NASA/TM--2003-212110

Computation

Using CE/SE

of Pressurized

Method

Gas Bearings

Sorin Cioc, Florin Dimofte, and Theo G. Keith, Jr.

University of Toledo, Toledo, Ohio

David P. Fleming

Gleim Research Center, Cleveland, Ohio

February 2003

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NASA/TM--2003-212110

Computation

Using CE/SE

of Pressurized

Method

Sorin Cioc, Florin Dimofte, and Theo G. Keith, Jr.

University of 'Ibledo, 'Ibledo, Ohio

David P. Fleming

Glenn Research Center, Cleveland, Ot4o

Bearings

Prepared for the

2002 Ann.ual. Meeting

sponsored by the Society" of Tribologists and Lubrication Eigineers

Houston, _I-exas, May 19-23, 2002

National Aeronautics and

Spa ce Administration

Glelm Research Center

February 2003
NASA Center for Aerospace Information

71121Standard Drive

Hanover, MD 211076

Available frorn

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.zrc.nasa.gov
Computation of Pressurized Gas Bearings Using CE/SE Method

Sorin Cioc, Florin Dimofte, and Theo G. Keith, Jr.

University of Toledo

Toledo, Ohio 43606

David P. Fleming

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

The space-time conservation element and solution element (CE/SE) method is extended to

compute compressible viscous flows in pressurized thin fluid films. This numerical scheme has

previously been used successfully to solve a wide variety of compressible flow problems, including

flows with large and small discontinuities. In this paper, the method is applied to calculate the pressure

distribution in a hybrid gas journal bearing. The formulation of the problem is presented, including the

modeling of the feeding system. The numerical results obtained are compared with experimental data.

Good agreement between the computed results and the test data were obtained, and thus validate the

CE/SE method to solve such problems.

NOMENCLATURE

a

aF,bF,cF

aG,bG,cG

C

D

e

f

g

h

r,f:

L

n

Elw

ns

P

P

Po

P,

0

0

R

Radius of feeding orifice

Coefficients in Taylor series expression of functionf

Coefficients in Taylor series expression of function g

Radial clearance

Diameter of feeding pocket

Eccentricity

J_ + g/_ Vector flux term

ircumferential flux term, see Eq. (5)

Axial flux term, see Eq. (5)

Film thickness (dimensional)

h/C Non-dimensional film thickness

unit vectors in circumferential and in axial directions, respectively

Length of bearing

Adiabatic coefficient

Number of waves (for wave bearings)

Number of feeding holes

Fluid pressure

P/Po Non-dimensional pressure

Atmospheric (reference) pressure

Supply pressure

Mass flow rate through one feeding hole

Non-dimensional mass flow rate through one feeding hole (see Eq. (17))

Journal bearing radius

NASA/TM--2003-212110 1
b/

U

U

X

L

z

7

Ax

At

_W

_t

P

P

po

gO

Time

, P° (C12 Non-dimensional time

h-_ Dependent variable in governing equation

o_R Velocity in circumferential direction

t.tc0 Non-dimensional circumferential velocity

2rtp0

Circumferential coordinate

x/(2rtR) Non-dimensional circumferential coordinate

Position of maximum film thickness, measured in negative direction of axis

Position of wave peak (for wave bearings), measured in negative direction of axis Y

Axial coordinate

z/(2rtR) Non-dimensional axial coordinate

Isentropic exponent; 7 = 1.405 for air

Grid spacing in longitudinal direction

Time step

Weight parameter characterizing one form of artificial dissipation, or

e/C Eccentricity ratio

Wave amplitude (for wave bearings)

Fluid viscosity

Fluid density

P/Po Non-dimensional fluid density

Atmospheric (reference) density

Angular velocity of the joumal bearing

Subscripts and superscripts, other than shown above

()0

()i

( )x,()z,(

Value of the variable at the point 0

Supply pocket index

Partial derivative with respect to x, z, and t, respectively

Time step

INTRODUCTION

Gas lubrication, though preceded by occasional experimental work since the mid 19th century,

experienced a strong development in the "Golden Era" of gas lubrication, which started in the last years

of World War II, and ended in the first part of the 1970's (Pan, 1990). In this period much important

work was published: Fuller (1956), Ltmd (1964, 1967), Castelli and Pirvics (1968), Constantinescu

(1969). More recently, significant advancements were reported in gas film modeling including

rarefaction effects (Wu and Bogy, 2001), in numerical methods applicable to high-speed bearings (Faria

and San AndrOs, 2000), and particularly in the treatment of complex geometries, including

discontinuities (Bonneau, Huitric and Toumerie, 1993).

NASA/TM--2003-212110 2
The space-time conservation element and solution element (CE/SE) method was proposed by

Chang and To (1991). Over the past several years it has been utilized in a number of fluid flow

applications that involve shock waves, contact discontinuities, acoustic waves, vortices and chemical

reactions. One of its main features is that it can simultaneously capture small and large discontinuities

(such as sound waves and shock waves) without introducing numerical oscillations into the solution, as

shown by Chang, Wang, and Chow (1999), and Chang et al. (1998).

Compressible viscous flow in pressurized thin fluid films, with application in hybrid gas bearings,

can encounter large pressure gradients due to the feeding system or to the large peripheral velocities of

the bearing. In these conditions, the computational methods based on standard finite difference methods

or classic finite volume methods may prove inadequate, having convergence problems or inducing

oscillations into the solution. Thus, a method that is conceptually simple, is second order accurate for

the entire domain, and is able to naturally deal with large gradients and/or discontinuities in the solutions

without introducing numerical oscillations or smearing, is welcome.

ANALYSIS

The two-dimensional, transient, Reynolds equation, written for a Newtonian compressible fluid in

laminar flow is,

(_ph _.__.._ I phg ph 3 c_p + c_ ph 3 = (1)

0t 0x_ 2 12_t _x _z 12_t _z 0

The flow is considered polytropic, i.e.,

= const., (2)

pn

where the polytropic exponent n = 1 for isothermal flow, n = 1.405 for adiabatic flow, or can have other

values for general polytropic flows.

A more suitable form of the Reynolds equation for numerical formulation is obtained using a

new variable u that is the product of the non-dimensional film thickness and the pressure, i.e.,

u = h-_ (3)

In terms of u, in non-dimensional variables, the Reynolds equation can be written as

c_u +c_ f +c_g=o, (4)

0_- 0_ 0g

where the flux termsfand g are,

NASA/TM--2003-212110 3
f m

g m

B

2 48rc2_-n 1

48rc2_-n 1

(5)

All partial derivatives considered in Eqs. (4) and (5) are carried out relative to non-dimensional variables

f, 2, g. In the following, in order to simplify the expressions, the non-dimensional notation (upper bar)

will be dropped, and all variables will be implicitly considered in non-dimensional form.

Consider a triangular mesh that covers the (x, z) spatial domain. One triangle BCD and its three

neighboring elements are shown in Fig. 1. Point A is the centroid of the triangle BCD, while points E, F

and G are the centroids of the neighboring triangles BCH, CDI and BDJ, respectively. The CE/SE

1

17+-

method calculates the values of the dependent variables u,ux,u z for point A at the time step t = t 2

using the corresponding values of the same variables for the points E, F and G at the time step t = tn . In

order to calculate the three unknowns at the new time step, a system of three equations will be derived.

Consider the quadrilateral ABEC. Simultaneously integrating Eq. (4) over the surface of this

1

n+-

quadrilateral and in time, between time steps t _ and t 2 (see Fig. 2), yields

_,+1 _,+1

III 0Udtdcy+ I _ 0z]

ABEC t_' C_ l t_' A

(6)

Performing the time integration for the first term and transforming the surface integration into a contour

integration for the second term (using Green's theorem) produces

_+1

II _u 2-u" dcy+ /_._dsdt=0, ABEC t _' ABEC

(7)

where _ is the unit vector normal to the contour, oriented outwards and

? =fT+g (s)

is the vector in the (x,z) plane characterized by the Cartesian unit vectors ([,/_). Functionsfand g are

given by Eq. (5). Equation (7) implies conservation of flux in the three-dimensional space (x, z,t).

Functions u,f g are next written with linear approximations using first order Taylor expansions, i.e.,

. +(.x)o(X- Xo)+(.z)o(Z- Zo)+(.,)o(, -,o), (9)

(lO)

NASA/TM--2003-212110 4
g _-go + _,c_ux)o

0 0

(11)

Substituting Eq. (9) into Eqs. (10) and (11) yields linear expressions forfand g as functions of

(x,z,t). In Eqs. (10) and (11), coefficients ar,br,cr,ac,bc,c C are considered constant when

integrating Eq. (7), and are known as functions of Uo,(Ux)o,(Uz) o .

calculated as function of the space derivatives using Eq. (3), i.e.,

(Ut)o =-(f_)o -(gz)o =-ar(U_)o -ac(Uz)o. (12)

Equations (8-12) are then substituted into Eq. (7). Point

1

?/÷-

In Eq. (9), the time derivative can be

XA,,ZA,,t is used as the Taylor

expansion point for the expressions of u 2, f and g on the contour segments AB and CA, while point

(xe,, ze,, t ") is used as the Taylor expansion point for the expressions of u ", f and g on the contour

segments BE and EC. Thus, a first equation with three unknowns, the values u, u_,u z at the new half

time step XA,,ZA,, t 2 , is obtained. The coordinates of points A' and E' are selected in a suitable

way, as will be shown later. The equation is linear and has the general form

l l l

alu A, +b l u_ ,2 +Cl UzjA, +dlU_' +el u_ ,+ Uz e, + gl =O. (13a)

Two other similar equations are obtained using the same procedure for the conservation elements

ACFD andADGB. These equations have the general form

l l l

a2u A, +b 2 ux A'2+c2 UzxA, +d2u_' +e2 u_ r' + Uz r' +g2=O, (13b)

l l l

a3uA n+-- ' 2 + b3 ( Ux )n+-- A' 2 + C3 ( b/z )n+-- A' 2 _}_ d3u_ ' + e3 ()nf3()n Ux o' + Uz o' + g3 = 0. (13c)

The linearized system formed by Eqs. (13a), (13b), and (13c) can be solved using an iterative

1 1 1

method; note that the coefficients ai, bi, c_,i = 1, 2, 3 are functions of the unknowns uA, , _u_JA' , _UzJA' •

An alternate approach is to choose the Taylor series expansion pointA' as the center of the hexagon

BECFDG. The other three Taylor series expansion points E', F', G' can be chosen arbitrarily; however,

in order to maintain consistency, they are selected as the centers of the corresponding hexagons formed

around the neighboring triangular elements.

Adding Eqs. (13a), (13b), and (13c) yields a new equation that represents the flux conservation

over the hexagon and over one half time step. When point A' is the centroid of the hexagon BECFDG,

this equation has a simpler form given by

NASA/TM--2003-212110 5
1

a_mUA' +dluE' +el ()Ux E' + Uz E' +d2uF' +e2 ()Ux F' + hi Uz F'

d3u_' + e3 ()ux c' + s3()Uz c' + g,,m = O,

(14)

where asum is given by

asu m = AABEC + AACFD + AADCB = ABECFDC. (15)

1

17÷ m

Equation (14) has only one unknown, uA, 2, and can easily be solved explicitly. It is also

important to note that all the coefficients in Eq. (14) depend only on the geometry (coordinates of the

points) and the values of the dependent variables at the previous half time step so that an iterative

1

17÷-

method is not needed. After calculating the value u A, 2, the values of the other two dependent variables

1 1

n+-- n+--

(ux)A, 2 and (Uz)_, 2 can be calculated using any two of the Eqs. (13a), (13b), and (13c). This is also

called the a scheme.

Note that the a scheme introduces "no significant damping" (Chang, Wang, & Chow, 1999;

Chang et al., 1998), so that some form of artificial dissipation is generally necessary. The scheme can be

simplified and simultaneously stabilized by calculating the space derivatives in a different way. The

scheme thus obtained is called the a - _ - ot - 13 scheme. In this scheme, the derivatives are calculated as

weighted averages between the derivatives calculated from the governing equations, as shown above

(the a scheme), the derivatives calculated using 2-D central difference finite difference formulae (weight

parameter _) and the derivatives using 2-D side finite differencing (weight parameter 13). Parameter ot is

the power index used in the computation of the non-linear weighted average that employs 2-D side finite

differencing. The complete formulation of the way the derivatives are calculated in the a- _- ot- 13

scheme can be found in Chang et al. (1998). It is important that, for a certain value of the weighting

parameter _ (_=0.5), the a-_-ct-13 scheme eliminates the necessity of calculating the space

derivatives from the governing equations, thus the method becomes purely explicit.

The feed system, formed by a number ns of orifices with the general geometry shown in Fig. 3,

can be modeled using a generally accepted formula (Lund, 1964, 1967), that links the mass flow ratio to

the feed system geometry and the pressures at the ends of the feed system considering both effects of the

orifice restrictor and the inherent restrictor, i.e.,

7_a2_ _

l+_,dh)

m

where the non-dimensional mass flow Q is a fraction CD of the ideal mass flow,

NASA/TM--2003-212110 6
/ 7+1

"Q=CDJ ] 2[- T lq T

[/ 27 (P'lT[1 (P'lT-[ for(2--_l vl <Pi<-l.

 77 70L- 70 j ,r+,j p,

(17)

When no restricting orifice is present (i.e., an inherent restrictor), Eq. (16) becomes

0--=ah 0 (18)

The feeding system is introduced into the bearing computation through the boundary conditions.

Thus, on each feeding pocket contour, the mass flow rates calculated from the bearing equations and

from the feeding system, respectively, must be equal

(Oi)be,,_,g = (Oi)feed_,g, i = 1,2,...,us. (19)

Equation (19) represents a nonlinear set of ns equations, where the pressures in the bearing at the

feeding holes are the unknowns p_,i =l,2,...,ns. This system is solved iteratively using Newton's

method.

RESULTS AND DISCUSSION

A computer code has been developed based on the described method, using the a-e-c_- 13

scheme with e = 0.5. This code has been tested for some simple cases where experimental data were

available, considering isothermal flow (n = 1).

The first case considered was the circular gas bearing without any feeding system. The results are

shown in Fig. 4, where the same bearing with aspect ratio L/(2R)= 2.0 is examined for two bearing

numbers A = (6gcoR 2 j/(PoC2) and for two eccentricities. There are no significant differences between

/

the experimental and calculated pressure distributions, at least in the middle plane where the

experimental results are available. The relative differences between the experimental and calculated

non-dimensional loads _ =Load/(Po2LR ) are 3.5% and 4.0%; both represent improvements over the

theoretical results shown by Constantinescu (1969).

The second case considered is the wave journal bearing (Dimofte, 1995) without any feed system.

The fluid film thickness in an aligned wave bearing is given as

h-: 1+ e cos 2rt(Y + T0)+ ew cos[2rmw (_ + _)]. (20)

In the last term on the right hand side of Eq. (19), % is the wave amplitude, n Wis the number of waves

and x W is the wave origin relative to the origin of the Y axis. For the present case, the bearing diameter

NASA/TM--2003-212110 7
andbearinglengthare50 mm, the radial clearanceis 0.02mm, n W= 3, _,_ = 0.3, and the relative

minimum film thickness is h-mm-- 0.3 for all cases. The bearing number A is 3.566.

The results are compared with a finite difference based code built by Dimofte (1995). Different

bearing positions (x W) have been tested and the results (calculated load magnitude and position) are

presented in Table 1. The results show very good agreement between the two methods.

The third case considered is a pressurized gas bearing without rotation and with zero eccentricity.

Details of the bearing geometry and working conditions are presented in Table 2. Figures 5a and 5b

show the calculated and experimental pressure distributions for two values of the radial clearance,

corresponding to the subsonic flow regime (obtained when C -- 12.7_tm ), and the choked flow regime

(obtained when C--31.75 _tm ) in the feed system, in two longitudinal planes situated at the jet position

(Fig. 5a) and at half distance between jets (Fig. 5b). Pressure peaks are visible at the jet positions; also

nearly constant pressure is obtained between the supply planes. The predicted pressure peaks at the

position of jet are higher than the experimental values (this difference is more visible for subsonic inlet

flow, however it is present in both cases). This is due to the difficulty of measuring the local pressure at

the feeding orifice position. The value for the correction factor CD is 0.8 for subsonic inlet flow, and

0.85 for choked flow.

Finally, the code was applied to different configurations where experimental or computed data

were not available.

Figure 6 shows the pressure distribution (sections at the jet and at the mid-jet positions) obtained

for five supply planes for the same bearing as in the previous case for C = 12.7_un. The supply planes

are equally distanced with respect to each other and with the bearing ends. It is seen that, although the

external pressure is the same for all supply holes, the pressure that develops in the pockets (inside the

bearing) is not the same for all supply planes. The pressure distribution between two consecutive feeding

planes has an almost linear form. Furthermore, at the central supply plane, the peak pressures are not as

prominent as the peak pressures at the other supply planes. This suggests that the central supply plane

does not have an important contribution for the general pressure distribution inside the bearing.

Figure 7 shows the pressure distribution in a half bearing for the same bearing geometry as in the

previous two cases, but in a running condition (40,000 rpm, A--25.76), with a relative eccentricity

= 0.5. The value CD = 0.8 was used. The pressure distribution is very complex. It can be seen that for

many feeding holes the flow is inverted, i.e., is directed from the bearing towards the feeding system,

because the pressure inside the bearing is higher than the supply pressure.

CONCLUSIONS

The CE/SE computational method has been extended for the first time to calculate compressible

viscous flow in pressurized thin fluid films with restrictors in series. Formulation of the method was

presented along with numerical results obtained for both non-pressurized and pressurized bearings. The

results were compared with existing experimental and computed data. The results demonstrate the

ability of the method to accurately predict pressure distribution in such flows.

While most numerical methods handle space and time differencing terms in the governing

equations separately, this relatively new scheme treats them in a unified way. This results in very good

NASA/TM--2003-212110 8
accuracyeventhoughtheunknownsareconsideredlocallylinearandevenwhenarelativelycoursegrid

is used.Also, the entirestructureof the method,aswell asthe developedcode,is relatively simple.

Unlikemostnumericalschemes,nospecialtreatmentis necessaryfor discontinuities,providingthatthe

governingequationsarewrittenin strongconservativeform.However,themethodis limited to time

dependentequations.Steadystateresults,suchasthosediscussedin this work, canbeobtainedonly

afterthestabilizationof theunsteadysolutionstartingfrominitial conditions.

Future work will focus on predicting dynamiccharacteristicsof pressurizedgas bearings.

Improvingthe feedingsystemwith restrictorsin seriesmodelingwill alsobe considered,aswell as

improvingthecomputationtime.Casesincludingdiscontinuitieswill alsobeconsidered.

REFERENCES

Bonneau, D., Huitric, J., Toumerie, B., (1993), "Finite Element Analysis of Grooved Gas Thrust

Bearings and Grooved Gas Face Seals", J. of Tribology, Trans. ASME, 115, 3, pp 348-354.

Castelli, V., Pirvics, J., (1968), "Review of Numerical Methods in Gas Bearing Film Analysis", J.

of Lubrication Technology, Trans ASME, vol. 90, October, pp. 777-792.

Chang, S.C., To, W.M., (1991), "A New Numerical Framework for Solving Conservation Laws -

The Method of Space-Time Conservation Element and Solution Element", NASA TM-104495, NASA,

Cleveland, OH.

Chang, S.C., et al., (1998), "Fundamentals of CE/SE method", NASA/TM--1998-208843, PDF

file http://www.grc.nasa.gov/WWW/microbus/cese/ltxjcp2d.pdf, Cleveland, OH.

Chang, S.C., Wang, X.Y., Chow, C.Y., (1999), "The Space-Time Conservation Element and

Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for

solving Conservation Laws", J. of Comp. Phys., 156, 1, pp. 89-136.

Constantinescu, V.N., (1969), "Gas Lubrication", American Society of Mechanical Engineers, Ed.

Robert L. Wehe, New York, pp. 455-467.

Dimofte, F., (1995), "Wave Journal Bearing with Compressible Lubricant; Part I: The Wave

Bearing Concept and a Comparison to the Plain Circular Bearing", STLE Tribology Trans., Vol. 38, 1,

pp.153-160.

Faria, M.T.C, San AndrOs, L., (2000), "On the Numerical Modeling of High-Speed Hydrodynamic

Gas Bearings", Trans. ASME, vol. 122, 1, pp. 124-130.

Fuller, D.D., (1956, 1984), "Theory and Practice of Lubrication for Engineers", New York, Wiley.

Lund, J.W., (1964), "The Hydrostatic Gas Journal Bearing With Journal Rotation and Vibration",

J. of Basic Engineering, Trans. ASME, vol. 86, June, pp. 328-336.

NASA/TM--2003-212110 9
Lund, J.W., (1967), "A Theoretical Analysis of Whirl Instability and Pneumatic Hammer for a

Rigid Rotor in Pressurized Gas Journal Bearings", J. of Lubrication Technology, Trans. ASME, vol. 89,

April, pp. 328-336.

Pan, Coda H.T., (1990), "Gas Lubrication (1915-1990)", Achievements in Tribology. on the 75th

Anniversary of the ASME Research Committee on Tribology, Trib-Vol. 1, pp. 31-55.

Wu, L., Bogy, D.B., (2001), "Numerical Simulation of the Slider Air Bearing Problem of Hard

Disk Drives by Two Multidimensional Upwind Residual Distribution over Unstructured Triangular

Meshes", J. of ComputationalPhysics, 172, 2, pp. 640-657.

Table 1. - Comparison between calculated data with CE/SE method and with finite

difference method

Eccentricity

0.502

O_ W

(deg)

40

Load CE/SE

(N)

177

Load FD

(N)

177

Load angle

CE/SE (deg)

40.49

Load angle

FD (deg)

40.18

0.460 32 170 170 39.09 38.8

0.433 24 166 165 36.37 36.11

0.416 16 163 162 32.94 32.69

0.404 8 158 158 29.30 29.06

0.400 0 154 154 25.61 25.38

0.404 -8 151 151 22.08 21.86

0.416 -16 149 148 18.95 18.75

Table 2. - Pressurized gas bearing geometry and working conditions

Bearing length (mm)

Bearing diameter (mm)

Supply planes

Supply plane position (mm)

Holes/supply plane

117.5

60.4

12.7

14

Orifice diameter (ram) 0.16

Pocket diameter (mm)

Supply pressure (Pa)

Injection angle (deg)

0.9

5.514x10 _

90

NASA/TM--2003-212110 10
H,

B

C

I

Fig. 1 - Triangular mesh element and its neighbors.

E1

_

gl

/................... \

x

Fig. 2 - Conservation volume in (x, z,t) space.

NASA/TM--2003-212110 11
y

Fig. 3 - Feeding orifice geometry.

115

11

105

P/Po

O95

O9

O85

O8

- CE/SE->_=o I

.... CE/SE -> z=zmax/4 I /.."-'"._

-" CE/SE -> z=zmax/2 I .,_" \t

-- CE/SE -> z=zmax*3/4l ,,_"*" .,."°-'N _'

_h ] O Experiment-> z=O [_*S',." _,

- s,s •

18

16

14

P/Po

12

CE/SE -> z=O

.... CE/S E -> z=zmax/4

-" CE/SE -> z=zmax/2 pf"-,

-- CE/S E -> z=zmax*3/4 ,_,Z

o Experiment -> z=O ,_/ t-",

, /

50 100 150 200 250 300 350 50 100 150 200 250 300 350

Circumferential position [deg] Circumferential position [deg]

A = 0.45, Sexv = 0.4 ], _exv = 0. ]496, 0e_v = 650 A = ].569, Se_v = 0.5 ], _e_P= 0.545, 0e_v = 37.50

S o_p = 0.40, _o_p = 0.1548, 0o_p = 69 ° S o_p = 0.5 l, _o_p = 0.567, 0o_p = 38.00

Fig. 4 - Comparison between numerical and experimental results for pressure distribution

in gas journal bearing.

NASA/TM--2003-2121 l0 12
_3

o

_2

Oq

0

Position of Jet

w

CE/SE,

x Experiment,

--_CE/SE,

Experiment,

A

v

C=12.7

C=12.7

C=31.75

C=31.75

A

v

0.I 0.2 0.3

Z/L

(a) Jet position.

Mid Jet Position

0.4

_,m

_m

_m

_m

0.

_3

o

_2

_____xCE/SE,

x Experiment,

--CE/SE,

0 Experiment,

C=12.7 _m

C=12.7 _m

C=31.75 _m

C=31.75 _m

e------

0q

0 0.I 0.2 0.3 0.4 0

Z/L

(b) Mid-jet position.

Fig. 5 - Comparison between calculated and experimental pressure distributions.

NASA/TM--2003-212110 13
5

4

o)

2

1/

O0 0'.1 0'.2 0'3 0'.4 0.5

Z2L

Fig. 6 - Pressure distribution for circular bearing without eccentricity; five supply planes.

iiiiiiiiiiiiiiiiiii_ a[ eoord naie [z_ _R_]  ̧ ̧ i  ̧ ̧ ̧ ̧ i  ̧ ̧i

Fig. 7 - Pressure distribution for circular bearing with eccentricity; five supply planes.

NASA/TM--2003-212110 14
Form Approved REPORT DOCUMENTATION PAGE

OMB No. 0704-0188

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3, REPORT TYPE AND DATES COVERED

February 2003 T'echnical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Computation of Pressurized Gas Bearings Using CE/SE Method

& AUTHOR(S)

Sorin Cioc, Florin Dimofte, Theo G. Keith, Jr., and David P. Fleming

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Resem'ch Center at Lewis Field

Cleveland, Ohio 44135 - 3191

WBS-22-274-00-02

8. PERFORMING ORGANIZATION

REPORT NUMBER

E----13764

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

National Aeronautics and Space Administration

Washington, DC 20546-0(101 NASA TM------2003-212110

11. SUPPLEMENTARY NOTES

Prepared for tile 2002 Annual Meeting sponsored by the Society of Tribologists and Lubrication Engineers, Houston,

'Texas, May 19-23, 2002. Sorin Cioc, Florin Dimofte, and Theo G. Keith, Jr., University of Toledo, Department of

Mechanical Engineering, Toledo, Ohio 43606; David P. Fleming, NASA Glenn Research Center. Responsible persom

David P. Fleming, organization code 5950_ 216-433-6013.

12a, DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category: 37 Distribution: Nonstandard

Available electronicaJly at bttp://_lt:.-s._rc.nasa.aov

_l-his pnblication is available from the NASA Center for AeroSpace In*brmadon, 301-621-0390.

12b. DISTRNBUTION CODE

13. ABSTRACT (Maximum 200 words)

The space-time conservation element and solution element (CE/SE) method is extended to compute compressible viscous

flows in pressurized thin fluid films. This numerical scheme has weviously been used successfully to solve a wide

variety of compressible flow problems, including flows wilh large and small discontinuitites. In this paper, the method is

applied to calculate ihe pressure distribution in a hvbrid_ gas_ journal bearing. The formulation of the problem is presented,

including the modeling of the feeding system, the numerical results obtained m'e compared with experimental data. Good

agreement between tile computed results and tile test data were obtained, and thus validate the CE/SE method to solve

such problems.

14. SUBJECT TERMS

Bearings; Gas; Hydrostatic lubrication; Flow rate

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540-01-280-5500

15. NUMBER OF PAGES

20

16. PRICE CODE

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