Advanced Modelling of Elastohydrodynamic Lubrication PetraBrajdicPhD2005

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Doctorate Thesis

Advanced Modelling of Elastohydrodynamic

Lubrication

Petra Brajdic-Mitidieri

Tribology Section and Thermofluids Section

Department of Mechanical Engineering

Imperial College London

November, 2005

Abstract

The work presented in this thesis concerns computational modelling of lubrication

processes in various types of bearings using Computational Fluid Dynamics.

To describe the flow of a lubricant in a bearing, the Reynolds Equation is widely

used. This equation is deduced from the Navier-Stokes equations under certain as-

sumptions. In most cases, it can accurately predict the characteristics of the flow in

the lubricant film. However, if one wants to look at the region further away from the

narrow gap or if the surface roughnesses are of order of magnitude of the film thickness,

the Reynolds Equation may no longer be appropriate, and the need for using the full

set of Navier-Stokes equations becomes apparent.

In this work, order of magnitude analysis is conducted on the governing equations of

flow of lubricant in the two regions of the bearing; the contact region and the region far

from the contact. It is concluded that in order to accurately model the entire domain,

one needs to use the full Navier-Stokes equations.

The Finite Volume Method is introduced as it will be the discretisation method

employed in this work.

The Computational Fluid Dynamics is validated as a suitable means to compu-

tationally model the flow of a lubricant in simple converging bearings, for which the

analytical solution of the Reynolds Equation exists. By doing that, it is determined

that the CFD can accurately model lubrication problems.

In order to computationally model the geometry of a roller bearing, cavitation must

be addressed. A computational model for cavitation is introduced and tested.

Finally, the cavitation model is applied to the complex geometry of a pocketed pad

bearing, and a reduction in the friction coefficient is noted due to the pocket.

2

3

Acknowledgements

I would like to express my gratitude to my supervisors, Prof. H.A. Spikes, Prof. A.D.

Gosman and Prof. E. Ioannides for their continuous guidance and support during this

study.

I would also like to thank the past and present members of the Tribology and

Thermofluids Sections for their cooperation and support and for creating a pleasant

atmoshpere for work.

The text of this Thesis has benefited from valuable comments from Dr. H. Jasak

whose help is greatly appreciated.

I would also like to thank Mrs. Chrissy Stevens for the help with many adminis-

trative matters and for being a great friend.

My special thanks go to my family, especially to my husband Dario, my daugh-

ter Mara and my parents, Krunoslava and Mladen, for their enormous patience and

support.

The financial bursary provided by SKF, Netherlands is gratefully acknowledged.

Contents

1 Introduction 17

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1.1 Fluid Film Lubrication . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Previous and Related Studies . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.1 Numerical Work in Hydrodynamic and Elastohydrodynamic Lu-

brication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.2 Use of CFD in Fluid Film Lubrication . . . . . . . . . . . . . . 21

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Governing Equations of Fluid Flow 26

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Governing Equations of Continuum Mechanics . . . . . . . . . . . . . . 27

2.2.1 Mass and Momentum Conservation . . . . . . . . . . . . . . . . 27

2.2.2 Constitutive Relations for Newtonian Fluids . . . . . . . . . . . 28

2.2.3 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Order of Magnitude Analysis . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.1 Non–Dimensional Variables . . . . . . . . . . . . . . . . . . . . 31

2.3.2 Characteristic Values . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Non–Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Non–dimensional Equations with Characteristic Values . . . . . . . . . 35

2.5.1 Outside the Contact . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.2 Dominant Terms Outside the Contact . . . . . . . . . . . . . . . 36

4

Contents 5

2.5.3 Inside the Contact . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5.4 Dominant Terms Inside the Contact . . . . . . . . . . . . . . . . 39

2.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Finite Volume Discretisation 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Components of a Numerical Solution Method . . . . . . . . . . 41

3.1.2 Properties of The Numerical Solution Method . . . . . . . . . . 43

3.2 Spatial Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Discretisation of the Governing Equations . . . . . . . . . . . . . . . . 46

3.3.1 Convection Term . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.2 Diffusion Term . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.3 Source Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.4 Temporal Discretisation . . . . . . . . . . . . . . . . . . . . . . 53

3.3.5 Implementation of Boundary Conditions . . . . . . . . . . . . . 55

3.4 System of Linear Algebraic Equations . . . . . . . . . . . . . . . . . . . 58

3.5 Discretisation of Navier-Stokes Equations . . . . . . . . . . . . . . . . . 61

3.5.1 Derivation of the Pressure Equation . . . . . . . . . . . . . . . . 62

3.5.2 Pressure-Velocity Coupling . . . . . . . . . . . . . . . . . . . . . 63

3.5.3 The PISO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.4 The SIMPLE Algorithm . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.6.1 Richardson Extrapolation . . . . . . . . . . . . . . . . . . . . . 66

3.7 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Validation of CFD Approach Using Simple Converging Bearings 69

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Theoretical Considerations in Isoviscous-Rigid Hydrodynamic Lubrication 70

Contents 6

4.2.1 Simplifications Leading to Reynolds Equation . . . . . . . . . . 70

4.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.3 Reynolds Equation . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.4 Infinitely Long Bearings . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Analysis of an Infinitely Long Linear Wedge . . . . . . . . . . . . . . . 74

4.3.1 Reynolds Solution . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.2 Numerical Results and Mesh Selection . . . . . . . . . . . . . . 75

4.4 Analysis of the Step Bearing . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.1 Reynolds Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.2 Mesh Selection and Numerical Results . . . . . . . . . . . . . . 82

4.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Computational Modelling of Cavitation 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Previous Work in Cavitation . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Physical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Cylinder on a Flat Surface . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5.1 Reynolds Solution for Full Sommerfeld Condition . . . . . . . . 95

5.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5.3 Mesh Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Low Friction Pocketed Pad Bearing 105

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Linear Bearings with Pockets . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.2 Load and Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Contents 7

6.2.3 Mesh Study and Selection . . . . . . . . . . . . . . . . . . . . . 108

6.3 2D Bearing with the Single Pocket . . . . . . . . . . . . . . . . . . . . 109

6.3.1 Varying Pocket Height . . . . . . . . . . . . . . . . . . . . . . . 110

6.3.2 Varying Location of the Pocket . . . . . . . . . . . . . . . . . . 111

6.3.3 Varying Convergence Ratio . . . . . . . . . . . . . . . . . . . . 116

6.3.4 Varying Size of the Pocket . . . . . . . . . . . . . . . . . . . . . 121

6.4 2D Bearing with Multiple Pockets . . . . . . . . . . . . . . . . . . . . . 121

6.4.1 Four Pockets Covering 25% of Total Area . . . . . . . . . . . . . 122

6.4.2 Eight Pockets Covering 50% of Total Area . . . . . . . . . . . . 122

6.5 3D Linear Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.5.1 3D Linear Wedge with One Pocket Covering 25% of Total Area 124

6.5.2 3D Linear Wedge with Two Pockets Covering 25% of Total Area 127

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.7 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 Summary and Conclusions 132

7.1 Governing Equations of the Flow of Lubricant . . . . . . . . . . . . . . 133

7.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3 Validation of CFD Solver On Simple Hydrodynamic Bearings . . . . . . 134

7.4 Modelling of Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.5 Low Friction Pocketed Pad Bearing . . . . . . . . . . . . . . . . . . . . 136

7.6 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

List of Figures

2.1 A schematic picture of a contact in fluid film lubrication. . . . . . . . 32

3.1 Control volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Face Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Two neighbouring nodes, P and N in a non-orthogonal mesh . . . . . . 51

3.4 Over-relaxed approach in non-orthogonality treatment. . . . . . . . . . 53

3.5 Parameters at the face boundary . . . . . . . . . . . . . . . . . . . . . 56

4.1 Two generalised surfaces in relative motion . . . . . . . . . . . . . . . . 70

4.2 Linear wedge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 Mesh structure in the computational domain of the linear wedge. . . . 76

4.4 Variation of maximum pressure and total load with the mesh refinement

in (a) the x and z directions, (b) x direction, (c) z direction. . . . . . 77

4.5 Pressure distribution along the bottom wall of the linear wedge. . . . . 78

4.6 The x - component of the velocity vector at three different locations in

the linear wedge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.7 Velocity profiles on three different locations along the linear wedge. . . 79

4.8 The Rayleigh step bearing. . . . . . . . . . . . . . . . . . . . . . . . . 80

4.9 Mesh and block structure in the computational domain of the step bear-

ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.10 Pressure distribution along the bottom wall of the step bearing. . . . . 83

4.11 The x - component of the velocity vector at the inlet and the outlet of

the step bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8

List of Figures 9

4.12 Velocity profile at the three positions in the step bearing. . . . . . . . . 85

5.1 Pressure-density relationship with different values for ag. . . . . . . . . 93

5.2 Bearing geometries used in cavitation model. . . . . . . . . . . . . . . . 94

5.3 Pressure distribution along the bottom wall of the roller bearing; . . . . 95

5.4 Density and pressure distributions along the bottom wall and bottom

half of the cylinder; (a) and (b) for rolling case, (c) and (d) for sliding

case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5 Density distribution and isolines in exit region. . . . . . . . . . . . . . . 98

5.6 Pressure–density dependence for the half-cylinder case . . . . . . . . . 99

5.7 Streamlines in cavitation cases. . . . . . . . . . . . . . . . . . . . . . . 100

5.8 Velocity profiles for the half-cylinder case in the converging region lead-

ing to the contact at x = 0 m. . . . . . . . . . . . . . . . . . . . . . . . 101

5.9 Velocity profiles for the full cylinder case in the converging region leading

to the contact at x = 0 m. . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.10 Density distribution and the velocity profile at the reformation point at

x = 40e-5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.11 Mesh structure in the computational domain of the cylinder on the flat

geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.1 Linear wedge with the pocket. . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Block structure used in pocketed bearing analysis. . . . . . . . . . . . . 109

6.3 Pressure and shear stress distribution along the bottom wall of the bear-

ing for various heights of the pocket. . . . . . . . . . . . . . . . . . . . 112

6.4 Dependence of friction coefficients on the height of the pocket. . . . . . 113

6.5 Velocity distribution at the beginning, middle and the end of the pocket. 113

6.6 Pressure and velocity distributions at: (a) the inlet of the pocket, (b)

the exit of the pocket. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

List of Figures 10

6.7 Pressure distribution along the bottom wall for bearings with varying

location of a pocket. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.8 Friction coefficient ratio µh3/µ0 for different positions of the pocket. . . 115

6.9 Pressure distribution at the bottom wall of the bearing for varying con-

vergence ratio, K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.10 Density distribution at the beginning of the pocket for K = 0.001. . . 119

6.11 Rate of convergence of solution for density in the pocket inlet region. . 120

6.12 Pressure distribution at the beginning and the end of the pocket for

K = 0.001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.13 Linear wedge with multiple pockets. . . . . . . . . . . . . . . . . . . . . 122

6.14 Pressure distribution for the cases with four pockets covering 25% of

total area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.15 Top view of a 3D linear wedge with the pocket. . . . . . . . . . . . . . 125

6.16 Pressure distribution across the bottom wall for the case with B1 = 5 mm.

126

6.17 Pressure distribution along the bottom wall for y = 0 m. . . . . . . . . 126

6.18 Top view of a 3D linear wedge with two pockets. . . . . . . . . . . . . . 127

6.19 Pressure distribution across the bottom wall for the case with two pock-

ets and B1 = 7 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.20 Pressure distribution along the bottom wall for y = 0 m. . . . . . . . . 129

List of Tables

1.1 Overview of numerical work using CFD in fluid film lubrication. . . . 24

2.1 Characteristic values inside and outside the contact in a bearing. . . . . 32

4.1 Mesh refinement in x- and z- directions. . . . . . . . . . . . . . . . . . . 76

4.2 Mesh refinement in both x- and z- directions. . . . . . . . . . . . . . . . 83

5.1 Values of the fluid properties used in the cavitation model. . . . . . . . 93

5.2 Number of points in the x− and the z− direction for each block. . . . . 103

6.1 Mesh study for 2D pocketed bearing. . . . . . . . . . . . . . . . . . . . 109

6.2 Influence of pocket height on bearing performance. . . . . . . . . . . . 111

6.3 Influence of pocket position on bearing performance. . . . . . . . . . . 116

6.4 Influence of the convergence ratio on bearing performance. . . . . . . . 118

6.5 Influence of the pocket size on bearing performance. . . . . . . . . . . . 121

6.6 Cases of linear wedge with four pockets covering 25% of total area. . . 123

6.7 Cases of linear wedge with eight pockets covering 50% of total area. . . 124

6.8 Effect of pocket position for 3D single pocket. . . . . . . . . . . . . . . 125

6.9 Cases of 3D linear wedge with two pockets covering 25% of total area. . 128

11

Nomenclature

Latin Characters

a – general vector property

aN – matrix coefficient corresponding to the neighbour N

aP – central coefficient

B – bearing width

Co – Courant number

d – vector between P and N

E – exact error

et – Taylor Series Error estimate

F – mass flux through the face, friction of the bearing

f – face, point in the centre of the face

fi – point of interpolation of the face

g – body force

h – mesh size, height of the bearing

I – unit tensor

K – convergence ratio

12

Nomenclature 13

k – non-orthogonal part of the face area vector

L – bearing length

N – point in the centre of the neighbouring control volume

n – outward pointing face area vector

p – pressure

P – point in the centre of the control volume

RP – right-hand-side of the algebraic equation

Re – Reynolds number

Sφ – source term

SMi – component of the source term in the Cartesian coordinates

SP – linear part of the source term

Su – constant part of the source term

t – time

U – velocity vector

u – component of the velocity vector U in x-direction in the Cartesian coordinates

V – volume

v – component of the velocity vector U in y-direction in the Cartesian coordinates

VP – volume of the cell

W – load support of the bearing

w – component of the velocity vector U in z-direction in the Cartesian coordinates

Nomenclature 14

x – position vector

x – component of x in the Cartesian coordinates

y – component of x in the Cartesian coordinates

z – component of x in the Cartesian coordinates

Greek Characters

α – pressure viscosity coefficient, under-relaxation factor, void fraction

∆ – orthogonal part of the face area vector

Γφ – diffusivity

η – dynamic viscosity

ν – kinematic viscosity

µ – friction coefficient

ρ – density

σ – stress tensor

τ – shear stress

Φ – exact solution

φ – general scalar property

Superscripts

q – mean

Nomenclature 15

Subscripts

q0 – characteristic value

q∗ – dimensionless value

Abbreviations

Bi-CG – Bi-Conjugate Gradient

CFD – Computational Fluid Dynamics

CG – Conjugate Gradient

CV – Control Volume

EHL – Elasto-Hydrodynamic Lubrication

FD – Finite Differencing

FV – Finite Volume

FVM – Finite Volume Method

HL – Hydrodynamic Lubrication

ICCG – Incomplete Cholesky Conjugate Gradient

NS – Navier-Stokes

PISO – Pressure-Implicit with Splitting of Operators

RE – Reynolds Equation

SIMPLE – Semi-Implicit Method for Pressure-Linked Equations

UD – Upwind Differencing

16

Chapter 1

Introduction

1.1 Background

The overall objective of this study is to develop and apply an efficient, CFD–based

simulation of hydrodynamic lubricated contact and apply this to study the performance

of pocketed bearings. Once developed, the numerical model will also be highly suited to

investigate the flow of lubricant in complex geometries of hydrodynamic bearings and

will give a good starting point for further numerical modelling of elastohydrodynamic

contacts.

1.1.1 Fluid Film Lubrication

Whenever there are two surfaces in rubbing contact, there is a friction between them.

This friction is associated with energy dissipation and often mechanical damage to the

rubbing surfaces so it is generally desirable to reduce it as much as possible. One way

of reducing friction is to separate two surfaces with a liquid lubricant. The lubricant

film should satisfy two requirements. Firstly, it should have a low shear strength to

obtain a low friction. Secondly, it should be strong enough to carry the entire load in

the direction perpendicular to the surfaces, to prevent direct contact between surfaces.

There are two main types of fluid film lubrication: hydrodynamic lubrication (HL)

and elastohydrodynamic lubrication (EHL). In hydrodynamic lubrication, the surfaces

form a shallow, converging wedge, so that, as their relative motion causes lubricant

entrainment into the contact, the lubricant becomes pressurised and so able to sup-

17

Chapter 1. Introduction 18

port load. The film thickness depends on the surface shapes, their relative speeds, and

the properties of the lubricant. Generally, film thickness is of the order of microme-

ters, supporting applied pressure of the order of Mega Pascals. This pressure is not

high enough to significantly deform the rubbing surfaces nor to increase the lubricant

viscosity.

The existence of a pressurised lubricant film was first noted by Tower [64]. In re-

sponse to his work, Reynolds [55], in 1886, developed a theory to explain fluid film

formation. He simplified the Navier-Stokes (NS) equations, assuming small film thick-

ness relative to the contact length, non-varying pressure across the film thickness, and

the dominance of certain viscous terms. The equation obtained relates fluid pressure

to the rate of gap convergence, surface velocities and lubricant viscosity. It is referred

to as the Reynolds Equation.

There were two other important fundamental equations derived around that time.

In 1881, Hertz [37] published his study of the contact between two spherical bodies, to

show how the surfaces deform due to high, local pressure. In 1892, Barus [7] determined

how the viscosity of oils increases as a function of pressure.

A detailed overview of the history of lubrication can be found in the History of

Tribology by Dowson [18].

Reynolds equation rapidly became a useful tool in bearing analysis and design.

However, when in 1916, Martin [50] and Grumbel [32] tried to apply Reynolds equation

to the lubrication of gear teeth, the film thickness they predicted was far too small, in

comparison to the surface roughness, to explain the long term successful operation of

gears. The difference between this problem and the previous ones, was that, in gears,

there is a non-conformal (concentrated) contact.

When the contact between the surfaces is a line or a point (non conformal contact),

the load is concentrated over a small contact area, and thus generates much higher

pressure of the order of Giga Pascals. Such a high pressure has two beneficial effects


not taken account of in hydrodynamic lubrication: it elastically flattens the surfaces,

creating a locally conforming contact, and it greatly increases the viscosity of the

lubricant in contact.

It took another 30 years until the work of Ertel [21] and Grubin and Vinogradova

[31] combined Reynolds equation with the two effects that Hertz and Barus determined;

the elastic deformation and viscosity increase due to the high pressure, to provide an

elegant semi–analytical solution to this problem.

Since EHL comprised of three key equations: Reynolds equation, elastic deforma-

tion, and viscosity dependence on pressure, the problem became much more compli-

cated to solve than HL problem, and the role of a digital computer and numerical

solutions in its analysis was far larger.

1.2 Previous and Related Studies

1.2.1 Numerical Work in Hydrodynamic and

Elastohydrodynamic Lubrication

Reynolds equation is a second order differential equation and thus not amenable to

analytical solution. This means that, from the time of Reynolds through to the 1940s,

almost all solutions of HL were based on the analytical solution of simplified forms of

RE, notably approximations in which pressure variation in one direction was assumed

to be zero.

In 1949, Cameron and Wood [11], and Sassenfeld and Walther [60], produced the

first computer–based numerical solutions in HL. No thermal or non-Newtonian effects

were included. For a historical review of various contributions in HL numerical work

the reader is referred to Cameron [12] and to Dowson [18].

In recent times, most numerical work in HL has involved the use of Reynolds equa-

tion and the finite difference method, although recently, as discussed in the next section,

some work has been carried out using CFD approach.


The first numerical solution of EHL was obtained in 1951 by Petrusevich [54], and

predicted a strange singularity in the pressure distribution: the ’pressure spike’. In the

1960s, the foundation of modern numerical solutions of the EHL problem was laid by

Dowson and Higginson [19], who solved the line contact problem for a variety of oper-

ating conditions, and providing a film thickness equation based on these calculations.

In 1976, advances in computer technology allowed Hamrock and Dowson [35] to solve

the circular contact problem. The numerical method was a direct approach based upon

the simple point Gauss–Seidel scheme, and even though it had a slow convergence, pre-

dictions of central and minimum film thicknesses were obtained for a comprehensive

set of conditions The method was, however, inadequate for pressures higher than about

0.5 GPa. In 1981, Evans and Snidle [24] [25], successfully extended the inverse method

to the point contact problem and solutions up to a maximum Hertzian pressure of 1.5

GPa were obtained.

Extensive research continued to improve numerical methods, but convergence was

still very slow and also computation of the elastic deformations was a very time-

consuming process. In 1977, Brandt [9] introduced the multigrid technique as a way to

accelerate drastically the convergence of non–linear elliptical equations. Lubrecht et al.

[48] [49] were the first to develop line and point contact solutions using this technique.

Highly loaded simulations were still limited by numerical instabilities.

Great progress has been made in recent years to understand the numerical problems.

Venner [68] has been a main contributor to these developments. He indicated the two

major difficulties that needed to be overcome if a good convergence at high pressure

was to be achieved. Numerical problems are all dependent on the changes in the nature

of Reynolds equation throughout the computational domain and on the manner used to

treat the Reynolds and elasticity equations with a high number of discretization points.

In the high–pressure region the problem behaves as an integral problem and the elastic

deformation integral is dominant. As a result, pressure changes tend to accumulate


when relaxing Reynolds equation for pressure. Venner showed that this effect can be

controlled by using a distributive relaxation scheme. The second problem, present

only in point contact, comes from the loss of coupling of the Reynolds equation in the

direction transverse to the flow in the high–pressure region. This problem requires the

use of line relaxation. However, to address both the problem of the accumulation of

change and the loss of coupling, the development of a distributive line relaxation is

required.

By integrating these numerical schemes into a multigrid technique, and by using

multilevel multi–integration technique developed by Brandt and Lubrecht [10], Venner

laid the foundations for fast EHL solvers.

The other method of handling the elastic deformation numerically is differential

deflection, given by Evans and Hughes [26]. The effect of the pressure distribution in

this method is shown to be extremely localised in comparison with direct evaluation

of the deflection. This reduces computing time significantly.

Even though major progress has been made in recent decades in numerical methods

in lubrication, the majority of work is still based on Reynolds equation and the finite

difference method.

1.2.2 Use of CFD in Fluid Film Lubrication

In this Section an overview of published CFD work in fluid film lubrication is presented

in chronological order, excluding the work relating to the cavitation and to the recessed

hydrodynamic bearings. In Chapter 5, previous studies related to cavitation will be

discussed, whilst in Chapter 6 an overview of the work on recessed hydrodynamic

bearings will be given.

Solution using CFD differs from finite difference work in the discretization method.

In the finite difference method, discrete approximations for the differential operators of

the governing differential equation are used. For the HL and EHL problem, this results


in a discrete representation of the Reynolds equation or its equivalent. However, ap-

plication of this discrete equation to a discretized solution domain does not necessarily

ensure that mass conservation exists. This has been exhibited in solutions by Hamrock

and Jacobsen [36].

CFD work started in the late 1970s with the application of the control volume

method as the discretisation tool. This technique was described by Patankar [52] in

1980. The major advantage that is offered by this technique is that the fundamental

conservation from which the governing differential equations are derived is maintained

in the discrete solution, regardless of the level of solution domain discretisation. This

provides the capability of more accurately modelling the phenomena being examined.

In 1985, Blahey [8] used the control volume method to examine thermal effects

in elliptical EHL contact and presented a method for the numerical solution of this

problem. He solved the set of simplified NS equations, using his own code. The

solution including thermal effects significantly lowered the height of the pressure spike

compared to the isothermal case. Thermal effects also reduced the fluid film thickness

considerably (25% less film thickness).

In 1992, Chang [14] analyzed the elliptical thermal EHL contact by using a control

volume method similar to Blahey’s. A novelty in his work is that he included a study

of the effects of non-Newtonian behaviour.

In 1997, Zhang and Rodkiewicz [74] used a CFD technique to examine the effects of

changing the height and the length of the fore-region (the groove) in hydrodynamically–

lubricated thrust bearings. They found that fluid inertia needs to be taken into consid-

eration when modelling the fluid behaviour in the fore-region. However, by setting the

groove depth equal to zero, the results showed negligible fluid inertia (as assumed by

Reynolds equation). It was also found that the depth of the groove had little influence

on the bearing performance, but the length of this region had a profound influence.

In 1998, Chen and Hahn [15] studied the suitability of computational fluid dynamics


for solving steady state hydrodynamic lubrication problems. The geometries studied

were slider bearings, step pad bearings, journal bearings and squeeze-film dampers.

The relevance of inertia and viscous terms neglected in the derivation of the Reynolds

number were investigated, and it was shown that the generally neglected viscous terms

have negligible effect.

In 2000, Schafer et al. [61] used CFD to show that the application of Reynolds

equation is permissible for the case of pure rolling in EHL line contact, but not when

considering partial or pure sliding. However, they used a Newtonian fluid model and

assumed isothermal conditions, which resulted in unrealistically high shear stresses in

the lubricant film.

In 2000, Almqvist [2] developed a thermo-hydrodynamic (THD) model for a lu-

bricated, pivoted thrust bearing based on CFD. The bearing could tilt both radially

and circumferentially, allowing for three–dimensional temperature distribution in the

oil film and in the pad, as well as two–dimensional temperature variation in the run-

ner. Viscosity and density were treated as functions of both temperature and pressure.

Fairly good agreement between theoretical and experimental investigations was found.

In 2001, Almqvist and Larsson [1] investigated the use of the NS equations in the

solution of thermal, smooth line contact EHL problems. Cavitation was simulated by

modifying the density, using the Dowson-Higginson expression, see Hamrock [34], when

the pressure is above a specified cavitation pressure pcav. When the pressure fell below

pcav, a second order polynomial was used to interpolate the density down to zero. They

also investigated the presence of the singularity in the pressure gradient.

In 2003, van Odyck and Venner [67] used the Stokes equation (NS equation minus

inertia terms) to solve the EHL problem. In order to handle non-rectangular boundaries

they transformed the independent variables from Cartesian to curvilinear coordinates.

They also discussed the difference between the two-phase (TP) cavitation model and

the Reynolds (RR) cavitation model. With the TP model it is possible to simulate a


cavitated region inside the contact, which is not possible with the RR model. In the

numerical solver, multigrid method [69] was implemented.

Table 1.1 summarises the main work using CFD in fluid film lubrication to date.

Year Author Reference Area of Set of Thermal Viscosity Remarks

work equations effects Density

1985 Blahey, A. G. [8] EHL reduced Yes η = f(p, T ) used his

NS eqs. ρ = f(p, T ) own code

1992 Chang [14] EHL reduced Yes η = f(p, T ) Non-Newtonian

NS eqs. ρ = f(p, T ) rheology

1997 Zhang [74] HL full No η = const. fore-region in

Rodkiewicz NS eqs. ρ = const. thrust bearings

1998 Chen [15] HL full No η = const. CFD validation

Hahn NS eqs. ρ = const. for HL problem

2000 Shafer [61] EHL reduced No η = f(p) rolling vs.

et al. NS eqs. ρ = const. sliding

2000 Almqvist [2] HL reduced Yes η = f(p, T ) experiment vs.

NS eqs. ρ = f(p, T ) theory

2001 Almqvist [1] EHL full Yes η = f(p, T ) validation of CFD

Larsson NS eqs. ρ = f(p, T ) in EHL

2003 van Odyck [67] EHL Stokes eq. No η = f(p) Two–phase model

Venner ρ = f(p) for cavitation

Table 1.1: Overview of numerical work using CFD in fluid film lubrication.

1.3 Thesis Outline

The use of CFD/NS to study lubrication problems has a number of significant ad-

vantages compared to the conventional FD/RE approach. Firstly, it enables a single

model to be set up for a whole, macro–scale lubrication domain. Thus it is possible

to simulate a full roller bearing including the far inlet, near inlet, contact and outlet

regions. This would not be possible using RE since the far inlet region involves high

Reynolds number. In principle, the fluid flow in a whole engineering component might


be simulated.

Another potential advantage is in rough surface lubricated contact, where the lu-

bricant film thickness is comparable to the scale of the roughness. In such conditions,

some of the assumptions which lead to RE may no longer hold, e.g. neglect of inertia.

There are also some bearing geometries to which RE cannot be applied, e.g. rapidly

diverging regions within a pad bearing, where recirculation may take place. Other

systems which CFD can model relevant to film fluid lubricant include two phase flow

such as cavitation, or lubrication by emulsions or solid dispersions.

The overall objective of this research is to develop and validate a CFD–based solu-

tion to the HL and EHL problems. The specific aims are as follows:

• To develop a CFD methodology for complex geometries in the HL research;

• To expand the computational domain well outside the contact zone;

• To include the two-phase flow (i.e. cavitation) in the analysis;

• To develop a CFD methodology for smooth and rough surface EHL simulation

based on finite volume and multigrid techniques;

• To use this to test the validity of the assumption of constant pressure across the

thickness of the film.

Chapter 2

Governing Equations of Fluid Flow

2.1 Introduction

In this Chapter the governing equations of continuum mechanics will be introduced.

Since the analysis of the fluid flow will be done at macroscopic length scales (> 10−9 m),

the molecular structure of matter and individual molecular motions will be ignored.

The behaviour of the fluid will be described in terms of macroscopic properties, e.g.

velocity, pressure and density, and their space and time derivatives. All the cases in

this study will be regarded as isothermal and therefore the energy equation will not be

included.

Order of magnitude analysis will be carried out on the governing equations with

the characteristic values for lubrication. To analyse the equations, non–dimensional

variables are introduced. By doing this, it will be possible to assess the relative sig-

nificance of the various terms in each equation for different regions in the domain and

geometries of interest.

In the contact region, where the aspect ratio (ratio between characteristic length and

characteristic height) is large (≈ 103), the Reynolds number is very small (Re ≈ 10−3)

and certain terms from the NS equation may be neglected. The full set of governing

equations then reduces to the Reynolds Equation. However, this is valid only in the

contact region and assuming there is no steep surface roughness present.

If one wants to expand the domain further outwards into the inlet and outlet regions,

26

Chapter 2. Governing Equations of Fluid Flow 27

the convective terms in NS equations cease being negligible. In this Chapter, order of

magnitude analysis is conducted both for the contact and far regions, demonstrating

the need to use the full NS equations in the latter.

2.2 Governing Equations of Continuum Mechanics

All the equations in this Chapter will be presented both in vector notation and in the

expanded form in the Cartesian (x, y and z) coordinates. The vector notation will

be used in Chapter 3 as a suitable form in which we will describe the Finite Volume

Method. The expanded form of the equations in the Cartesian coordinates will be used

in this Chapter to perform the order of magnitude analysis.

2.2.1 Mass and Momentum Conservation

The governing equations of fluid flow represent mathematical statements of the con-

servation laws of physics (Versteeg, [70]):

• The mass of a fluid is conserved, i.e.

∂ρ

∂t+ ∇ · (ρU) = 0, (2.1)

or,

∂ρ

∂t+

∂ρu

∂x+

∂ρv

∂y+

∂ρw

∂z= 0. (2.2)

• The momentum is conserved:

– Conservation of linear momentum:

∂ρU

∂t+ ∇ · (ρUU) = ρg + ∇·σ, (2.3)

– Conservation of angular momentum:

∂ρ(x × U)

∂t+ ∇ · [ρ(x × U)U] = ρ(x × g) + x × (∇·σ). (2.4)


The conservation of momentum can be also written in Cartesian components as:

x-direction ρDu

Dt=

∂(−p + σxx)

∂x+

∂σyx

∂y+

∂σzx

∂z+ SMx, (2.5)

y-direction ρDv

Dt=

∂σxy

∂x+

∂(−p + σyy)

∂y+

∂σzy

∂z+ SMy, (2.6)

z-direction ρDu

Dt=

∂σxz

∂x+

∂σyz

∂y+

∂(−p + σzz)

∂z+ SMz, (2.7)

where

→ ρ is density,

→ x is the position vector,

→ x, y and z are the Cartesian components of the position vector x,

→ U is the velocity vector,

→ u, v and z are the Cartesian components of the velocity vector U,

→ σ is the stress tensor,

→ p is pressure, i.e. normal stress,

→ σij is the viscous stress component which acts in the j-direction on a surface

normal to the i-direction,

→ g is the body force,

→ SMi is the source term which includes contributions due to body forces,

acting in the i-direction.

The conservation laws expressed by Eqs. (2.1 to 2.7) are valid for any isothermal

continuum. The number of unknown quantities is, however, larger than the number of

equations in the system, making the system indeterminate.

2.2.2 Constitutive Relations for Newtonian Fluids

In order to close the system, it is necessary to introduce additional, so-called con-

stitutive relations. These depend on the properties of the continuous medium in


question. In the case of Newtonian fluids under isothermal conditions, the following

set of constitutive relations can be used:

• Dependence of density on pressure. One of the relationships used in the

EHL for mineral oil is given by Gohar [29]:

ρ(p) = ρ0

(

1 +0.6p

1 + 1.7p

)

, (2.8)

where p is pressure in GPa and ρ0 is the lubricant atmospheric density.

• Dependence of viscosity on pressure. In this study the Barus law (Cameron

[12]) is used:

η(p) = η0 exp(αp), (2.9)

where η(p) is the viscosity at gauge pressure p, η0 is the viscosity at atmospheric

pressure and α is a constant depending on oil, called the pressure-viscosity coef-

ficient.

• The Newton’s law of viscosity in vector notation:

σ = −

(

p +2

3η∇·U

)

I + η [∇U + (∇U)T ], (2.10)

or written in the expanded form:

σxx = 2η∂u

∂x−

2

3η∇U, σyy = 2η

∂v

∂y−

2

3η∇U, σzz = 2η

∂w

∂z−

2

3η∇U,

σxy = σyx = η

(∂u

∂y+

∂v

∂x

)

, σxz = σzx = η

(∂u

∂z+

∂w

∂x

)

,

σyz = σzy = η

(∂v

∂z+

∂w

∂y

)

, (2.11)

where the constant of proportionality between the viscous stress and the rate of

deformation is the dynamic viscosity η.

2.2.3 Navier-Stokes Equations

The constitutive relations given above, together with the governing equations in Sec-

tion 2.2.1 for a continuum create a closed system of partial differential equations for

Newtonian fluids:


• Continuity equation in vector notation:

∂ρ

∂t+ ∇ · (ρU) = 0, (2.12)

or in the expanded form:

∂ρ

∂t+

∂ρu

∂x+

∂ρv

∂y+

∂ρw

∂z= 0. (2.13)

• Navier-Stokes equations in vector notation:

∂ρU

∂t+ ∇ · (ρUU) = ρg −∇

(

P +2

3η∇ ·U

)

+ ∇ · [η (∇U + (∇U)T )], (2.14)

or in the expanded form:

x-direction∂ρu

∂t+ div(ρuu) = div(η gradu) −

∂p

∂x+ SMx (2.15)

y-direction∂ρv

∂t+ div(ρuvu) = div(η gradv) −

∂p

∂y+ SMy (2.16)

z-direction∂ρw

∂t︸︷︷︸

temporal

+ div(ρwu)︸︷︷︸

convective

= div(η gradw)︸︷︷︸

diffusive

−∂p

∂z+ SMz

︸︷︷︸

source

(2.17)

– In order to simplify the momentum equations, smaller contributions to the

viscous stress terms have been included in the momentum source, SMi.

In Eqs. (2.15 to 2.17), the Navier-Stokes equation is shown with four terms: the

temporal derivative, convective, diffusive and source terms. The temporal term governs

the rate of change of the property (in this case, velocity component) in time. The

convective term represents the rate of property change due to the flow through the

control volume. Diffusive term governs diffusion effects caused by the gradients (e.g.

shear stress) in the field. All the terms that cannot be grouped as convection or

diffusion form the source term.


2.3 Order of Magnitude Analysis

2.3.1 Non–Dimensional Variables

In order to perform an order of magnitude analysis on equations from Section 2.2,

non–dimensional variables are introduced. Non-dimensional variables are generally

used in numerical analysis. Their advantage is that the results have generality and the

problems of different systems of units are removed.

Assuming the characteristic values, the non–dimensional variables, denoted by sub-

script ∗, are shown as a ratio between the dimensional variable and their characteristic

value, denoted by subscript o:

Pressure p∗ =p

p0,

Density ρ∗ =ρ

ρ0

,

x - Cartesian coordinate x∗ =x

x0,

y - Cartesian coordinate y∗ =y

y0

,

z - Cartesian coordinate z∗ =z

z0,

velocity component in x-direction u∗ =u

u0

, (2.18)

velocity component in y-direction v∗ =v

v0,

velocity component in z-direction u∗ =w

w0

,

Dynamic viscosity η∗ =η

η0,

Pressure - viscosity coefficient α∗ =α

α0.

2.3.2 Characteristic Values

In both hydrodynamic and elastohydrodynamimc lubrication, we encounter geometries

where the aspect ratio (i.e. ratio between the length and the height) in contact zones

is of order of magnitude 103. Figure 2.1 shows a schematic picture of a contact divided

into two regions:


(a) outside a contact in the far zone, with aspect ratio ≈1;

(b) within a typical contact zone, with aspect ratio ≈103.

(a)(b)

(a)z

y

x

Figure 2.1: A schematic picture of a contact in fluid film lubrication.

The behaviour of lubricant in the two regions is significantly different, partly be-

cause of higher pressures in the contact zone, which may in turn cause elastic defor-

mation of the surfaces and changes in viscosity. The Reynolds number also varies by

several orders of magnitude between the contact zone and the far region. Table 2.1

contains characteristic values of the variables inside and outside of the contact.

Variable Outside the contact Inside the contact

x0 [m] 10−2 10−3

y0 [m] 10−2 10−3

z0 [m] 10−2 10−7

u0 [m/s] 1

v0 [m/s] 1

w0 = z0

x0

u0 [m/s] 1 10−4

t0 = x0

u0

= y0

v0

= z0

w0

[s] 10−2 10−3

η0 [Pas] 10−2

ρ0 [kg/m3] 103

α0 [Pa−1] 10−8

p0 [Pa] 105 108

Rex = u0x0ρ0

η0

103 102

Rey = v0y0ρ0

η0

103 102

Rez = w0z0ρ0

η0

103 10−2

Table 2.1: Characteristic values inside and outside the contact in a bearing.


2.4 Non–Dimensional Equations

In order to determine which terms in the equations are dominant, dimensional variables

in the governing equations will be replaced by the product of their non-dimensional

variable (see Eqn. 2.18) and the characteristic value. Different regions of the bearing

will be studied with their respective characteristic values. An exponential, Barus-type

viscosity-pressure relationship, Eqn. (2.9), is assumed.

Momentum equation. The Navier-Stokes equations, Eqs. (2.15 to 2.17), were writ-

ten in simplified form where the smaller viscosity terms were included in the source

term, SMi. However, in this Section the order of magnitude analysis will be conducted

on the equations where all the viscous terms are shown. The x-component of the

momentum equation is:

∂ρ∗u∗

∂t∗+

∂(ρ∗u∗u∗)

∂x∗

+v0

u0

x0

y0

∂(ρ∗v∗u∗)

∂y∗+

w0

u0

x0

z0

∂(ρ∗w∗u∗)

∂z∗−

21

Rex

exp(αp)∂2u∗

∂x2∗

−

1

Rexexp(αp)

∂

∂y∗

[(x2

0

y20

∂u∗

∂y∗+

v0

u0

x0

y0

∂v∗∂x∗

)]

−

1

Rexexp(αp)

∂

∂z∗

[(x2

0

z20

∂u∗

∂z∗+

w0

u0

x0

z0

∂w∗

∂x∗

)]

+ (2.19)

2

3

1

Rex

exp(αp)∂

∂x∗

[(∂u∗

∂x∗

+v0

u0

x0

y0

∂v∗∂y∗

+w0

u0

x0

z0

∂w∗

∂z∗

)]

= −p0

ρ0u20

∂p∗∂x∗


Similarly, the y-component can be written in a non-dimensional form as:

∂ρ∗v∗∂t∗

+

u0

v0

y0

x0

∂(ρ∗u∗v∗)

∂x∗

+∂(ρ∗v∗v∗)

∂y∗+

w0

v0

y0

z0

∂(ρ∗w∗v∗)

∂z∗−

1

Reyexp(αp)

∂

∂x∗

[(y2

0

x20

∂v∗∂x∗

+u0

v0

y0

x0

∂u∗

∂y∗

)]

−

21

Reyexp(αp)

∂2v∗∂y2

∗

−

1

Rey

exp(αp)∂

∂z∗

[(y2

0

z20

∂v∗∂z∗

+w0

v0

y0

z0

∂w∗

∂y∗

)]

+ (2.20)

2

3

1

Rey

exp(αp)∂

∂y∗

[(u0

v0

y0

x0

∂u∗

∂x∗

+∂v∗∂y∗

+w0

v0

y0

z0

∂w∗

∂z∗

)]

= −p0

ρ0v20

∂p∗∂y∗

,

while the z-component is:

∂ρ∗w∗

∂t∗+

u0

w0

z0

x0

∂(ρ∗u∗w∗)

∂x∗

+v0

w0

z0

y0

∂(ρ∗v∗w∗)

∂y∗+

∂(ρ∗w∗w∗)

∂z∗−

1

Rezexp(αp)

∂

∂x∗

[(z20

x20

∂w∗

∂x∗

+u0

w0

z0

x0

∂u∗

∂z∗

)]

−

1

Rezexp(αp)

∂

∂y∗

[(z20

y20

∂w∗

∂y∗+

v0

w0

z0

y0

∂v∗∂z∗

)]

−

21

Rezexp(αp)

∂2w∗

∂z2∗

+ (2.21)

2

3

1

Rezexp(αp)

∂

∂z∗

[(u0

w0

z0

x0

∂u∗

∂x∗

+v0

w0

z0

y0

∂v∗∂y∗

+∂w∗

∂z∗

)]

= −p0

ρ0w20

∂p∗∂z∗

.

The continuity equation can be written in non-dimensional form as:

∂ρ∗

∂t∗+

∂ρ∗u∗

∂x∗

+x0

y0

v0

u0

∂ρ∗v∗∂y∗

+x0

z0

w0

u0

∂ρ∗w∗

∂z∗= 0, (2.22)

while the viscosity dependence on pressure is:

η∗(p) = exp(αp). (2.23)

Non-dimensional form of the density dependence on pressure is written as:

ρ∗(p) = 1 +0.6p0

1 + 1.7p0. (2.24)


2.5 Non–dimensional Equations with

Characteristic Values

Characteristic values for the region of interest are now substituted in Eqs. (2.19 to

2.24), to identify the dominant terms for each equation. Printed in bold are the relative

magnitudes of each term.

2.5.1 Outside the Contact

The x-component of the momentum in the outside the contact region is:

1∂ρ∗u∗

∂t∗+

1∂(ρ∗u∗u∗)

∂x∗

+ 1∂(ρ∗v∗u∗)

∂y∗+ 1

∂(ρ∗w∗u∗)

∂z∗−

10−3 exp(αp)∂2u∗

∂x2∗

−

10−3 exp (αp)∂

∂y∗

(

1∂u∗

∂y∗+ 1

∂v∗∂x∗

)

−

10−3 exp (αp)∂

∂z∗

(

1∂u∗

∂z∗+ 1

∂w∗

∂x∗

)

+ (2.25)

10−3 exp (αp)∂

∂x∗

(

1∂u∗

∂x∗

+ 1∂v∗∂y∗

+ 1∂w∗

∂z∗

)

= −102∂p∗∂x∗

,

the y-component is written as:

1∂ρ∗v∗∂t∗

+

1∂(ρ∗u∗v∗)

∂x∗

+ 1∂(ρ∗v∗v∗)

∂y∗+ 1

∂(ρ∗w∗v∗)

∂z∗−

10−3 exp (αp)∂

∂x∗

(

1∂v∗∂x∗

+ 1∂u∗

∂y∗

)

−

10−3 exp (αp)∂2v∗∂y2

∗

−

10−3 exp (αp)∂

∂z∗

(

1∂v∗∂z∗

+ 1∂w∗

∂y∗

)

+ (2.26)

10−3 exp (αp)∂

∂y∗

(

1∂u∗

∂x∗

+ 1∂v∗∂y∗

+ 1∂w∗

∂z∗

)

= −102∂p∗∂y∗

.


The z-component of the Navier-Stokes equation outside the contact is:

1∂ρ∗w∗

∂t∗+

1∂(ρ∗u∗w∗)

∂x∗

+ 1∂(ρ∗v∗w∗)

∂y∗+ 1

∂(ρ∗w∗w∗)

∂z∗−

10−3 exp (αp)∂

∂x∗

(

1∂w∗

∂x∗

+ 1∂u∗

∂z∗

)

−

10−3 exp (αp)∂

∂y∗

(

1∂w∗

∂y∗+ 1

∂v∗∂z∗

)

−

10−3 exp (αp)∂2w∗

∂z2∗

+ (2.27)

10−3 exp (αp)∂

∂z∗

(

1∂u∗

∂x∗

+ 1∂v∗∂y∗

+ 1∂w∗

∂z∗

)

= −102∂p∗∂z∗

.

The continuity equation with the characteristic values for the outside of the contact is:

1∂ρ∗

∂t∗+ 1

∂ρ∗u∗

∂x∗

+ 1∂ρ∗v∗∂y∗

+ 1∂ρ∗w∗

∂z∗= 0, (2.28)

while viscosity and density are constant:

η∗(p) = exp(αp) = 1, (2.29)

ρ∗(p) = 1. (2.30)

2.5.2 Dominant Terms Outside the Contact

Since the pressures outside the contact region are of order of p = 105 Pa, non-

dimensional viscosity and density are effectively constant. From Eqs. (2.25 to 2.27),

it can be seen that the diffusive terms have an order of magnitude three times smaller

than the convective terms in the momentum equations outside the contact. This is

expected, considering that the Reynolds number is Re ≈ 103.


2.5.3 Inside the Contact

The x-component of the momentum in the inside the contact region is:

1∂ρ∗u∗

∂t∗+

1∂(ρ∗u∗u∗)

∂x∗

+ 1∂(ρ∗v∗u∗)

∂y∗+ 1

∂(ρ∗w∗u∗)

∂z∗−

10−2 exp (αp)∂2u∗

∂x2∗

−

10−2 exp (αp)∂

∂y∗

(

1∂u∗

∂y∗+ 1

∂v∗∂x∗

)

−

10−2 exp (αp)∂

∂z∗

(

108∂u∗

∂z∗+ 1

∂w∗

∂x∗

)

+ (2.31)

10−2 exp (αp)∂

∂x∗

(

1∂u∗

∂x∗

+ 1∂v∗∂y∗

+ 1∂w∗

∂z∗

)

= −105∂p∗∂x∗

.

According to the dominant terms from Eqn. (2.31), the dimensional form of the x-

momentum reduces to:

η∂2u

∂z2= −

∂p

∂x. (2.32)

Similarly, the y-component of the momentum, expressed with the characteristic values

for the inside of the contact is:

1∂ρ∗v∗∂t∗

+

1∂(ρ∗u∗v∗)

∂x∗

+∂(ρ∗v∗v∗)

∂y∗+ 1

∂(ρ∗w∗v∗)

∂z∗−

10−2 exp (αp)∂

∂x∗

(

1∂v∗∂x∗

+ 1∂u∗

∂y∗

)

−

10−2 exp (αp)∂2v∗∂y2

∗

−

10−2 exp (αp)∂

∂z∗

(

108∂v∗∂z∗

+ 1∂w∗

∂y∗

)

+ (2.33)

10−2 exp (αp)∂

∂y∗

(

1∂u∗

∂x∗

+ 1∂v∗∂y∗

+ 1∂w∗

∂z∗

)

= −105∂p∗∂y∗

.

According to the dominant terms from Eqn. (2.33), the dimensional form of the y-

momentum reduces to:

η∂2v

∂z2= −

∂p

∂y, (2.34)


while the z-component is:

1∂ρ∗w∗

∂t∗+

1∂(ρ∗u∗w∗)

∂x∗

+ 1∂(ρ∗v∗w∗)

∂y∗+

∂(ρ∗w∗w∗)

∂z∗−

102 exp (αp)∂

∂x∗

(

10−8∂w∗

∂x∗

+ 1∂u∗

∂z∗

)

−

102 exp (αp)∂

∂y∗

(

10−8∂w∗

∂y∗+ 1

∂v∗∂z∗

)

−

102 exp (αp)∂2w∗

∂z2∗

+ (2.35)

102 exp (αp)∂

∂z∗

(

1∂u∗

∂x∗

+ 1∂v∗∂y∗

+ 1∂w∗

∂z∗

)

= −1013∂p∗∂z∗

.

In Eqn. (2.35), the magnitude of the scaling factor of the r.h.s. is 11 orders of magni-

tude greater than that of the l.h.s. In order for the l.h.s. to be equal to the r.h.s., the

following must hold:

∂p

∂z≈ 0. (2.36)

The continuity equation inside the contact region is:

1∂ρ∗

∂t∗+ 1

∂ρ∗u∗

∂x∗

+ 1∂ρ∗v∗∂y∗

+ 1∂ρ∗w∗

∂z∗= 0, (2.37)

while viscosity dependence on pressure becomes:

η∗(p∗) = exp(αp), (2.38)

and density dependence on pressure, according to Eqn. (2.24), assumes the following

values:

ρ =

1.03 for p = 0.1 GPa,

1.2 for p = 1 GPa.

(2.39)


2.5.4 Dominant Terms Inside the Contact

In the contact, the pressure can rise from the order of magnitude of 107 Pa to 109 Pa.

That implies that the non-dimensional viscosity, Eqn. (2.38), can assume values from

1 to 104. One has to bear in mind that the Barus law for viscosity dependence on

pressure, Eqn. (2.9), generally becomes inaccurate above 0.5 GPa and even more so if

the ambient temperature is high.

From Eqs. (2.31 to 2.35), it can be seen that inside the contact, the diffusive terms

dominate in the momentum equations. This is expected, considering that the Reynolds

number is Re ≈ 10−2. Due to the very high local pressure in the contact, fluid viscosity

(2.38) and density (2.39) are no longer constant, but dependent on pressure.

2.6 Closure

In this Chapter the governing equations for the flow of a lubricant in a bearing were

given. These equations were then non-dimensionalised and order of magnitude analysis

was carried out for the two regions of the bearing: the contact region and the region

outside the contact.

The order of magnitude analysis has shown that the Reynolds number based on film

height varies over the full computational domain from 10−2 to 103. Because of such a

big variation, different terms dominate the governing equations for the fluid flow in the

two regions. In the region outside the contact, convective terms are dominant, whereas

in the contact region diffusive terms are greater. This is one of the reasons why it is

necessary to use the full set of Navier-Stokes equations when solving the fluid flow over

the entire bearing and not only in the contact region.

In the high-pressure, contact region, the aspect ratio between film thickness, z0,

and the characteristic length, x0 is of the order of 10−4. In that region, if the surfaces

are so smooth that the minimum film thickness in the problem is large compared

to the surface roughness, Navier-Stokes equations simplify to Eqs. (2.32, 2.34 and


2.36), from which the Reynolds equation is obtained (see Chapter 4). However, if the

characteristic length of the surface roughness (x-dimension) is comparable to the local

film thickness, the local aspect ratio becomes much higher and certain viscous terms

in Navier-Stokes equations cannot be neglected (van Odyck [66]). Reynolds number is

still Rez � 1 which means that the convective terms remain negligible. The Reynolds

equation, therefore, cannot be used for calculation of the flow parameters in the contact

region with surface roughnesses, but Navier-Stokes equations can be reduced to Stokes

equations (i.e., the convective terms may be omitted).

Chapter 3

Finite Volume Discretisation

3.1 Introduction

In Chapter 2, fluid flow for situations relevant for lubrication, was described by a set

of partial differential equations which cannot be solved analytically. To obtain approx-

imate solutions numerically, we must use a discretisation method which approximates

the differential equations by a system of algebraic equations. The solution of this

system produces values at discrete locations in space and time.

One has to bear in mind that the numerical results are always approximate. The ac-

curacy of numerical solution depends on the quality of discretisations used. The errors

arise from the fact that differential equations describing the flow may contain approx-

imations and idealisations; from the approximations in the discretisation method, and

from convergence criteria, which may stop the iterations before the exact solution of

discretised equations is found.

3.1.1 Components of a Numerical Solution Method

The steps towards the numerical solution method are as follows:

Mathematical Model. It is the set of equations and boundary conditions that de-

scribe a particular problem. As mentioned before, this model may include simplifica-

tions and idealisations of the exact conservation laws. The solution method is usually

designed for a particular set of equations.

41

Chapter 3. Finite Volume Discretisation 42

Coordinate System. Different coordinate systems (e.g. Cartesian, cylindrical, spher-

ical, etc.) may be used depending on the form of the governing equations and on the

geometry of the problem. In this work, the coordinate system employed will be Carte-

sian.

Discretisation Method. One has to select an appropriate discretisation method

for the mathematical model chosen. The Finite Volume Method (FVM) will be used

in this work, which consists of the discretisation of the solution domain and equation

discretisation (Muzaferija [51]). The FVM consists of the following steps:

• Spatial Discretisation. Since the Finite Volume Method is used in the work,

the solution domain has to be divided into a finite number of subdomains, called

control volumes (CV). Each control volume can be of any polyhedral shape with

variable number of neighbours, but not overlapping with them. The grid used in

this study will be a block-structured grid, with a two level subdivision of solution

domain. The domain is divided coarsely into large segments, or blocks, which are

then subdivided into control volumes. By doing that, one can easily introduce a

much finer grid in areas where a high resolution is required.

• Finite Approximations. According to the grid type and discretisation method,

approximations used in the discretisation process must be selected. In the FVM,

one must select the methods of approximating surface and volume integrals. Some

of them will be briefly mentioned in the following Sections. The choice of ap-

proximation greatly influences the accuracy of the numerical solution. The more

accurate an approximation it is, the more computational work it requires. In this

study, the second-order approximation method will be used.

• Convergence Criteria. The convergence criteria must be set for the iterative

method. In the solution algorithms that will be used in this work, there are

two levels of iteration: inner iterations, within which the linear equations are


solved, and outer iterations, that deal with the non-linearity and coupling of the

equations. It is important to consider both the accuracy and efficiency when

deciding to stop the iterative process on each level.

Solution Method Methods of solving the system of algebraic equations can be di-

vided into two categories:

1. Solution of the linear system;

2. Solution methods for handling multiple coupled equations (not in a single linear

system).

The problems which are encountered in CFD work use the latter methods. The two of

such methods, which will be used in this work are:

• SIMPLE algorithm, for steady-state, laminar, non-cavitating flows,

• PISO algorithm, for transient flows and flows with cavitation.

3.1.2 Properties of The Numerical Solution Method

In order for the solution method (in this work, FVM) to be acceptable, it must possess

certain properties. They are listed below:

Consistency. For a method to be consistent, the truncation error, i.e. the difference

between the discretised equation and the exact one, must become zero when the mesh

spacing tends to zero. Truncation error is usually proportional to a power of the grid

spacing ∆x and/or the time step ∆t. The method is called an n-th order approximation

if the leading term in the truncation error is proportional to (∆x)n. In this work, we

are going to use second order approximation throughout.

One has to bear in mind that even if the approximations are consistent, it does not

necessarily mean that the solution will become exact as ∆x → 0. For this to happen,

the solution must also be stable, as defined below.


Stability. A numerical solution method is stable if it does not increase the errors

that appear in the numerical solution process. For temporal problems, the stable

method will produce a bounded solution whenever the solution of the exact equation is

bounded. For iterative methods, a stable method is one that does not diverge. Stability

can often be difficult to investigate. However, it is known that many solution schemes

require the time step to be smaller than a certain limit or that under-relaxation must

be used.

Convergence. A numerical method is convergent if the solution of the discretised

equations tends to the exact solution of the differential equation as the grid spacing

tends to zero. Convergence is usually checked using numerical experiments, i.e. re-

peating the calculation on a series of successively refined grids. If the method is stable

and if all approximations used in the discretisation process are consistent, the solution

usually converges to a grid-independent solution.

Conservation. The numerical method, both on a local and a global basis, should

respect the conservation laws that differential equations represent. The finite volume

method used in this work is conservative both for each individual control volume and

for the solution domain as a whole.

Boundedness. Numerical solutions should lie within proper bounds. That means

that physically non-negative quantities (e.g. density) must always be positive; other

quantities, e.g. concentration must lie between 0% and 100%.

Realisability. In this work, we will have to solve the system in which cavitation

occurs. That problem is too complex to be treated directly, and the method designed

must instead guarantee physically realistic solution. This, itself, is not a numerical

issue but models that are not realisable may result in unphysical solutions or cause

numerical methods to diverge.


Accuracy. Numerical solutions of fluid flow are only approximate solutions. They

always include three kinds of systematic errors (Ferziger and Peric [28]):

• Modeling errors, i.e. the difference between the actual flow and the exact solution

of the mathematical model;

• Discretisation errors, i.e. the difference between the exact solution of the con-

servation equations and the exact solution of the algebraic system of discretised

equations;

• Iteration errors, i.e. the difference between the iterative and exact solutions of

the algebraic equations systems.

It is important to be aware of these errors and to try to distinguish one from another.

For example, modeling errors are negligible in case of laminar flows, since the

Navier-Stokes equations represent an accurate model of the flow. However, in cavi-

tating flows, the modeling error may be very large, making the exact solution of the

numerical model qualitatively wrong. This type of error is also introduced by simplify-

ing the geometry of the solution domain, simplifying boundary conditions, etc. These

errors are not known a priori; they can only be evaluated by comparing numerical so-

lutions in which the discretisation and convergence errors are negligible with accurate

experimental results.

3.2 Spatial Discretisation

In the Finite Volume Method, discretisation of the solution domain produces a number

of discrete points on which the governing equations are solved. It is done by dividing

the domain into a finite number of control volumes (CV), and the conservation equa-

tions are applied to each CV. Control volumes do not overlap and completely fill the

computational domain.


Even though the control volume can be a general polyhedron, in the present study

all the CV-s are of a hexahedral shape, as shown in Figure 3.1. The computational point

P is the centroid of the control volume at which all the variable values are calculated.

��

��

z

y

x

f

n

P

Nf

fdS

f

VP

Figure 3.1: Control volume

The control volume is bounded by a set of flat faces and each face is shared with

only one neighbouring CV. The face area vector nf is constructed for each face in such

a way that it points outwards from the ”owner” and towards the ”neighbour” cell, is

normal to the face and has the magnitude equal to the area of the face, Sf . For the

shaded face in Fig. 3.1, the owner and neighbour cell centres are marked with P and

N respectively. For simplicity, all the faces of the control volume will be marked with

f , which also represents the point in the middle of the face (see Fig. 3.1).

The finite volume method can accomodate any type of grid, making it suitable for

complex geometries. The mesh can be refined locally, with the computational points

added only in parts of the domain where higher resolution is necessary.

3.3 Discretisation of the Governing Equations

In the previous Chapter the three components of the momentum equation were given

by Eqs. (2.15 to 2.17). It is useful to introduce a general variable φ, so that the


conservative form of all fluid flow equations (including also energy equation, etc.) can

be written as follows:

∂ρφ

∂t︸︷︷︸

temporal

+∇ · (ρUφ)︸︷︷︸

convection

−∇ · (ρΓφ∇φ)︸︷︷︸

diffusion

= Sφ(φ)︸︷︷︸

source

. (3.1)

In words

Rate of change Net rate of flow Rate of change Rate of change

of φ in fluid + of φ in or out of - of φ due to = of φ due to

element fluid element diffusion sources or sinks

Eqn. (3.1) is called a transport equation for property φ. It is used as the starting

point for computational procedures in the FVM. By setting φ equal to 1, u, v and w (if

thermal effects are included, also T ) and selecting appropriate values for the diffusion

coefficient Γ and source terms, Eqs. (2.13 to 2.17) are obtained.

The FVM uses the integral form of Eqn. (3.1) as the starting point:

∂

∂t

∫

VP

ρφ dV +

∫

VP

∇ · (ρUφ)dV −

∫

VP

∇ · (ρΓφ∇φ)dV

=

∫

VP

Sφ(φ)dV (3.2)

Volume integrals in the convective and diffusive terms are re-written as integrals

over the bounding surface of the CV by using the Gauss’ divergence theorem:

∫

CV

∇·a dV =

∮

∂V

dn · a, (3.3)∫

CV

∇φ dV =

∮

∂V

dnφ, (3.4)∫

CV

∇ a dV =

∮

∂V

dna, (3.5)

where ∂V is the closed surface bounding the volume V and dn represents an infinites-

imal surface element with associated outward pointing normal on ∂V .


Since the diffusion term includes the second derivative of φ in space, this is a second-

order equation. To ensure consistency, the order of the discretisation must be of equal

or higher order than the order of the equation that is being discretised.

If we assume that φ = φ(x, t) varies linearly in space and time around the point P ,

it can be written:

φ(x) = φP + (x − xP ) · (∇φ)P , (3.6)

φ(t + ∆t) = φt + ∆t

(∂φ

∂t

)t

, (3.7)

where

φP = φ(xP ), (3.8)

φt = φ(t). (3.9)

The discretisation in this study is therefore second-order accurate in space and time,

since the most dominant term of the truncation error in Taylor series is proportional

to (x− xP )2, which for a 1-D situation is equal to the square of the size of the control

volume:

φ(x) = φP + (x − xP ) · (∇φ)P +1

2(x − xP )2 · (∇∇φ)P + ...

︸︷︷︸

truncation error

, (3.10)

The equivalent analysis shows that the truncation error in Eqn. (3.7) is proportional

to ∆t2, resulting in the second-order temporal accuracy.

From Eqn. (3.6), it follows that:∫

VP

φ(x) dV =

∫

VP

[φP + (x − xP ) · (∇φ)P ] dV

= φP

∫

VP

dV +

[∫

VP

(x − xP )dV

]

· (∆φ)P

= φPVP , (3.11)

where VP is the volume of the cell, since the point P is the centroid of the control

volume:∫

VP

(x − xP ) dV = 0. (3.12)


In order to obtain a discretised form of the Gauss’ theorem, Eqn. (3.3) can be

transformed into a sum of integrals over all faces:

∫

CV

∇·a dV =

∮

∂V

dn · a,

=∑

f

(∫

f

dn · a

)

. (3.13)

If we assume that |a| varies linearly in space, the face integral in Eqn. (3.13) can be

written as:

∫

f

dn · a =

(∫

f

dn

)

· af +

[∫

f

dn(x − xf)

]

· (∇a)f

= n · af (3.14)

By combining Eqs. (3.11, 3.13 and 3.14), the following is obtained:

(∇·a)VP =∑

f

n · af , (3.15)

where af is the value of the variable a in the middle of the face f and n is the outward-

pointing face area vector.

3.3.1 Convection Term

The discretisation of the convection term is obtained using Eqn. (3.15):

∫

VP

∇·(ρUφ)dV =∑

f

n·(ρUφ)f

=∑

f

n·(ρU)fφf

=∑

f

Fφf , (3.16)

where F is the mass flux through the face:

F = n·(ρU)f (3.17)

Convection Differencing Scheme

From Eqn. (3.16), it is clear that the approximation to the integral form of the con-

vection term requires the values of variables at locations other than computational


nodes (i.e. CV centres). These values will have to be expressed in terms of the nodal

values by interpolation. Numerous schemes are available, but in this study differencing

schemes using only the nearest neighbours of the control volume will be employed.

In order for the scheme to be valid, the boundedness of the discretised convection

term must be preserved, i.e. if φ initially varies between two bounds, the convection

term will never produce the values of φ that fall outside of the interval between those

values.

φP

φN

f

dNP

φf

Figure 3.2: Face Interpolation

Assuming the linear variation of φ between P and N , as shown in Fig. 3.2, the face

value is calculated according to:

φf = fxφP + (1 − fx)φN , (3.18)

where fx is the ratio of distances fN and PN :

fx =fN

PN(3.19)

The differencing scheme described by Eqn. (3.18) is called Central Differencing

(CD). One of the major inadequacies of the CD scheme is its inability to identify flow

direction. The value of the property φ at the cell face is always influenced by both

φP and φN using this scheme. In a strongly convective flow from P to N , the above

method is unsuitable because the cell face should receive much stronger influence from

node N than from node P . That causes unphysical oscillations in the solution for

convection-dominated problems (Patankar [52]), thus violating the boundedness of the


solution. A more thorough description of the CD scheme can be found in Ferziger and

Peric [28].

Boundedness can be secured with the Upwind Differencing (UD) scheme. This

scheme takes into account the flow direction when determining the value at a cell face;

the convected value of φ at a cell face is taken to be equal to the value at the upstream

node:

φf =

φf = φP for F ≥ 0,

φf = φN for F < 0,

(3.20)

Even though boundedness of the solution is guaranteed under UD scheme (Patankar

[52]), the accuracy is affected by implicitly introducing the numerical diffusion term.

This term violates the order of accuracy of the discretisation and can severely distort

the solution.

3.3.2 Diffusion Term

The diffusion term will be discretised using the assumption of linear variation of φ and

Eqn. (3.15):

∫

VP

∇ · (ρΓφ∇φ)dV =∑

f

n · (ρΓφ∇φ)f

=∑

f

(ρΓφ)fn · (∇φ)f . (3.21)

P N

n

f

d

Figure 3.3: Two neighbouring nodes, P and N in a non-orthogonal mesh

In case of the orthogonal mesh, vectors d and n in Fig. 3.3 are parallel and the


following stands:

n · (∇φ)f = |n|φN − φP

|d|. (3.22)

An alternative would be to calculated the cell-centred gradient for the two cells sharing

the face as:

(∇φ)P =1

VP

∑

f

nφf , (3.23)

and interpolate it to the face:

(∇φ)f = fx(∇φ)P + (1 − fx)(∇φ)N . (3.24)

Although both of these methods are second-order accurate, Eqn. (3.24) uses more

computational time and its truncation error is four times larger than in the first method.

However, the first method described with Eqn. (3.22) cannot be used on non-orthogonal

meshes which are normally present in CFD.

In order to make use of the higher accuracy of Eqn. (3.22), the product n·(∇φ)f is

split into two parts:

n · (∇φ)f = ∆ · (∇φ)f︸︷︷︸

orthogonal contribution

+ k · (∇φ)f︸︷︷︸

non-orthogonal contribution

, (3.25)

where the two vectors ∆ and k must satisfy the following:

n = ∆ + k. (3.26)

Vector ∆ is chosen to be parallel to d, which allows for use of Eqn. (3.22) on the

orthogonal contribution, limiting the less accurate method only to the non-orthogonal

part. There are many ways in which n can be deconstructed. One of them, the over-

relaxed approach is shown in Figure 3.4, in which both orthogonal and non-orthogonal

contributions are increased. Different approaches are appropriate for different problems

(Jasak [42]), with the final form being the same for all of them:

n · (∇φ)f = |∆|φN − φP

|d|+ k · (∇φ)f . (3.27)


NP

n

df ∆

k

Figure 3.4: Over-relaxed approach in non-orthogonality treatment.

3.3.3 Source Term

All terms in the transport equation, except diffusion, convection and temporal terms,

are included in the source term, Sφ(φ). Before discretising it, the FVM approximates

the source term by means of a linear form:

Sφ(φ) = Su + SP φ, (3.28)

where Su and SP can depend on φ. The volume integral of Eqn. (3.28) is written as:

∫

VP

Sφ(φ)dV = SuVP + SP VP φP . (3.29)

3.3.4 Temporal Discretisation

So far, in this Section, convection, diffusion and source terms have been discretised.

Using Eqs. (3.17, 3.27 and 3.29), and assuming that the control volumes do not change

in time, the integral form of transport equation, Eqn. (3.2) can be written as:

∂

∂t(ρP φP VP ) = −

∑

f

Fφf +∑

f

(ρΓφ)f n · (∇φ)f

+ SuVP + SP VP φP (3.30)

For simplicity, the r.h.s. of the Eqn. (3.30) will be denoted by M(t):

∂

∂t(ρP φP VP ) = M(t). (3.31)

Temporal discretisation of Eqn. (3.31) can be conducted in various ways, some of which

are presented below:


Euler Explicit Method In this method the face values of φ and ∆φ are determined

from the old-time field (superscript o). The linear part of the source term is also

evaluated using the old-time value. That is why the r.h.s. is denoted with M(to):

ρnP φn

P − ρoP φo

P

∆tVP = M(to). (3.32)

The new value of φP can be calculated directly - it is not necessary to solve the system

of linear equations. The disadvantage of this method is the Courant number limit. The

Courant number is defined as:

Co =Uf · d

∆t, (3.33)

where Uf is the velocity interpolated on the face f . If the Courant number is larger

than unity, the explicit system becomes unstable. This puts a severe limitation in

choice of the size of the time step, ∆t.

Euler Implicit Method In this method, the temporal discretisation is the same as

in the previous one, but the face-values are expressed in terms of the new time-level

cell values, hence the r.h.s. is denoted by M(to):

ρnP φn

P − ρoP φo

P

∆tVP = M(tn). (3.34)

This is still a first-order accurate method, but unlike the explicit, this method creates

a system of equations which have this form:

aP φnP +

∑

N

aNφnN = RP . (3.35)

The coupling of the system is much stronger than in the explicit approach, and the

system is stable even if the Courant number limit is violated. Unlike the explicit

method, this form of temporal discretisation guarantees boundedness.

Crank-Nicholson Method The temporal discretisation is the same as in the pre-

vious two methods, and the face-values are expressed as an average between new- and


old-time values:

ρnP φn

P − ρoP φo

P

∆tVP =

1

2[M(to) + M(tn)]. (3.36)

This method is second-order accurate in time and stable, but the boundedness of the

solution is not guaranteed.

If M(t) in Eqn. (3.36) is substituted by the r.h.s. from Eqn. (3.30), the following

expression is obtained:

ρnP φn

P − ρoP φo

P

∆tVP +

∑

f

Fφnf

∑

f

(ρΓφ)f n · (∇φ)nf

+∑

f

Fφof

∑

f

(ρΓφ)f n · (∇φ)of

= SuVP + SP VP φnP + SPVP φo

P (3.37)

3.3.5 Implementation of Boundary Conditions

In order to solve a discretised form of transport equation, we must determine φf and

nf ·(∇φ)f , i.e. the value of the variable φ and its normal gradient on face f .

Previously in this Section, it was shown how to find those values for the internal

mesh faces. However, for the faces located on the boundary of the domain, those values

are calculated from the boundary conditions.

There are two basic types of boundary conditions; a Dirichlet boundary condition

sets the value of the variable at the boundary, while a Neumann boundary condition

sets the value of its normal gradient at the boundary. A control volume with face

b located at the boundary of spatial domain is shown in Fig. 3.5. Vector db = Pb

connects the centroid P with the centre of the face b, while vector dn is parallel to

normal vector nb and defined as follows:

dn = (nb · db)nb. (3.38)

It is further assumed that the value which is set at the boundary face is constant over

that entire face. The boundary conditions are implemented into discretised equations

in the following way:


P

b

bb

n

S

n

d d

b

Figure 3.5: Parameters at the face boundary

• Dirichlet boundary condition

The value of variable φ is given at the boundary face b as φ = φb, which has to be

respected when discretising convection and diffusion term at the boundary face.

– Convection Term. This term is discretised according to Eqn. (3.16):

∫

VP

∇·(ρUφ)dV =∑

f

Fφf .

Since φf = φb at face b, the convection term at the boundary face is:

∫

VP

∇·(ρUφ)dV = Fbφb, (3.39)

where Fb is the face flux.

– Diffusion Term. This term is discretised according to Eqn. (3.21):

∫

VP

∇ · (ρΓφ∇φ)dV =∑

f

(ρΓφ)fn · (∇φ)f .

The normal gradient at face b is given as:

nb · (∇φ)b = Sb ·φb − φP

|dn|(3.40)

It follows that the diffusion contribution at the boundary face b is:

(ρΓφ)b Sbφb − φP

|dn|(3.41)


• Von Neumann boundary condition The normal gradient of the variable φ is

prescribed at the boundary face as:

nb·(∇φ)b = gb. (3.42)

Depending on which term this conditions is applied, that is done in the following

way:

– Convection term. In order to obtain the discretised value of the convection

term, it is necessary to find the value of φ at the boundary face:

φb = φP + |dn| gb, (3.43)

which means that the convection contribution at the boundary face will be

the following:

Fb (φP + |dn| gb). (3.44)

– Diffusion term. From Eqn. (3.42), it follows that the diffusion term at

the boundary face is:

(ρΓφ)b Sb gb. (3.45)

Physical Boundary Conditions

For incompressible flow, physical boundary conditions used in this study, are as follow-

ing:

• Inlet boundary. At the inlet boundary the velocity field has a fixed value, while

the pressure has the fixed zero gradient boundary condition.

• Outlet boundary. Mass conservation has to be satisfied when considering the

outlet boundary. There are two possibilities:

– The velocity distribution at the boundary is the same as in the row of cells

adjacent to the boundary. The pressure has again zero gradient condition.


This approach can become unstable if there is an inflow through the face

specified as the outlet. This can be handled with additional modifications.

– The pressure distribution is specified and there is zero gradient boundary

condition on velocity. The mass conservation is guaranteed by the solution

of pressure equation.

• Symmetry plane boundary. The component of the gradient normal to the

boundary should be fixed to zero. The components parallel to the boundary are

projected to the boundary face from the inside of the domain, i.e. it is treated

by considering a mirror image cell beyond the boundary face.

• No-slip at the walls. The velocity of the fluid on the wall is equal to the

velocity of the wall itself. Since the flux through the solid wall is zero, the

pressure gradient condition is zero gradient.

3.4 System of Linear Algebraic Equations

In Eqn. (3.37), φnf and nn

f·(∇φ)n

f are dependent on the values of variable φ in the

neighbouring control volumes at the time tn. This relationship can be written as a

linear algebraic equation:

aP φnP +

∑

N

aNφnN = RP , (3.46)

where aP is a diagonal coefficient, aN a neighbouring coefficient, and RP on r.h.s.

contains all the other components of the transport equations which are not treated

explicitly. For each control volume there is one linear equation of the form of Eqn.

(3.46). Thus assembling the system of linear algebraic equations:

[A ] {φ} = {R }. (3.47)

The matrix [A ] contains coefficients aP on its diagonal, while above and below are

coefficents aN . Vector {φ} contains the values of φ for all the control volumes in the


mesh, while vector R represents the source term. The matrix [A ] is a sparse matrix in

which most of the matrix coefficients are equal to zero.

This system of equations can be solved in many different ways which fall into two

categories: direct and iterative methods. Direct methods are suitable for small systems,

since the number of operations necessary to reach a solution is, at best, equal to the

number of the equations squared, thus making it too expensive for large systems.

In this study, iterative methods are used, which start with an initial guess and

improve it until solution tolerance criteria are met. Iterative methods are more eco-

nomical than direct ones. However, they require diagonal dominance of the matrix [A ]

to guarantee convergence. A matrix is diagonally equal if, in each row, the magnitude

of the diagonal coefficient is equal to the sum of magnitudes of off-diagonal coefficients:

|aP | =∑

N

|aN |. (3.48)

In order to be diagonally dominant, the following must stand for at least one row of

the matrix:

|aP | >∑

N

|aN |. (3.49)

The solver convergence is improved with the increase of diagonal dominance of the

system. Each component of the transport equation has the following influence on the

diagonal dominance of the system:

• Temporal term. Discretisation of the temporal term contributes only to the

diagonal coefficient and the source term of the system, and therefore increases

diagonal dominance. From Eqn. (3.37), it follows that the discretisation of the

temporal term contributes withρnV n

P

∆tto the diagonal coefficient. Therefore, with

the decrease of the time step ∆t, diagonal dominance of the system is increased.

• Convection term. The convection term creates a diagonally equal matrix only

for the Upwind Differencing (UD) scheme. In case of the Central Differencing

scheme, the matrix does not guarantee boundedness. There are various methods


to improve the quality of the matrix for higher-order differencing schemes. One

of them is a deferred correction implementation (Khosla and Rubin [45]) in which

any differencing scheme is treated as an upgrade of UD. Here, the part of the

convection term corresponding to UD is built into the matrix, while the other part

is added into the source term. This, however, does not guarantee boundedness

even though the matrix is now diagonally equal.

• Diffusion term. Discretisation of the diffusion term on an orthogonal mesh

produces a diagonally equal matrix which guarantees boundedness. In case of

a non-orthogonal mesh, the matrix remains diagonally equal, but, since non-

orthogonal correction contributes to the source term of the system, diagonal

equality is not sufficient for the boundedness of the solution.

• Source term. If SP < 0, in Eqn. (3.28), the diagonal dominance is increased

and the term SP VP is part of the diagonal coefficient. If SP > 0, the entire source

term is part of the r.h.s. of the system.

From the above discussion it is seen that the only terms that enhance the diagonal

dominance are the linear part of the source and the temporal derivative. In steady-

state calculations, the influence of the temporal derivative does not exist, and the

diagonal dominance is enhanced through under-relaxation. If we again consider the

original system of equations, Eqn. (3.46):

aP φnP +

∑

N

aNφnN = RP ,

the diagonal dominance is increased by addition of the artificial term to both left and

right-hand side of Eqn. (3.46):

aP φnP +

1 − α

αaP φn

P +∑

N

aNφnN = RP +

1 − α

αaP φo

P , (3.50)

i.e.,

aP

αφn

P +∑

N

aNφnN = RP +

1 − α

αaP φo

P , (3.51)


where φoP is the value from the previous iteration and α is the under-relaxation factor

(0 < α ≤ 1). When steady-state is reached, the value of φ stays the same in the

consecutive iterations (φoP = φn

P ) and the additional terms cancel out.

In this study, the Conjugate Gradient (CG) method, proposed by Hestens and

Steifel [38] is used. This method guarantees the the number of iterations needed to

obtain the solution is less or equal to the number of equations in the system. For

symmetric matrices, the ICCG solver (Jacobs [41]) is used, while asymmetric matrices

will be solved with Bi-CGSTAB method by van der Vorst [65].

3.5 Discretisation of Navier-Stokes Equations

A closed system of partial differential equations for Newtonian fluid flow was presented

in Chapter 2, by Eqs. (2.13 to 2.17). The incompressible form of this system can also

be written as:

∇ · U = 0, (3.52)

∂U

∂t+ ∇·(UU) −∇·(ν∇U) = −∇p. (3.53)

When solving this system, one has to pay attention to non-linear term in the momentum

equation, i.e. ∇ · (UU), and to the pressure-velocity coupling.

There are two ways of treating the non-linear term in the momentum equation

(Jasak [42]) - either by using a solver for non-linear systems, or by linearising the con-

vection term. The linearisation of this term is appropriate for steady-state calculations,

where, when the solution has converged, the fact that a part of the non-linear term

has been lagged is not significant. In transient problem, one can either iterate over

non-linear terms or neglect the non-linearity effects. If the time-step is large, itera-

tion greatly increases computational costs. However, in order to resolve the transient

problem well, one needs to use a small time-step, which keeps the iteration costs low

and insures that the non-linear system is fully resolved for each time-step. This holds

because the effect of lagged non-linearity is insignificant.


3.5.1 Derivation of the Pressure Equation

Assuming that all the source terms, except the pressure gradient, are contained in the

H(U), the momentum equation can be written in a semi-discretised form:

aPUP = H(U) −∇p. (3.54)

Eqn. (3.54) has been obtained from the integral form of momentum equation and

divided by the volume in order to enable face interpolation of the coefficients.

The H(U) term contains the matrix coefficients multiplied by corresponding veloc-

ities and the source part apart from the pressure gradient. In this case there are no

additional source terms except the transient term:

H(U) =∑

N

aN(U)N +U0

∆t. (3.55)

From Eqn. (3.54), it follows:

UP =H(U)

aP

−∇p

aP

, (3.56)

and the expression for the velocities on the cell face, Uf , can be obtained by interpo-

lation:

Uf =

(H(U)

aP

)

f

−

(1

aP

)

f

(∇p)f , (3.57)

The discretised form of the continuity equation is obtained from Eqn. (3.15):

∇ · U =∑

f

n · Uf = 0. (3.58)

When Eqn. (3.57) is substituted into Eqn. (3.58) the following form of the pressure

equation is obtained:

∇ ·

(H(U)

aP

)

= ∇ ·

(1

aP∇p

)

. (3.59)

By using Eqn. (3.15), the Laplacian terms in Eqs. (3.54 and 3.59) are discretised,

leading to the following form of Eqs. (3.52 and 3.53):

aPUP = H(U) −∑

f

n · (p)f , (3.60)

∑

f

n ·

(H(U)

aP

)

f

=∑

f

n ·

[(1

aP

)

f

(∇p)f

]

. (3.61)


Fluxes are obtained using Eqn. (3.57):

F = n · Uf = n·

[(H(U)

aP

)

f

−

(1

aP

)

f

(∇p)f

]

, (3.62)

which are guaranteed to be conservative if Eqn. (3.61) is satisfied.

3.5.2 Pressure-Velocity Coupling

There are two ways of treating linear dependence of velocity on pressure and vice-versa

shown in the discretised form of the Navier-Stokes system, Eqs. (3.60 and 3.61):

• Simultaneous algorithms, in which the complete system of equations is solved

simultaneously over the entire domain. This procedure does not handle non-

linearity and is comparatively costly, since the resulting matrix is several times

larger than the number of computational points.

• Segregated approach, in which the equations are solved in sequence. PISO

(Issa [40]) and SIMPLE (Patankar [52]) and their derivatives are the most popular

methods of solving inter-equation coupling in the pressure-velocity system. Their

iterative nature allows the non-linearity in the velocity equation to be handled

in the same framework, which means that they have lower storage requirements.

They are the methods used in this study.

3.5.3 The PISO Algorithm

The PISO algorithm has been developed by Issa [40] for solving transient flow calcu-

lations described by the discretised Navier-Stokes system for incompressible flow, Eqs.

(3.60 and 3.61). It can be described in the following steps:

1. Momentum Predictor. The momentum equation is solved first, using the pres-

sure field from the previous time-step. The solution of the momentum equation,

Eqn. (3.60), gives an approximation of the new velocity field.


2. Pressure Solution. Using the predicted velocities, operator H(U) is assembled

and the pressure equation can be formed. The solution of the pressure equation

gives the first estimate of the new pressure field.

3. Explicit Velocity Correction. With the new pressure field, a set of conser-

vative fluxes is found. The velocity field is also corrected as a consequence of

the new pressure distribution. Velocity correction is done explicitly using Eqn.

(3.56).

4. Update of H(U) term. Since Eqn. (3.56) consists of two parts, there are

also two parts of the velocity correction - a correction due to the change in

the pressure gradient ( 1aP∇p) and the transported influence of corrections of

neighbouring velocities (H(U)aP

). Since the velocity correction is explicit, the latter

part is neglected - it is assumed that the entire velocity error comes from the error

in the pressure term. This is not true and it is therefore necessary to correct the

H(U) term, formulate the new pressure equation and repeat the procedure.

In other words, the PISO loop consists of an implicit momentum predictor followed

by a series of pressure solutions and explicit velocity corrections. The loop is repeated

until a pre-set tolerance is reached.

The dependence of H(U) on the flux field is, on the other hand, not taken into

consideration. It means that even though after each pressure solution, with a new set

of conservative fluxes is available, the term H(U) is not recalculated. It is assumed

that the non-linear coupling is less important than the pressure-velocity coupling and

the coefficients in H(U) are kept constant through the entire correction sequence and

will be changed only in the next momentum predictor.

3.5.4 The SIMPLE Algorithm

When solving a steady-state problem iteratively, non-linearity of the system becomes

more important since the effective time-step is much larger.


The SIMPLE algorithm, developed by Patankar [52] has the following steps:

1. To initiate the SIMPLE calculation a pressure field is guessed. From the momen-

tum equation, using the guessed pressure field, the velocity field is solved. The

equation is under-relaxed using Eqn. (3.50), with the velocity under-relaxation

factor αU .

2. The pressure equation is solved to obtain the new pressure distribution.

3. A new set of fluxes is calculated using Eqn. (3.62). The new pressure field includes

both the pressure error and convection-diffusion error. In order to obtain a better

approximation of the pressure field, the coefficients of H(U) are recalculated with

the new set of conservative fluxes.

4. The pressure solution is under-relaxed in order to include the velocity part of the

error:

pn = po + αp(pp − po), (3.63)

where

• pn is the approximation of the pressure field which will be used in the next

momentum predictor,

• po is the pressure field used in the momentum predictor,

• pp is the solution of the pressure equation,

• αp is the pressure under-relaxation factor.

The recommended values of under-relaxation factors are (Peric [53]):

• αp = 0.2 for the pressure,

• αp = 0.8 for the velocity.


3.6 Error Analysis

As already mentioned in Subsection 3.1.2, numerical solutions of fluid flow problems

are only approximate solutions, with systematic errors which fit into three categories:

modeling errors, discretisation errors and iteration errors. A more detailed description

of error analysis is given by Jasak [42].

The major source of numerical errors is the discretisation of the governing equations.

In this study, discretisation is second-order accurate in space and time, i.e. it is assumed

that the variation of the function over each control volume is linear. If a better solution

is needed, control volumes should be chosen in such a way that the assumption about

the linear variation becomes acceptable. In this study, the method chosen to examine

the discretisation error will be Richardson extrapolation.

3.6.1 Richardson Extrapolation

The error in the numerical solution is defined as the difference between the exact

solution of the governing equation Φ(x, t), and the numerical solution of the discrete

system φ:

E = Φ − φ. (3.64)

Richardson extrapolation is the most popular form of error estimation based on the

Taylor series truncation error analysis. The basic idea of Richardson extrapolation is

to obtain an approximation of the leading term in the truncation error from suitably

weighted solutions on two meshes with different cell size (Muzaferija [51]). The spatial

variation of the exact solution on two meshes with spacing h1 and h2 can be symbolically

written as (Muzaferija [51]):

Φ(x) = φ(x, h1) + hp1C(x) + O(x, hq

1), (3.65)

Φ(x) = φ(x, h2) + hp2C(x) + O(x, hq

2), (3.66)

where


• φ(x, hi) is the approximate solution on the mesh with spacing hi,

• hpi C(x) is the leading term of the truncation error,

• h = h(x) is the local mesh size calculated as the ratio of cell volume and surface

area:

h =VP

∑

f |S|, (3.67)

• p is the order of accuracy of the discretisation method,

• O(x, hqi ) is the rest of the truncation error.

From Eqs. (3.65 and 3.66), C(x) can be approximated as:

C(x) =φ(x, h2) − φ(x, h1)

hp1 − hp

2

. (3.68)

The estimate of C(x), Eqn.(3.68), can be used to improve the fine mesh solution

φ(x, h2). The improved (qth order accurate) solution is:

φ(x, 0) = φ(x, h2)

(h1

h2

)p

(h1

h2

)p

− 1− φ(x, h1)

1(

h1

h2

)p

− 1. (3.69)

This improved solution can be used to estimate the error in φ(x, h2).

The Richardson extrapolation error estimate is calculated from the difference be-

tween the improved solution and the solution from the fine mesh:

et(φ) = |φ(x, 0) − φ(x, h2)| =|φ(x, h2) − φ(x, h1)|

(h1

h2

)p

− 1. (3.70)

For second-order accurate discretisation, this yields:

et(φ) =|φ(x, h2) − φ(x, h1)|

(h1

h2

)2

− 1. (3.71)


3.7 Closure

In this Chapter, the Finite Volume Method of discretisation has been described. The

spatial discretisation, explained in Section 3.2, allows for the use of the arbitrary control

volumes, making it appropriate for complex geometries. In Section 3.3, discretisation of

each term of the governing equations has been described. Implementation of boundary

conditions has been briefly discussed.

In Section 3.4, a system of linear algebraic equations, obtained from the discreti-

sation of governing equations has been analysed. The influence of each term in the

transport equation on the system matrix has been studied.

The discretisation procedure for the Navier-Stokes systems has been presented in

Section 3.5. The pressure equation has been derived and two methods of solving the

pressure-velocity system have been described: the PISO algorithm for the transient

calculations and the SIMPLE method for steady-state flows.

Finally, in Section 3.6, Richardson Extrapolation, one of the methods of finding

numerical error was presented. This method will be used throughout this study.

Chapter 4

Validation of CFD Approach UsingSimple Converging Bearings

4.1 Introduction

In Chapter 3 we have briefly described the Finite Volume discretisation. By discretising

the governing equations for a particular problem, we obtain a system of algebraic

equations. When that system is solved, it produces an approximate solution at a

number of points in the domain. The quality of the numerical solution depends on the

chosen mesh as well as the applied discretisation practice.

This Chapter investigates the applicability of CFD as a suitable tool for solving lam-

inar hydrodynamic lubrication problems. Such applicability needs to be established for

lubrication problems because the fluid film thickness is several orders of magnitude less

than its longitudinal or transverse dimension, whereas the CFD method was developed

and has been shown to be an accurate numerical process only for relatively much thicker

films.

In this Chapter, two geometries (infinitely long linear wedge and infinitely long step

bearing) for which analytical solutions of the Reynolds Equation exist, are analysed.

The Reynolds Equation, itself, is an approximation which holds well in certain hy-

drodynamic conditions. Since both are hydrodynamic problems with smooth surfaces,

the Reynolds Equation should be applicable, so computational results using full NS

equation should, in principle, compare closely the analytical solution.

69

Chapter 4. Validation of CFD Approach Using Simple Converging Bearings 70

4.2 Theoretical Considerations in Isoviscous-Rigid

Hydrodynamic Lubrication

In isoviscous-rigid hydrodynamic lubrication (HL) the two surfaces are arranged in a

shallow wedge as shown in Figure 4.1. Their relative motion drags the lubricant in

U1

V1

W1

V2

W2 U2

hx

zy

Surface 2

Surface 1

Figure 4.1: Two generalised surfaces in relative motion

between them, creating pressures of up to 200 MPa, high enough to support external

loads. The lubricant film is usually thicker than the biggest surface roughness, which

means that there is negligible solid/solid contact. Pressures encountered in isoviscous-

rigid HL are not high enough to significantly deform the rubbing surfaces nor to increase

the lubricant viscosity. Mathematical analysis is based on series of approximations in

NS equations which yields the Reynolds Equation.

4.2.1 Simplifications Leading to Reynolds Equation

In Chapter 2, a full set of governing equations for fluid flow was given. In fluid film

lubrication, certain simplifications due to the properties of the lubricant and geometry

of the bearing can be made, leading to the derivation of the Reynolds Equation from

NS equations.

By assuming that:

• viscosity and density are constant across and through the fluid film,

• the isothermal conditions hold,

• the problem is steady-state,


the governing equations for any fluid flow, Eqs. (2.13 to 2.17) reduce to:

1

ρ

∂p

∂x= ν

(∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2

)

−

(∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

)

(4.1)

1

ρ

∂p

∂y= ν

(∂2v

∂x2+

∂2v

∂y2+

∂2v

∂z2

)

−

(∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z

)

(4.2)

1

ρ

∂p

∂z= ν

(∂2w

∂x2+

∂2w

∂y2+

∂2w

∂z2

)

−

(∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z

)

(4.3)

∂ρ

∂t+

∂ρu

∂x+

∂ρv

∂y+

∂ρw

∂z= 0 (4.4)

To obtain the Reynolds Equation, it is further assumed that:

• the pressure does not vary across the film, i.e.∂p

∂z= 0;

• inertial forces are negligible compared with viscous forces;

• the only relevant viscous terms are ν∂2u

∂z2and ν

∂2v

∂z2.

By taking into account these assumptions and since ν = ηρ, where ν is kinematic

viscosity, Eqs. (4.1 to 4.3) simplify to:

∂p

∂x= η

∂2u

∂z2, (4.5)

∂p

∂y= η

∂2v

∂z2, (4.6)


∂p

∂z= 0. (4.7)

The Eqs. (4.5 to 4.7) were also obtained through order of magnitude analysis for the

region inside the contact of the bearing in Chapter 2.

4.2.2 Boundary Conditions

For both cases studied in this Chapter, pressure at the inlet and outlet boundaries of

the bearing is set to zero and zero velocity gradient in the direction normal to sliding

is assumed. The entire domain is fully flooded.

At the solid walls, the ’no-slip’ boundary condition is assumed for the flow equations,

i.e. the velocity field has a fixed value, while the pressure field has a zero gradient

boundary condition.

4.2.3 Reynolds Equation

If we integrate Eqn. (4.5) over the boundary conditions at solid walls, z1 = 0 u1 = U1,

z2 = h u2 = U2, the x-component of the velocity profile in across the film is obtained:

u =1

2η

∂p

∂x· (z2 − zh) + (U2 − U1)

z

h+ U1. (4.8)

The velocity consists of three terms; the first term on r.h.s. in Eqn. (4.8) describes the

flow due to the pressure gradient, the second term describes the flow due to the mean

surface velocity and the third term is the velocity of the bottom surface. Similarly, the

y-component of the velocity profile is obtained from Eqn. (4.6).

By calculating the flow in each direction and applying the law of continuity of

volume flow (equivalent to the law of the continuity of mass, because of the constant

density), the Reynolds Equation for 2D -geometry is obtained:

∂

∂x

(h3

η

∂p

∂x

)

+∂

∂y

(h3

η

∂p

∂y

)

= 12

{(U1 + U2

2

)∂h

∂x+

(V1 + V2

2

)∂h

∂y+ (W2 − W1)

}

.

(4.9)


4.2.4 Infinitely Long Bearings

Let us assume that the bearing surfaces do not move in the transverse direction (i.e.

W1 = W2 = 0). If the flow in the transverse direction can be neglected, strictly

speaking valid only for infinitely long bearings, although it is often assumed to be true

even for bearings with aspect ratios (width-length ratios) as low as 5 (Chen and Hahn

[15]), the pressure is a function of x only and Eqn. (4.9) simplifies to:

∂

∂x

(h3

η

∂p

∂x

)

= 6U1∂h

∂x, (4.10)

which has an analytical solution for both geometries studied in this Chapter. The

pressure distribution is obtained by the integration over x.

Load

Load per unit length, W/L, is obtained by further integration of pressure equation:

W

L=

∫ B

0

pdx. (4.11)

Friction

Total friction is given by:

F =

∫ L

0

∫ B

0

τdydx, (4.12)

with shear stress τ at the bottom wall being:

τ =

(

η∂u

∂z

)

z=0

. (4.13)

Friction Coefficient

The friction coefficient is defined as the ratio between the total friction and the total

load on the bearing:


µ =F

W. (4.14)

4.3 Analysis of an Infinitely Long Linear Wedge

In this study the FOAM computer package [73] was used to obtain numerical solutions

for the full NS equations. FOAM is an open source CFD code based on the Finite

Volume discretisation and unstructured, polyhedral, body-fitted grids.

The infinitely long linear wedge investigated is shown in Figure 4.2.

h0h

B

z

x

h1

U1

Figure 4.2: Linear wedge.

The film thickness at any point x is given as:

h(x) = h1 − (h1 − h0)x

B= h0

{h1

h0+

(

1 −h1

h0

)x

B

}

. (4.15)

Dimensional values used in this case were:

- Bearing width - B = 20 mm

- Maximum height - h1 = 2 µm

- Minimum height - h0 = 1 µm

- Velocity of the bottom wall - U1 = 1m/s

- Density of the lubricant - ρ = 103 kg/m3

- Dynamic viscosity of the lubricant - η = 10−2 Pas

The Reynolds number for this case is Re = 10−1, indicating laminar flow.

4.3.1 Reynolds Solution

Substituting Eqn. (4.15) into the 1D Reynolds equation, Eqn. (4.10) and integrating

over x, the pressure distribution is obtained for an infinitely long linear wedge (Cameron


[12]):

p =6UηB

h20

K xB

(1 − x

B

)

(2 + K)(1 + K − K x

B

)2 , (4.16)

where

K =h1 − h0

h0

. (4.17)

Velocity is obtained from Eqn. (4.8):

u =3UK

h20 (2 + K)

1 + K − xB

(2 + K)(1 + K − Kx

B

)3 · (z2 − zh) − U1z

h+ U1, (4.18)

where h is given by Eqn. (4.15).

Substituting Eqn. (4.16) into Eqn. (4.11), the load per unit length is obtained:

W

L=

6UηB2

h20

1

K

(ln(1 + K)

K−

2

2 + K

)

. (4.19)

Total friction per unit length is

F

L=

BηU

h0

(4ln(1 + K)

K−

6

(2 + K)

)

. (4.20)

4.3.2 Numerical Results and Mesh Selection

In case of infinitely long linear wedge, the mesh consisted of one block with a uniform

grid as shown in Figure 4.3. The number of grid points in the longitudinal (Nx) and

transverse (Nz) direction was varied, and the resultant variations of maximum pressure

and total load were observed. Table 4.1 shows the results for the mesh refinement.

The mesh study was done in three ways. Firstly, the initial grid size 100 x 10 cells

was refined four times by factor 2 in both directions. The error in the numerical solution

for the maximum pressure along the bottom wall, et(pmax), was found using Richardson


Nx

Nz

Figure 4.3: Mesh structure in the computational domain of the linear wedge.

Nx x Nz pmax et(pmax) W/L F/L

[MPa] [MPa] [kN/m] [kN/m]

Reynolds 50.00 - 635.5 0.1545

50 x 5 46.38 - 622.6 0.1507

100 x 10 49.04 0.88 622.6 0.1531

200 x 20 49.76 0.24 632.3 0.1544

400 x 40 49.94 0.06 634.7 0.1545

800 x 80 49.98 0.01 635.3 0.1544

50 x 10 49.11 - 621.3 0.1518

100 x 20 49.77 0.22 632.0 0.1533

100 x 40 49.98 - 634.7 0.1532

100 x 10 49.04 0.88 622.6 0.1531

200 x 10 49.03 - 622.9 0.1537

400 x 10 49.02 - 623.1 0.1539

Table 4.1: Mesh refinement in x- and z- directions.


Extrapolation, Eqn. (3.71). However, the grid refinement can also be done only in one

direction, leading to much smaller grids and less computational time. Figure 4.4 shows

the variation of maximum pressure along the bottom wall with the refinement done in

both x and z directions and separately only in x and only in z direction.

From the mesh study conducted it can be concluded that the refinement has to be done

in z direction. The mesh used from now onwards was of the grid size 100 x 20.

Load and Friction

Numerical values for total load and total friction along the bottom wall of the bearing

were obtained by using Simpson’s rule:

Φ =

Nx/2−1∑

i=0

1

3∆x (f2i + 4f2i+1 + f2i+2), (4.21)

where

• ∆x is the spacing between the two neighbouring points in the x-direction,

• Φ =W

Lfi = pi adjacent to lower surface - for calculation of load,

• Φ =F

Lfi = τi adjacent to lower surface - for calculation of friction.

0 2 4 6 8Grid refinement factor

0

0.5

1

1.5

2

Var

(pm

ax)

[%]

x-z refinementx refinementz refinement

Figure 4.4: Variation of maximum pressure and total load with the mesh refinementin (a) the x and z directions, (b) x direction, (c) z direction.


Pressure

In Figure 4.5 numerical results for the pressure distribution along the bottom wall

of the bearing are shown alongside those obtained from the Reynolds approximation,

Eqn. (4.16). The maximum error in pressure is max(

p−pRey

pRey

)

= 0.10%, which confirms

good accord between analytical and numerical results.

0 0.005 0.01 0.015 0.02x [m]

0

10

20

30

40

50

p [M

Pa]

Reynolds’ solutionNumerical results

Figure 4.5: Pressure distribution along the bottom wall of the linear wedge.

Velocity

In Figure 4.6, the analytical results from Eqn. (4.18) for the x-component of velocity in

three different locations of the bearing are compared with the numerical results. The

maximum error is 0.7%.

In Figure 4.7, the velocity profiles at the inlet, in the middle and at the outlet of the

linear wedge are shown. The difference between the velocity profiles at the inlet and

the outlet is due to different pressure gradients, i.e. the pressure gradient is positive

at the inlet and negative at the outlet of the bearing.


0 0.2 0.4 0.6 0.8 1u [m/s]

0

0.5

1

1.5

2z

[µm

]

ReynoldsNumerical results

0 0.2 0.4 0.6 0.8 1u [m/s]

0

0.5

1

1.5

2

z [µ

m]


0 0.2 0.4 0.6 0.8 1u [m/s]

0

0.5

1

1.5

2

z [µ

m]


(a) (b) (c)

Figure 4.6: The x - component of the velocity vector at (a) x = 0.02e − 2 m,(b) x = 1.334e − 2 m and (c) x = 1.98e − 2 m.

Figure 4.7: Velocity profiles on three different locations along the linear wedge.

4.4 Analysis of the Step Bearing

The other geometry of interest is an infinitely long step bearing as shown in Figure 4.8.

What makes this problem interesting is that the analytical solution of the Reynolds

Equation exists, and at the same time, the steep geometry of the step creates the need

for local refinement of the computational mesh in order to produce good numerical

results.


B1 B2

B

h1h0z

x U1

Figure 4.8: The Rayleigh step bearing.

The film thickness at any point x is given as:

h(x) = h1 for 0 ≤ x < B1,

h(x) = h0 for B1 < x ≤ B2. (4.22)

Dimensional values used in this case were:

- Bearing width, left of the step - B1 = 10 mm

- Bearing width, right of the step - B2 = 10 mm






The Reynolds number for this case is Re = 10−1, indicating laminar flow.

4.4.1 Reynolds Solution

Pressure

The pressure distribution for this case is analytically obtained from the oil flow rate

equation (Cameron [12]), assuming the pressure gradient is constant due to the geom-

etry (i.e. flow between parallel walls) and positive on the left side of the step, and

constant and negative on the right side of the step. After equating the flow rate from

the left and from the right side at the step, the maximum pressure is obtained:


pmax =6ηU1(h1 − h0)

h31/B1 + h3

0/B2. (4.23)

Velocity

Since the pressure gradient is constant on the right hand side and on the left hand

side of the step, an analytical solution exists for the velocity profile across the bearing

despite the discrete geometry at the step.

The maximum pressure is calculated from Eqn. (4.23) and the pressure gradients

for the left and the right side of the step are obtained from:

∂p

∂x=

2pmax

Bfor 0 ≤ x < B1,

−2pmax

Bfor B1 < x ≤ B2.

(4.24)

Eqs. (4.1 to 4.4) are reduced to:

ν∂2u

∂z2=

1

ρ

∂p

∂x. (4.25)

After integrating twice over dz, using appropriate boundary conditions (z1 = 0 u1 = U1,

z2 = h1 u2 = 0 for the inlet, and z1 = 0 u1 = U1, z2 = h0 u2 = 0 for the outlet) and

substituting the pressure gradient from Eqn. (4.24), the equation for the x-component

of velocity is obtained:

u =

1

η

pmax

Bz2 −

(ηU1 + pmax

Bh2

1

)

h1ηz + U1 for 0 ≤ x < B1,

−1

η

pmax

Bz2 −

(ηU1 + pmax

Bh2

0

)

h0ηz + U1 for B1 < x ≤ B2.

(4.26)


block 1

block 2

block 3

Figure 4.9: Mesh and block structure in the computational domain of the step bearing.

Load

The total load per unit length is merely pmaxB/2 as the pressure distribution is trian-

gular:

W

L= pmax

B

2=

6ηU1B

2

(h1 − h0)

h31/B1 + h3

0/B2(4.27)

Friction

The friction is calculated from Eqn. (4.12):

F

L=

pmax

2(h1 − h0) + U1η

(B1

h1+

B2 − B1

h0

)

(4.28)

4.4.2 Mesh Selection and Numerical Results

As shown in Fig. 4.9, the computational domain is divided into three blocks, each

having Nx points in x-direction and Nz points in z-direction. The mesh is uniform in

z-direction and non-uniform in x-direction with the largest cell (i.e. the one furthest

away from the step), ∆xmax being 3000 times bigger than the smallest (i.e. the one

closest to the step), ∆xmin. Values for ∆xmax and ∆xmin for each mesh are shown in

Table 4.2. The grid refinement around the step is necessary to capture the occurrence

of the recirculation and other flow changes due to the sudden change in geometry.

The error was estimated using Eqn. (3.71). The mesh used from now onwards has

dimensions for each block of 50 x 10 control volumes.


Nx x Nz ∆xmax ∆xmin pmax et(pmax) W/L F/L

[m] [m] [MPa] [MPa] [kN/m] [kN/m]

Reynolds - - 66.67 - 666.7 183.33

25 x 5 2.8e-3 9.5e-7 64.94 - 649.4 182.47

50 x 10 1.5e-3 5.0e-7 66.23 0.43 662.2 183.11

100 x 20 7.8e-4 2.6e-7 66.56 0.11 665.5 183.28

200 x 40 3.9e-4 1.3e-7 66.65 0.03 666.4 183.32

Table 4.2: Mesh refinement in both x- and z- directions.

0 0.005 0.01 0.015 0.02x [m]

0

10

20

30

40

50

60

70

p [M

Pa]

Numerical resultsReynolds

Figure 4.10: Pressure distribution along the bottom wall of the step bearing.

Pressure

In Figure 4.10, the analytical results for pressure distribution along the bottom wall of

the bearing are plotted against numerical results. The maximum error in pressure is

max(

p−pRey

pRey

)

= 0.09%, which confirms good accord between analytical and numerical

results.

Load and Friction

Because the mesh is non-uniform in x-direction, Simpson’s rule, Eqn. (4.21), could

not be used in the calculation of load and friction. The following equation was used


instead:

Φ =

Nx/2−1∑

i=0

1/2 (x2i+1 − x2i) (f2i+1 + f2i), (4.29)

with

• Φ =W


• Φ =F


Numerical results for load and friction are shown in Table 4.2.

Velocity

In Figure 4.11, the analytical results from Eqn. (4.26) for the x-component of velocity

on the left and the right of the step of the bearing are compared with the numerical

results. The maximum error is 0.5%.

In Figure 4.12, the velocity profile is shown in three different locations. An interest-

ing phenomenon of a backflow from the step all the way back to the inlet can be seen

(Figure 4.11 a.), which is also predicted in the analytical solution (4.26). The backflow

is due to the adverse pressure gradient (Figure 4.10) in the inlet region.

4.5 Closure

This Chapter investigated the applicability of CFD as a tool for solving laminar hy-

drodynamic lubrication problems. The study was done on two different geometries:

an infinitely long linear wedge and an infinitely long step bearing. Reynolds approxi-

mation holds for both of those geometries. For both cases in this Chapter, it can be

seen that the numerical CFD results are in good agreement with the analytical ones

(the maximum error for pressure ≤ 1%). This validation of the computational method

was necessary, because CFD usually deals with cases with very different geometry to

that present in bearings. The code used has proved to be robust enough to handle big

aspect ratios (≈ 103) present in these cases. The following conclusions can be drawn

from these two cases:


0 0.25 0.5 0.75 1u [m/s]

0

0.5

1

1.5

2

z [µ

m]


(a)

0 0.2 0.4 0.6 0.8 1u [m/s]

0

0.5

1

1.5

2

z [µ

m]


(b)

Figure 4.11: The x - component of the velocity vector at (a) x = 0.5 mm and(b) x = 19.5 mm.

Figure 4.12: Velocity profile at the three positions in the step bearing.


1. The CFD software has been shown to be capable of accurately handling steady

state hydrodynamic lubrication problems.

2. CFD results have confirmed the assumption of constant pressure across the film

(apart for the rapidly converging step in step bearing).

3. CFD results have confirmed that the neglect of inertia and certain viscous terms

in Reynolds equation is justified in the bearings analysed.

Based on this, we can safely use CFD in hydrodynamic lubrication.

Chapter 5

Computational Modelling ofCavitation

5.1 Introduction

In Chapter 4 the applicability of the CFD to solving HL problems was examined

using two hydrodynamic problems for which the Reynolds Equation has an analytical

solution. From the two geometries studied, it was shown that, for simple, converging

geometries, the CFD is a suitable tool for solving HL problems.

In the case of a roller bearing, however, the pressure drops below the atmospheric

pressure in the diverging region, so cavitation occurs. The physical situation in the case

of cavitation is extreme in several respects. The density ratio between two phases is very

large, roughly 10,000:1. The aspect ratio of the computational cells in the cavitating

region is of the order of 1000:1, which means that individual cavitation bubbles could

span several computational cells. These extremes make this a challenging problem to

model. Special steps are required to ensure stability when modelling a large change in

density over the space of a few computational cells.

In this Chapter, a cavitation model will be tested on a case of an infinitely long

roller bearing. It will be assumed that there is no elastic deformation, no piezo-viscous

effects and that the problem is iso-thermal.

87

Chapter 5. Computational Modelling of Cavitation 88

5.2 Previous Work in Cavitation

In CFD, cavitation has been modelled numerically by two main methods, interface

tracking and continuum methods. Interface tracking is a numerical technique for two-

phase flows. In this case each phase is treated separately and there is only one phase

present at any location in the solution domain. Normally, there are separate equations

for conservation of mass and momentum of each phase, which makes the computational

cost of modelling at least six times the cost of continuum methods, as shown by Rider

and Kothe [57]. The authors also noted that the maximum density ratio between the

liquid and vapour phase, with this method can be 1,000:1.

In a continuum method model, both phases are considered to be the same fluid,

where the density reflects the liquid and vapour content of the cell. The computational

cost of this model is lower than of the interface tracking because continuum method

requires no more equations than single-phase flow. The works of Kubota et al. [47],

Delannoy and Kueny [17], Chen and Heister [16], and Avva et al. [5] are all examples

of continuum methods.

Kubota et al. [47] developed one of the first continuum models of cavitating flow.

Their model was based on the assumption that the fluid was a uniform mixture of

liquid and very small, spherical bubbles. They used relations for bubble radius, average

cell density, and pressure to close the set of equations. Because of the extreme density

ratio between the liquid and vapour, they chose to neglect the mass of the vapour. The

authors were limited to moderate void fractions because of severe stability problems.

Delannoy and Kueny [17] used a simpler method for closure of the hydrodynamic

equations. They assumed a barotropic equation of state in which density was a function

of pressure. The two constant densities for liquid and vapour were joined by a sine

function whose maximum slope was chosen to represent roughly the speed of sound

of a two-phase mixture. Their scheme was based on the SIMPLE algorithm, which

models incompressible flows. One of the limitations of their model is that they used


the vapour-to-liquid density ratio of 1:100, rather than a realistic ratio of 1:105. Their

model was later employed to model the cavitation in HL contacts by van Odyck [66]

but he was limited to use the unrealistically high vapour density.

Chen and Heister [16] rejected the idea of a simple one-to-one mapping of pressure

and density. They argued that the pressure field should be related to the density

history. However, they assumed that the bubble number per unit mass was constant

and that each cell was uniformly filled with small, spherical bubbles. This technique

worked well for large-scale flows, but was not appropriate for small-scale geometry, like

a roller bearing.

Avva et al. [5] used an energy equation for closure. They started with an energy

equation for a two-phase mixture and assumed homogeneous flow, no slip, and ther-

modynamic equilibrium. With these assumptions, the energy equation was simplified

to a single fluid energy equation based on a mean cell density. The model was limited

to the density ratio of 3,000:1. Avva’s model forms the basis of the one used in the

current study.

5.3 Physical Modelling

An important limitation of all the studies already mentioned is that the pure phases

are considered to be incompressible. The model which is used in the current study,

considers the compressibility of both pure phases. Thermal non-equilibrium effects will

be neglected, as will hydrodynamic non-equilibrium.

Suitable average properties (density, viscosity) are determined and the mixture

is treated as a pseudofluid that obeys the usual equations of single-component flow.

Differences in velocity, temperature, and chemical potential between the phases will

promote mutual momentum, heat and mass transfer. Often these processes proceed

very rapidly, particularly when one phase is finely dispersed in the other, and it can

be assumed that equilibrium is reached. In this case the average values of velocity,


temperature, and chemical potential are the same as the values for each component

and the flow is homogeneous equilibrium flow . This method is a convenient way of

modeling the flow because it requires no more equations than the single-phase flow.

In this study, cavitation was addressed by considering the lubricant to be a two-

phase fluid, transforming from entirely liquid at high pressure, through a mixed liq-

uid/vapour phase at intermediate pressure, to entirely vapour phase at low pressure.

This enables a suitable fluid density-pressure relationship to be derived which spans

the liquid and vapour states and thus permits cavitation based on vapour formation

to occur as the pressure falls in a divergent region (Dellanoy and Kueny [17]). In the

current study, the isentropic phase transformation approach was employed (Avva [5]).

The conservation of energy equation, assuming homogeneous equilibrium of the two

phases and including the viscous dissipation term, Φ is written as:

ρDh

Dt=

DP

Dt+ ∇k∇T + Φ. (5.1)

Assuming that the rapid change between liquid and vapour phases during cavitation

is isentropic and that enthalpy and pressure work terms dominate (i.e. they are at least

five orders of magnitude greater than the conduction and viscous terms), the energy

equation reduces to:

ρDh

Dt=

DP

Dt. (5.2)

The remaining terms describe reversible work and Eqn. (5.2) can be written as an

isentropic model equation:

a2 Dρ

Dt=

DP

Dt. (5.3)

Eqn. (5.3) is considered an isentropic model of the phase change, where the speed

of sound of the mixture, a, is given by the homogeneous equilibrium model (HEM)

(Wallis [71]):

1

a2= ρ ·

(α

ρga2g

+1 − α

ρla2l

)

, (5.4)


where ρg, ag, ρl, al are the density and speed of sound of the vapour and liquid phases

respectively, ρ is the density of the phase mixture and α is the volume fraction of

vapour or ”void fraction” defined by:

α =ρ − ρl

ρg − ρl. (5.5)

The densities of both the superheated vapour and the subcooled liquid were set to vary

linearly with pressure according to:

(i) Liquid phase; ρl = ρlSat+

p − psat

a2l

(ii) Vapour phase; ρg = max

(p

a2g

, ρgsat

)

, (5.6)

where ρlSat and ρgSat are the densities of the liquid and vapour at the saturation

pressure, pSat. (The second term in Eqn. (5.6 ii) prevents the density becoming

negative at negative pressures).

The density of the phase mixture is calculated using the continuity equation, Eqn.

(2.2), and the local vapour fraction can then be obtained via Eqn. (5.5).

The dynamic viscosity, η of the phase mixture is assumed to be linearly proportional

to the density, i.e.;

η = ρ · (ανg + (1 − α) νl) . (5.7)

5.4 Fluid Properties

From the equation of state for the ideal gas, the dependence of ρgsaton psat is derived:

ρgSat=

psat · M

R · T, (5.8)

where M is the molar mass, T temperature, and R is the Boltzmann constant.

In Faber [27], the speed of sound in vapour for an ideal gas is expressed by:

a2g =

γp

ρ=

γRT

M, (5.9)

where γ is the specific heat ratio cp/cv.

For a lubricant (in this case, values taken are for n-hexadecane), typical values are:


• M = 0.3 kg/mol,

• γ = 1.2 and

• pSat = 300 Pa.

The temperature is set to be T = 353 K. With these values:

• from Eqn. (5.8), it follows that ρgsat= 0.03 kg/m3,

• from Eqn. (5.9), it follows that ag = 108 m/s.

Speed of sound in liquid was obtained from Ball and Trusler [6], while the kinematic

viscosity and saturation density of liquid were chosen to give a dynamic viscosity of

0.01 Pas.

Each of the fluid parameters clearly has its influence on the pressure-density re-

lationship, i.e. how steep is the curve which links the fluid phase with the vapour

phase in graphical representation of ρ(p). In order to determine that influence, the

pressure-density relationship of the model was obtained analytically (Schmidt [62]) by

integrating Eqn. (5.3):

ρ = 10p−pSat

pgl ·ρlρga

2g (ρg − ρl) + ρ2

l

(ρga

2g − ρla

2l

)

ρga2g (ρg − ρl) + 10

p−pSatpgl ρl

(ρga

2g − ρla

2l

). (5.10)

A critical feature of the cavitation model was found to be the speed of sound in vapour,

ag and too low values gave computational instability. This can be seen from Figure 5.1,

which shows the plots of Eqn. (5.10) for three different values of ag, whilst the other

parameters were kept at the constant, previously stated, values. From this Figure, it

can be seen that the gradient of change between the vapour and the liquid phase is the

biggest for ag = 100 m/s, which leads to the non-convergence of the numerical results.

For this reason a value of 200 m/s was chosen, which is somewhat higher than the ideal

gas theory value of 108 m/s.

The fluid properties employed in the present calculations are listed in Table 5.1.


-2500 -2000 -1500 -1000 -500 0 500p [Pa]

0

200

400

600

800

1000

ρ [k

g/m

3 ]

ag = 100 m/s

= 200 m/s = 300 m/s

Figure 5.1: Pressure-density relationship with different values for ag.

Saturation pressure psat = 300 Pa

Saturation density of fluid ρlsat= 1000 kg/m3

Saturation density of vapour ρgsat= 0.03 kg/m3

Speed of sound in fluid al = 1340 m/s

Speed of sound in vapour ag = 200 m/s

Kinematic viscosity of liquid νl = 10−5 m2/s

Kinematic viscosity of vapour νg = 2.6−5 m2/s

Table 5.1: Values of the fluid properties used in the cavitation model.

5.5 Cylinder on a Flat Surface

To test that the above model successfully handles cavitation in diverging regions, two

geometries shown in Figure 5.2 were employed: a flat wall sliding against a half cylinder

and a full cylinder rolling between the two walls. In both cases, an infinitely long

geometry was assumed.

In both cases, the width of the domain is L = 120 mm, the radius of the cylinder is

R = 10 mm, and the minimum gap between the cylinder and the wall is h0 = 100 nm.

The centre of the cylinder has coordinates (0,0,R). The values of fluid properties are

in Table 5.1.

Two cases are considered:


z

L

h0

x=0x

h

R

U1

φ

L

h0

x=0

h

R

U1

U1

U2

h0

φ

(a) (b)

Figure 5.2: Bearing geometries used in cavitation model; (a) Pure sliding, half-cylindercase, (b) Pure rolling, full-cylinder case.

• Pure sliding, where U1 = 1 m/s, and

• Pure rolling, where U1 = U2 = 0.5 m/s.

The film thickness (i.e. the distance between the bottom wall and the line z = R) at

a distance x from the minimum gap is given by:

h(x) = h0 + R (1 − cos φ) for |x| ≤ R,

h(x) = h0 + R for |x| ≥ R. (5.11)

In case of a roller bearing, high pressures are localised around the minimum film thick-

ness, where φ has a small value. Therefore, higher terms in the cosine expansion can

be neglected:

cos φ = 1 −φ2

2+

φ4

4−

φ6

6+ . . . ≈ 1 −

φ2

2. (5.12)

By combining Eqs. (5.11 and 5.12), the approximation of the film thickness for |x| ≤ R

can be written as:

h(x) = h0 + R ·φ2

2= h0 +

x2

2R= h0

(

1 +x2

2Rh0

)

. (5.13)


-0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003x [m]

-40

-30

-20

-10

0

10

20

30

40p

[MPa

]

Reynolds SolutionCFD - no cavitation

-0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003x [m]

-40

-30

-20

-10

0

10

20

30

40

p [M

Pa]

CFD - cavitationCFD - no cavitation

(a) Full Sommerfeld (b) Full Sommerfeld vs. Cavitation

Figure 5.3: Pressure distribution along the bottom wall of the roller bearing; (a) fora full Sommerfeld solution near the centre of the roller bearing; (b) numerical resultsobtained with and without the cavitation model

5.5.1 Reynolds Solution for Full Sommerfeld Condition

By substituting Eqn. (5.13) into the Reynolds Equation, Eqn. (4.10), and integrating

over the boundary conditions p = 105 Pa for x = ±∞, we obtain (Cameron [12]):

p(x) = −2U1ηx

h20

(

1 +x2

2Rh0

)2 + 105 for |x| ≤ R

p(x) = −2U1ηx

(h0 + R)2 + 105 for |x| ≥ R (5.14)

This pressure distribution, also known as Full Sommerfeld Condition, is anti- sym-

metrical, producing, therefore, unrealistic negative pressures and zero load support. In

order to obtain a realistic result, cavitation must be included in the numerical model.

Figure 5.3 (a) shows the pressure distribution obtained from Eqn. (5.14) alongside the

numerical results using the SIMPLE algorithm. In Figure 5.3 (b), numerical results

for pressure distribution with and without cavitation model are plotted alongside each

other for comparison.


5.5.2 Numerical Results

Graphs in Figure 5.4 show expanded views of the CFD pressure and density distri-

butions near the centre of the contact for the two cases, comparing the values at the

bottom wall and bottom half of the cylinder. Graphs (a) and (c) show all the pressure

and density values, whilst the graphs (b) and (d) show the values of pressure around

the atmospheric pressure (patm = 0.1 MPa). The density peak at the position of maxi-

mum pressure indicates the compressibility of the liquid phase. It can be seen that the

cavitation does not occur instantaneously as the pressure drops below the saturation

pressure. According to the model employed, the fluid can be over-expanded by reduc-

ing its pressure to below the vapour pressure, and even to negative pressures, until the

liquid reaches the limit of its tensile strength and cavitates (Apfel [3]). The reformation

takes place in a 0.5 mm zone just downstream of the minimum film thickness.

Figure 5.5 shows density distribution in the diverging part of the bearing for both

the pure sliding and the pure rolling case. In this figure, the geometry is distorted

(minimum gap is increased to z0 = 10−4 m) when showing density distribution. Figures

5.4 and 5.5 show lots of similarities between the pure rolling and the pure sliding cases.

However, the density variation in the immediate diverging region of the contact, shows

that for the pure rolling case, the fluid cavitates next to the cylinder surface. That

might be expected since the curvature of this surface will provide a slightly greater

local level of expansion. In the pure sliding case, the fluid cavitates initially close to

the centre of the fluid film but evaporates more rapidly from the sliding surface than

from the stationary one.


-0.0004 -0.0002 0 0.0002 0.0004x [m]

-40

-20

0

20

40

p [M

Pa]

pwall

pcylinder

-0.0004 -0.0002 0 0.0002 0.0004x [m]

0

500

1000

1500

2000

ρ [k

g/m

3 ]

ρwall

ρcylinder

-0.0004 -0.0002 0 0.0002 0.0004x [m]

-0.1

0

0.1

0.2

0.3

p [M

Pa]

pwall

pcylinder

-0.0004 -0.0002 0 0.0002 0.0004x [m]

0

500

1000

1500

2000

ρ [k

g/m

3 ]

ρwall

ρcylinder

(a) (b)

-0.0004 -0.0002 0 0.0002 0.0004x [m]

0

500

1000

1500

2000

ρ [k

g/m

3 ]

ρ wallρ cylinder

-0.0004 -0.0002 0 0.0002 0.0004x [m]

-40

-20

0

20

40

p [M

Pa]

p wallp cylinder

-0.0004 -0.0002 0 0.0002 0.0004x [m]

0

500

1000

1500

2000

ρ [k

g/m

3 ]

ρ wallρ cylinder

-0.0004 -0.0002 0 0.0002 0.0004x [m]

-0.1

0

0.1

0.2

0.3

p [M

Pa]

p wallp cylinder

(c) (d)

Figure 5.4: Density and pressure distributions along the bottom wall and bottom halfof the cylinder; (a) and (b) for rolling case, (c) and (d) for sliding case


(a)

(b)

Figure 5.5: Density distribution and isolines in exit region; (a) for the pure rolling, full-cylinder contact, (b) for the pure sliding, half-cylinder contact (Note that the verticalz-scale varies across the region shown).


In Figure 5.6, the pressure-density dependence obtained from numerical results is

compared to the one obtained from Eqn. (5.10). There is a distinction between the

formation of cavitation and the subsequent re-formation of the fluid. The numerical

results do not correspond exactly to the analytical solution because according to the

barotropic relation pressure and density are not assumed to be in equilibrium.

-5000 0 5000 10000p [Pa]

0

200

400

600

800

1000

ρ [g

/m3 ]

Cavitation forms - cylinderLiquid reforms - cylinderCavitation forms - wallLiquid reforms - wallTheory - a

g = 200 m/s

Figure 5.6: Pressure–density dependence for the half-cylinder case (magnified aroundcavitation values).

Figures 5.7 (a) and 5.7 (b) show streamlines for the cavitation cases. The geometry

is distorted by increasing the apparent minimum gap to 10−3 m. Backflow can be

noticed both in the converging and diverging regions.

In Figure 5.8 it is shown that the backflow starts at x = -5e-5 m in the converging

region of the half-cylinder case and it goes all the way back to the inlet.

Figure 5.9 shows that, for the full cylinder case, the backflow in the converging

region starts at x = -15e-5 m at the bottom wall and at x = 15e-5 m at the top wall.

In the full cylinder, as expected, the flow is antisymmetrical with respect to x = R.

The backflow in the inlet region occurs because not all of the fluid can go through the

small gap, only a small amount gets dragged into the diverging region. Figure 5.10

shows the density isolines and the velocity profile for the half-cylinder and the full-

cylinder cases in the diverging region. It can be noticed that the backflow starts at


Figure 5.7: Streamlines in cavitation cases; (a) the pure-sliding, half-cylinder, (b) thepure-rolling, full-cylinder (distorted geometry - minimum gap increased to 10−3 m).


Figure 5.8: Velocity profiles for the half-cylinder case in the converging region leadingto the contact at x = 0 m.

(a)

(b)

Figure 5.9: Velocity profiles for the full cylinder case in the converging region leadingto the contact at x = 0 m; (a) near the bottom wall, (b) near the top wall.

x = 40e-5 m and it goes all the way to the outlet. The backflow in the outlet region

occurs because of the negative pressure in the cavitating zone which draws the fluid

towards itself. The backflow zones at the inlet and the outlet are in the same positions

in the full-cylinder as in the half-cylinder case.

5.5.3 Mesh Selection

The computational domain is divided into blocks with non-uniform grid in the x-

direction. Figure 5.11 shows the bottom right quarter of the bearing. The mesh is

symmetrical in respect to x = 0 and z = R. The aspect ratio of the mesh ranges from


(a)

(b)

(c)

(d)

Figure 5.10: Density distribution and the velocity profile at the reformation point atx = 40e-5 m; (a) density isolines for the half cylinder, (b) velocity profile for the halfcylinder, (c) density isolines for the full cylinder, (d) velocity profile for the full cylinder.


0.7Rx=0

1

0.05R6R

z=R

z=0.3R

z=−h0

z=0

R

2

4

3

Figure 5.11: Mesh structure in the computational domain of the cylinder on the flatgeometry (bottom right quarter).

the order of 103 near the center to the order of 1 in the far regions from the contact. In

order to successfully model the cavitation we must have a very fine mesh in the center

of the bearing. A coarse mesh at the inlet/outlet region is sufficient.

The number of cells in each block is given in Table 5.2. Blocks numbered 1-3, have

the same number of points in the z-direction, Nz3, whilst block number 4 has Nz4

points in the z-direction.

Case Nx1 Nx2 Nx3 Nx4 Nz3 Nz4

1 500 150 100 100 15 10

Table 5.2: Number of points in the x− and the z− direction for each block.

The width of the block number 1 is x = 0.05·R = 5·10−4 m. In block 2, the x

dimension of the cell adjacent to block 1, was 250 times smaller than the one adjacent

to the block 3 (a uniform expansion of ca. 1.04). From Figure 5.4, it can be seen that

the steep pressure and density gradients occur only in the block 1, nearest to the centre.

That is why the mesh is the finest in that region. Further towards inlet and outlet,

the flow is parallel to the moving wall and there are no steep gradients in pressure and

density distributions, therefore the mesh can be coarser.


5.6 Closure

In this Chapter two cases were studied: the pure rolling, full cylinder and the pure

sliding, half cylinder. In both cases the cavitation occurs in the divergent region of

the bearing, which requires a suitable computational model. The computational model

used in this work is a continuum method model with both the vapour and the liquid

phases considered to be the same fluid. The density of the fluid determines wheather

the fluid is a vapour, liquid or a mixture of the two.

Numerical results have shown many similarities between the pure sliding and the

pure rolling case. In both cases, there was occurrence of the backflow in the converging

and diverging region. The difference between them was that the fluid cavitated next

to the cylinder surface in the pure rolling case, whereas in the pure sliding case, it

started cavitating in the centre of the fluid film but evaporated more rapidly close to

the sliding surface.

It should be noted that the cavitation model used in this work should not be

taken to be quantitatively accurate. Firstly, it is unlikely that the assumed liquid

to vapour transition is the only process of cavitation in bearings; it is equally likely

that cavitation occurs via the escape of dissolved gases in low pressure liquid regions.

Also, the isentropic evaporation cavitation model adopted is one developed primarily

for behaviour in higher Reynolds number flows and may not be fully appropriate to

expansion in the pocket entry. However, the predictions made from it are likely to be

at least qualitatively correct.

Chapter 6

Low Friction Pocketed Pad Bearing

6.1 Introduction

Chapter 4 confirmed the applicability of CFD in treating HL problems on two geome-

tries: an infinitely long linear wedge and an infinitely long step bearing. It was seen

how the CFD solver can handle the steep geometry of the step bearing and how results

coincided with those obtained with Reynolds equation.

In Chapter 5, the geometry of a roller bearing was modelled, introducing the need

for cavitation modelling in the diverging region of the bearing. The results, even though

obtained by a simplified model, gave a realistic distribution of pressure and density.

This Chapter addresses the behaviour of lubricant in closed pockets or recesses in

bearings. Although pockets have been widely used for many years in hybrid hydrody-

namic/hydrostatic bearings, there are normally placed at the bearing inlet to promote

fluid entrainment (entry pockets) or are connected to an oil supply port or channel.

The use of isolated pockets in the bearing surface has, until recently, received little

attention.

In the last few years, however, it has been found that the friction coefficient of

very low or zero convergence ratio bearings (whose surfaces are almost parallel), such

as those present at piston skirts and face seals, may be very significantly reduced if

one of the bearing surfaces is indented with many tiny, closed pockets (Ronen et al.

[58], Ryk [59], Etsion [23], Goloch [30]). Theoretical analysis of such textured surface

105

Chapter 6. Low Friction Pocketed Pad Bearing 106

systems has been carried out using Reynolds equation, which suggests that cavitation

may occur at the divergent entrance to the pockets, followed by a local increase in

hydrodynamic pressure (Hamilton [33], Etsion [22]). Typical suggested dimple sizes

are 50 to 100 µm diameter, 5 to 10 µm depth, with a pocket density such as to cover

about 10 to 25% of the total bearing area.

More recently, experimental work has indicated that the main role of these pockets

may be to promote fluid entrainment at low speeds and thus cause the bearing to enter

the full-film hydrodynamic regime (with concomitantly reduced friction) at a lower

sliding speed than for smooth surfaces (Wang [72]), (Kovalchenko [46]). This may

imply that the dimples may be acting in the inlet as entry-pockets, as suggested by

Tonder [63]. Very recently, CFD has been employed to show that roughness, where

the depth and width scales of the roughness features are of similar order, can produce

additional lift due to convective inertia (Arghir et al. [4]).

In this study, a simple linear pad bearing was analysed in which the stator had one

or more shallow, macroscale pockets. A wide range of convergence ratios, from values

of unity, typical of plain thrust bearings, to very small values more representative of

piston skirt/liner contacts are considered.

In 6.2 general information about the fluid properties, boundary conditions and the

mesh study will be presented.

In Section 6.3, a 2D bearing with a single pocket will be studied. In this study, 2D

is used to mean a bearing which is infinitely long transverse to the direction of surface

motion, while a 3D one is a bearing of finite length transverse to the sliding/rolling

direction. The effect of varying the pocket height, position, size and convergence ratio

on total load support and total friction will be examined. The case with very low

convergence ratio will be briefly discussed because of the possible cavitation effects.

In Section 6.4, similar analysis will be conducted for a 2D bearing with more than

one pocket. Section 6.5 will conclude the Chapter with the study of a 3D bearing with


pockets.

6.2 Linear Bearings with Pockets

Throughout this Chapter, density of liquid is set to ρ = 103 kg/m3, dynamic viscosity

to η = 10−2 Pas and the speed of the bottom wall of the bearing is U1 = 1 m/s. The

bearing width is B = 20 mm (see Figure 6.1).

The convergence ratio K, of the bearing is given by:

K =h1 − h0

h0

. (6.1)

In most work in this Chapter, the convergence ratio, K, was kept at a typical engineer-

ing value for pad bearings of K = 1. This convergence ratio gives almost optimally

high pressures in a linear bearing and in no case at K = 1 was any cavitation observed

to take place in the bearing pocket.

6.2.1 Boundary Conditions

The pressure at the inlet and outlet boundaries was set to atmospheric (105 Pa) and

zero velocity gradient in the normal direction was assumed for the velocity field. This

defines a fully developed flow approximation through these boundaries. It is important

to set the boundaries relatively far from the region of interest, so that they do not

overly influence the numerical solution. The domain was fully flooded.

At the solid walls, the ’no-slip’ boundary condition was assumed for the momentum

equations. In practice this involved setting the velocity of the face of each fluid cell

adjacent to the wall to have the same velocity as that of the wall itself.

6.2.2 Load and Friction

Total load and friction were obtained using Eqn. (4.29):

Φ =

Nx/2−1∑

i=0

1/2 (x2i+1 − x2i) (f2i+1 + f2i),

with


• Φ =W


• Φ =F


Friction coefficient is the ratio of the bearing friction per unit length to the bearing

load per unit length, as shown in Eqn. (4.14):

µ =F

W.

6.2.3 Mesh Study and Selection

In order to find an appropriate mesh for the bearing pockets, it was necessary to carry

out a grid refinement study. The bearing geometry shown in Figure 6.1 was used with

the pocket size BP = 5 mm and its entrance located at x = 10 mm. The convergence

ratio was K = 1. To model each pocket, an eight block mesh was used as shown in

Figure 6.2. A uniform mesh was employed in blocks 1,2,7 and 8 but a non-uniform

mesh was used within the pocket itself. In blocks 3,4,5 and 6, the x dimension of the

cells adjacent to the entry and exit of the pocket were 400 times smaller than those of

the cells in the centre of the pocket (a uniform expansion of ca. 1.06).

h1 h

..

B2BpB1B

h3

x

z y

U

h0

Figure 6.1: Linear wedge with the pocket.

Table 6.1 summarises various cases in which the number of cells in both the x and

the z direction were varied. As with no pocket, the solution is more sensitive to grid


h1

..

Bp

x

z y

U

h0

4 6

81 2 3 5 7

0.01Bp

Figure 6.2: Block structure used in pocketed bearing analysis.

refinement in the z than in the x-direction. Based on Table 6.1, the mesh from option

2 was employed in further work in this study. In that case, cells immediately adjacent

to the pocket inlet and exit have width 0.4 µm.

Option Block Block Blocks Blocks Blocks pmax W/L

1 8 2 & 7 3 & 6 4 & 5 MPa kN/m

1 50 x 10 25 x 10 20 x 10 100 x 10 100 x 100 45.15 608.3

2 50 x 20 25 x 20 20 x 20 100 x 20 100 x 200 45.82 617.4

3 50 x 30 25 x 30 20 x 30 100 x 30 100 x 300 46.00 619.2

4 50 x 40 25 x 40 20 x 40 100 x 40 100 x 400 46.12 620.0

5 50 x 20 25 x 20 10 x 20 50 x 20 50 x 200 45.84 617.1

6 50 x 20 25 x 20 20 x 20 100 x 20 100 x 200 45.82 617.4

7 50 x 20 25 x 20 40 x 20 200 x 20 200 x 200 45.83 617.4

Table 6.1: Mesh study for 2D pocketed bearing.

6.3 2D Bearing with the Single Pocket

The cases studied in this section have the geometry as shown in Figure 6.1. The pocket

of height h3 is constructed on the stationary, top wall. The height, size and position of

the pocket will be varied, as well as the convergence ratio of the bearing. Dependence


of the friction coefficient on those parameters will be discussed.

6.3.1 Varying Pocket Height

Whilst the height of the pocket, h3, was varied, the other values were kept constant:

- Bearing width, left of the step - B1 = 10 mm

- Pocket width - BP = 5 mm

- Bearing width, right of the step - B2 = 5 mm






Figures 6.3a and 6.3b compare pressure and shear stress profiles at the lower, mov-

ing surface across the bearing for a range of h3/h0 ratios from zero (no pocket) to 40

(40 µm deep pocket). As the depth of the bearing pocket increases, the pressure field

becomes flatter and the shear stress lower, although the latter does not approach zero

even for the deepest pocket. This is because the boundary layer does not expand to

span the pocket, so that a high velocity gradient is maintained close to the moving

wall even within deep pockets. Because of the flatter pressure distribution, the bearing

load support is reduced by up to 4% for the deepest pockets, as indicated in Table 6.2.

However, the friction is reduced by up to 24%, with the result that the pocket pro-

duces a considerable reduction in the friction coefficient, as seen in Table 6.2. Figure

6.4 shows the dependence of the friction coefficient on the pocket height.

To obtain a true estimate of the performance benefit of the pocket, comparison

should be made with a non-pocketed bearing having the same load support. This

can be done simply by considering a slightly more viscous lubricant in the pocketed

bearing. However, since load support and friction of an isoviscous pad bearing are

both proportional to viscosity (Cameron [13]), the friction coefficient of the 40 µm


deep pocketed bearing is still only 0.76 of the non-pocketed bearing.

Pocket depth Load/unit length Friction/unit length Friction coeff. µh3/µ0

h3 [µm] W/L [kN/m] F/L [kN/m] µh3

0 634 0.153 0.000242 1

5 619 0.139 0.000225 0.926

10 613 0.126 0.000206 0.845

20 611 0.115 0.000189 0.779

30 611 0.113 0.000185 0.762

40 610 0.112 0.000184 0.759

Table 6.2: Influence of pocket height on bearing performance.

Figure 6.5 shows the velocity distribution at the beginning, middle and the end of

the pocket of width 5 mm and height 20 µm. It illustrates the overall fluid circulation

pattern. Figure 6.6 shows expanded views of the velocity field and lower surface pres-

sure at the pocket inlet and exit regions. Figure 6.6 (a) illustrates how the recirculating

flow impacts the incoming fluid and the subsequent development of a boundary layer

next to the moving surface. Just upstream of the pocket entrance the pressure falls

slightly, before rising as the incoming meets the recirculating fluid. The change is only

a very small proportion of the pressure for the cases with K = 1 and, as will be shown

below, is more clearly seen at lower convergence ratios. Figure 6.6 (b) shows that there

is a sudden increase in pressure just before the bearing exit. This is the well-known

”ram effect” that occurs at rapidly converging constrictions such as the entrance to

pad bearings and in step bearings (Rhim [56]). At K = 1, it makes only a very slight

contribution to pressure.

6.3.2 Varying Location of the Pocket

In this set of cases the height of the pocket was kept constant, h3 = 20 µm, but the

location of the pocket (B1, see Figure 6.1) was varied. The results for calculated load,


0 5 10 15 20x [mm]

0

10

20

30

40

50

60

p [M

Pa]

no pocketh

3/h

0 = 3.5

10 20 40

(a) Pressure

0 5 10 15 20x [mm]

0

0.0025

0.005

0.0075

0.01

τ [M

Pa]

no pocketh

3/h

0 = 3.5

10 20 40

(b) Shear Stress

Figure 6.3: Pressure and shear stress distribution along the bottom wall of the bearingfor various heights of the pocket.


0 10 20 30 40h

3/h

0

0.8

0.85

0.9

0.95

1

µ h3/µ

0

Figure 6.4: Dependence of friction coefficients on the height of the pocket.

Figure 6.5: Velocity distribution at the beginning, middle and the end of the pocket.


9.99 10.00 10.01x [mm]

45.02

45.04

45.06

p [M

Pa]

(a)

14.99 15.00 15.01x [mm]

45.74

45.76

p [M

Pa]

(b)

Figure 6.6: Pressure and velocity distributions at: (a) the inlet of the pocket, (b) theexit of the pocket.


friction, and friction coefficient are shown in Table 6.3. Pressure distribution along the

bottom wall is shown in Figure 6.7. From Table 6.3 and Figure 6.8 it can be seen that,

as might be expected, the optimum pocket position is centred just downstream of the

centre of the bearing, in the high pressure region.

0 5 10 15 20x [mm]

0

10

20

30

40

50

60

p [M

Pa]

no pocketB

1 = 2.5 mm

= 5.0 mm = 7.5 mm = 10 mm = 12.5 mm

Figure 6.7: Pressure distribution along the bottom wall for bearings with varyinglocation of a pocket.

0 2 4 6 8 10 12 14B

1 [mm]

0.7

0.8

0.9

1

µ h3/µ

0

Figure 6.8: Friction coefficient ratio µh3/µ0 for different positions of the pocket.


Pocket position Load/unit length Friction/unit length Friction coeff. µh3/µ0

B1 [mm] W/L [kN/m] F/L [kN/m] µh3

2.5 402 0.101 0.000252 1.036

5.0 521 0.102 0.000212 0.873

7.5 560 0.108 0.000193 0.766

10.0 611 0.115 0.000189 0.779

12.5 567 0.110 0.000195 0.800

Table 6.3: Influence of pocket position on bearing performance.

6.3.3 Varying Convergence Ratio

Figure 6.9 compares the pressure profiles at the lower surface across five bearings of

different convergence ratio, K, all having a single, 20 µm deep pocket of width 5 mm,

starting in the middle of the bearing. As expected, there is a very large reduction in

the pressure generated and thus in load support, as K is reduced. Several features

of interest can be seen. Within the pocket at low convergence ratios, there appears

to be a significant hydrodynamic pressure build-up. This build-up actually occurs

at all convergence ratios and can also be seen for K = 1 in Figure 6.6a. However, as

convergence ratio is reduced, the pressure generated in the pocket becomes comparable

in magnitude, and eventually exceeds, the pressure generated in the surrounding pad

regions.

In the K = 0.1 to 0 profiles, a pressure drop is seen to occur within the first

converging region, upstream of the pocket inlet: in effect the front land is beginning to

behave as an independent bearing. Eventually, at K = 0.001 and K = 0, the pressure

falls below zero at the pocket inlet and cavitation ensues. This is illustrated clearly in

Figures 6.10 (a) and 6.10 (b) which show the fluid density in the vicinity of the pocket

entrance step for the K = 0.001 bearing. Figure 6.10 (b) shows a magnified view of the

area of interest. Cavitation begins upstream of the pocket entrance at the stationary

wall and extends into the pocket. With the cavitation model used, this results in a


0

30

60p

[MPa

]K = 1

0

5

10

15

p [M

Pa]

K = 0.1

0

1

2

p [M

Pa]

K = 0.01

0

0.5

1

p [M

Pa]

K = 0.001

0 5 10 15 20x [mm]

0

0.5

1

p [M

Pa]

K = 0

Figure 6.9: Pressure distribution at the bottom wall of the bearing for varying conver-gence ratio, K.


two-phase fluid over most of this region, but there is almost full vaporisation around

the sharp entrance tip. This is shown in Figure 6.11 which plots the density across

the pocket inlet, along a line parallel to the lower surface but just touching the tip

of the inlet. Figure 6.11 also shows how the density calculation converges over time

eventually to reach a stable, oscillatory solution.

Pressure converged more rapidly and Figure 6.12 shows the lower wall pressure on

an expanded scale at both the entrance and exit of the pocket. At the entrance, the

pressure falls below zero and the fluid cavitates (although it only falls to a density of

700 kg/m3 at the lower wall and thus does not reach the full vapour state). There is a

very clear ram effect at the pocket exit.

Table 6.4 summarises the effect of the pocket on load support, friction and friction

coefficient for the five convergence ratios studied.

K=0 K=0.001 K=0.01 K=0.1 K=1

pocket no pocket no pocket no pocket no pocket no

pocket pocket pocket pocket pocket

W/L [kN/m] 4.232 - 4.770 1.997 17.60 19.70 159.0 173.0 610.9 635.5

F/L [kN/m] 0.0595 - 0.0603 0.200 0.0596 0.199 0.0662 0.191 0.116 0.155

µ 0.014 - 0.013 0.10 0.0034 0.010 0.0004 0.0011 0.00019 0.00024

Table 6.4: Influence of the convergence ratio on bearing performance.

It can be seen the pocket produces a very large reduction in friction coefficient at low

convergence ratios compared to the linear, plain bearing. At intermediate convergence

ratios, the origin of this effect is similar to that found with K = 1 in the previous Section

of this Chapter, i.e. the reduced friction in the pocket is greater than the reduced load

support due to the pocket. (The proportionate reduction in friction increases with

decreasing K since the average gap decreases as K is reduced at fixed h0, increasing

the friction for the unpocketed bearing K). However, when K becomes very small, a

second effect comes into play in that the load support of the pocketed bearing becomes

greater than that of the unpocketed bearing. This is because, at very low convergence,


(a)

(b)

Figure 6.10: Density distribution at the beginning of the pocket for K = 0.001; (a) forthe entire height of the pocket, (b) for the magnified region around the step.


9.95 10 10.05 10.1x [mm]

0

200

400

600

800

1000ρ

[kg/

m3 ]

t = 0.6 s = 0.8 s = 1.0 s = 1.2 s = 1.4 s = 1.6 s = 1.8 s = 2.0 s

Figure 6.11: Rate of convergence of solution for density in the pocket inlet region.

9.95 10 10.05x [mm]

-0.02

-0.01

0

0.01

0.02

p [M

Pa]

14.95 15 15.05x [mm]

0.69

0.7

0.71

0.72

0.73

p [M

Pa]

(a) (b)

Figure 6.12: Pressure distribution at the beginning and the end of the pocket forK = 0.001; (a) beginning, (b) end.


the pressure build-up in the pocket itself, which acts as an independent step bearing,

exceeds the very small pressure generation in the rest of the bearing. For shallower

pockets this effect would be expected to become significant at higher convergence ratios.

6.3.4 Varying Size of the Pocket

In order to determine which size of the pocket will give the most reduction in the

friction coefficient, the pocket was kept at the constant position with B1 = 10 mm

and with the constant height of h3 = 20µm. Table 6.5 shows load per unit length, the

friction per unit length and the friction coefficient for various sizes of the pocket. It

can be seen that the optimum size of the pocket is 25% of the bearing size.

Pocket size Load/unit length Friction/unit length Friction coeff. Coeff. ratio

BP [mm] W/L [kN/m] F/L [kN/m] µh3 µh3/µ0

no pocket 634 0.153 0.000242 1

0.3 620 0.135 0.000218 0.901

0.4 620 0.133 0.000215 0.888

0.5 611 0.115 0.000189 0.779

0.6 610 0.120 0.000197 0.814

0.7 588 0.113 0.000192 0.793

Table 6.5: Influence of the pocket size on bearing performance.

6.4 2D Bearing with Multiple Pockets

In the previous Section, the linear wedge with one pocket was analysed. It was shown

that, in order to obtain maximum reduction in the friction coefficient, the pocket area

was to be of ca. 25% of the total bearing width, that the pocket must not be placed too

close to the beginning of the bearing and that the friction coefficient reduces sufficiently

with the pocket height of h3 = 20 µm.

In this Section, it will be shown how CFD code can successfully model the bearings


with multiple pockets and that similar conclusions about the size and the position of

the pockets can be made as in the case with only one pocket. This Section will study

the cases with the general geometry shown in Figure 6.13. The block structure will be

B

B1 Bp Bf Bp B2Bp

U

h3

hh1

x

yzh0

Figure 6.13: Linear wedge with multiple pockets.

similar to the one of the bearing with the single pocket (see Figure 6.2), with the mesh

being the finest at the beginning and the end of each pocket.

6.4.1 Four Pockets Covering 25% of Total Area

In previous cases, the linear wedge had one pocket, covering 25% of the total area. In

this subsection, the bearing has four pockets which cover 25% of total area (see Figure

6.13). The cases studied are shown in Table 6.6. The first case has even spacing with

the following dimensions: B1 = B2 = 1.875 mm, Bp = 1.25 mm, Bf = 3.75 mm. In

other two cases, B1 is varied and Bp = Bf = 1.25 mm. In Figure 6.14, the pressure

distribution across the bottom wall is shown. From the results in Table 6.6 and pressure

plots in Figure 6.14, we see that placing the pocket too close to the inlet does not lead

to the reduction in friction coefficient.

6.4.2 Eight Pockets Covering 50% of Total Area

In this subsection, the bearing has eight pockets which cover 50% of total area (see

Figure 6.13). The pocket depth is varied, whilst the other parameters are kept at

constant values: B1 = B2 = 0.625 mm, Bp = Bf = 1.25 mm. From Table 6.7, it can

be seen that such geometry is not adequate for reducing friction coefficient. We can


Position of the Load/unit length Friction/unit length Friction coeff. Coeff. ratio

first pocket

B1 [mm] W/L [kN/m] F/L [kN/m] µh3 µh3/µ0

no pocket 634 0.153 0.000242 1

1.875 487 0.112 0.000230 1.052

even spacing

2.5 472 0.102 0.000216 0.893

7 580 0.109 0.000188 0.777

Table 6.6: Cases of linear wedge with four pockets covering 25% of total area.

0 5 10 15 20x [mm]

0

10

20

30

40

50

p [M

Pa]

B1 = 2.5 mm

= 7 mm = 1.875 mm (even)2D Reynolds

Figure 6.14: Pressure distribution for the cases with four pockets covering 25% of totalarea.


see, however, that as in the cases with one pocket, the increase of pocket height leads

to the reduction of the friction coefficient.

Pocket depth Load/unit length Friction/unit length Friction coeff. Coeff. ratio

h3 [µm] W/L [kN/m] F/L [kN/m] µh3 µh3/µ0

no pocket 634 0.153 0.000242 1

5 354 0.115 0.000326 1.347

10 333 0.103 0.000309 1.277

20 325 0.088 0.000271 1.120

40 322 0.079 0.000245 1.012

Table 6.7: Cases of linear wedge with eight pockets covering 50% of total area.

6.5 3D Linear Wedge

In previous Section, all the cases studied assumed that the bearing is infinitely long,

which in reality, is not true. In order to show the applicability of CFD also on modelling

of 3D geometry, this Section looks into effects of pockets in a 3D square linear pad

bearing.

The pressure at all the four edges (inlet, outlet, y = ±L/2) of the bearing was set to

atmospheric (105 Pa) and zero velocity gradient in the normal direction was assumed

for the velocity field. At the solid walls, the ’no-slip’ boundary condition was assumed

for the momentum equations as in the previous cases.

6.5.1 3D Linear Wedge with One Pocket Covering 25% of

Total Area

A 20 mm square 3D linear pad bearing with a square pocket was examined as shown

in Figure 6.15. As for the 2D bearings analysed, the pocket area was 25% of the total

(Bp = Lp = 10 mm) and K = 1 and h0 = 1 µm.

The position of the pocket is varied, whilst the other parameters are kept at constant

values. The pocket height is h3 = 20 µm. Table 6.8 summarises the load and friction


Outlet

B1 Bp

B

L1

L

x

y

y=0Lp

Inlet

Figure 6.15: Top view of a 3D linear wedge with the pocket.

performance for various positions of the pocket. The same conclusion can be drawn as

in previous cases; the optimum position of the pocket is in the high pressure region.

It can be seen that the friction coefficient reduction is less than for 2D cases but it is

still considerable. Figure 6.16 shows the pressure field for the case with the maximum

reduction of the friction coefficient, with B1 = 5 mm. In Figure 6.17, the pressure

distribution along the bottom wall at y = 0 is shown.

Location of front Load/unit length Friction/unit length Friction Coefficient

of pocket coefficient ratio

B1 [mm] W/L [kN/m] F/L [kN/m] µh3 µh3/µ0

No pocket 276 0.150 0.000544 1

5 220 0.119 0.000540 0.992

7.5 242 0.116 0.000481 0.884

7.5 242 0.116 0.000481 0.884

8.0 241 0.113 0.000479 0.881

8.5 235 0.115 0.000488 0.896

Table 6.8: Effect of pocket position for 3D single pocket.


Figure 6.16: Pressure distribution across the bottom wall for the case with B1 = 5 mm.

0 5 10 15 20x [mm]

0

10

20

30

40

p [M

Pa]

3D - no pocketB1 = 5 mm 7.5 mm 7.7 mm 8 mm 8.5 mm

Figure 6.17: Pressure distribution along the bottom wall for y = 0 m.


6.5.2 3D Linear Wedge with Two Pockets Covering 25% ofTotal Area

In this subsection, 3D linear wedge bearing with two pockets is studied. The top view

of the bearing (x-y plane) is shown in Figure 6.18.

B1

B

L1

L

x

y

Lp

Bp Bp

Inlet Outlet

B2

y=0

Figure 6.18: Top view of a 3D linear wedge with two pockets.

The positions of both pockets, B1 and B2, are varied while the other parameters

are kept at constant values: B = L = 20 mm, Lp = 2Bp = 10 mm, L1 = 5 mm. Table

6.9 shows the numerical results for pockets in various positions.

In Figure 6.19, pressure field is shown for the case with B1 = 0.7 ·10−2 m. In Figure

6.20, the pressure distribution along the moving wall for y = 0 is shown.

6.6 Conclusions

The work in this Chapter has produced information about fluid behaviour in sharply

pocketed bearings at two levels. It has provided some details of how lubricants flow

and cavitate within fluid film bearings and, in particular, in sharp, closed pockets of

a type not convincingly analysed using Reynolds equation. It has also highlighted the

practical impact that these pockets have on bearing frictional performance.


Position of the Gap between Load/ Friction/ Friction Coefficient

first pocket the pockets unit length unit length coefficient ratio

B1 [mm] B2 [mm] W/L [kN/m] F/L [kN/m] µh3 µh3/µ0

No pocket 0 276 0.150 0.000544 1

5 2.5 230 0.117 0.000508 0.935

6 1 236 0.117 0.000497 0.914

6.5 1 238 0.117 0.000491 0.903

6.5 1.5 234 0.116 0.000496 0.912

7 1 237 0.116 0.000490 0.902

8 0.5 233 0.115 0.000493 0.906

Table 6.9: Cases of 3D linear wedge with two pockets covering 25% of total area.

Figure 6.19: Pressure distribution across the bottom wall for the case with two pocketsand B1 = 7 mm.


0 5 10 15 20x [mm]

0

10

20

30

40

p [M

Pa]

3D - no pocketsB

1 = 5 mm, B

2 = 2.5 mm

= 6 mm, = 1 mm = 6.5 mm, = 1 mm = 6.5 mm, = 1.5 mm = 8 mm, = 0.5 mm = 7 mm, = 1 mm

Figure 6.20: Pressure distribution along the bottom wall for y = 0 m.

There appear to be two different ways that the pockets studied can lead to a sig-

nificant friction coefficient reduction. One, which can operate in medium to high

convergence, high-pressure bearings, is achieved by positioning pockets so that they

reduce friction more than they reduce load support. To do this, the pockets must be

located in the high pressure region at the rear of the bearing. They must not be so close

to the inlet as to severely limit hydrodynamic pressure build-up. This approach may

be a practical means of significantly reducing friction in steadily-loaded journal and

thrust bearings. The second mechanism only becomes significant in low convergence,

low load bearings, where the amount of pressure build-up over most of the bearing is

negligible. In this case, the pocket provides a convergent step within the bearing within

which hydrodynamic pressure can develop. This effect is helped by the occurrence of

cavitation at the pocket inlet but, as seen for the K = 0.001, begins to occur even

when no cavitation occurs. This mechanism is essentially the same as that suggested

by Hamilton et al. [33] and later by Etsion et al. [22], based on a Reynolds analysis

of textured parallel surfaces.


This work does not rule out the possible contribution of inertial effects to load

support recently reported by Arghir et al. [4] in bearings with a high density of small

surface roughness features. These effects were, however, evident when the amplitude

and wavelength of roughness were of the same order of magnitude while the current

analysis, the ratio of pocket depth to width was typically only 0.005.

This work is clearly only the first step in the study of textured and pocketed hydro-

dynamic bearings. The pockets studied were always stationary, but there is interest

in the effect of pockets entering and then passing through a contact. The pockets in

this work have very simple, vertical edges, which, as seen in Figure 6.10, are especially

likely to promote cavitation, but which may be difficult to achieve in practical bear-

ings. Despite the limited scope, the work has shown the power of applying a modern

CFD package to explore the behaviour of lubricants in non-conventional hydrodynamic

bearing geometries.

6.7 Closure

A CFD model with cavitation has been applied to analyse the flow behaviour, load

support and friction of linear, convergent pad bearings having a closed pocket. This has

shown the development of a boundary layer and fluid recirculation behaviour within

the pocket, and a ram effect at the pocket exit.

It has been found that cavitation occurs only at very low bearing convergence ratios

and it takes place at the immediate pocket entrance, centred on the sharp recess edge.

A closed pocket of the type studied can produce a reduction in friction coefficient by

two different mechanisms. At high to medium convergence ratios, a suitably positioned

pocket in the high pressure region of the bearing reduces shear stress within the pocket

more than its pressure build-up. The result is lower friction and thus a reduced friction

coefficient. This effect increases with pocket height but levels out for a pocket to film

thickness ratio greater than ≈ 20. As the convergence ratio is reduced, an additional


mechanism for friction coefficient comes into play in which the pocketed bearing has

higher load support (and thus lower friction coefficient) than the non-pocketed case.

This is because there is so little hydrodynamic pressure rise in the non-recessed

regions of the bearing that the pressure rise generated within the pocket due to its

convergent step shape makes a significant, and eventually an overwhelming contribution

to the overall load support of the bearing.

Chapter 7

Summary and Conclusions

The main concern of this study is finding a suitable computational model to describe the

flow of lubricant in hydrodynamic bearings. Although Computational Fluid Dynamics

has been an area of intensive study for the last three decades, it is only recently that

it has started to be applied in the field of lubrication.

The Reynolds equation has been widely used in lubrication for calculating the

pressure, film thickness, load support and other flow parameters in the high pressure

areas of the bearing. However, its restrictions become evident in the regions far from

the contact zone and if surface roughnesses of the size comparable to the film thickness,

are present. In those cases, the full set of Navier-Stokes equations needs to be solved.

The geometries found in lubrication problems vary significantly from those usually

modelled with the CFD; the aspect ratios found in lubrication can be of order of 103,

while the aspect ratios encountered in typical CFD problems are of order of 1. This

meant that the CFD solver had to be validated before applying it to complex geometries

for which there is no analytical solution of the Reynolds equation.

The next step was to create the computational model for the cavitation which occurs

in the divergent region of the bearing, where the pressure drops below the saturation

pressure.

Finally, the suitable solver was employed to computationally model pad bearings

with one or more pockets, the geometry which could not be examined with the Reynolds

equation.

132

Chapter 7. Summary and Conclusions 133

In the following Sections, the summary and conclusions of each Chapter will be

given.

7.1 Governing Equations of the Flow of Lubricant

The governing equations of the flow for a Newtonian fluid under isothermal conditions

consist of the conservation of mass and the conservation of momentum. The conser-

vation of momentum, or Navier-Stokes equation has four terms: transient, convective,

diffusive and source term. Depending on the type of the flow, different terms are dom-

inant. If the Reynolds number is low (Rez < 0.1), the diffusive terms are significantly

higher than the convective ones, and vice versa for higher Reynolds numbers.

Due to the high pressures, the viscosity and density stop being constant. The

dependence of density on pressure is relatively weak, but the viscosity can increase by

several orders of magnitude as the pressure gets into the EHL range (> 108 Pa).

In the high pressure region, convective terms and certain viscous terms drop out

from Navier-Stokes equations and the pressure is assumed to be constant across the film

thickness. That leads to the Reynolds equation, which is a suitable tool for solving cases

with smooth geometry in the area near the contact. However, if the surface roughnesses

with the characteristic length comparable to the film thickness are present, the local

aspect ratio becomes much higher and the viscous terms cannot be neglected from

Navier-Stokes equations.

According to order of magnitude analysis, Reynolds equation is not adequate for

solving the fluid flow in the areas far from the contact region or if the surface rough-

nesses are present.

7.2 Discretisation

The Finite Volume method is a well-established numerical procedure, suitable for fluid

flow simulations because of its conservative properties. It is based on a discretisation


method which splits the computational domain into polyhedral control volumes with

a variable number of faces. The governing equations of continuum mechanics are con-

sequently discretised in the integral form over each control volume. The discretisation

is performed in real space, using a fixed Cartesian coordinate system on meshes that

do not change in time. The discretisation uses a colocated variable arrangement, with

all fields sharing the same control volumes.

The Finite Volume discretisation described above has been applied to fluid flow

problems. The solution algorithm for the Navier-Stokes system is summarised. The

pressure-velocity coupling is treated by the SIMPLE algorithm for steady-state calcu-

lations. For transient problems, the PISO algorithm has been used.

The Finite Volume method is an approximation of the full solution of the governing

equations. Therefore, there exists an error in the numerical simulations obtained using

this method. The error which arises from the discretisation procedure is estimated by

Richardson Extrapolation. The discretisation error is calculated from the leading term

of the truncation error from the Taylor series.

7.3 Validation of CFD Solver On Simple

Hydrodynamic Bearings

In hydrodynamic lubrication, there are geometries for which the Reynolds equation has

an analytical solution. Two of them are an infinitely long linear wedge and an infinitely

long step bearing. The analytical solution is useful to validate the numerical results

obtained with the CFD solver. The pressure field and the x-component of the velocity

vector obtained using CFD and the Reynolds equation, have been compared directly,

with the maximum difference being < 1%. The error in the numerical solution for the

maximum pressure is found using Richardson Extrapolation, and it shows a decrease

as the mesh gets finer in both x and z directions. However, the error decreases more

with grid refinement in the z direction only. Total load and friction per unit length are

calculated from the analytical solution and from the numerical results and again, they


are in good agreement (the maximum difference is < 0.1% for the finest mesh).

For the linear wedge the mesh consists of one block and it is uniform in both

directions. The geometry is smooth, i.e. it does not have any steep gradients. However,

in the case of step bearing there is a steep geometry of the step, which creates the need

for local refinement of the computational mesh around the step.

The conclusions to be drawn from the study of these two geometries were that the

CFD software was capable of accurately handling steady state hydrodynamic lubrica-

tion problems. The work also confirmed the assumption of constant pressure across the

film thickness and that the neglect of inertia and certain viscous terms in the Reynolds

equation is justified.

7.4 Modelling of Cavitation

In the case of a roller bearing, the cavitation occurs in the diverging region when the

pressure of the fluid drops below the saturation pressure. As the fluid evaporates, the

density ratio between liquid and vapour phases is of the order of 105. The sudden

change in density makes this a challenging problem to model.

The numerical cavitation method used in this work is a continuum method, where

both phases are considered to be of the same fluid and only one set of governing

equations needs to be solved. This lowers the computational cost considerably. The

conservation of energy equation is used for closure, and the speed of sound of the

mixture is given by the homogeneous equilibrium model. Both liquid and vapour

phase are treated as compressible.

By integrating the energy equation, one obtains the equation for density dependence

on pressure. The critical feature in determining the gradient of the slope which links

the liquid and vapour phase is the speed of sound in vapour. In order for the model to

be numerically stable, the value of the speed of sound in vapour had to be somewhat

higher than the realistic one.


The two cases studied are the pure rolling, full-cylinder and the pure sliding, half-

cylinder. When modelled without the cavitation, the numerical results were in agree-

ment with the Full Sommerfeld solution, as expected. With the cavitation model in-

cluded, the results for the two cases show many similarities. In both cases the pressure

drops below zero in the diverging region and the cavitation does not occur instanta-

neously but gradually over the space of ≈ 0.02R. The difference between the two cases

is that for the pure rolling case, the fluid begins to cavitate closer to the cylinder, while

for the pure sliding case, the fluid starts cavitating in the middle of the film thickness

but evaporates more rapidly from the moving wall. Backflow is present in both cases

in the converging and the diverging region.

The cavitation model used in this work produced results which are qualitatively

likely to be accurate. However, the results should not be considered to be quantitatively

correct. That is mainly because of the assumption that the liquid to vapour transition

is the only process of cavitation, because the speed of sound in vapour has unrealistic

value and because the possibility of dissolved gasses in the liquid phase is neglected.

7.5 Low Friction Pocketed Pad Bearing

The use of CFD enables us to examine more complex geometries, which could not be

solved with the Reynolds equation. In this study one or more pockets are placed on

the top wall of the simple pad bearing. Their height, size and position are varied, as

well as the convergence ratio of the bearing. The effect of the pockets on the reduction

of the friction coefficient is examined and the following conclusions are made:

• As the height of the pocket increases, both load support and friction decrease.

However, the reduction in load support is much smaller than the reduction in

friction, thus reducing the friction coefficient.

• The optimum location for the beginning of the pocket is in the middle of the

bearing. If the pocket is too close to the inlet, the pressure build-up is reduced,


making the load support smaller which has negative effects on the reduction of

the friction coefficient.

• The size of the pocket which gives the biggest reduction in the friction coefficient

is 25% of the total area of the bearing.

• The presence of the pocket with the optimum height, position and size, reduces

the friction coefficient for the pad bearing with all the convergence ratios stud-

ied in this work. However, the pad bearing with smaller convergence ratio can

support smaller load. For the convergence ratio K = 0.001, the fluid cavitates at

the beginning of the pocket. For that case, the pocket acts as a separate bearing

producing a pressure build up which is significant in comparison to the pressure

in the inlet region of the bearing.

We can conclude that there are two ways that a closed pocket in pad bearing can reduce

the friction coefficient. In bearings with high convergence ratios, a suitably-positioned

and sized pocket reduces friction more than the load support, and in bearings with

low convergence ratios the pocketed bearing has higher load support than in the non-

pocketed case, leading again to the reduction in friction coefficient.

7.6 Suggested Future Work

The results in this study are obtained using several simplifications which are not real-

istic:

• Isothermal conditions were assumed throughout;

• The bearings were assumed to have smooth surfaces;

• In the cavitation model, the isentropic phase transformation approach was em-

ployed, and it was assumed that the fluid to vapour transition is the only process

of cavitation;


• The surfaces of the roller bearing were assumed to be infinitely rigid, not allowing

for the elastic deformations due to the high pressures;

• The lubricant was assumed to be isoviscous.

In future work these simplifications need to be eliminated. One of possible directions in

which to continue this work is to improve the present cavitation model. As discussed

in Chapter 5, there are various types of cavitation models used in CFD. The most

elaborate are full multiphase models, which, given the initial sizes and distribution of

cavitation nucleii within the fluid, can calculate the evolution of vapour field, including

growth/collapse, transport by the fluid motion, etc. Employing a model of that type

would, most likely, improve the quantitative aspect of the result.

The next step towards the EHL solution could be the inclusion of elastic deforma-

tion of the surfaces due to the high pressures. There are two main ways of including

elastic deformation in numerical simulations of the EHL problem: multigrid methods

(Venner and Lubrecht [69]) and differential deflection (Evans and Hughes [26]). Both

of these methods reduce computational time needed to calculate the flow parameters

and the elastic deformation. In multigrid method, one uses several grids of different

sizes to smooth the iterative errors. The central point is that error components with

wavelengths which are comparable to the grid size are reduced efficiently, and those

with large wavelengths compared to the mesh size converge slowly. The basis of multi-

grid solutions method is to use coarser grids to solve the smooth error components,

which reduces computational time significantly. In differential deflection, the explicit

deflection equation is replaced by a second-order differential equation. In numerical cal-

culations the deflection can be reduced to a sum of coefficients multiplied by pressures

in each point of the domain. In the same way, the second-order differential equation

of the deflection can be constructed as a sum of coefficients multiplied by the corre-

sponding pressures. The coefficients in second-order differential deflection are shown

to decay at a much higher rate as the modulus of the index increases from zero than


do the coefficients of the explicit deflection equation.

The inclusion of piezo-viscous effects could be the next step, followed by the intro-

duction of thermal effects. This would mean that the algorithm for the EHL problem

with piezo-viscous and thermal effects is developed. This CFD-solver could then be

applied to various geometries with surface roughnesses.

140

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List of Publications

Brajdic-Mitidieri, P., Gosman A. D., Ioannides, E., Spikes, H.A.: “CFD Analysis of a

Low Friction Pocketed Pad Bearing”, Journal of Tribology, Transactions of the ASME,

127, pp. 803-812, 2005.

Documents

Advanced Modelling of Elastohydrodynamic Lubrication PetraBrajdicPhD2005