following formulation is found:
r,, = pu2r (9)
where the shear stress velocity is dependent on the region where the calculated velocity occurs,
i.e. either in the viscous sublayer or the full turbulent layer. Details can be seen in [10l,
This theory leads to the friction factor of the seal. The Reynolds number is defined
Re = pc=H (1 O)
21,t
Results of the calculated friction factor in relation to other methods can be seen in fig. 8.
For all the methods there can be observed a tendency of decreasing friction factors with
increasing Reynolds numbers. For the calculated method the friction factor reaches an
asymptotic value, whereas in the other eases the factors are still decreasing.
0,1
0,01
0,001
0,0001
" ;g = 8 ='0 0 0 0
4- 4- 4- 4- 4- 4- 4-
LU ILl U.I 1.1.1 I1.1 ttl tl.I
° 8 8 ° ° °o . q . q q
Re
Dorfman
...... Serkov
....... Childs et al.
..... Lee el al.
-- Calculation
Fig. 8. Comparison of friction factors obtained by various methods and authors.
216
Comparison between Theory and Experiment
The cross coupled stiffness is an important input value for calculations of the rotordynamic
stability limit. For three differencies of inlet and outlet pressure maximum preswirl and
constant rotating speed the calculated results are compared to experimental results (fig. 9). The
calculated results are obtained by the bulk-flow theory presented before including the friction
factors contained in fig. 8.
The calculation exceeds slightly the experiment whereas the tendency is in good agreement.
The almost proportional increase of the calculated cross force with the inlet-outlet pressure
difference cannot be confirmed totally by experiment. The measured cross force is about linear
to eccentricity.
25
20
15
105
° -2o1S'1°t
-25
J
02 0,4
Eccentricity [rnm]
# 160000Pa/m
160000Pa/c
• 300000Pa/m
....... 300000Pa/c
• 460(0 Pa/m
460000Pa/c
Fig. 9. Comparison of calculation and experiment in dependence of the pressure difference
( n = 6000 rpm; m - measurement, c -c,aleulation )
Conclusions
Measured normal and cross forces for a fourteen cavity staggered labyrinth are presented.
The forces are obtained by the circumferential pressure distribution in each cavity when the
rotor is in an eccentric position relative to the casing. The influence of inlet swirl, pressure ratio
and rotor speed has been investigated.
The restoring force acts in direction of eccentricity and can decrease indirectly the stability
limit. The cross force is linear to eccentricity and increases with the pressure ratio. Despite the
fact that the length of the seal is large the effect of inlet swirl has a higher effect than the
rotating speed. The absolute value of circumferential velocity is not decisive for the magnitude
of the cross force which, is the highest in the first cavities.
The calculation of the labyrinth forces is very sensitive to friction factors. When the friction
factors, which are obtained by turbulent flow computation are used as input quantities in a
bulk-flow theory calculation the agreement with experimental data is fairly good.
217
References
[1] Baumgartner M.: Evaluation of Exciting Forces in Turbomachinery Induced by Flow
in labyrinth Seals. ASME Conf. on Mech. Vib. and Noise, Boston, Mass., USA, Sept.
1987, pp. 337-346.
[2] Childs D., Nelson C., Nicks C., Scharrer J., Elrod D., Hale K.: Theory versus
,Experiment for the Rotordynamic CoeffÉcients of Annular Gas Seals." Part 1 - Test
Facility and Apparatus. ASME J Tribology, Vol. 108, 1986, pp. 426-432.
[3] Childs D., Scharrer J.: An lwatsubo-based Solution for Labyrinth Seals. Comparison to
Experimental Results. J. Eng. for Gas Turbines and Power, Vol. 108, 1986, pp. 599-
604.
[4] Hauck L.: Measurement and Evaluation of Swirl-Tip Flow in Labyrinth Seals of
Conventional Turbine Stages. NASA Conference Publication 2250, 1982, pp. 242-259.
[5] Kwanka K.: Stability Behavior of a Three Journal Bearing Rotor System Excited by a
Lateral Force. ISROMAC-3, Dynamics (Editor J.H. Kim and W.J. Wang), Honolulu,
Hawaii, USA, April 1990, pp. 387-400.
[6] Leie B., Thomas H.-J.: Self-excited Rotor Whirl Due to Tip-Seal Leakage Forces.
NASA Conference Publication 2133, 1980, pp. 303-316.
[7] Nordmann R., Massmann H.: Identification of Stiffness, Damping and Mass
Coefficients for Annular Seals. IMechE, Paper C 280/84, 1984, pp. 159-166.
[8] Nordmann, R., Weiser, P.: Evaluation of Rotordynamic Coefficients of Look- Trough
Labyrinth by Means of a Three Bulk Flow Model. NASA Conference Publication 3122,
1990, pp. 147-163.
[9] Nordmann R., Dietzen, F.: A 3D Finit-Difference Method for Calculating the Dynamic
Coe[J_cients of Seals. ISROMAC-2, Honolulu, Hawaii, 1988
[10] Ortinger W.: The Calculation of Friction Factors in a l_byrinth Seal by Means of
Turbulent Flow Computation. 1SROMAC- 4, Honolulu, Hawaii, USA, April 1992, pp.
340-349.
[11] Steckel J.: Experimentelle Untersuchungen zum Durchflufl- und Radialkraftverhalten
einer Labyrinthdichtung. Dissertation TU Miinchen, 1993.
[12] Thieleke G., Stetter H.: Experimental lnvestigaaons of Exciting Forces Caused by Flow
in Labyrinth Seals. NASA Conference Publication 3122, 1990, pp. 109-134.
218


Calculation and measurement of the influence of flow parameters on rotordynamic coefficients in labyrinth seals

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