Do convective inertia forces affect turbulent bearing characteristics?

Do convective

inertia forces affect turbulent

bearing characteristics? Z.S. Safar and G.S.A. Shawki*

Ever increasing rotational shaft speeds and the use of unconventional lubricants such as water and liquid metals has prompted growing interest in turbulent lubrication theory t . High surface velocities, large bearing dimensions, and low kinetic viscosities of lubricants give high values of Reynolds number and, eventually, super-laminar flow.

A p p e n d i x 1 - Analysis Notation

radial bearing clearance fluid film thickness axial length of bearing rotational journal speed load applied per unit projected bearing area fluid-f'dm pressure radius of journal circumferential velocity of journal surface U, V, W fluid velocity in x, y and z directions respectively W total load acting on bearing X, y, Z coordinate system non-dimensional quantity known as load A criterion P6 2/~/V shear stress T 8 relative bearing clearance = c/R eccentricity ratio = e/.c eddy diffusivity ¢2rn 0,,02 values of 0 at beginning and end of operative film respectively coefficient of viscosity of lubricant kinematic viscosity of lubricant (= X/p) 0 p lubricant density ca) angular speed of journal rotation C

h L N P P R U

An analytical study has been carried out at the University of Cairo on the effect of convective inertia on film pressure and load capacity of full infinitely long journal bearings. A computer program, based on the analysis outlined in Appendix 1, was developed to yield analytical results for fluid film pressure and bearing load capacity under widely varying conditions. Results were obtained for laminar and turbulent regimes. Results and discussion

=

Analytical curves obtained for the pressure distribution around the bearing (Fig 1) show that consideration of inertia effects gives increased pressure. The effect of inertia seems, however, to be mainly confined to the last third of the operative fluid film. Since inclusion of inertia effects in the analysis gives increased film pressure, it is not surprising that bearing load capacity also increases (Fig 2). Although favourable, this effect seems rather insignificant.

y.,C/-y

y*

0

*Facult.v o f Engineering, Cairo University, Cairo, Egypt

Fig I (Left) Pressure distribution in a journal bearing

700

600

500

L/D: 09 o •

Fig 2 (Below) Bearing load capacity is affected by convective inertia forces

//~

With inertia Without inertia

// ,//~

"~, I I ~I

~.

%

,,,..,oooo

i o ,'/// . y .2" "

L

. o, t-_'bY, o ,Y/Y

400

g

O 300 C E C5

I

./Lo / /

o. p Iii Y,¢°°°

,8,,

200

! !/

0 5

I,,,,1!1, , , , / I I /

o

~l// i/~I//

e----- ~

o

Considering inertia

Disregardinginertia

1111 E/Z/ lO0 Ol 0

248

58

76 114 Angular displacement O*

152

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0

I I00

I 200

1 300

I 400

I 500

1 600

Load criterion, A = P ~ 2 / k / l

0301-679X/78/1104-0248 $01.00 © 1978 IPC Business Press

This analysis assumes that the lubricant is an incompressible Newtonian fluid. Steady fully developed turbulent flow is considered. The Boussinesq's eddy diffusivity is described by Reichardt's formula using 'the law of the wall'. Under turbulent flow conditions, the Navier-Stokes equations assume the form:

-- OUI" = ~ ( v ~ -

UI" ~¢i

ui' uj')

i~x---it

1 aft

[ aXi

(1)

where Di/is the shear stress tensor. Should the law of the wall be chosen to relate turbulent shear to mean velocity gradient, we obtain the relationship:

em ( aUi aUI" Dii =X(1 +-~-) ~ + ax i )

(2)

The coefficient of eddy viscosity (-~)is a function o f the dimensionless distance from the wall reaching a constant value in the turbulent core 2. Thus: y* era= k [ y * - 6 1 * t a n h - ] (3) v 61" with k = 0.4, 61" = 10.7. The dimensionless distance from the nearest wall y* is based on the local shear stress. For the turbulent core we may put 3 : h,/ (o)c~ = 0.07. v , ,

inertia forces is used, ie considering velocity profile in the inertialess case, and substituting the velocity distribution given by Safar et al 4 into equations (5), the right hand side of this equation vanishes; we are thus only left with the integral

fh u2 H dr~ 0 to evaluate. This integral is approximately equal s to

H(flu dr/) 2 0 Thus, the dimensionless pressure equation may be put in the form: H6 ( 112 )2 (aP*)2 H4 11 ( ~p*. 41r" -~o - 1 2 ".~0 " +-J-~ (I12 - I2Io) /0 - 2) /1-~ H~ + Io I2 = fl

v

aO ]h Oy 0

p [ ~_ff_fh U Wdy+ ~z ? W ' d y ] oz o o

r/dr/

o ~ ( l + em) 0

Io

=

fI o

dr/

a(1 + e r a )

(8)

0

(5)

and c~ is a constant to be evaluated from the Reynolds boundary conditions. References

I. Salbel F_.A.and blacken N.A. Nonlaminar Behavior in

=

aWlh0

2.

Equations (5) can be made dimensionless by the following substitutions:

3.

- a f~f h +- [ Xy( l + ae m )zo

r/2 drl 0

(4)

ot a_fl_fh ~2 dy + ~ fh W U dy] = ox o oz o + IX(1 + -e-m)

(7)

o Or(1 + cm)

Using the basic assumptions of classical hydrodynamic theory, and including inertia terms, the integrated momentum equations may be written in the form:

-ai°h ax

cl

in which

11 = fl

Irul+(d) 2p

=

x = RO

y = hr/

z =~ ~

h =cH R ww

X =~q'a o p* = XiN(e)2p

U=R~

4. (6)

For the case of steady two dimensional flow, the assumption that velocity profiles are not strongly influenced by

5.

Bearings: A Critical Reviewof the Literature, Trans. ASME, Vol. 96, Series F (Jan. 1974), pp. 174-181 Eltod I-LG.and Ng C.W.A Theory of Turbulent Fluid Films and its Application to Bearings, Tran& ASML, Vol. 89, Ser. F, [July 1967), pp. 346-362 HinzeLO. Turbulence,McGraw-Hill Book Company Inc, New York, 1959 SafatZ. and Szeri A. ThermohydrodynamicLubricationin Laminar and Turbulent Regimes, Tran& ASME, Vol. 96, Series E (1974), pp. 48-57 SmalleyA.J., Vohr J.H., CastelliV. and MaehmannC. An Analyticaland ExperimentalInvestigationof Turbulent Flow in BearingFilmsincludingConvectiveFluid Inertia Forces, Tran~ ASME, VoL 96, Series F, (1974/, pp. 151-157

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