[American Institute of Aeronautics and Astronautics 37th Joint Propulsion Conference and Exhibit - Salt Lake City,UT,U.S.A. (08 July 2001 - 11 July 2001)] 37th Joint Propulsion Conference

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-34221At A A

AIAA 2001-3482

ANALYSIS OF A COMPLIANT GAS FOIL SEAL WITHTURBULENCE EFFECTS

Mohsen Salehi, Ph.D.(Member ASME/AIAA/STLE)

Hooshang Heshmat, Ph.D.(Fellow ASME/STLE)

Mohawk Innovative Technology, Inc.®Albany, New YorkTel. (518)862-4290 Fax. (518) 862-4293 e-mail:[email protected]

37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit8-1 Uuly, 2001

Salt Lake City, Utah

permission to copy or to republish, contact the copyright owner named on the first page.For AIAA-held copyright, write to AIAA Permissions Department,

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2001-3482ANALYSIS OF A COMPLIANT GAS FOIL SEAL WITH TURBULENCE EFFECTS

Mohsen Salehi, Ph.D.Member AIAA, STLE, ASME Associate Member

Hooshang Heshmat, Ph.D.Fellow ASME, STLE

Mohawk Innovative Technology, Inc. (MiTi)Albany, New York

Tel. (5.18)862-4290 Fax. (518) 862-4293 e-mail: [email protected]

ABSTRACTA method is presented to include the turbulence effects influid flow analysis of a high speed compliant gas foil seal(CFS). The method takes into account certain well-established facts concerning turbulent shear flow. Thecompressible fluid flow field is assumed to depend onlocal film thickness, surface velocity, pressure gradientand surface compliance. The non-linear effects due tocompliance of the flow boundaries, compressibility of thefluid, and coupling of the shear induced circumferentialflow and pressure driven axial flow are considered. Theturbulence effect is accounted for using non-linearcoefficients in the coupled compressible governingequations of the flow pressure and film thickness. Thecomputational method employs the successive over-relaxation (SOR) method for solving the governingequations of the flow field and fluid film thickness. Nooptimization study has been conducted for the rate ofconvergence in the numerical analysis. The relaxationfactor is found to be a key parameter in convergencesolution when the compliant foil seal is operating at highspeed, high eccentricity ration or low differential pressureacross the seal. It is found that due to non-symmetricboundary condition, the method of Column Matrix whichis used for compliant foil bearing did not result in aconvergence solution.

INTRODUCTIONCompliant foil bearings are used in a wide variety ofrotating machinery, including air cycle machines,cryogenic turbo-expanders and gas turbine engines toname a few. Their application offers performance andreliability improvements through the elimination of oiland the lubricant supply system. A compliant foil seal(CFS), similar to a compliant foil bearing (CFB), is a self-

acting hydrodynamic mechanical components that developa high pressure gas film. This generated gas filmultimately has sufficient load capacity to separate the sealtop surface from the rotating journal, thus permittingcontinuous non-contacting operation. The basicconfiguration of a CFS is shown schematically in Figure 1.The thin high pressure gas film is formed between thesurface of a rotor (or a journal) and the top foil. With theproper choice of structural compliance, the seal surfacedeformation can be tailored to deform in a manner thatenhances the seal performance. For example, thecompliant seal design can be tailored to form a variableaxial converging wedge shape for improved sealingperformance.In the previous work by Salehi et al. (1999) computationalanalysis of a CFS was addressed. The combinedhydrodynamic and structural governing equations weresolved simultaneously using finite difference method.This coupled structural and hydrodynamic analysisexcluded the stiffness contribution of the top foil but didaccount for the influence of the top foil on individuallycalculated nodal deflections due to pressure. Since the

Foil y^i-i | j j 1 1 ) 11^<

A<

Fig. 1 schematic of Compliant foil seal

Copyright© Mohawk Innovative Technology® Inc. 1American Institute of Aeronautics and Astronautics


n ' Poiseuille Flo)

Fig. 2 Couette and Poiseuille flows in a CFS

top foil is a continuous medium supported by stiffnesselements at discrete locations and the developed pressureis distributed over top foil, any localized or discrete nodedeflection is affected by the stiffness and deflection of theneighboring node points. Thus to compensate for thecoupling influence of the top foil on the film thicknessand pressure profile, each discrete node point deflectionincluded the linear stiffness contribution of the fourneighboring nodes. The analysis was primarily based ona laminar flow condition, in both axial andcircumferential directions. The current analysis takes intoaccount certain well-established facts concerningturbulent flow. The flow in a CFS is a combination of theshear flow in the direction of rotation (Couette flow) andthe pressure flow in the axial direction (Poiseuille flow).The two flow's configuration is shown in Fig. 2. Thethickness of the gas film is normally in the range of 15-100 \JLUI. The gas flow through the seal can beapproximated by a flow between two cylinders with smallspacing where the inner cylinder is rotating at high speed.Since the layer of gas is thin, an isothermal flowcondition across the gas layer can be assumed. In anisothermal flow with a pressure gradient incircumferential and axial directions, the instability arisesin one or both of the following forms:a. Centrifugal InstabilityThe controlling non-dimensional parameter is the Taylornumber defined as

R (1)

where Re=/to/i/K Taylor (1923) was able to calculatethe critical number for the instability. The minimumnumber suggested was Tc=1707. It is addressed by Cole(1965, 1976) that the value of the critical Taylor numberdepends on the clearance ratio (C0/R) and not the aspect

2001-3482ratio (L/C0). As the Taylor number increases above itscritical value, the axisymmetric Taylor vortices turn tononaxisymmetric disturbances.

b. Parallel flow instabilityThe flow condition is similar to that of a Poiseuille flow.The instability is characterized by propagating waves.The viscous force and inertial forces are stabilizing anddestabilizing forces, respectively. The controlling non-dimensional parameter is the axial Reynolds numberdefined as

(2)

where Dh is the hydraulic diameter. The parallel flowinstability normally occurs at Rep > 1200.

ANALYSISCompressible form of Reynolds equation is used todetermine the performance of a compliant foil seal. Therelevant governing equation for laminar region with thefollowing dimensionless variables

can be presented as

ae ae

(3)

(4)

The film thickness variation, due to deflection of bumpsunder hydrodynamic pressure and eccentricity can bepresented as:

h= (A) = €.cos(0 -

In a CFS, the flow is under the combination of bothcentrifugal and parallel instabilities. The objective of thestudy here is to include the turbulence effects in thegoverning equation of the fluid. By unwrapping the topfoil and considering a range of speed and differentialacross the seal, the bulk fluid flow model is similar to oneshown in Fig. 3. The above model geometry is used forsolving the numerical governing equations.

American Institute of Aeronautics and Astronautics


0,,,

Qin ~

y i i iI l II ^ . 1 1• Compression , ,

1 i 1

A ! A, ! !Y 'Y e]h.toQ..U.

Y

Expansion

Ain Y 0=360° i

X=2nR|

> Q.HU

Z-oP*PM

Fig. 3 Bulk flow directions in a CFS

Turbulence Models for Thin Film FlowsVarious approaches to turbulence effects in a thin filmhydrodynamic lubrication have been taken. The twodistinctive categories are based on the characteristics ofthe flow close to the wall or boundary (law of the wall)and the second one is the bulk flow model. The first oneemploys various turbulence theories, and describes fluidvelocity in a turbulent flow in detail, as a superposition ofthe time-averaged flow velocity and the fluctuationvelocity. The second approach attempts in linearmodeling of the flow using the bulk characteristics. Thetwo categories are briefly discussed here.Constantinescu (1962) developed one of the earliestmodels of turbulent lubrication. He followed the mixinglength theory and expressed the Reynolds stress as

(6)

with the mixing length / assumed proportional to the ycoordinate. This approach resulted in the modifiedReynolds equation for turbulent flow:

Uch(7)

2001-3482Another approach was proposed by Ng and Pan (1965) intheir linearized theory. They utilized the fact that for aturbulent shear flow which is steady and experience ashear stress TW on an impermeable wall at y=0, the meanvelocity profile in a region near the wall can be describedby the following universal correlation known as "the lawof the wall"

(9)

where U*=U/UT, y+==uTy/v, and

w^ =_f rw\ P

(10)

is known as the friction velocity. According to this theorythe Reynolds stress is presented as

-u'w'=endu

'dy (11)

where e m is known as eddy viscosity. Richardts'empirical formula for the eddy viscosity was used.Neglecting the variation of stress along the surfacecoordinates and based on the variation of the meanpressure based on simplified equations of the mean flowpressure (Szeri, 1980), they reached the followingequations for the average velocities

1u = —

V

fGx and Gz are turbulence functions which depend on thelocal Reynolds number (Uh/v) and the characteristics ofthe flow (shear flow dominated, pressure flow dominated,etc.). For strong Couette flows Constantinescu gives

——=12+0.0260Re08265

——=12+0.0198Re0741

G-

(8)

where the Txy(h/2) and 1 (11/2) are integration constantswhich are determined by imposing the followingconditions

w(A)=0(13)

By assuming that the flow is a small perturbation ofturbulent Couette flow, linearizing the flow, and



substituting the Eqs. (12) into the continuity equation, Ngand Pan (1965) reached the equation for the pressuredistribution of a lubricant in a journal bearing that isformally identical to the Eq. (7) derived byConstantinescu (steady state incompressible fluid flow).However, the resulting expressions for turbulencefunctions Gx and G7:

1 -=12+0.0136Re096

——=12+0.0043Rea9

G.

(14)

(15)

differ from the ones proposed by Constantinescu (cf. Eqs.8). The approach of Ng and Pan was later generalized byElrod and Ng (1967) in their nonlinear theory whichdoes not make an assumption of Couette dominatedturbulent flow.The bulk flow theory for turbulent lubrication wasdeveloped by Hirs (1973). Instead of analyzing thedetails of turbulent flow, he took advantage of anexperimental observation that the shear stress at a surface(whether stationary or moving) is only weakly dependenton the character of the flow (Couette vs. Poiseuille) andthat it can be related to the local Reynolds number as

0 10"Xo - —— - CONSTANTINESCU

NG AND PAN— - — HIRS

103 104 10'REYNOLDS' NUMBER

Fig. 4 Comparison of turbulent theories (Taylorand Dowson, 1974)

2001-3482follows:

(16)

where Re = h um /v is the Reynolds number based onmean (bulk) velocity um and m and n are constantsdetermined from experiments. This observation makes itpossible to derive the governing equations for turbulentflow. Although formally different, these equations can becast in the form of Eq. (7) for comparison purposes. Theresulting expressions for Gx and Gz for dominant Couetteflow are:

1 -=0.0687Re0'75

——=0.0392Rea75

(17)

with the assumption that the above values are not allowedto be greater than 12.

Turbulence Effects in Combined Couette-Poiseuille FlowFrom the models so far, the models by Ng and Pan(1965), Elrod and Ng (1967), and bulk theory of Hirs(1973) give more reliable results than the pioneering workof Constantinescu (1962). In most lubricationengineering applications, the Couette flow is dominantand all of the models discussed predict similar values forthe turbulence functions of Gx and Gz, with the differenceonly for moderate Reynolds numbers where transitionfrom laminar to turbulent flow takes place (see Fig. 4).The concern one should have about using the models istheir applicability for the compressible flow and forcondition where the Poiseuille flow is the dominant flow.Although Ng and Pan specifically state that their theoryis applicable to incompressible films, their theory hasbeen successfully applied to model compressible flows atmoderate compressibility numbers. However, theassumption of dominant Couette flow makes the theoryinapplicable to the CFS flow situation. To alleviate thisproblem, Constantinescu suggested that in cases whenpressure flow may be important, the values of Gx and Gzcalculated from the linearized theory (Eqs. 14,15) shouldbe compared with the corresponding values determinedfrom



G Gx G.0.681

(18)

2001-3482as: m = -0.25, n = 0.066. The turbulence coefficients Gxand Gz can be determined from

where

=—VpJUV

(19) HU dx.(21)

is the Reynolds number based on the local pressuregradient. The values of Gx and Gz computed based onCouette flow (Eqs. 14,15) can be used as long as they arelower than the values from Eq. (18). Otherwise, Eq.(18)should be used. In a CFS, mainly the magnitudes of therotor speed, initial clearance, rotor diameter anddifferential pressure determines which of the flowregimes, Couette or Poiseuille, is dominant. The modelswhich are only developed based on the Couette dominantflow and the turbulent functions are only based on theCouette Reynolds are not applicable in the entire range ofoperation of the CFS's. It can be presumed that themodels in which the differential pressure is accounted forin estimation of the turbulence functions provide a moreaccurate answer. These models include the nonlinearmodel of Elrod and Ng and Bulk flow theory of Hirswhich is usually used in flow analysis of turbulent flow inseals (Childs and Dressman, 1985; Nelson, 1985;Scharrer and Nelson, 1991; San Andres, 1991;Venkataraman and Palazzolo, 1997). In Elrod and Ngtheory, the turbulence functions Gx and Gz depend notonly on the local Reynolds number, but on the localpressure gradient as well, and solution must be iterative.Despite this difficulty, some researchers chose this pathto analyze incompressible and compressible turbulentflow in seals (Frene, 1992; Lucas et al., 1994,1996).In model by Hirs, the basic equations for the pressuregradient in a 2-D incompressible flow are (Hirs, 1973)

(20)/JU GX.

Compressible Turbulent Flow in a Compliant Foil SealTo account for the turbulence effects in a CFS, thecompressible Reynolds equation for turbulent conditioncan be expressed as

= 6U a* (22)

and the film thickness is represented by the followingequation

h=C y (p-Pr (23)

The term Ky (p-pr) represents the effect of bumps stiffnesson the deformation of a node. Due to continues formationof the top foil, deformation of a node is assumed to beeffected by the four neighboring nodes and these effectsare summed and averaged. The applied boundaryconditions area. p(x=0)=p(x=L) (periodic boundary condition)b. p(z==0) = ph and p(z=L)=p1, (axial boundary

conditions)By including the factors Gx and Gz, the following equationcan be used

(24)

where U is the sliding velocity, Ux and U: are averagefluid velocities normalized with respect to U, and m andn are constants, which were determined experimentally

Once the pressure field and film thickness are found inthe entire flow domain, the leakage or side flow can beestimated from the following equation



\2n\dz(25)

The reaction forces in y and x directions (0 = 0 and 0 =90, respectively- see Fig. 1) can be obtained from thefollowing equation

(26)

The non-dimensional force and the overall load capacitycan be found from the following correlations

F =- (27)

The power loss can be found from the speed and torque.The following correlations represent the non-dimensionaltoque and power loss

(29)

2001-3482where p* = p/pr, h*=h/C0, e* = e/C0, a(l represent thecompliancy, and (p is the load angle.A finite difference method (FDM) computer program waswritten that solves for the pressure field and film thicknesssimultaneously. The numerical method was acombination of successive over-relaxation (SOR) anditeration method. The number of grids circumferentiallyand axially and also the relaxation parameters weretreated as variables in order to assure convergence incases where the film thickness was very small or thepressure gradients were large. Convergence was assumedto be reached when the relative deviation for twosuccessive steps of the solution for the pressure and filmthickness was less than 5%. Depending on themagnitudes of the parameters such as speed, initialclearance, stiffness and pressure gradient across the seal,the number of iteration varied in the range of 1000-3000.With a grid network shown in Fig. 5, the finite differenceequations was solved similar to the procedure describedin detail in Salehi et al (1999).Computation of the coefficients are based on thefollowing Reynolds numbers

Rec =hRcop (32)

where

RdO

NUMERICAL PROCEDURE AND METHOD OFSOLUTIONThe governing equations of the pressure and filmthickness are solved simultaneously in order to obtain thepressure and film thickness for each computational grid.Prior to solving the equations, in order to increase theefficiency of computation, the equations are non-dimensionalized. The following two non-dimensionlessequations are proposed for the pressure and filmthickness

(31)

The non-dimensional forms of the parameters canrepresented as

1/2

M<£>2*£>2) ' Re*=*e/(*/OW(34)

where Rer*=Co3p/Rjiv.The selected values for the Gx and Gz from the equationsis based on the minimum value calculated. This can be putin the following form

(36)

G: = Min[G.(Re),Gp(Rep)] (35)



2001-3482here , the turbulent coefficients are calculated based onthe pressure gradient and speed . The sample plots of theGx and Gz are shown in Figs. 6 and 7. The plots arescaled for 0-1.The numerical solution starts with identification of theknown initial values such as speed, dimensions of theseal, initial clearance, compliance, etc. These values arethen used in finite difference equations of the pressureand film thickness in order to find the pressure and filmthickness field. Based on these values, the pressuregradients, Reynolds number and finally the Turbulencefactors are obtained. These factors and the initial valuesare employed in order to find new pressure and filmthickness field. The above iterations continues until aconvergency in the magnitude of the coefficients or thepressure field is obtained. The flow chart for theprogram is shown in Fig. 8.

Gas Inertia EffectsWhen a relatively low viscosity fluid such as air is used,the inertial pressure drop at sharp contraction or suddenchange of cross section area should be accounted for. Ina CFS, normally the sharp edges are eliminated and thetop foil and the rotor sleeve are very smooth, thereforethe effect of inertia is minimal. In short length CFS's orwhere multi section top foil is used, the effect should beconsidered. The inertia effect can be estimated bycalculating the velocity and considering a loss coefficientbased on the geometry and velocity. The generalequation for the velocity in the film can be expressed asfollowing

Z=L, I=M

"tr:"6j

Z=0,I=l,J=l Z=0,I=1,J=N

Fig. 5 Grid network for finite difference solution

Re =<arh/v

Fig. 6 Turbulent coefficients variation with CouetteReynolds

^-\J (37)

and the pressure drop can be calculated through thefollowing expression

Rco . (38)

where fL is the loss coefficient. The above correlationshould be used when the scalar multiplication of thevelocity vector and vector normal to the surface is greaterthen zero. In this work, the effect of the inertial is notconsidered.

i.o0.9 -

0.8 -

0.7 -

0.6 -

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -

0.0 -

Fig. 7 Turbulent coefficient variation with PoiseuilleReynolds



2001-3482

Init ial parameters _ ___....._.. ....(G-l ) Table 1 Input parameters used for the base case

_, Finite diff. Eqsofmottoi

y<p, &",)-.-

t Turbulent: coefficients

,————————-—"——I———, Convergencej Couette and Poiseuille Reynolds I Criteria met (?)

i >T NO jCalculate:

j Leakagei Power lossj Load capacityi Top foil new form

Fig. 8 Numerical solution flow chart

RESULTS AND DISCUSSIONIn this section selected results of numerical analysis for acompliant foil seal with initial parameters shown in Table1 is presented. The values in Table 1 are chosen as abase line for the performance evaluation. For parametricstudy, each specific parameter is varied and its effect isshown in the corresponding plots. Due to normaloperation of the compliant foil seal at high speeds, andpresence of a thin layer of gas, heat is generated in thefilm, however, the effect of the temperature rise on theviscosity of the fluid is not accounted for in this study.The presentation of results start with Fig. 9 where asample of 3-D pressure profile in a CFS is shown. Thehigh pressure and low pressure of the seal are kept at 448and 102 kPa, respectively (65, and 15 psi). As shownthere, the combination of the axial pressure andhydrodynamic pressure can be seen. The hydrodynamicpressure contributes in altering the axial pressure gradientand load capacity. As can be seen, due to eccentricity, atthe end section of the seal the pressure falls below theambient pressure.The variation of non-dimensionalized mass flow vs thedifferential pressure is shown in Fig. 10. The non-dimensional mass is represented by the followingEquation

r=- mju(39)

For this analysis, the rotational speed of N = 40,000 rpmis constant. This parameter does not include the speedparameter and as it will be seen later, the speed effect is

Din.

8

Lin.

1

N(rpm)

40000

Cmil.

2

€

.5

a

.2

AP(psi)

50

A

21

Fig. 9 Sample pressure profile in a CFS

10

i 4

2

Poiseuille Flow Dominant

Couette FIowDomi

20 100 12040 60 80

AP(psi)Fig. 10 No-ndimensional mass flow rate with Ap

minimal on this parameter (Fig. 12). A semi-linearbehavior can be seen from the variation of the non-dimensional mass flow and the differential pressure. Asthe analysis predicts, the Couette flow is dominant up toabout 75 Psi differential pressure after which thePoiseuille flow will be dominant. However, at this stagethe choking and shock phenomena is not considered andthe future studies should address the separation andchoking of the flow through the thin film. However, thevelocity gradient at the surface for the high pressure (Eqs.20 and 21) indicates of high shear stress.

8American Institute of Aeronautics and Astronautics


2001-3482

1£u

1"I

0.6

0.0

Op (Poiscuillc flow)Gz (Couette flow)Gx (Couetlc flow)

20 40 60 80 100 120

Fig. 11 Variation of turbulence coefficients with Ap

The average estimated values of the turbulent coefficientsare shown in Fig. 11. It is clear that the Poiseuillecoefficient at the high pressure drops and it is theminimum of all turbulent coefficients.Variation of the non-dimensional mass flow with thespeed is shown in Fig. 12. In the most flow domain theCouette flow is dominant. The analysis estimates aminimum variation of the mass flow rate with the speed.This is due to the fact that the circumferential pressuregradients are mainly affected by the increase in therotational speed. Since the governing equations ofmotions are non -linear and the pressure at the boundariesshould be satisfied, the minimum effect on the axialpressure gradient and boundaries is seen. The numericalanalysis also depicts similar behaviors in the variation ofthe pressures. Variation of the estimated average valuesfor the turbulent coefficients for this condition is shownin Fig. 13.The ratio of turbulence to laminar power loss in the sealis shown in Fig. 14. The prediction are shown for theCFS at a constant speed where the differential pressureacross the seal is changed. For more accurate predictionof the power loss more studies on the variation of theviscosity with the pressure and temperature should beconducted.

CONCLUSIONSIn this study the previous analysis of the compliant foilseal was expanded to include the effect of turbulence.

g9I

N * 40000 r]U - 8 inL - I in

In moct domain the Couette flow is dominant

10000 20000 30000 40000 50000 60000 70000 80000 90000Rotor Speed (rpm)

Fig. 12 Variation of non-dimensional mass with therotor speed

121£

3 1-0-£

1

•a 0.6-j

- Gp (Poiseuille flow)- Gz (Coeutle flow)- Gx (Couette flow)

10000 20000 30000 40000 50000 60000 70000 80000 90000

Rotor Speed (rpm)

Fig. 13 Variation of estimated G coefficient withrotor speed

N = 40000 rpmD = 8 inL = l i n

0 20 40 60 80

AP (Psi)

Fig. 14 Turbulent to laminar power loss

100 120



Some facts on the turbulence analysis of the flow in thinfilms are discussed and some models were presented.The current analysis uses a semi-linear model where theturbulence effects are included as coefficients in thegoverning equations of pressure and film thickness. Thecompressible Reynolds equation was modified for thispurpose. The Turbulence coefficients are function of twodifferent Reynolds number and the pressure gradients inthe axial and circumferential directions. The twoReynolds number are associate with the circumferentialflow (Couette flow) and the axial flow (Poiseuille flow).It is found that depending on the operational parametersone of the flow modes (Couette or Poiseuille) can havethe dominant effect. The variation of the turbulenceinfluence coefficients with speed and differential pressureis also shown. With defining a non-dimensional massflow, it was found that its variation is mainly dominatedwith the differential pressure rather than the speed of therotor. This indicates that the mass flow rate should bemore dominated by the differential pressure across theseal (Poiseuille effects) rather than the Coeuette effects.The turbulent to laminar power loss vs differentialpressure for a preselected seal geometry is also shown.

ACKNOWLEDGMENTThe authors would like to thank NASA, specially Dr.Bruce Steinetz and Margaret Proctor for their support ofthe research study. Thanks are reserved for MiTi®'s staffspecially Dr. Piotr Hryniewicz for his help on thediscussion of turbulent models and literature search.

NOMENCLATUREC0 initial clearanceCFS compliant foil sealD j ournal diameter (2 * R)Dh hydraulic diametere eccentricityfL loss coefficientGx, Gz turbulence functionsh film thicknessh* non-dimensional film thicknessi unit linear vector in x directionj unit linear vector in z direction1 mixing lengthL lengthN rotational speed, (rpm)p pressurep' pressure fluctuationp* non-dimensional pressureR journal radius

TTau, v,w

2001-3482Re local Reynolds number for the shear flow(Uh/v)Rep local Reynolds number for the pressure flow

torqueTaylor number (C0/R)*Re2

fluid velocity components in the x, y, and zdirections

u', v', w' fluctuation velocity componentsmean (bulk) velocity in the x and z directionsmean velocity in x and z directionsjournal linear velocity (Rco)non-dimensional velocity (Eq. 9)axial velocity, widthcoordinate along the perimeter (x=R 0)coordinate across the filmnon-dimensional coordinate yaxial coordinatenon-dimensional axial coordinate (z/R)

ux,uzUx,UzU

wX

yy+zz*Symbolsa04>a>

complianceangular coordinateload angleTiN/30, angular velocity

(i kinematic viscosityv dynamic viscositya bump foil complianceem eddy viscosityA Bearing number, compressibility numberF non-dimensional mass flow rate (EQ. 39)p densityT shear stresse eccentricity ratio (e/C)Sub/Super scriptsw wallr reference

REFERENCESChilds, D. W., Dressman J. B., 1985, "Convergent-Tapered Annular Seals: Analysis and Testing forRotordynamics Coefficients", Trans ASME, J. ofTribology, Vol. 107, pp. 307-317.

Cole, J.A., 1965, "Experiments on Taylor Vorticesbetween eccentric Rotating Cylinders", in Proceedings ofthe second Australian Conf. On Hydraulics and FluidMechanics, pp. B313-320.

Cole, J.A., 1976, "Taylor-Vortex Instability and AnnulusLength Effects", J. of Fluid Mech., Vol. 75., pp. 1-15.



Constantinescu, V. N., 1962, "Analysis of BearingsOperating in Turbulent Regime", Trans. ASME, J. ofBasic Engineering., Vol. 84, p. 139.

Elrod, H.G., Ng, C. W., 1967, "A Theory for TurbulentFluid Films and its Application to Bearings", Trans.ASME, J. of Tribology, Vol. 89, pp. 346-362.

Frene J., 1992, "Analysis for Incompressible Flow inAnnular Pressure Seals", Trans ASME, J. of Tribology,Vol. 114, pp. 431-438.

Hirs G. G., 1973, "A Bulk-Flow Theory for Turbulencein Lubricant Films", Trans. ASME, J. of LubricationTechnology, Vol 95, pp. 137-146.

Lucas V., Danaila, S., Bonneau O., Frene J., 1994,"Roughness Influence on Turbulent Flow ThroughAnnular Seals", ASME, J. of Tribology, Vol. 116, pp.321-329.

Lucas V., Bonneau O., Frene J., 1996, "RoughnessInfluence on Turbulent Flow Through Annular SealsIncluding Inertia Effects", Trans ASME, J. of Tribology,Vol. 118, pp. 175-182.

Nelson C. C., 1985, "Rotordynamics Coefficients forCompressible Flow in Tapered Annular Seals", TransASME, J. of Tribology, Vol. 107, pp. 318-325.

Ng. C. W., Pan, C.H., 1965, "A Linearized TurbulentLubrication Theory", Trans. ASME, J. of BasicEngineering., Vol 87, pp. 675-682.

Salehi, M., Heshmat H., Walton, J. F., Cruzen S., 1999,"The Application of Foil Seals to a Gas Turbine Engine",AIAA paper 99-2821, 35th AIAA/ASME/SAE/ASEEjoint Propulsion Conference& Exhibit, June 20-24, LosAngeles, CA.

San Andres L.A., 1991, "Analysis of Variable FluidProperties, Turbulent Annular Seals", Trans ASME, J. ofTribology, Vol. 113, pp. 694-702.

Scharrer, J. K., Nelson, C. C., 1991, "RotordynamicCoefficients for Partially Tapered Seals. Trans ASME, J.of Triboiogy, Vol. 113, pp. 53-57.

Szeri, A. Z., 1980, "Tribology", Hemisphere Publishingcorporation. Washington DC.

2001-3482Taylor, G.I., 1923, "Stability of a Viscous LiquidContained Between Two Rotating Cylinders", Philos.Trans. R. Soc. London Ser. A., Vol. 223, p. 289-343.

Taylor, C. M., Dowson, D., 1974, "Turbulent LubricationTheory — Application to Design", Trans. ASME, J. ofLubrication Technology, Vol 96, pp. 36-47.

Venkataraman B., Palazzolo A. B., 1997,"Thermodynamic Analysis of Eccentric Annular SealsUsing Cubic Spline Interpolations", TribologyTransactions, Vol. 40, pp. 183-194.


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[American Institute of Aeronautics and Astronautics 37th Joint Propulsion Conference and Exhibit - Salt Lake City,UT,U.S.A. (08 July 2001 - 11 July 2001)] 37th Joint Propulsion Conference