Davim J.P. Tribology for Engineers: A Practical Guide

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109

Lubrication and roughness

similar behaviour. But it should be borne in mind that the

plots are given for a single time moment T (Figs 3.16(b),

3.17(b)) and a single coordinate point Z (Figs 3.16(a),

3.17(a)); animated pictures are needed for studying the

overall situation.

The pressure distribution is not periodically identical along

the X-axis (as was the case of sinusoidal roughness with the

same wave numbers on both surfaces), while along the T-axis

such identity is evidently preserved. Thus examination of a

single wavelength does not yield complete information about

the surface behaviour as a whole.

Effect of asperity height, inter-surface and

intra-surface wave ratios on hydrodynamic

pressure

Roughened surfaces with the data of Table 3.3 were studied.

The reference data yield to the dimensionless cavitation

pressure threshold p

cav

= 4.86·10

–2

at the studied value of

the dimensionless roughness was 0.15. Here we present the

calculations for the case of unequal asperity heights and

wavelengths on the surfaces and unequal wave numbers in

the coordinate directions of each surface, the equal case

being a particular one of the general solution.

As can be seen from eq. [3.61], the hydrodynamic pressure is

governed by six parameters – the scaling factor, asperity height

and wave number (both for the upper surface), asperity ratio,

intra- and inter-surface wave ratios. The scaling factor is taken

constant and equal to 1, which means equal evaluation lengths

in the X- and Z-directions. Further, the dimensionless pressure

is related to the relative gap and through it to the asperity

height on the lower surface, so that varying the latter parameter

affects the dimensionless cavitation pressure threshold; besides,

variation of the asperity height on the lower surface entails that

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Tribology for Engineers

of its upper counterpart at the same relative gap. To avoid this

complication, all calculations were based on a constant asperity

height A

. By this means the number of factors to be studied

was reduced from six to four.

Now the pattern of maximal pressures at different

values of the mentioned parameters can be examined. By this

means, we can ﬁ nd the excess of the pressure over the

cavitation threshold and check for the presence of a positive

hydrodynamic effect.

Figure 3.18 shows the dependence of the maximal pressure

on the asperity height ratio A

in the range 2/3–2 for

asperity heights A

= 0.1, 0.15, 0.2; inter-surface wave ratios

 = 3, 1, 0.5, 0.33; A

= 0.15 and k

= 3. These dependences

are presented for intra-surface wave ratios 

= k

= 1/3,

1, 2 – Fig. 3.18(a), (b) and (c) successively. As can be seen, for

Reference values used in calculations

Table 3.3

Parameter Notation Reference value Dimensionality

Sliding velocity u 8.9 m/s

Fluid dynamic

viscosity

3cPa

Clearance

6.4

Control cell

evaluation length

L 0.2 mm

Asperity height on

lower surface

Relative gap

Δ + 2R

= 1

12.4

Inter-surface wave

ratio



1/3 . . . 2 dimensionless

Wave number on

upper surface,

X-direction

1 . . . 9 units

Intra-surface wave

ratio, upper

surface



1/2 . . . 3 dimensionless

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111

Lubrication and roughness

Figure 3.18

Maximal pressures versus asperity height ratio

at different intra-surface wave numbers

; asperity height of lower surface 0.15; and

intra-surface ratios: (a) 2/3, (b) 1, (c) 2

the wave ratio 3, the maximal positive pressure goes through

a very weak maximum at asperity ratio above 1.3, but as a

whole it decreases as the asperity ratio increases. Reduction

of the asperity height on the upper surface reduces the

maximal pressure at all studied values of the intra ratio. The

different intra ratios evidence that when the wave number in

the Z-direction is greater than in the X-direction (Fig. 3.18(a))

the maximal pressure is higher than in the opposite case,

(Fig. 3.18(c)) or in that of equal wave numbers (Fig. 3.18(b)).

This also demonstrates that inclusion of the second dimension

strongly inﬂ uences the calculated values of the hydrodynamic

pressure, yielding a more reliable result.

Further, as can be seen from Fig. 3.18, the positive

hydrodynamic effect is observed only in the case of intra ratio

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Tribology for Engineers



= 1/3 and inter ratio 3; as a whole, absence of the positive

effect in the studied triangular version conﬁ rms that in reality

the roughnesses are not a positive factor in lubrication.

Figure 3.19 presents the dependences of the maximal

pressure for inter ratios in the range  = 1/3–4, at three

different intra ratios 

= 1/3, 1 and 2; it is actually a

reconstruction of Fig. 3.18 with additional data for the lower

surface wave number k

= 6. As can be seen, beginning with

wave ratio 2 the maximal pressures are lower than the

cavitation threshold, so that the positive effect is absent at



> 1 for all studied roughness parameters.

The interesting fact here is that the P

max

() curves pass

through an inﬂ ection point where asperity height change

does not inﬂ uence the maximal pressure, while beyond

Figure 3.19

Maximal and cavitation pressures versus

inter-surface wave ratio at lower surface wave

numbers k

= 3 and 6, asperity height A

= 0.15;

for intra ratios: (a) 2/3, (b) 1, (c) 2

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Lubrication and roughness

this point a larger asperity height leads to lower maximal

pressure as should be expected – larger asperities increase

lubrication resistance to the relative motion of the surfaces.

Before the inﬂ ection point the asperity height on the upper

surface is small (asperity ratios 2/3 and 1) and so is the

number of asperities there (in our calculations k

varies in

the range 1–6 at k

= 3), which makes for higher load support

at smaller asperity heights. With larger wave numbers on

the lower surface, k

= 6, the inﬂ ection point apparently

jumps to larger wave ratios; as a whole, as higher pressures

are obtained at smaller wave ratios, it is preferable to produce

surfaces with fewer waves and small amplitudes – which is

unrealistic.

Comparison results for sinusoidal and triangular asperities

(in the case of equal inter- and intra-roughness parameters

= 1, 

= 

= 1) shows, Fig. 3.20, that the triangulars

make for a maximal pressure about 25 per cent less than the

sinusoidals in the range of small wave numbers; at larger

wave numbers the discrepancies decrease up to less than

1 per cent. Comparison at A

≠ A

shows two pressures lower

(by one-half and less) for the triangulars in the wave number

range 2–19 with A

= 2; the maximal pressures were higher

than the cavitation threshold at wave numbers between 1 to

3 only. All that is additional evidence of the detrimental role

of roughness in ﬂ uid ﬁ lm hydrodynamics – seeing that the

triangular conﬁ guration is the more realistic model.

All calculations for triangular roughness were carried out

with the aid of a specially written Matlab program covering

different input values of the roughness, roughness ratio, and

intra- and inter-surface wave ratios. The program operated

with sparse matrices with a view to economy in recourse to

memory, and increased computing capacity at the larger

wave ratios. Summarizing the reported observations for

triangular roughness:

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Tribology for Engineers

■

The maximal support effect is reached at upper surface

wave numbers three times those of the lower.

■

The dependence of the maximal pressure on the surface

wave number has an inﬂ ection point before which the

maximal pressure is higher for lower wave ratios at smaller

asperity heights.

■

For larger asperity heights the load support decreases

and the maximal hydrodynamic pressure is below the

cavitation threshold.

■

Above the intra surface wave ratio 2, the wave number in

the Z-direction is double that in the X-direction, all

pressures are below the cavitation threshold, so that no

positive hydrodynamic effect is possible here.

Figure 3.20

Cavitation threshold and maximal hydrodynamic

pressure versus wave number; 

= 

= 1,

= A

= 0.15 for triangular, and 0.25 for

sinusoidal roughnesses

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Lubrication and roughness

3.7 Nomenclature

A – dimensionless asperity height of surface roughness

(amplitude)

H, h – dimensionless and dimensional ﬁ lm thicknesses

P, p – dimensionless and dimensional hydrodynamic pressure

L – dimensional slider length/width

B, C, D – matrices for searching constants in pressure

distribution equation

, e – dimensionless and dimensional journal eccentricity

c – dimensional radial clearance

, C

– ﬁ rst and second constant of integration

k – number of waves on surface along coordinate; also,

wedge coefﬁ cient in Section 3.5.1

K – vector in matrix used for ﬁ nding constant of integration,

triangular asperities

 – wave ratio



– bearing scale factor



– inter-surface scale factor

, L

– dimensional surface length and width

, M

– dimensional mass ﬂ ow rates per unit length

R – dimensional journal radii

– dimensional asperity height (amplitude)

S – dimensionless local roughness step

, V

– ﬂ uid ﬁ lm velocities along x, y, z directions

T, t – dimensionless and dimensional time

u, v – dimensional velocitiy of sliding surface, along

coordinate x and y respectively

(X, Y, Z), (x, y, z) – dimensionless and dimensional Cartesian

coordinates

w – dimensional normal load, otherwise dimensional ﬂ uid

ﬁ lm carrying capacity

Δ – dimensional minimal ﬁ lm thickness/gap between surfaces

– gap line unitary functions

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Tribology for Engineers

– wavelength of surface proﬁ le along coordinate dimension,

dimensional

– dimensional dynamic viscosity of ﬂ uid

– angular coordinate (bearings) or combined coordinate

(roughness)

– dimensional kinematic viscosity of ﬂ uid

– angular velocity

– dimensional ﬂ uid ﬁ lm shear stresses near surface

f – dimensional forces acting on a ﬂ uid ﬁ lm

– friction force

Φ,

– dimensionless and dimensional phase shift

(displacement) of sinusoidal roughness of upper surface

relative to that of lower

3.8 Subscripts

0 – relative/initial value

1 and 2 – moving(upper) and stationary (lower) surface,

respectively

A or a – amplitude

b – bearing

c – constant

cav – cavitation

e – extreme point of gap proﬁ le

f – friction

i – extreme point number

max – maximal value

t – time-dependent value

x, z – coordinate directions

Superscript (i) is identical with subscript i.

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Lubrication and roughness

3.9 Acknowledgement

I wish to thank Mr. E. Goldberg, former resident scientiﬁ c

editor of the Technion, for valuable help in editing the chapter.

3.10 References

Anno, J. N., Walowit, J. A. and Allen, C. M. (1968),

‘Microasperity lubrication’, ASME Journal of Lubrication

Technology 91(2): 351–5.

Anno, J. N., Walowit, J. A. and Allen, C. M. (1969), ‘Load

support and leakage from microasperity – lubricated face

seals’, ASME Journal of Lubrication Technology 92(4):

726–31.

ANSI/ASME Standard B46.1–1985 (1985) ‘Surface Texture’,

New York, American Society of Mechanical Engineers.

Arghir, M., Roucou, N., Helene, M. and Frene, J. (2003),

‘Theoretical analysis of the incompressible laminar ﬂ ow

in a macro-roughness cell’, ASME Journal of Tribology

125(2): 309–18.

Bernard, J. H., Steven, R. S. and Bo, O. J. (2004),

Fundamentals of Fluid Film Lubrication, second edition,

Marcel Dekker Inc.

Booker, J. F. (1965), ‘A table of the journal-bearing integrals’,

J Basic Eng, Transactions of the ASME 87(2): 533–5.

Booser, E. R. (1988), CRC Handbook of Lubrication: Theory

and Design, 2, CRC Press, Boca Raton.

Bronshtein, I. N. and Semendyayev, K. A. (2007), Handbook

of Mathematics, 5th edition, Springer, 1164.

Burstein, L. (2006), ‘Two-sided surface roughness and

hydrodynamic pressure distribution in lubricating ﬁ lms’,

Lubrication Science 19(2): 101–12.

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Tribology for Engineers

Burstein, L. (2008), ‘Effect of sinusoidal roughened surfaces

on pressure in lubricating ﬁ lm’, Int J Surface Science and

Engineering 2(1–2): 52–70.

Burstein, L. (2009), ‘Sinusoidal roughness in hydrodynamic

lubrication. New transient 2D solution’, in Davim,

J. P., Tribology Research Advances, New York, Nova

Publishers.

Burstein, L. (2010), ‘Hydrodynamic lubrication of roughened

surfaces with different asperity heights and wave numbers’,

Journal of Tribology and Surface Engineering 1 (in

print).

Burstein, L. and Ingman, D. (1999), ‘Effect of pore ensemble

statistics on load support of mechanical seals with pore-

covered faces’, ASME Transactions, Journal of Tribology

121(44): 927–32.

Burstein, L. and Ingman, D. (2000), ‘Pore ensemble statistics

in application to lubrication under reciprocating motion’,

Tribology Transactions 43(2): 205–12.

Dowson, D., Taylor, L. M., Godet, M. and Berthe, D. (1978),

‘Surface Roughness Effect in Lubrication. Leeds–Lyon

Symposium on Tribology’ (4th Symposium, 1977, Lyon,

France), London, Mechanical Engineering Publications.

Labiau, A., Ville, F., Sainsot, P., Querlioz, E. and Lubrecht,

T. (2008), ‘Effect of sinusoidal surface roughness under

starved conditions on rolling contact fatigue’, Proceedings

of the IMechE, Part J: Journal of Engineering Tribology

222(3): 193–200.

Lai, T. W. (1994), ‘Development of non-contacting spiral

groove liquid face seals’, Lubrication Engineering 50(8):

625–40.

Letallear, N., Plouraboue, F. and Prat, M. (2002), ‘Average

ﬂ ow model of rough surface lubrication: ﬂ ow factors for

sinusoidal surfaces’, ASME Transactions, Journal of

Tribology 124(3): 539–46.

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