Prediction of carry over coefficient for fluid flow through teeth on rotor labyrinth seals using computational fluid dynamics

Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

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Prediction of Carry- over Coefficient for Fluid Flow through
Teeth on Rotor Labyrinth Seals Using Computational Fluid
Dynamics
Dr.Saba Yassoub Ahmed, (Assis. Prof.)
Department of Mechanical Engineering/College ofEngineering/University of Babylon/Babylon-Iraq.
E-mail:saba_ya@yahoo.com
Dr.Basim Ajeel Abass, (Assis. Prof.)
E-mail: drbajab62@yahoo.com
Nuhad Abd-Allah Hamza, (Assis. Lect.)
Abstract
The analysis presented in this work deals with the computational fluid dynamics (CFD) based modeling for the
carry-over coefficient of flow through teeth on rotor straight through labyrinth seal. A two dimensional model
for the flow through the seal was built in FLUENT 6.3.26. A grid independence study was carried out showing
the influence of grid refinement on the fluid mass flow rate. The effect of turbulence modeling, seal geometry
and shaft rotational speed have been taken into consideration. Three CFD models to predict the carry-over
coefficient considering the effect of the parameters mentioned before have been presented. The prediction
models have been validated against experimental works conducted by Scharrer (1988) and Eser (2004). The
results obtained in the present work for the effect of different parameters, namely, (Number of teeth, Reynolds
number, rotational speed and seal geometry) for two labyrinth seals with (0.127mm) and (0.33mm) tooth
clearance have been presented. A prediction equation for the carry-over coefficient of the labyrinth seal has been
proposed through this work. Verification for some of the results obtained through this work with that published
by other workers has been carried out. The results obtained through the present work found to be in a good
agreement with the results of other workers.
Key words: Labyrinth seal, Carry-over coefficient, CFD.
1-Introduction
The main function of the labyrinth seals is to reduce the leakage flow between components of different pressure
and prevent the rotor from contacting the stator at a very high speed, because any contact between the rotor and
stator causes seal damage, deterioration of seal performance and engine failure. Labyrinth seals have proved to
be quite effective and long-lived as evidenced by their widespread use, chiefly on rotating machinery such as gas
compressors and turbines. The straight type labyrinth seal is the most used in turbomachinery because of its
greater kinetic energy carry over in comparison with other types of seals. Most previous works show that the
modeling of seals was mainly based on the bulk flow model (Child, D.W and Scharrer, J. (1986), Esser, D.,
Kazakia, J.Y., (1995), Yilmaz, D. and Esser, D., (2004), Gamal, A.M., (2007)).The main drawback of this
model, due to the simplifying assumptions, is that sometimes it fails to predict the flow of labyrinth seal. Due to
the bulk flow model limitation, its claimed by Pugachev, (2012) that the computational fluid dynamics (CFD)
method for solving Navier- Stokes equations were applied to obtain more satisfactory predictions for various seal
types at different boundary and operating conditions. The previous studies on the application of (CFD) methods
focus on modeling the fluid flow through labyrinth seals of small clearances with teeth on the stator, Saikishan
and Morrison (2009), Anand and Woo (2011) and Sunil,(2011) (2010). Vamish (2011) studied the effect of shaft
rotation on flow parameters without modeling carry-over coefficient. A model for labyrinth seal with large
clearance has been developed. In the present work a (CFD) model for the carry-over coefficient of teeth on rotor
straight labyrinth seal has been developed in order to propose a numerical correlation for the carry over
coefficient of the seal. Effects of flow parameters and seal geometry with and without rotational speed effect, for
compressible and incompressible fluid flow on carry-over coefficient of labyrinth seal have been investigated.
2. Mathematical Modeling
2-1 Labyrinth Seal Geometry and Coordinates.
The analysis presented here is applicable to straight labyrinth seal with teeth on the rotor. Figure (1) shows the
geometry and coordinates system for a single cavity with two teeth on rotor labyrinth seal and axisymmetric flow
(i.e., two dimensional flow simulation in radial and axial direction).

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W
C

Pressure outlet

Pressure Inlet

stator

H

rotor

S

R
ω
X

Fig.(1) Seal geometry and coordinates system.
W is the tooth width (mm). C is the clearance (mm). H is the tooth height (mm), S is the tooth pitch (mm)
2-2 Governing Equations.
The main governing equations used to model the problem of the present work are:
Continuity Equation:
The continuity equation for incompressible fluid flow can be written as(Chain Nan .Y (1988)):
+
+ =0
(1)

For compressible fluid flow the effect of compressibility must be considered; hence, the continuity equation
can be rewritten as (Chain Nan .Y (1988)):
(

)

+

+

=0

(2)

Momentum equations
The momentum equation for 2-D axisymmetric geometry in axial direction can be written as:(Chain
Nan .Y (1988))
(
) +
(
)–
−
=−
+
+
−
(rμ (∇. v
!"))
(3)
While the momentum equation in radial direction can be written as:(

)+

(

)−

−

=−

+

+

+

#$

−

%

$

−

(rμ (∇. v
!"))
(4)
The governing equations of fluid flow for a steadily rotating frame can be written as follows:
Conservation of mass:
&. ' = 0
!!!"
(5)
Conservation of momentum:
"
&. ( ' '
!!!" ")+ (( ∗ ' = −&* + &+,-- +.
!" ")
(6)
The fluid velocities can be transformed from the stationary frame (stator) to the rotating frame (rotor) using
the following relations:
!!!!" = ' − /
'
" !!!!"
(7)
/ =(∗ "
!!!!" !"
(8)
Where:
( = angular velocity relative to a stationary reference frame (rpm).
!"
!!!!" = the velocity viewed from the rotating frame ( relative velocity).
'
' = the velocity viewed from the stationary frame ( absolute velocity) .
"
/ = velocity due to the moving frame (m/sec).
!!!!"
" = position vector from the origin of rotating frame (rotor) to the axis of rotation.
Turbulence model
The standard k- ε turbulence model is used. The turbulence kinetic energy, k, and the specific dissipation energy,
ε, are obtained from the following transport equations [Vamish (2011)]:
0

( 1) +

2

( 1/3 ) =

2

45 +

%6

78

9

:

;

< + =: + => − ? − @A + BC

52

(9)

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0

( D) +

Where

2

( D/3 ) =

2

45 +

%6
7E

9

F

;

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<+H

F
F : (=:

+ H F => ) − H

LLLLL
Gk is the production of κ and is modeled as Gk = − /I /̀K
̀

F$
F C

+ BF

(10)

M;
2

Gb represents generation of κ due to bu
buoyancy.
:
@A represents compressibility effects on turbulence and is modeled as YM = 2 ?
.
COP
BC and BF are user-defined source terms.
defined
G: and GF are the turbulent Prandtl numbers for k and D, respectively, and have default values of 1.0 and 1.3
ault
respectively.
C1ε and C2ε are constants with default values of 1.44 and 1.92.
2.3 Solution procedure
Grid generation
Initially the seal geometry is meshed by using GAMBIT 2.3.16. A long entrance and exit regions
.
before and after the seal are considered to insure an equilibrium condition before the flow enters the cavity. The
r
mesh has been adapted with maximum pressure gradient set to 0.1, near the rotor surface and the stator wall to
,
insure that the Y+ values are near or less t
than 5 to resolve the laminar sub-layer as shown in Fig.(2).
layer

Prediction mass flow rate
(kg/sec)

Fig(2). Computational mesh.
A grid independence was obtained by observing the mass flow rate predicted for a given pressure ratio with
various number of grids. It can be shown from fig.(3) that the change in mass flow rate was less than 0.5% as the
fig.(3)
number of grids increase more than 20000
0.002
0.0018
0.0016
0.0014
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0
0

10000

20000

30000 40000
Number oF Nodes

50000

60000

70000

.
Figure (3). Accuracy of mass flow rate prediction against number of nodes.
Equations from 1to 8 are solved by TDM with residual for each property equal to (10-4) using different under
relaxation factor from 0.1 to 0.8.
Computational Method
The above model is numerically solved in order to calculate the effect of flow and geometric parameters on the
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carry over coefficient. The solution is based on the finite volume method by using the commercial (CFD) code
the
FLUENT 6.3.26 .The continuity and momentum equations have been discretized using standard κ-ε turbulence
model which has been proved to be accurate for modeling the flow through seals with and wit
without rotational
effect(Morrison and Al-Ghasem, 2007) .
Ghasem,
Carry over Coefficient
The carry over coefficient can be defined as the portion of the undissipated kinetic energy of the fluid exiting at
the seal clearance that is converted to the next cavit (Woo, 2011). The carry over coefficient, γ, and the
cavity
.
percentage of the kinetic energy carried over into the next cavity, χ, are defined by (Hodkinson, 1939) as
follows:
(11)
Q =
Where:

χ=

RS

U

UVW XYZ[

(12)

Where β is the divergence angle defined by noting the position on the downstream tooth where the radial
downstream
velocity is zero Saikishan and Morrison (2009) Jeng, (2011), and Sunil,(2011). The exact position needs to be
(2009),
accurately determined for accurate determination of (β). Hence Tecplot 360 software has been used for this
purpose after suitable magnification and limiting the range of measurement. The probe tool was used to detect
ter
and extract regions of least radial velocity which correspond to stagnation points in the cavity. Referring to
Fig.(4) the angle (β) can be calculated as fo
follows:
`

= tanR ( $)
`a

(13)

Fig.(4):Shows The Contour Plots of The Radial Velocity and Measurement of β.
The above definition of γ assumes one major recirculation zone in the cavity. This assumption is valid
recirculation
until rotor shaft speeds impart significant tangential velocities which produce a body force and a secondary
recirculation zone which lead to an invalid definition of γ. The carry over coefficient ca
cannot be less than one
according to this definition. Since it is difficult to define γ when there is no dividing streamline that separates
the flow in the cavity from the flow above tooth, the carry over coefficient (γ) is assumed to be unity when
(γ
stream line is carried out into the cavity due to centrifugal effects.
3. Results and Discussions
In the present work a number of simulations, as presented in table (1), are performed with fixed seal geometry
and varying flow conditions. The carry over coefficient is calculated for each case. The effects of different flow
parameters (Reynolds number, pressure ratio,…etc) on kinetic energy carry over are studied. Also, the effect of
mber,
shaft rotation on the carry over coefficient is studied by respective simulations. The initial cases considered are
for a single cavity (two teeth) seal. While a seal with multiple cavities will be considered later. The seal
seal
geometry has been fixed so as to consider only the effect of flow parameters. The initial analysis utilizes water to
eliminate compressibility effects and considers a stationary shaft. Results obtained are compared with that
obtained
obtained for labyrinth seal with air flows operating under the same conditions. The Reynolds number based on

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the seal clearance can be evaluated as in (Dereli and Eser ( 2004)).
2d
e
(14)
bc =
f gh
Where:
Dm = (D+2B+C) .
D = Shaft diameter (m).
B = Tooth height (mm).
C = Clearance between stator and rotor (mm).
m
= 0.001003 l. $
no pqrc
h
µ= Dynamic viscosity of fluid i
z
m
Rx
= 1.7894 × 10 l. $
no qy
h
The effect of different flow parameters affecting the carry-over coefficient of straight through labyrinth seals
working with incompressible fluid flow have been discussed as follows:
3.1. Effect of Reynolds number
A prediction model for carry over coefficient of incompressible fluid, needs a study to the relation between the
carry over coefficient and Reynolds number at different exit pressures. Fig (5) shows such study for seal
geometry case (1) in table (1).
1.14
1.12

Carry over coefficient

1.1
1.08
1.06

Pe=1 atm

1.04

Pe=2 atm

1.02

Pe=5 atm
1
0.98
0

2000

4000

6000
8000
10000
Reynolds number

12000

14000

16000

Figure (5). Carry over coefficient vs. Reynolds Number for water at different
exit pressures for case (1) in Table (1).
The effect of the Reynolds number on the carry over coefficient is effectively modeled using a power law curve
fit of the following form
•a
γ = C } ~Re + C ‚$ ~ ƒ „ $
(15)
Where C1 and C2 are possibly functions of seal geometry and other flow parameters yet to be determined. The
•a

presence of C ‚$ in the model ensures that γ becomes 1 when the Reynolds number is 0. This is essential as the
minimum value of γ is 1. For this baseline geometry (case 1 in table (1)), C1=0.9322 and C2=0.019.
3.2. Effect of seal geometry
Effect of the following main geometrical parameters on carry over coefficients has been studied as follows:
3.2.1. Effect of clearance
Seal clearance represents one of the major geometric parameter affecting the carry over coefficient with the
change of Reynolds number. Hence the constants C1 and C2 depend on the clearance as shown in Figure (6).

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1.2
C2

C1

1

C1 , C2

0.8
0.6
0.4
0.2
0
0

0.005

01
0.01

0.015

0.02

0.025

0.03

0.035

0.04

C/S
Figure (6). Variation of C1 and C2 with c/s ratio.
igure

Figure (7). Reynolds number VS. carry over coefficient for cases (1- 4) of Table (1)
It can also be seen from figure (7) that the carry over coefficient increases as the seal clearance increases for
the same Reynolds number. This is can be attributed to the higher inertial force of the j The lower viscous
jet.
force at the tip of the tooth causes smaller divergence angle of the jet. A large portion of the jet is travelling the
cavity and passing above the downstream tooth. A relatively smaller portion of the kinetic energy is dissipated in
the cavity producing a higher carry over coefficient. In order to access the effect of c/s ratio, seals with four
carrydifferent clearances are simulated for a range of Reynolds numbers with a fixed pitch corresponding to the cases
(1 to 4) presented in Table (1). The power law given by equation (15) is applied to each data series and constants
C1 and C2 are found for the best fit.
It is well known that the carry over coefficient approaches 1 as Re approaches 0 according to equation (12). In
order to satisfy this, C1 must be equal to (1) while C2 should be equal to zero at c/s = 0 (as zero clearance implies
no flow and zero Re). Considering the above observation, C1 and C2 in equation (15) are expressed in the form:
†
C1= 1-R1… ˆ
(16)
‡
and

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†

C2=R2 … ˆ
(17)
‡
The constants R1 and R2 are determined as 9.67 and 2.6277 for the RMS of relative error which equals 1.08%.
Substituting equations (16) and (17) in equation (15), leads to the following model for the carry over coefficient:
†

‰ = Š1 − 9.436565 5 9• (bc + (1 − 9.436565(Ž/•))-1/2.6277(C/S))2.6277(C/S)
‡

(18)

3.2.2.Effect of Tooth Width
The effect of (w/s) on the carry over coefficient for the cases (2, 5, 6 and 7) of table (1) can be shown in
figure (8). It can be deduced from this figure that the tooth width has a little effect on the carry over coefficient,
spatially for Reynolds number higher than 2000. It has been found that the relationship between the carry over
coefficient and the Reynolds number can be represented by power law relation as follows
γ = c1(Re+R0 )C2 .
The relation between constants C1 and C2 and tooth width (w/s) for a certain (c/s=0.015) has been studied as
shown in figure (9).
1.3
w/s=0.0075
w/s=0.1
1.25

w/s=0.25
w/s=0.5

γ

1.2

1.15

1.1

1.05

1
0

2000

4000

6000

Re 8000

10000

12000

14000

Figure(8). Relationship between carry over coefficient and Reynolds number for different tooth widths
(for c/s = 0.015).

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1

constant c1,c2

0.75

0.5

c1

c2

0.25

0

0.00

0.20

0.10

0.30

0.40

0.50
50

0.60

w/s
Figure (9) .Variation of c1, c2 with tooth width (for c/s = 0.015).
The effect of tooth width can be added to the previous model of carry over coefficient presented in equation ( 18)
as follows:
‘
γ = (1- 9.436565 … ˆ-R3f1) (Re +RO)(2.677 (C/S)+R4 f2)
m
Where:
‘

R( .“”” („/Ÿ)V ¡ ¢$ )
(

Ro= 51 − 9.436565 … ˆ − R f 9
(19)
m
Both f1 and f2 represent the functions of w/s. R3 and R4 are constants for a given c/s, whic are functions of c/s.
which
The variation of C1 and C2 with tooth width for different values of c/s can be shown in figure (11) for the cases
(1, 4,8 and 9 ) explained in Table (1).
1.20

c/s=
=0.0075,c1
c/s=
=0.0075,c2
0.80
C1, C2

c/s=
=0.015,c1
c/s=
=0.015,c2
c/s=
=0.0375,c1

0.40

c/s=
=0.0375,c2

0.00
0.00

0.
.20

0.40

0.60

0.80

1.00

W/S

Figure ( 10) . Variation of c1and c2 with tooth width for different value of c/s.
values
The effect of tooth width on the carry over coefficient can be included in equation (19) as follows:
•

‘

› • ¤ ¥

γ = (1- 9.436565 … ˆ-A(c/s)a(w/s)b)(Re + RO) 5 .“””5–9V£5œ9 5 – 9 9 (20)
m
The a, b, c and d have been calculated using SPSS software. Hence equation (20) can be re written as:
‘
γ = (1-9.436565… ˆ-0.401307(c/s) 0.2958546(w/s)0.91784897) *
(Re+RO)

m
•
›
¤
5 .“ ””5 9V—. ˜x™š˜ 5 9 ¦.§¨§©¦ª¡ 5 9 ¦.ª§¡ª$a¨¦¨ 9
–

œ

–

(19)

Ro = (1 − 9.436565(c/s) − 0.401307(c/s)0.2958546(w/s)0.91784897)

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−1
Ž
Ž
p
2.6277 + 0.2954893( )—.“š“ —˜™ ( )—.˜“™˜ š—š
•
•
•
The above modification equation has been proved to be a better equation for the model used since the relative
error reduced from 1.07% to 0.9%.
3.2.3. Effect of Shaft Rotational Speed
A shaft rotational speed may changes the flow pattern within the seal which influences the carry over
coefficient. The effect of rotational speed on the carry-over coefficient of the labyrinth seal has been studied by
considering different shaft speeds at a given flow condition and seal geometry for the cases 1, 2 and 4 of table
(1). Reynolds number is assumed to be 1000 as shown in figure (12). It can be seen from this figure that the
carry over coefficient is independent of shaft speed. The carry over coefficient seems to be constant for the range
of shaft speeds (0 -10000) rpm as shown in figure (11-a). For the range of shaft speeds (11000-20000) rpm the
fluid is subjected to higher radial pressure gradient within the cavity leading to detachment of the streamlines
that exit at the preceding tooth and being drawn into forming a secondary vortex. Hence for cases like that in
figure (11-b) the carry over coefficient cannot be defined by Hokinson’s equation and the carry over coefficient
is taken as the least possible value ( γ=1 in this case).
ŽŽ =

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Table (1). Seal geometries used for simulation
No.
Clearanc
Pitch
Tooth
of Teeth e (mm)
(mm)
Width
(mm)
2
0.03
4
0.03
2
0.06
4
0.03
2
0.09
4
0.03
2
0.15
4
0.03
2
0.06
4
0.4
2
0.06
4
1
2
0.06
4
2
2
0.03
4
1
2
0.15
4
2

Case No.

Tooth
Height
(mm)
4
4
4
4
4
4
4
4
4

Shaft
Diameter
(mm)
60
60
60
60
60
60
60
60
60

1
2
3
4
5
6
7
8
9
3.3. Validation
The CFD models developed in the present work are validated against previous experimental and theoretical
works conducted by other workers. Figure (12) shows a comparison between the results obtained by the leakage
mass flow rate of air through the straight - through labyrinth seal model with that obtained experimentally by
Scharrer (1988) , Derli and Eser (2004) . The seal geometry and the operating conditions used in validation
process are summarized in table (2). The root mean square error between the results obtained with that for
Scharrer and Eser was to be 5% and 11%, respectively.
Table (2) . Seal Geometries and Operating Conditions
Seal Geometry
Operating Conditions
Number of teeth = 5,10,15,16

Pin = 7.00E+5 & 3.08 E+5

N/m2

Radius of Shaft = 0.0756 m

Pout = 1.01 E+5

N/m2

Height of Teeth =0.03175 m

Temperature

Clearance = 0.000127 & 0.00033 m

Gas Constant 287.06

Width of teeth = 2 E-4

Rotational velocity 16000 & 20000 RPM

Pitch = 0.002175 m & 0.003175 m

Fluid Air

=300

0

K

Nm/kg .0 k

Leakage mass flow (kg/sec)

0.06
0.05
0.04
0.03

Present work
0.02

Y.Dereli and D.Eser
0.01

J.K. Scharrer

0
0

5

10

15

20

Number of teeth
Figure (12). Comparison of leakage flow rate at Pin=7.00E+5 N/m2.
Also the results obtained for the carry-over coefficient of labyrinth seal with air as a working fluid have been
compared to that obtained by Vamshi, Y. (2011), as shown in figure (13).
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1.55


1.5

Present work (C/S=0.02)
Vamshi K.,(2011), (C/S=0.02)

1.45


1.4

Vamshi K.(2011), (C/S=0.03)

1.35
1.3
1.25
1.2
1.15
0

50

100
150
Shaft Speed (m/sec)

200

250

Figure (13 ) . Effect of shaft rotation on carry over coefficient at
Re = 1000 of Air .
The carry over coefficient for the straight through labyrinth seal with a water as working fluid has been
compared with that obtained by Vamshi, Y. (2011), as shown in figure (14).


1.25

1.225
Vamshi K.,(2011), (C/S=0.02)

1.2
Vamshi K.(2011), (C/S=0.03)

1.175

1.15
0

100

200

300

400

Shaft Speed (m/sec)
Figure (14). Effect of shaft rotation on carry over coefficient at
Re = 1000 of water .
The relative error for the result in Figure (13) was found to be 2.8 %. While for figure (14) was found to be
0.4 % .
4. Conclusion
It is clear from above results that the carry over coefficient affected by the seal geometry and flow parameters
such as Reynolds number. The results obtained show that the carry over coefficient increased with increasing the
clearance to pitch ratio and decreased with increasing the tooth width to pitch ratio. Effect of shaft rotation on
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carry over coefficient has been studied and show that the carry decreased after a certain value of rotational speed
(10000 rpm) due to the secondary recirculation zone which appear at higher rotational speed prevent the fluid to
leakage were dissipation energy increased .
Acknowledgement
We would like to express our deep thanks to Saikishan, S., in Texas A&M for his suggestion to complete our
research.
Nomenclature

Symbol
A
B
C
Cε1
Cε2
¬D
Re
s
U
V
w

Latin Symbols
Definition
Clearance area
Height of tooth
Radial clearance
k-ε turbulence model constant, 1.44
Shaft diameter
Reynolds number based on clearance
Tooth pitch
Axial velocity
Radial velocity
Tooth width m

Units
m2
m
m
m
m
m/sec
m/sec
m

Greek symbols
Symbol

Definition

Units

®
β
¯

Flow coefficient
Divergence angle of jet
Percentage of kinetic energy carried over

radian

ε
γ
k

Dissipation of Turbulent Kinetic energy
Kinetic energy carry over coefficient
Turbulent Kinetic energy

m2/sec3
m2/sec3

°

Fluid density at tooth inlet

kg/m3

References
• Chain Nan,Y., ’’Numerical Axisymmetric Turbulent Flow in Combustors and Diffusers’’, NASA - CR4115, 1988.
• Esser, D. and Kazakia, J. Y., 1995, “Air Flow in Cavities of Labyrinth Seals,International Journal of
Engineering Science, 33 (15), pp. 2309-2326.
• FLUENT 6.3.26 User’s Guide, Fluent Inc., Lebanon, NH, USA.
• Gamal, A.M.,“ Leakage and Rotodynamic Effects of Pocket Damper Seals and See-Through Labyrinth
Seals,” Ph.D. dissertation. Texas A&M University, College Station, 2007.
• Hodkinson, B. “Estimation of the Leakage Through a Labyrinth Gland .” Proc. Inst. Mech. Engrs., vol.
141, pp. 283-288, 1939.
• Jeng, W. , 2011, “Analysis Of Compressible And Incompressible Flows Trough See – Trough
Labyrinth Seals” M.S., Texas A&M University , College station .
• Morrison, G.L. and Al-Ghasem, A., 2007, “Experimental and Computational Analysis of a Gas
Compressor Windback Seal,” GT2007-27986, Proceedings of ASME Turbo Expo 2007, Montreal,
Canada, May 14-17.
• Pugacheve, A., CFD Predicted Rotordynamic Coefficients For a 20 Teeth on Stator Labyrint Seal at
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