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Hydrodynamic bearings allow the various parts of mechanical devicesto be driven easily while ensuring a reliability which eliminates any riskof rupture or premature wear. When the operating conditions are severe(high or quickly varying loads, great rotational frequency) it becomesdifficult to achieve this double goal without the assistance of powerfulprediction digital models.

This 4-volume series of books brings together the elements necessaryfor the realization and the validation of these tools.

This volume, the third in the series, presents several thermo-hydrodynamic models. Many examples related to the bearings of theinternal combustion engines illustrate the possibilities of these tools.The definition and the digital modeling of the problems ofthermoelastohydrodynamic lubrication complete this volume.

These books are of great interest both for researchers wishing to extendtheir knowledge of the behavior of connecting rod and crankshaftbearings of internal combustion engines and for engineers aiming todevelop equipment that reduces engine energy losses while increasingtheir reliability. The analysis, modeling and solution methods presentedhere are sufficiently general to be applied to all reciprocating systems,such as, for example, piston compressors.

Dominique Bonneau is Professor Emeritus specializing in the numericalmodelization of lubrication problems who has worked as a teacher-researcher at the IUT of Angoulême and at the Institute PPRIME(Laboratory of Mechanics of Solids) of the University of Poitiers-CNRS-ENSMA in France.

Aurelian Fatu is Professor and a researcher at the Institute PPRIMEspecializing in the modeling of problems of lubrication for enginebearings and for systems’ sealing devices.

Dominique Souchet specializes in the modeling of thermo-hydrodynamic lubrication of journal and thrust bearings withNewtonian or non-Newtonian fluids. He is a university lecturer andresearcher at the Institute PPRIME.

Thermo-hydrodynamicLubrication in

Hydrodynamic Bearings

Dominique Bonneau Aurelian Fatu

Dominique Souchet

NUMERICAL METHODS IN ENGINEERING SERIES

W683-Bonneau 3.qxp_Layout 1 02/07/2014 12:12 Page 1

Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings

Series EditorPiotr Breitkopf

Thermo-hydrodynamicLubrication in

Hydrodynamic Bearings

Dominique BonneauAurelian Fatu

Dominique Souchet

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, aspermitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,or in the case of reprographic reproduction in accordance with the terms and licenses issued by theCLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at theundermentioned address:

ISTE Ltd John Wiley & Sons, Inc.27-37 St George’s Road 111 River StreetLondon SW19 4EU Hoboken, NJ 07030UK USA

www.iste.co.uk www.wiley.com

© ISTE Ltd 2014The rights of Dominique Bonneau, Aurelian Fatu and Dominique Souchet to be identified as the authorsof this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014942901

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-683-9

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY

Contents

PERFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

NOMENCLATURE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

CHAPTER 1. THERMO-HYDRODYNAMICLUBRICATION . . . . . . . . . . . . . . 1

1.1. Global thermal balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Energy equation for the lubricant film . . . . . . . . . . . . . . . . . . . 41.2.1. Particular case of non-filled film zones . . . . . . . . . . . . . . . 5

1.3. Fourier equation inside the solids. . . . . . . . . . . . . . . . . . . . . . 61.4. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.1. Supply ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.2. External walls of solids . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3. Surfaces at solid truncations . . . . . . . . . . . . . . . . . . . . . . 91.4.4. Interfaces between film and solids . . . . . . . . . . . . . . . . . . . 91.4.5. Supply orifices and grooves . . . . . . . . . . . . . . . . . . . . . . 111.4.6. Axial extremities of the lubricant film . . . . . . . . . . . . . . . . 17

1.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

CHAPTER 2. THREE-DIMENSIONALTHERMO-HYDRODYNAMICMODEL . . . . . 19

2.1. Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2. Discretization of the film energy equation . . . . . . . . . . . . . . . . . 202.2.1. Stationary case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2. Transient case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3. Discretization of Fourier equation in the solids . . . . . . . . . . . . . . 382.4. Assembly of discretized equations for the film and the solids . . . . . . 402.5. Numerical behavior of the THD finite element model . . . . . . . . . . 432.5.1. Definition of reference problems. . . . . . . . . . . . . . . . . . . . 43

vi Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings

2.5.2. Behavior for a stationary case . . . . . . . . . . . . . . . . . . . . . 452.5.3. Behavior for a transient case . . . . . . . . . . . . . . . . . . . . . . 572.5.4. Behavior in the case of a variation in the axial direction of the filmthickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.5.5. Evaluation of the global thermal method (GTM) . . . . . . . . . . . 70

2.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

CHAPTER 3. SIMPLIFIEDTHERMO-HYDRODYNAMICMODELS . . . . . . . . . . 73

3.1. Simplified THD model based on the Rhode and Li assumptions. . . . . 733.1.1. Expression of the pressure and reduced Reynolds equation . . . . . 733.1.2. Velocity components . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1.3. Energy and Fourier equations . . . . . . . . . . . . . . . . . . . . . 763.1.4. Discretization of equations . . . . . . . . . . . . . . . . . . . . . . . 773.1.5. Evaluation of the method based on Rhode and Li assumptions . . . 82

3.2. Simplified models for cyclic regimes. . . . . . . . . . . . . . . . . . . . 853.2.1. Model with the temperature averaged on thefilm thickness (ATM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.2.2. Model with a parabolic temperature profile acrossthe film thickness (PTM). . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.3. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

CHAPTER 4. COMPUTING THETHERMOELASTICDEPENDENCYMATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.1. Computing the thermoelastic dependency matrices to beused for the three-dimensional and Rhode and Li models . . . . . . . . . . . 1044.2. Computing the thermoelastic dependency matrices to beused for the simplified models . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2.1. Equation setting for compliance matrices when the thermalboundary layer is modeled by a transfer coefficient . . . . . . . . . . . . . 1064.2.2. Equation setting for compliance matrices when the thermalboundary layer is modeled by a Fourier series . . . . . . . . . . . . . . . . 107

4.3. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

CHAPTER 5. GENERALALGORITHM AND SOFTWARE FOR SOLVINGBEARINGLUBRICATION PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1. Parameters and equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1.1. The parameters that must be known before computing . . . . . . . 1115.1.2. The unknown parameters, objective of the computation . . . . . . . 1135.1.3. The equations to be solved . . . . . . . . . . . . . . . . . . . . . . . 114

5.2. General algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Contents vii

5.3. Solving finite element discretized EHD problemwith the Newton–Raphson method . . . . . . . . . . . . . . . . . . . . . . . 1175.3.1. Constitutive equations for the EHD problem . . . . . . . . . . . . . 1175.3.2. Discretized equations for the EHD problem . . . . . . . . . . . . . 1195.3.3. Solving algorithm for the EHD problem . . . . . . . . . . . . . . . 129

5.4. Techniques for reducing the computation time . . . . . . . . . . . . . . 1315.4.1. Non-systematic evaluation of the Jacobian matrix . . . . . . . . . . 1315.4.2. Decomposition of the hydrodynamic pressure . . . . . . . . . . . . 132

5.5. Mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.5.1. Principle of the refinement method . . . . . . . . . . . . . . . . . . 1385.5.2. Computation of the local compliance matrix . . . . . . . . . . . . . 1405.5.3. Expression of the shell surface deformation . . . . . . . . . . . . . 141

5.6. Architecture of software for bearing lubrication computation . . . . . . 1435.7. An example of TEHD computation software: ACCEL . . . . . . . . . . 1455.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Preface

This volume constitutes the third part of a series dedicated to hydrodynamicbearings.

Volume 1 [BON 14a] describes in detail the physical properties of lubricants thatplay an essential role in the hydrodynamic process, followed by the equations ofhydrodynamic lubrication and the models for numerical solving. The description of“elastohydrodynamic” (EHD) models also forms part of the content of the firstvolume.

Volume 2 [BON 14b] is dedicated to the study of mixed lubrication. The roleplayed by the roughness of surfaces – in terms of hydrodynamics on the one hand,and in terms of contact of the surface asperities on the other hand – is analyzed indetail and completed by a presentation of the corresponding numerical techniques.This volume also deals with the problem of surface wear in this context.

In bearings under extreme operating conditions, increases in temperature causedby the shearing of the lubricant can reach several tens of degrees Celsius. This, dueto the combined effect of a decrease in viscosity and a dilation of the solids, leads toimportant changes in the thickness of the lubricating film. A new shear-viscosity-temperature interdependence substitutes the pressure-thickness of the film of theEHD problem’s interdependence. In this case, the equation of the energy in the filmis associated with the Reynolds equation, both equations constituting the basis of the“thermo-hydrodynamic” (THD) problem. Apart from the global thermal model, theTHD models are complex. Chapter 1 of this third volume gives a detaileddescription of these models. They must consider the spatiotemporal variation of thetemperature in the lubricant film and in the surroundings solids. The Fourierequation for the solids is added to the system of the Reynolds equation and energyequation. The conditions at the interfaces and the boundaries with the lubricantsupply ducts and the external ambient medium are also discussed in this chapter.

x Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings

Chapter 2 presents a three-dimensional (3D) finite element modelization of theTHD problem. Steady-state as well as transient situations are considered. Thenumerical behavior of the 3D-THD finite element model is analyzed throughnumerous simulations. For these, a rigid bearing with a circumferential profile that issimilar to the profile obtained for a heavily loaded deformable bearing is definedanalytically. This bearing can be used as a reference for the evaluation of new THDmodels.

The 3D-THD model presented in Chapter 2 is cumbersome to put into practice,and its application takes up vast amounts of central processing unit (CPU) time,which is unacceptable for industrial applications. In order to put powerful tools atthe disposal of industrial research and development (R&D) departments, simplifiedmodels that are more efficient in terms of computation time are detailed in Chapter3. Their performances are evaluated by comparison between their results and thosegiven by the 3D-THD model.

The association of situations described in Volumes 1 and 2 and in the previouschapter – EHD, mixed lubrication, THD – defines thermoelastohydrodynamic(TEHD) problems as they are presented in Chapter 4. The simultaneousconsideration of all the problems requires sophisticated and effective calculationtechniques, which are described in detail. The digital tools developed by the authorsof this book and their collaborators are primarily dedicated to the study of thebearings of internal combustion engines, and more generally to the study of allrelated systems (piston compressors). However, as experience has shown, thechanges needed to deal with problems of bearings designed for other applicationsare minor, which underlines the general character of the presented algorithms.

Volume 4 [BON 14c] completes this series. In this volume, the problems specificto the calculation of engine and compressor bearings are described in detail. Thisfinal volume also contains a chapter on the techniques of optimization for thecalculation of bearings, with applications for the calculation of a connecting rodbearing in an internal combustion engine.

Bibliography

[BON 14a] BONNEAU D., FATU A., SOUCHET D., Hydrodynamic Bearings, ISTE, London andJohn Wiley & Sons, New York, 2014.

[BON 14b] BONNEAU D., FATU A., SOUCHET D., Mixed Lubrication in Hydrodynamic Bearings,ISTE, London and John Wiley & Sons, New York, 2014.

[BON 14c] Bonneau D., FATU A., SOUCHET D., Internal Combustion Engine BearingsLubrication in Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, NewYork, 2014.

Nomenclature

Points, basis, repairs, links and domains

M point inside the lubricant filmM1 point on wall 1 of the lubricant filmM2 point on wall 2 of the lubricant filmO origin point of lubricant film repair (developed bearing)x, y, z Cartesian basis for the film (developed bearing)xc, yc, zc Cartesian basis for the housingΩ, ΩF film domainΩ0, Ωr film domain, non-active zoneΩp film domain, active zoneΩS domain occupied by a solid∂Ω0 boundary of a non-active zone parallel to the circumferential direction∂Ω1 part of an active zone boundary where the pressure is imposed∂Ω2 part of an active zone boundary where the flow rate is imposed∂Ωam up-flow boundary for a non-active zone∂Ωav down-flow boundary for a non-active zone∂ΩS boundary of a solid

Non-dimensional numbers

Nu 0,41Re Prmc Nusselt number

Pr /Cp kμ Prandtl number

xii Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings

ℜ , Re Uhρμ

Reynolds number

*ℜCR

ℜ modified Reynolds number

Pe pC ULk

ρ Péclet number

Scalars

B m bearing half-widthC m bearing radial clearanceCm lubricant / external gas mixture densityCp J kg-1 °C-1 specific heatD Pa ; m universal variable representing p else r – hEc discretized contact equationEi discretized Reynolds equation relative to node iERx, ERy discretized load equationsEMOx, EMOy discretized moment equationsF kg m-2 Couette flow rate factorG kg (Pa.s)-1 Poiseuille mass flow rate factorH Wm-2 °C-1 thermal transfer coefficientH1 m level of wall 1 at point with x, z projected coordinatesH2 m level of wall 2 at point with x, z projected coordinatesJ, J2 m Pa-1 s-1 integrals on film thicknessL m bearing widthN interpolation functionQm kg s-1 lubricant mass flow rateQv m3 s-1 lubricant volume flow rateQC m3 s-1 lubricant volume flow rate per cycle passing through the

bearing extremitiesQC

+ m3 s-1 lubricant volume flow rate per cycle outing through thebearing extremities

QC¯ m3 s-1 lubricant volume flow rate per cycle entering through thebearing extremities

R m bearing radiusT ° C temperatureU m s-1 shaft peripherical velocity for a bearingU1 m s-1 velocity of wall 1 in x direction at point (x, H1, z)

Nomenclature xiii

U2 m s-1 velocity of wall 1 in x direction at point (x, H2, z)V m s-1 squeeze velocity for a bearingVF m s-1 circumferential velocity for boundary ∂Ωam or ∂Ωav

V1 m s-1 velocity of wall 1 in y direction at point (x, H1, z)V2 m s-1 velocity of wall 2 in y direction at point (x, H2, z)W m s-1 shaft axial velocity for a bearingW weighting functionW1 m s-1 velocity of wall 1 in z direction at point (x, H1, z)W2 m s-1 velocity of wall 2 in z direction at point (x, H2, z)dtS m contribution of the thermoelastic deformation of solid S to

the film thicknessh m lubricant film thicknessk Wm-1 °C-1 thermal conductivityne element numbernne node number per elementnpi number of integration points per elementnx, nz number of elements for the film mesh in circumferential

and axial directionsp Pa pressure in the lubricant filmp Pa mean pressure in the film (mixed lubrication)pa Pa ambient pressure out of the bearingpal Pa supply pressure for a bearingqm kg s-1 m-1 mass flow rate per arc length unit for a curveqv m2 s-1 volume flow rate per arc length unit for a curver m lubricant filling in a non-active zonet s timeu m s-1 circumferential velocity component at a point into the filmuN, uT m normal and tangential displacements at a contact pointv m s-1 velocity squeeze component at a point into the filmw m s-1 axial velocity component at a point into the filmx m circumferential coordinate for a point into the filmxam m circumferential coordinate for a point of ∂Ωam

xav m circumferential coordinate for a point of ∂Ωav

y m coordinate in the thickness direction for a point into thefilm

z m axial coordinate for a point into the filmγ N m-1 lubricant surface tensionα relaxation coefficient

xiv Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings

α °Pa-1 piezoviscosity coefficientβ °C-1 thermoviscosity coefficientεx, εy relative eccentricity componentsδ1, δ2 m roughness heights for surfaces 1 and 2φf , φfs , φfp correction factor for the shear stress (mixed lubrication)φx, φz, φxx, φxz, φzx, φzx Poiseuille flow factorsφs, φsx, φsz Couette flow factorsζ parametric variableη parametric variableθ rad angular coordinate for a film point for a bearingλρ °C-1 thermal correction coefficient for the densityµ Pa.s lubricant dynamic viscosityµm Pa.s dynamic viscosity of the lubricant/gas mixture into the non-

active zonesξ parametric variableρ kg m-3 lubricant densityρ m kg m-3 density of the lubricant/gas mixture into the non-active

zonesσ m combined roughness of film wallsτxy, τzy Pa shear stress into the lubricant filmω rad s-1 shaft angular velocity with respect to the housingΔt s time stepΦ index function identifying active and non-active zones in

the lubricant filmΦ Wm-3 viscous dissipation

Dimensioned parameter

h h / σ

Vectors

de m elastic deformation normal to the film walldt m thermoelastic deformation normal to the film wallf N nodal forcesn unit vector orthogonal to a domain boundaryx unit vector in the direction of the shaft surface displacement

(developed bearing)

Nomenclature xv

xc , yc, zc unit vectors for a bearing; zc parallel to the bearing axisy unit vector in the direction of the film thickness (developed

bearing)z unit vector equal to x ∧ y{D} Pa ; m vector of nodal values for the universal variable Dp, {p} Pa vector of pressure nodal valuespc Pa contact pressurer residual of equations (Newton-Raphson process)Δs solution correction (Newton-Raphson process)

Torsors

pressureℑ pressure actions exerted on the housing

applied loadℑ loading for a shaft or thrust bearing

Matrices

[A] N Pa-1 integration matrix[C] m Pa-1 compliance matrix[C*] m Pa-1 compliance matrix with a band structure

C⎡ ⎤⎣ ⎦ m Pa-1 averaged compliance matrix

[Cthi] m Pa-1 thermoelastic compliance matrix for solid Si

[CTi] thermal compliance matrix for solid Si

[K] N m-1 stiffness matrix[P] projection matrix[S] m N-1 matrix of elementary solutions[SG] m N-1 matrix of elementary solutions for the refinement block

with the original mesh[SR] m N-1 matrix of elementary solutions for the refinement block

with the refined mesh[Ai] matrix of the problem i equation discretized by the finite

element method[J] Jacobian matrix[J*] Jacobian matrix with a band structure

xvi Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings

Indices

1, 2 surfaces delimiting the filmF film or lubricantS shaft, solidsupply lubricant supplyamb ambient medium

Acronyms

ATM averaged temperature methodCPU central processing unitEHD ElastohydrodynamicFE, FEM finite element methodFPS film partition searchGT, GTM global thermal (method)PTM parabolic temperature profile methodR&D research and developmentTEHD thermoelastohydrodynamicTHD thermo-hydrodynamic

Other notations

various values a priori known (boundary conditions)various values at the time step preceding the current time step

various values at two time steps before the current time step