NASA Technical Memorandum 88845

Thermohydrodynamic Analysis for Laminar Lubricating Films

Harold G. Elrod 14 Cromwell Court Old Saybrook, Connecticut

and

USAAVSCOM Technical Report 86-C-33

David E. Brewe

U. S.- ArH?i?A~&.ition Research and Technotogy Activity-A VSCQM Lewis Rt?sefZd# Ceffter -: _.

Cleveland, Ohio

- _ Propulsion Directorate -_

~

iNA5A-TM- 8384 ‘E) l H E R ff OH YDPOD Y NAMIC ANALYSIS N87- 1 1 1 24 FCB L A M I l i A E i L U E K I C A T I H G FILMS ( N A S A ) 1 4 p

CSCL 201 y 1 I U n c l a s A .: G3/34 U3865

Prepared for the Leeds-Lyon Symposium on Tribology Leeds, England, September 8-12, 1986

https://ntrs.nasa.gov/search.jsp?R=19870001691 2019-04-03T18:56:55+00:00Z

THEWOHYDRODYNAUIC ANALYSIS FOR LAUlNAR LUBRICATING PILUS

Harold G. Elrod 14 Cromwell Court

Old Saybrook, Connecticut 06475

and

David E. Brewe Propulsion Directorate

U.S. Army Aviation Research and Technology Activity - AVSCOn Lewis Research Center

Cleveland, Ohio 44135 USA

1. INTRODUCTION

The purpose of this paper is to present a new method for incorporating thermal effects into the calculated performance of laminar lubricat- ing films. There is enormous interest in the inclusion of such effects, as the recent reviews of Khonsari (1.2) attest. The reason for this

m interest is well founded, since the viscosity- = temperature dependence of typical lubricants is A such that the viscosity can vary many fold

across and along a bearing film, with attendant effects on load capacity.

remove the need here for a survey of prior lit- erature, and reference will be made principally to those works used for comparative purposes. Suffice to say that earlier theoretical contri- butions on the subject of thermohydrodynmic lubrication divide themselves roughly into two categories. In the first category are those which embody a full transverse (cross-film) treatment of the energy equation using finite- difference or finite-element methods, and in the second category are those which incorporate rather drastic approximations to the transverse phenomena, usually representing the local film temperature distribution by a single value. Both approaches certainly possess merit. the first approach obtains accuracy at the expense of computational speed, and the second obtains speed at the expense of accuracy.

We shall show here that if just two temper- atures, chosen at "Lobatto points", are used to characterize the transverse temperature distri- bution in a laminar lubricating film, the effects of that distribution can be surprisingly well predicted. The calculations we have so-far performed have been directed solely towards demonstrating this fact, and only Ndi- mentary numerical methods have been used in the plane of the film. computation times can yet be reported. believe the technique will prove to be quite suitable for practical calculations.

In the present analysis, fluid properties are taken as constant and uniform, except for the viscosity. laminar, with negligible inertia effects. The fluidity (reciprocal viscosity) is represented by a polynomial in terms of position across the film, with coefficients related to the local film temperature distribution. procedure permits a closed-form expression for the local lineal mass flux, albeit there is some difference between a fluidity profile which would everywhere correspond to the temperature profile and one which is thus approximated. The Lobatto-point temperatures, or mathematical

ln N

The up to date, extensive reviews in (1.2)

But

Accordingly, no meaningful But we

The flow is presumed to be

Use of this

equivalents, appear in two simultaneous partial differential equations obtained from the basic energy equation by a Galerkin procedure.

Implementation of the present approach has involved considerable tedious algebra, which, however, once done, causes no further embarrass- ment. The procedure should conveniently couple with cavitation algorithms, and preliminary testing indicates that no special handling is required to cope with moderate recirculation at film entry. Uoreover, it can accomnodate to some extent the temperature streaking from hot- oil carryover. We therefore expect to be able to exploit its use in a number of interesting directions.

2 .

CP

h

i

j

k

m

Pi

-

P

T

t

U

V

Vi

W

wi

X

Y

z

NOUENCLATURE

specific heat at constant pressure, J/kg-K

film thickness, m

Cartesian tensor index

Cartesian tensor index

thermal conductivity, J/m2-(K/m)

lineal mass flux, kg/m-s

Legendre polynomial, ith order (PO = 1; P1 = c; P2 = (3C2 - 1)/2 pressure, Pa

temperature, K

time, s

x-wise velocity, m/s

y-wise velocity, m/s

i-th component of fluid velocity vector, m/s

z-wise (cross-gap) velocity, m/s

Lobatto weight function for i-th quadrature position, Ci

lateral coordinate in direction of surface motion, m

lateral coordinate transverse to surface motion, m

coordinate perpendicular to gap midsurface, m

fluidity functions (see Table I)

fluidity functions (see Table I)

fluidity functions (see Table I)

fluidity functions (see Table I)

dimensionless coordinate transverse to film, 2z/h

fluid viscosity. N-s/m2

thermal diffusivity. m2/s

fluidity functions (see Table I)

fluidity (reciprocal viscosity), mlN-s

fluid density, kg/m3

fluidity functions (see Table I)

dissipation function, S/m3-s

fluidity functions (see Table I)

3 . BASIC EQUATIONS

In the absence of gravity, the momentum equation for a Newtonian fluid without dilational vis- cosity is:

where repeated subscripts imply sumnation. And the corresponding energy equation is:

where:

L a aT Dt = at + 'j d.j

is the time derivative following the fluid (Lagrangian derivative) and:

13.031

is the dissipation function.

momentum and heat is usually much less than transverse. Furthermore, inertia and pressure- energy effects are frequently negligible, and the transverse variation of pressure is small. Therefore, we adopt the following equations for laminar lubricating films.

In lubricating films, lateral diffusion of

and :

13.051

In these equations we shall treat the fluid vis- cosity as temperature dependent, and treat other fluid properties as constant.

To these equations must be added the mass continuity equation for an incompressible fluid. Thus:

+ + v . v = o 13.071

The coordinate system used with these equat- ions is defined in Fig. 1. For convenience, a reference surface is taken midway between the film walls. A local coordinate system is sub- stituted for a fixed Cartesian system, with the local x-y plane tangent to this reference surface. The film walls are rigid, but may be moving.

The Galerkin-style analysis used here involves the expansion of the temperature in a truncated series of Legendre polynomials. Satisfaction is required of as many moments of the energy equation as there are unknowns in this series. The ensuing partial differential equations for the Legendre components are then solved. In the present treat- ment, only two unknown components are used. And for these it is feasible to carry out explicit integration, as follows:

a at JT dz + /uT dz + $JT dz

13.091

All of the above integrals are taken from the "bottom" to the "top" of the film. The sub- scripts and -2 are used to denote the upper and lower walls, respectively. The coordinate "z" is measured from the midplane. Continuity has been used to convert transverse velocity constructs to lateral constructs, wherever possible.

4 . LOBATTO POINTS; DISTRIBUTION FORMULAS

An expression for the lineal mass flux can be developed directly from Eq. [3.051. (See, for example, Dowson and Hudson (3)). But the con- vective terms in Eq. 13.061 involve integrals of the velocity-temperature product, and so require detailed knowledge of the respective distributions. Such information for the veloc- ity is already available. For the temperature. we must develop our own expression.

to the cup-mean energy flow in the x-direction. Let the temperatures on the two walls be known, and the velocity be available where needed. If two sampling positions for the temperature -- and only two -- are to be allowed for estimating this integral. where should these positions be chosen?

Consider the integral luT dz which relates

Without knowledge of end-point values,

2

1

Location, Ci

Weight, Subscript wi

1/6 1 l& I 5/6 1 1 -1/&

1 516 -1 116 -2

l1 C4 dC = 0.4 (exact = 2/5) - c-

,--

The temerature distribution which passes through the Lobatto-point values is most easily expressed in terms of an expansion in Legendre polynomials. Thus, if we write:

3 T(C) = ykPk(O

k=O 14.011

then the Legendre coefficients are easily evaluated by integration.

Or :

The linear set of equations in 14.031 can be solved for the modes of description of the temperature distri- bution in the lubricating film. The Ti give us detail, and the Ti-give us overall proper- ties. In particular, To is the space-mean temperature at the point (x,y) and is the first moment.

Ti. providing us with two

For N = 2:

T2 + T-g - (T1 +

T3 = IT2 - T-2 - e (T1 - T-l) 4 14.071 ti - In these expressions, the wall temperatures T2 and T-2 are considered as known for purposes of the film calculation. It is then useful to note that:

T + T - T =- - - 14.081 2 2 TO

14.091

An expression for the fluidity might be developed a number of ways. Here we collate the fluidity with the temperatures at the Lobatto points; that is, at the walls and at two inte- rior points. The Legendre expansion for the fluidity is then developed in a manner com- pletely analogous to that for the temperature. For example,

yo = + 5€-1 + 5t1 + t2)/12 14.101

and the fluidity distribution is:

3

k=O 5 = c ^ikPk(C) 14.111

5. VELOCITY EXPRESSIONS; MASS FLUX

A double integration of Eq. 13.051 (with E ii 11n) gives the tanRentia1 velocity vector. Thus:

and :

15.031

In view of the Legendre expansion for t , 15.021 can be alternatively written as:

Now to obtain the lineal mass flux, the velocity expression 15.011 is integrated again across the passage, with the result: +

+ + h 2 - + h p = ( V + V ) - - - E A - - ' i ( T + 2 ? ) - 2 - 2 2 3 1 2 3 0 5 2 T1 = IT2 - T-2 + fi (T1 - T-l)l/ 4 14.051

15.051

3

Simplification gives the following more recog- nizable expression:

Here the fluidity parameters ri are the vehi- cle for the temperature-flow interaction. The parameter represents asymnetry of the fluidity distribution. It is interesting to note that for symmetric temperature (and fluid- ity) distributions. the & effect of temper- ature on the mass flow is through the arithmetic average of the fluidities at the Lobatto points. This result is a special case of the following formula applicable for any synunetric cross-film temperature distribution:

For such cases, Eq. 15.071 justifies the effec- tive viscosity concept, and shows the relative importance of the temperatures near the walls.

expression 15.061 leads to a generalized Reynolds equation, the divergence of the mass flux involving spatial derivatives of pressure. Thus:

Mass continuity applied to the mass flux

15.081 P

The first moment of the energy equation involves the velocity, w. With a little care, this velocity can be found via mass con- tinuity. We have:

aw + + 15.091 az

Transforming coordinates from (x.y.2) to (x,y,C) we find:

(Here the velocity vector is understood to con- sist only of (u,v).) Substituting this result into 15.091 and integrating, we get:

f5.111

Finally, integrating by parts, we obtain:

The expressions for the tangential and cross velocities will be essential for evaluating the contributions to convective heat transfer.

6. THE TEMPERATURE EQUATIONS

With the aid of Legendre series for the temper- ature and fluidity, we are now in a position to evaluate the integrals appearing in the zeroth and first moment of the energy equation; namely, Eqs. 13.081 and 13.091. Implementation is straightforward, but tedious. All requisite coefficients are given in Table I.

Equation 13.081 becomes:

+ - Co 16.011 pcP

where Yo" stands for the collection of terms:

3 $ 5 c s k + f +$J 16.021

k=O k=O

The temperature YO appearing here is the (-space mean temperature, and not the mixed- cup temperature.

The integral of the dissipation function is:

+ + 2y0A2 + Yl (2A B)

Equation 13.091 becomes:

where "C1" stands for the terms:

The moment of the dissipation function is:

4

Equations 16.011 and 16.041 are two simul- taneous partial-differential equations in the two variables, To and TI. Where they appear, T2 and y3 can be eliminated via Eqs. L4.081 and 14.091 in favor of these dependent variables. Coupled with the generalized Reynolds Eq. 15.071, they provide our approximate thermohydrodynamic treatment for laminar films.

As mentioned previously, our numerical tech- niques for dealing with these differential equa- tions have so far been extremely simple. The steady-state solutions to be presented in the next section were found by solving the foregoing equations in their transient form, and no study has been made of the possibilities for economy in the fluidity evaluations. Prior to this work, enough success has been obtained by others with analyses which neglect entirely any cross- film viscosity variations so that it seems unlikely that it will prove necessary to update the terms in Table I at every step of a solution.

-

7. RESULTS FOR THE INFINITELY-WIDE SLIDER BEARING

In 1963, Dowson and Hudson (3) performed some finite-difference calculations on the infinitely- wide flat slider bearing, including variable- property effects. They employed 100 increments along the length of the bearing, and a minimum of 20 increments transversely. These investiga- tors were concerned with the relative effects of density and viscosity variations upon load cap- acity, with the effects of solid-fluid thermal interaction, and with the possibilities for load support from parallel surfaces. Their findings serve as basis for our assumption of a constant- density fluid, and two special cases of their exploratory calculations will be used here for direct numerical comparisons.

Hudson were as follows (SI units):

Density: 1.7577~10~ kg/m3 Thermal diffusivity: 7.306~10-8 m2/s Viscosity: q = 0.13885*exp-0.045(T-311.11)) Pas Lubricant entrance temperature: 311.11 K

Wall temperatures: Uniformly at 311.11 K Runner velocity: 31.946 m/s Bearing length: 0.18288 m Uinimum gap: 0.00009144 m

In the first case for comparison, the film thickness ratio is 2/1. Figures 2 and 3 show the velocity and temperature profiles obtained by us at the entrance, at 0.65 of the length, and at the exit. Figure 4 shows the correspond- ing pressure profile along the length of the bearing. Included in this last figure is the profile that would result if the entrance value of viscosity persisted throughout the film. The circles were read from the Dowson-Hudson curves, and the agreement is almost within reading accuracy. ture contours.

In the second case, the bearing surfaces are parallel. The possibility of load support through a "viscosity wedge" is being explored. Figure 6 compares pressure distributions. Again, excellent agreement is achieved. We note parenthetically that Dowson and Hudson

The fluid properties used by Dowson and

Figure 5 compares predicted tempera-

demonstrated that, with more realistic wall boundary conditions for the temperature, the above-shown load support vanishes.

Finally, Figs. 7 to 9 show the velocity, temperature and pressure profiles for a flat slider with 411 film-thickness ratio. These calculations were performed to test the sen- sitivity of the analysis to flow reversal at the entrance. As mentioned earlier, such reversal can cause problems for point-by-point prediction methods. comparisons with other investigations.

8 . CONCLUSIOUS

A Galerkin-type analysis has been made of tem- perature effects in laminar lubricating films. The procedure capitalizes on the suitability of so-called "Lobatto points" for sampling of the temperature distribution. Preliminary indica- tions are that the use of just two such sampling points enables satisfactory prediction of bear- ing performance even in the presence of sub- stantial viscosity variation.

The procedure described herein yields two partial differential equations, one for the local space-mean temperature, and one for the first transverse moment of the temperature dis- tribution. These temperature equations are coupled to a generalized Reynolds equation. Results have been presented for the steady- state, infinitely-wide flat slider bearing, and comparisons with the earlier, detailed work of Dowson and Hudson are very encouraging. The procedure is quite general, and our intent is now to refine the numerical techniques, and to carry out calculations for more realistic con- figurations and boundary conditions.

9. ACKNOWLEDGUEUTS

It is a pleasure for HGE to express thanks for sponsorship as a Visiting Professor at the Technical Univ. of Denmark in 1977. during which time Dr. Jorgen Lund suggested thermohydro- dynamics as a fruitful area for research. Also, to acknowledge some subsequent in-house support from The Franklin Research Institute, Phila..

We have yet to make any

PA.

10 I

1.

2.

3.

USA

REFERENCES

Khonsari. U.U.. "A Review of Thermal Effects in Hydrodynamic Bearings. Part I: Slider and Thrust Bearings," ASLE Preprint 80. 86-M-2A-3, 41st Annual Ueeting, Toronto, Ont., lay 12-15, 1986. To be published in the Trans. of the ASLE, 1986.

Khonsari. U.M., "A Review of Thermal Effects in Hydrodynamic Bearings. Part 11: Journal Bearings." To be published in the Trans. of the ASLE. 1986.

Dowson. D. and Hudson, J.D., "Thermo-Hydro- dynamic Analysis of the Infinite Slider- Bearing: Part I. The Plane-Inclined Slider-Bearing. Part 11, The Parallel- Surface Bearing," in Lubrication and Wear, Kay 23-25, 1963, (Institute of Uechanical Engineers, London, 1964) 34-51.

5

4. Villadsen, J. and Michelsen, M.L., "Solu- 5. Abramowitz. I4. and Stegun, I.A.. "Handbook tion of Differential Equation Models by Polynomial Approximation," 1978, (Prentice- National Bureau of Standards, Applied Mathe- Hall, New York) p. 127.

of Mathematical Functions," 1964 (U .S .

matics Series No. 5 5 ) .

6

TABLE I. - FLUIDITY FUNCTIONS

a = 2(2y0 - yl + y2/5)[3 a = 2(r - 2y1/5 + y3/35)/3 a = 2(r - 2r /7)/15 a = 2(r - 2y3/3)/105

0

1 0

2 0 2

3 1

yo = 2(Y0 - Y2/5)/3 y1 = 2(5y0 - yl - 2y3/7)/15 y2 = 2(2Y0 - r2/7)/15 y = 2(1 - r /9)/35 3 1 3

6

6

52 = 2(4y1 - y3)/105

= 2(Y 13 - Y3/7)/5 = 2(-7 /5 + y /3 - 4Y2/35)/3

I = 2(r0 + r2/31/35

0 1

1 0 1

3

-----_____

u - FIGURE 1.- COORDINATE DEFINITIONS FOR SLIDER BEARING.

.2 .4 .6 .8 1 .o POSITION I N GAP, <

FIGURE 2.- X-VELOCITY VERSUS POSITION IN GAP.

350

340

Y

330 b

W cz 3 c U 3 320 SE w +

310

300 0

2.5

2.0 a a b

h I 2 1.5 X L

W cz

cn W cz a

b

1.0

.5

0

.2 . 4 .6 .8 P O S I T I O N I N GAP,<

FIGURE 3.- TEMPERATURE VERSUS P O S I T I O N I N GAP.

0 R E F . 3 (DOWSON AND HUDSON) PRESENT ANALYSIS

1 .o

.2 . 4 .6 .8 1,o P O S I T I O N I N BEARING, X/Q

FIGURE 4, - PRESSURE VERSUS P O S I T I O N I N BEARING,

REF. 3 (DOWSON AND HUDSON) I

I <x

7 n

I " m m I

n

1.5

. u) I

Ei .5 X CL

b

W w

CI) W w 0

8 0

-.5

-1

- U

FIGURE 5 . - TEMPERATURE CONTOURS, T/T,.

R E F . 3 (DOWSON AND HUDSON) PRESENT A N A L Y S I S

0 R E F . 3 (DOWSON AND HUDSON) PRESENT A N A L Y S I S

I '0 .2 .4 .6 .8 1 .o

P O S I T I O N I N BEARING, X/l?

FIGURE 6.- PRESSURE VERSUS POSITION I N BEARING.

40

30

V 0

> 10

0

1 .o -10 .2 .4 .6 .8

P O S I T I O N I N GAP, 5 FIGURE 7.- X-VELOCITY VERSUS P O S I T I O N I N GAP.

350

340

Y

. 330 I-

L

310

0 .2 .4 .6 .8 1 .o 300

P O S I T I O N I N GAP, 5 FIGURE 8.- TEMPERATURE VERSUS P O S I T I O N I N GAP.

15

12 a

‘ 9 9

p. . CD

X p.

W

v) v) W p.

. 3 6

3

0 .2 .4 .6 .8 1 .o POSITION I N BEARING, X / l

FIGURE 9.- PRESSURE VERSUS P O S I T I O N I N BEARING.

2. Government Accession No. '. Reprt No. NASA TM-88845

4. Title and Subtitle

USAAVSCOM-TR-86-C-33

7. Key Words (Suggested by Author@))

Thermal f i l m ; Thermohydrodynamic; Lubr ican ts ; Thermal e f f e c t s ; Laminar f i l m s ; S l i d e r bear ing; Lobat to p o i n t s

Thermohydrodynamic Analys is f o r Laminar L u b r i c a t i n g F i l m s

18. Distribution Statement

U n c l a s s i f i e d - u n l i m i t e d STAR Category 34

-

7. Author@)

Harold G. El rod and David E. Brewe

9. Security Classif. (of this report) 20. Security Classif. (of this page)

U n c l a s s i f i e d Unclas s i f i ed

9. Performing Organization Name and Address

NASA Lewis Research Center and Propuls ion D i rec to ra te , U.S. Army A v i a t i o n Research and Technology A c t i v i t y - AVSCOM, Cleveland, Ohio 44135

Nat iona l Aeronaut ics and Space Admin is t ra t ton Washington, D.C. 20546 and U.S. Army Av ia t i on Systems Command, S t . Louis, Mo. 63120

2. Sponsoring Agency Name and Address

21. No. of pages 22. Price'

3. Recipient's Catalog No.

5. Report Date

6. Performing Organization Code

505-63-81 8. Performing Organization Report No.

E-3235 10. Work Unit No.

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

5. Supplementary Notes

Prepared f o r t he Leeds-Lyon Symposium on Tr ibo logy , Leeds, England, September 8-12, 1986. Harold G. Elrod, Consul tant, 1 4 Cromwell Court, Old Saybrook, Connect icut 06475; David E. Brewe, Propuls ion D i rec to ra te , U.S. Army A v i a t i o n Research and Technology A c t i v i t y - AVSCOM.

6. Abstract

A Ga lerk in - type ana lys i s t o i nc lude thermal e f f e c t s i n laminar l u b r i c a t i n g f i l m s was performed. The l u b r i c a n t p roper t ies were assumed constant except f o r a tem- perature-dependent Newtonian v i s c o s i t y . es tab l i shed by c o l l o c a t i o n a t t he f i l m boundaries and two i n t e r i o r Lobat to po in ts . The i n t e r i o r temperatures are determined by r e q u i r i n g the zero th and f i r s t moment o f t he energy equat ion be s a t i s f i e d across t h e f i l m . i s fo rced t o conform t o a th i rd-degree polynomial app rop r ia te t o t h e Lobat to- p o i n t temperatures. Pre l im inary i nd i ca t i ons a re t h a t t he use o f j u s t two such sampltng p o i n t s enables s a t i s f a c t o r y p r e d i c t i o n o f bear ing performance even i n t h e presence o f subs tan t i a l v i s c o s i t y v a r i a t i o n .

The c r o s s - f i l m temperature p r o f i l e I s

The f l u i d i t y

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