a linear fluid inertia model for improved prediction of force ......viscous fluid force unsteady...

A Linear Fluid Inertia Model for Improved Prediction of Force Coefficients in Grooved Squeeze Film Dampers and Grooved Oil Seal Rings

Modern Lubrication – Appendix to Notes 11 & 13

Luis San Andres

Adolfo Delgado

Texas A&M University © 2009

Bearing & seal dynamic reaction forcesBearing & seal dynamic reaction forces

Stiffness coefficients

Damping coefficients

Typically: Small clearances (c/R < 0.001)Re*<<1No fluid inertia forces are included

YX

CCCC

YX

KKKK

FF

BYYYX

XYXX

BYYYX

XYXX

Y

X

2* cRe

Viscous fluid force

Unsteady flow fluid inertia forceSqueeze film Reynolds

number

rotor

X

Z

Bearing coordinate system

Y

X

Fluid inertia Fluid inertia –– When is it important?When is it important?

Added mass coefficients:- comparable to journal mass (i.e. can shift system natural frequency)- larger in grooves (i.e. large clearances)

-Large clearances and/or groove depths -Long axial length-High frequencies

Grooved Oil Seals & Squeeze Film Dampers.

Dynamic reaction forces:

Stiffness coefficients Damping coefficients

YX

MMMM

YX

CCCC

YX

KKKK

FF

SYYYX

XYXX

SYYYX

XYXX

SYYYX

XYXX

Y

X

Inertia coefficients

Oil seal:

SFD-CCOcentered:

0 0 00 0 0

X XX XX XX

Y YY YY YYS S S

F K C MX X XF K C MY Y Y

from centering spring

2

12* cRe

Fluid mass inside film is just a few grams, but…. since added mass is proportional to Diameter and 1/clearance MXX >> mass of solid steel journal [1]

M fluid D L c M steel steel D2

2 L

M XX D2

3

Lc 1

tanhLD

L

D

SHORT SEALM XX 2.91 kg0.2

M fluid 2.76 10 3 kg M XX

M steel0.67

M steel 4.34 kg

M XX 42.03 kg0.5

M fluid 6.9 10 3 kg M XX

M steel3.88

M steel 10.84 kg

LONG SEAL

Fluid inertia Fluid inertia –– How large is it?How large is it?

[1] Reinhardt, F., and Lund, J. W., 1975, “The Influence of Fluid Inertia on the Dynamic Properties of Journal Bearings,”ASME J. Lubr. Technol., 97(1), pp. 154-167.

L/D=0.2 L/D=0.5

Squeeze film damperSqueeze film damper

Known issue: Poor predictions of fluid film added masses for grooved SFDs

Common configurations include grooves and recesses to supply oil and to prevent oil starvation in squeeze film lands.

In aircraft gas turbinesand compressors,

squeeze film dampers aidto attenuate rotor

vibrations and providemechanical isolation.

Feed

groove

oil inlet

seal

housing

ournal

lubricant film

shaft

nti-rotation pin

ball bearing Discharge

groove

Lubricant

film

Added mass coefficient Added mass coefficient -- Literature reviewLiterature review

Grooved SFDsSan Andrés, 1992 (Predictions)

- SFD with a shallow groove behaves at low frequencies as a single land damper

- Groove influences dynamic force coefficients

Arauz and San Andrés, 1994 (Prediction and experiments)

- Dynamic pressure levels at the groove (cg/c < 10) and film land are of the same order of magnitude- Model underestimates the radial force and overestimates the tangential (damping) force at the damper film land.

Qingchang et al., 1998 (Predictions and experiments)- Experimental results show that radial (inertial) force is underpredicted by a factor of three.

San Andres and Delgado, 2006 (experiments)Added mass coefficients are underpredicted by classical model (5 times smaller)

Predictions of the inertia coefficient correlate well experimental data while damping coefficients are underestimated (two-stage liquid seal)

Kim and Lee, 2005 (Predictions and experiments)

Current models do not properly predict both damping and added mass coefficients in grooved SFDs

Lund et al., 2003 (Predictions and experiments)Fluid inertia force coefficient is overpredicted (up to 70 %) for increasing groove volumes

Added mass coefficient Added mass coefficient -- Literature review (cont.)Literature review (cont.)

Grooved SFDs

Oil seal ringsOil seal rings

Typical oil seal multi-ring assembly

- Locked oil seal rings can induce instability in compressors.

- Seals are grooved to reduce cross-coupled stiffness and lower lock-up forces.

In centrifugal compressorsoil seals are commonly used to prevent leakage of process gas into ambient.

Shaft

Outer seal

Inner seal

Oil supply (PS+P)

Process Gas (PS)Pa

Outer seal land

Inner seal land

Anti-rotation pinSeal loading spring

Shaft

Outer seal

Inner seal

Oil supply (PS+P)

Process Gas (PS)Pa

Outer seal land

Inner seal land

Anti-rotation pinSeal loading spring

[1] Graviss, M., 2005, “The Influence of a Central Groove on Static and Dynamic Characteristics of an Annular Liquid Seal with Laminar Flow,” M.S. Thesis, Texas A&M Univ., College Station, TX.

Parallel oil seal configuration [1]

17 mm

76c

c

25 mm

136c

Oil supplySeal lengt

Journal

Parallel seal configuration (balance thrust force due to pressure drop across the seals)

Includes ‘deep’ inlet (central) groove to feed seals

Parameter identification: FSEAL= 1/2 FTest conf.

Predictions do not consider groove or fluid inertia effects (Zirkelback and San Andrés 1996)

Smooth oil sealsSmooth oil seals-- Experimental resultsExperimental results

Childs et al., (2006, 2007)

Results Force coefficients are well predicted (C,K) except added mass coefficients

Large added mass coefficients (~15 kg)

Added mass predictions using Classical model (Reinhardt & Lund –1975) (single land- i.e. not including inlet groove) (2.84 kg)

2L

cJournal

Constant pressure

Compare smooth and grooved seals of short lengthCompare smooth and grooved seals of short lengthDamping, cross-coupled stiffness & added mass

3 3

3

18 ; ; 84 2 20XX XY XX XX

D L D LC K C Mc c

2-land seal: (deep groove divides lands)

3 3

3

1; ;4 2 20XX XY XX XX

D L D LC K C Mc c

L

c

30c

5c

3 3

3

12 ; ; 24 2 20XX XY XX XX

D L D LC K C Mc c

Coeffs are ¼ of original seal

flow

Grooved oil sealsGrooved oil seals-- Literature reviewLiterature review

- Grooves should reduce force coefficients by a factor of four, i.e.

Semanate and San Andrés, (1993)

Kxy (1 land)= 4 Kxy(2 lands)Cxx (1 land)= 4 Cxx(2 lands)

-Fluid inertia effects not predicted (considered negligible)

Baheti and Kirk, (1995)

Predictive models

- Reynolds and energy equation (Finite element solution)- Grooves should effectively isolate seal lands - Cross-coupled stiffness and damping coefficients are reduced by ~60 % for grooved configurations

- Bulk flow equation model

Childs et al., (2006)Single inner land groove and multiple groove oil seal (single clearance)

Large added mass coefficients (~30 kg)-

Higher than for ‘smooth’ seal

Childs et al., (2007)

Grooved oil sealsGrooved oil seals-- Experimental resultsExperimental results

One inner land groove with groove depths (5c,10c,15c)

Inner land groove does not effectively separate seal lands

Results Force coefficients are underpredicted (grooved seal)

[1] Graviss, M., 2005, “The Influence of a Central Groove on Static and Dynamic Characteristics of an Annular Liquid Seal with Laminar Flow,” M.S. Thesis, Texas A&M Univ., College Station, TX.

17 mm

76c

c

25 mm

136c

Oil supplySeal length

Journal

Parallel oil seal configuration [1]

Inner land groove

Test oil sealTest oil seal

Predictions Experiments≠

Inner land groove should reduce crossed-coupled stiffness and direct damping coefficients by a factor of four

Groove does not effectively separate seal lands


≠Null (neglected) added mass coefficients

Large added mass coefficients, increasing with increasing groove depth

Need for better predictive models

Inlet groove not considered (null dynamic pressure)- Null added mass coefficients

Large added mass coefficients

≠At most:


Ps- Pd >0

Pd :discharge pressure

yz

feed plenum groovemid-land groove

oil supply, Ps

Streamlines in axially symmetric grooved annular cavity.

PdPd

Fluid flow predictive modelFluid flow predictive model• Bulk flow, centered operation, incompressible fluid

• Qualitative observations of laminar flow field

• Boundary Conditions

• Characteristic groove depth

Groove effective depthGroove effective depth

Test seal

CFD simulations show: streamline separating flow regions IS a physical boundary delimiting the domain for squeeze film flow due to journal radial motions.

Inner land groove close up (CFD -Pressure driven flow)

Laminar flow

Ps= supply pressurePa= ambient pressure

10c

15c

Childs, D. W., Graviss, M., and Rodriguez, L. E., 2007, “The Influence of Groove Size on the Static and Rotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribol, 129(2), 398-406.

PaPs

, ,...I II N

Linear fluid inertia modelLinear fluid inertia model

3 3 12 6P Ph h h R hx x z z t x

n+1

Oil supply

zI zIIIzII zIV

1 2 3 4

zN

nIV NIIIIII

No fluid inertia advection

In each flow region:

2

22h h

t

with temporal fluid inertiaReynolds equation

0i t

X X Y YP = P + e P + e Pe

, ,...I II N

0

0

X Y

X Y

i tx x X x Y x

i tz z X z Y z

q = q + e q + e qe

q = q + e q + e qe

Perturbation flow analysisPerturbation flow analysis

Pressure field

Flow field

Axial

Circumferential

x= R

Y

X

e

h

R cos sini t

X Yh = c + e ( ) + e ( )e

,X Ye e < < c

Film thickness

For each individual flow region

View of rotating and whirling journal

Centered operation

n+1

Oil supply

zI zIIIzII zIV

1 2 3 4

zN

nIV NIIIIII

Boundary conditionsBoundary conditions

No generation of dynamic pressure

First-order pressures and axial flow rates must be equal

Null axial flow rate(geometrical symmetry)

PaPa

Ps= supply pressurePa= ambient pressure

Laminar flowPs

Journal centered operation

;

220

221 10 0

0

( )( ) co s( )i

2 2 - + i ( ) s in ( )

( )

L

XL R2 N NXX X X X X X

LYX YX YX X

X

R f z d zf z - + CK M d x d z

C g zK MR g z d z

( ) ( ) cos( ) ( ) sin( ) X X XP z f z g z

2

*3

2

3

( ) cosh sinh 12 1 Re

6( ) cosh sinh

X f f

X g g

Rz zf z c s i iR R c

Rz zg z c sR R c

Pressure field (Pressure field (PPXX))

Zeroth order (Static Pressure) 0P a s z

First order (Dynamic Pressure)

( , , , )f g f gc c s s

Obtained from boundary conditions

2

*Re12

c

Modified local squeeze film Reynolds number

Force Coefficients

First order solution

2 in (50.8 mm)

Shaft

Oil inlet

Discharge groove

Journal

Oil inlet plenum

Inlet groove

Squeeze film land

Test Squeeze film damperTest Squeeze film damper

Clearance= 127 m

Inlet groove

Film land

Dischargegroove

c

78 c

31 c

6 mm

4 mm

25 mm

I

II

III

y

z

Flow in

Sealed-end SFD assembly cut view.

Journal diameter: 127 mm

Mas

s co

effic

ient

[kg]

Dam

ping

coe

ffici

ent [

kN.s

/m]

Inlet groove-to-clearance ratio (cI/c)

201551

20151051

10

20

50

100

200

0

150

3010

Good correlation with experimental results for

both damping and added mass coefficients

Dischargegroove

Inlet groove

Film landc

78 c

31 c

6 mm

4 mm

25 mm

I

II

III

y

z Inlet groove

Film landc

78 c

31 c

6 mm

4 mm

25 mm

I

II

III

y

z

c

78 c

31 c

6 mm

4 mm

25 mm

I

II

III

y

z

Grooved SFDGrooved SFD

DynamicDynamic

pressurepressureExperiments

z

x=R

Inlet

Test grooved oil sealTest grooved oil seal


Clearance= 86 m

Journal diameter: 117 mm

Parallel oil seals Configuration [Childs et al]

17 mm

76c

c(0-15) c

25 mm

136c


Discharge plenum

Buffer sealJournal

2 mmx=R

Damping decreasesrapidly as groove effective

depth increases (~1/c

I II c

Mas

s co

effic

ient

[kg]

Dam

ping

coe

ffici

ent [

kN.s

/m]

Inlet groove-to-clearance ratio (cI/c)

20151051

20151051

Mass coefficient lesssensitive to groove

effective depth (~1/c)20

40

60

0

50

100

200

0

150

Smooth seal (inlet groove)Smooth seal (inlet groove)

Classical model prediction

Experiments

Experiments

Best predictions

Best predictions

136c

151051

151051

Mas

s co

effic

ient

[kg]

Dam

ping

coe

ffici

ent [

kN.s

/m]

Mid-land groove-to-clearance ratio (cIII/c)

0

20

40

60

0

50

100

200

150

I II III IV c

c12cI

c7cI

c12cI

c7cI

Grooved seal (Grooved seal (mid or inner land groovemid or inner land groove))

Best correlation withexperimental results for

groove effective depthat ~ 50% of groove

physical depth

Best predictions

Best predictions

cIIIc

cIII= 16c

Experiments

cIIIccIII= 16c

Experiments

4000 6000 8000 100000

20

40

60

Rotor speed [RPM]

I II III IV c

Oil seal (Oil seal (grooved and grooved and ungroovedungrooved))

Cross-coupled stiffness

Cro

ss-c

oupl

ed s

tiffn

ess

[kN

/m]

Model effectively predicts reduction of cross-coupled stiffness due to inner land groove.

(16c groove)

Kmodel ( =11 , =7I III

c c c c Kxy(model)

(no inner land groove)Kmodel ( =11 , =

I IIIc c c c Kxy(model)

kxy -kyx

cIII= cII (exp.) kxy -kyx

cIII= cII (exp.)

cIII=cII +15cII (exp.) kxy -kyx

Finite element solutionFinite element solutionOff-centered journal operation

x= R

Y

X

e

h

R e

Reynolds eqn. with temporal fluid inertia

0 cos sini tX Yh = h + e ( ) + e ( )e

Film thickness

3 3

22

212 6

P Ph hx x z z

h R h h ht x t

22

2h ht

z

x=R

Inlet

Test grooved oil sealTest grooved oil seal


Clearance= 86 mJournal diameter= 117 mm

Parallel oil seals Configuration [Childs et al]

17 mm

76c

c(0-15) c

25 mm

136c


Discharge plenum

Buffer sealJournal

2 mm

12

34

e

e

0h

Outlet plane

z

Inlet plane0 2

Grooves

x=R

Seal operating conditionsSeal operating conditions

-1.0

-0.8

-0.5

-0.3

0.0

0.3

0.5

0.8

1.0

-1.0 -0.8 -0.5 -0.3 0.0 0.3 0.5 0.8 1.0

x

y

smooth seal

Grooved seal

Load

Smooth seal

Grooved seal

Journal center locusindicates seal

operates with oilcavitation at the

largest testeccentricities

Journal locus

Shaft speed: 10,000 rpmStatic eccentricity ratio:

0, 0.3, 0.5, 0.7

Supply pressure: 70 bar

Test dataY

X

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 0.1 0.2 0.3 0.4 0.5 0.6

Static journal eccentricity ratio (e/c)

Leak

age

[kg/

s]Seal LeakageSeal Leakage

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 0.1 0.2 0.3 0.4 0.5 0.6


Leak

age

[kg/

s]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 0.1 0.2 0.3 0.4 0.5 0.6


Leak

age

[kg/

s]

Smooth seal- Predictions

Smooth seal- Experiments

Grooved seal- Predictions

Grooved seal- ExperimentsPredicted leakage

correlates well withexperiments for

both smooth landand grooved seal

Smooth SealGrooved Seal

(cg= 16c)

(c = 7c)

10,000 rpm, 70 bar




Grooved seal- Experiments

Direct dampingDirect damping

0

100

200

300

0 0.2 0.4 0.6


Dam

ping

[kN

.s/m

]

0

100

200

300

0 0.2 0.4 0.6

Static journal eccentricity ratio (e/c )

Dam

ping

[kN

.s/m

]

0

100

200

300

0 0.2 0.4 0.6


Dam

ping

[kN

.s/m

]

0

100

200

300

0 0.2 0.4 0.6

Static journal eccentricity ratio (e/c )

Dam

ping

[kN

.s/m

]

0

0

0

0 0.2 0.4

Static journal eccentricity ratio (e/c)0 0.2 0.4 0


Model predictsaccurately reduction

in direct damping dueto inner land groove.

Smooth Seal

Grooved Seal

Smooth Seal

Grooved Seal

Cyy

Cxx

10,000 rpm, 70 bar

(cg= 16c)

(c = 7c)

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]





CrossCross--coupled stiffnesscoupled stiffness

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Rotor speed (RPM)

Stiff

ness

[MN

/m]

Model effectivelypredicts reduction

in cross-coupledstiffness due to mid-

land groove.


Kxy

Kxy

Eccentricity ratio=0.3

Eccentricity ratio=0

= 0, 0.3


70 bar

(cg= 16c)

(c = 7c)

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5 0.6


Add

ed M

ass

[kg]

Added massAdded mass

5

10

15

20

25

30

35

40

Experimental data shows relatively large added mass coefficients. Predictions correlate well with experimental results.

Added mass coefficients are larger for grooved seal

Classical theory [1] predicts ~ 1/10 of test

value

[1] Reinhardt, F., and Lund, J. W., 1975, “The Influence of Fluid Inertia on the Dynamic Properties of Journal Bearings,” ASME J. Lubr. Technol., 97(1),

pp. 154-167.

Mxx





10,000 rpm, 70 bar

Smooth Seal

Grooved Seal

(cg= 16c)

(c = 7c)

• Damping and cross-coupled stiffness decrease rapidly as the effective groove depth increases(~1/c3).Added mass coefficients less sensitive to effective depth (~1/c).

• Predicted force coefficients (K,C,M) correlate well with experimental data for a narrow range of groove effective depths

• For shallow and short mid-land groove (oil seal) predicted (K,C,M) correlate best with test data when using 50% of actual groove depth.

• In oil seals, an inner land groove does not uncouple adjacent film lands!!

Conclusions:Conclusions:

The dynamic pressure field in deep grooves may be difficult to measure for most practical excitation frequency ranges. (fluid inertia pressures are proportional to 2).

Deep grooves do generate dynamic pressures of mainly inertial nature, which lead to large added mass coefficients.

There is a specific groove depth where the added mass coefficient peaks. This groove depth value for the studied configurations is < 10c.

Force coefficients in test configurations, like SFDs and oil seals, are a function of the ancillary geometries like feeding/discharge arrangements.

Conclusions (cont):Conclusions (cont):

a linear fluid inertia model for improved prediction of force ......viscous fluid force unsteady...

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