NUMERICAL METHODS IN ENGINEERING SERIES Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings Dominique Bonneau Aurelian Fatu Dominique Souchet
Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings
Series Editor Piotr Breitkopf Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings Dominique Bonneau Aurelian 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, as permitted 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 the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com ISTE Ltd 2014 The rights of Dominique Bonneau, Aurelian Fatu and Dominique Souchet to be identified as the authors of 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 Data A CIP record for this book is available from the British Library ISBN 978-1-84821-683-9 Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY
Contents PERFACE... NOMENCLATURE.... ix xi CHAPTER 1. THERMO-HYDRODYNAMIC LUBRICATION... 1 1.1.Globalthermalbalance... 1 1.2.Energyequationforthelubricantfilm... 4 1.2.1. Particular case of non-filled film zones... 5 1.3.Fourierequationinsidethesolids... 6 1.4. Boundary conditions... 7 1.4.1. Supply ducts... 7 1.4.2.Externalwallsofsolids... 8 1.4.3.Surfacesatsolidtruncations... 9 1.4.4.Interfacesbetweenfilmandsolids... 9 1.4.5. Supply orifices and grooves... 11 1.4.6. Axial extremities of the lubricant film... 17 1.5.Bibliography... 17 CHAPTER 2. THREE-DIMENSIONAL THERMO-HYDRODYNAMIC MODEL... 19 2.1.Modeldescription... 19 2.2.Discretizationofthefilmenergyequation... 20 2.2.1.Stationarycase... 20 2.2.2.Transientcase... 27 2.3.DiscretizationofFourierequationinthesolids... 38 2.4. Assembly of discretized equations for the film and the solids... 40 2.5.NumericalbehavioroftheTHDfiniteelementmodel... 43 2.5.1. Definition of reference problems.... 43
vi Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings 2.5.2. Behavior for a stationary case... 45 2.5.3.Behaviorforatransientcase... 57 2.5.4. Behavior in the case of a variation in the axial direction of the film thickness.... 69 2.5.5.Evaluationoftheglobalthermalmethod(GTM)... 70 2.6.Bibliography... 71 CHAPTER 3. SIMPLIFIED THERMO-HYDRODYNAMIC MODELS... 73 3.1. Simplified THD model based on the Rhode and Li assumptions.... 73 3.1.1.ExpressionofthepressureandreducedReynoldsequation... 73 3.1.2. Velocity components... 75 3.1.3.EnergyandFourierequations... 76 3.1.4.Discretizationofequations... 77 3.1.5.EvaluationofthemethodbasedonRhodeandLiassumptions... 82 3.2.Simplifiedmodelsforcyclicregimes... 85 3.2.1. Model with the temperature averaged on the film thickness (ATM)... 87 3.2.2. Model with a parabolic temperature profile across the film thickness (PTM).... 95 3.3.Bibliography... 101 CHAPTER 4. COMPUTING THE THERMOELASTIC DEPENDENCY MATRICES... 103 4.1. Computing the thermoelastic dependency matrices to be used for the three-dimensional and Rhode and Li models... 104 4.2. Computing the thermoelastic dependency matrices to be used for the simplified models... 105 4.2.1. Equation setting for compliance matrices when the thermal boundary layer is modeled by a transfer coefficient... 106 4.2.2. Equation setting for compliance matrices when the thermal boundary layer is modeled by a Fourier series... 107 4.3. Bibliography... 110 CHAPTER 5. GENERAL ALGORITHM AND SOFTWARE FOR SOLVING BEARING LUBRICATION PROBLEMS.... 111 5.1. Parameters and equations... 111 5.1.1. The parameters that must be known before computing... 111 5.1.2. The unknown parameters, objective of the computation... 113 5.1.3. The equations to be solved... 114 5.2. General algorithm... 115
Contents vii 5.3. Solving finite element discretized EHD problem with the Newton Raphson method... 117 5.3.1.ConstitutiveequationsfortheEHDproblem... 117 5.3.2. Discretized equations for the EHD problem... 119 5.3.3. Solving algorithm for the EHD problem... 129 5.4. Techniques for reducing the computation time... 131 5.4.1.Non-systematicevaluationoftheJacobianmatrix... 131 5.4.2. Decomposition of the hydrodynamic pressure... 132 5.5. Mesh refinement... 138 5.5.1.Principleoftherefinementmethod... 138 5.5.2. Computation of the local compliance matrix... 140 5.5.3. Expression of the shell surface deformation... 141 5.6. Architecture of software for bearing lubrication computation... 143 5.7.AnexampleofTEHDcomputationsoftware:ACCEL... 145 5.8.Bibliography... 147 APPENDIX... 149 INDEX... 153
Preface This volume constitutes the third part of a series dedicated to hydrodynamic bearings. Volume 1 [BON 14a] describes in detail the physical properties of lubricants that play an essential role in the hydrodynamic process, followed by the equations of hydrodynamic lubrication and the models for numerical solving. The description of elastohydrodynamic (EHD) models also forms part of the content of the first volume. Volume 2 [BON 14b] is dedicated to the study of mixed lubrication. The role played 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 in detail 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 caused by the shearing of the lubricant can reach several tens of degrees Celsius. This, due to the combined effect of a decrease in viscosity and a dilation of the solids, leads to important changes in the thickness of the lubricating film. A new shear-viscositytemperature interdependence substitutes the pressure-thickness of the film of the EHD problem s interdependence. In this case, the equation of the energy in the film is associated with the Reynolds equation, both equations constituting the basis of the thermo-hydrodynamic (THD) problem. Apart from the global thermal model, the THD models are complex. Chapter 1 of this third volume gives a detailed description of these models. They must consider the spatiotemporal variation of the temperature in the lubricant film and in the surroundings solids. The Fourier equation for the solids is added to the system of the Reynolds equation and energy equation. The conditions at the interfaces and the boundaries with the lubricant supply 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 the THD problem. Steady-state as well as transient situations are considered. The numerical behavior of the 3D-THD finite element model is analyzed through numerous simulations. For these, a rigid bearing with a circumferential profile that is similar to the profile obtained for a heavily loaded deformable bearing is defined analytically. This bearing can be used as a reference for the evaluation of new THD models. 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 at the disposal of industrial research and development (R&D) departments, simplified models that are more efficient in terms of computation time are detailed in Chapter 3. Their performances are evaluated by comparison between their results and those given by the 3D-THD model. The association of situations described in Volumes 1 and 2 and in the previous chapter EHD, mixed lubrication, THD defines thermoelastohydrodynamic (TEHD) problems as they are presented in Chapter 4. The simultaneous consideration of all the problems requires sophisticated and effective calculation techniques, which are described in detail. The digital tools developed by the authors of this book and their collaborators are primarily dedicated to the study of the bearings of internal combustion engines, and more generally to the study of all related systems (piston compressors). However, as experience has shown, the changes needed to deal with problems of bearings designed for other applications are minor, which underlines the general character of the presented algorithms. Volume 4 [BON 14c] completes this series. In this volume, the problems specific to the calculation of engine and compressor bearings are described in detail. This final volume also contains a chapter on the techniques of optimization for the calculation of bearings, with applications for the calculation of a connecting rod bearing in an internal combustion engine. Bibliography [BON 14a] BONNEAU D., FATU A., SOUCHET D., Hydrodynamic Bearings, ISTE, London and John 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 Bearings Lubrication in Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, New York, 2014.
Nomenclature Points, basis, repairs, links and domains M point inside the lubricant film M 1 point on wall 1 of the lubricant film M 2 point on wall 2 of the lubricant film O origin point of lubricant film repair (developed bearing) x, y, z Cartesian basis for the film (developed bearing) x c, y c, z c Cartesian basis for the housing Ω, Ω F film domain Ω 0, Ω r film domain, non-active zone Ω p film domain, active zone Ω S domain occupied by a solid Ω 0 boundary of a non-active zone parallel to the circumferential direction Ω 1 part of an active zone boundary where the pressure is imposed Ω 2 part of an active zone boundary where the flow rate is imposed Ω am up-flow boundary for a non-active zone Ω av down-flow boundary for a non-active zone Ω S boundary of a solid Non-dimensional numbers Nu m c1 Re Pr Pr μ Cp / k 0,4 Nusselt number Prandtl number
xii Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings R, Re * R Pe ρuh μ C R R CUL p ρ k Reynolds number modified Reynolds number Péclet number Scalars B m bearing half-width C m bearing radial clearance C m lubricant / external gas mixture density C p J kg -1 C -1 specific heat D Pa ; m universal variable representing p else r h E c discretized contact equation E i discretized Reynolds equation relative to node i E Rx, E Ry discretized load equations E MOx, E MOy discretized moment equations F kg m -2 Couette flow rate factor G kg (Pa.s) -1 Poiseuille mass flow rate factor H W m -2 C -1 thermal transfer coefficient H 1 m level of wall 1 at point with x, z projected coordinates H 2 m level of wall 2 at point with x, z projected coordinates J, J 2 m Pa -1 s -1 integrals on film thickness L m bearing width N interpolation function Q m kg s -1 lubricant mass flow rate Q v m 3 s -1 lubricant volume flow rate Q C m 3 s -1 lubricant volume flow rate per cycle passing through the bearing extremities + Q C m 3 s -1 lubricant volume flow rate per cycle outing through the bearing extremities Q C m 3 s -1 lubricant volume flow rate per cycle entering through the bearing extremities R m bearing radius T C temperature U m s -1 shaft peripherical velocity for a bearing U 1 m s -1 velocity of wall 1 in x direction at point (x, H 1, z)
Nomenclature xiii U 2 m s -1 velocity of wall 1 in x direction at point (x, H 2, z) V m s -1 squeeze velocity for a bearing V F m s -1 circumferential velocity for boundary Ω am or Ω av V 1 m s -1 velocity of wall 1 in y direction at point (x, H 1, z) V 2 m s -1 velocity of wall 2 in y direction at point (x, H 2, z) W m s -1 shaft axial velocity for a bearing W weighting function W 1 m s -1 velocity of wall 1 in z direction at point (x, H 1, z) W 2 m s -1 velocity of wall 2 in z direction at point (x, H 2, z) d ts m contribution of the thermoelastic deformation of solid S to the film thickness h m lubricant film thickness k W m -1 C -1 thermal conductivity ne element number nn e node number per element npi number of integration points per element nx, nz number of elements for the film mesh in circumferential and axial directions p Pa pressure in the lubricant film p Pa mean pressure in the film (mixed lubrication) p a Pa ambient pressure out of the bearing p al Pa supply pressure for a bearing q m kg s -1 m -1 mass flow rate per arc length unit for a curve q v m 2 s -1 volume flow rate per arc length unit for a curve r m lubricant filling in a non-active zone t s time u m s -1 circumferential velocity component at a point into the film u N, u T m normal and tangential displacements at a contact point v m s -1 velocity squeeze component at a point into the film w m s -1 axial velocity component at a point into the film x m circumferential coordinate for a point into the film x am m circumferential coordinate for a point of Ω am x av m circumferential coordinate for a point of Ω av y m coordinate in the thickness direction for a point into the film z m axial coordinate for a point into the film γ N m -1 lubricant surface tension α relaxation coefficient
xiv Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings α Pa -1 piezoviscosity coefficient β C -1 thermoviscosity coefficient ε x, ε y relative eccentricity components δ 1, δ 2 m roughness heights for surfaces 1 and 2 φ f, φ fs, φ fp correction factor for the shear stress (mixed lubrication) φ x, φ z, φ xx, φ xz, φ zx, φ zx Poiseuille flow factors φ s, φ sx, φ sz Couette flow factors ζ parametric variable η parametric variable θ rad angular coordinate for a film point for a bearing λ ρ C -1 thermal correction coefficient for the density µ Pa.s lubricant dynamic viscosity µ m Pa.s dynamic viscosity of the lubricant/gas mixture into the nonactive zones ξ parametric variable ρ kg m -3 lubricant density ρ m kg m -3 density of the lubricant/gas mixture into the non-active zones σ m combined roughness of film walls τ xy, τ zy Pa shear stress into the lubricant film ω rad s -1 shaft angular velocity with respect to the housing Δt s time step Φ index function identifying active and non-active zones in the lubricant film Φ W m -3 viscous dissipation Dimensioned parameter h h / σ Vectors d e m elastic deformation normal to the film wall d t m thermoelastic deformation normal to the film wall f N nodal forces n unit vector orthogonal to a domain boundary x unit vector in the direction of the shaft surface displacement (developed bearing)
Nomenclature xv x c, y c, z c unit vectors for a bearing; z c parallel to the bearing axis y 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 D p, {p} Pa vector of pressure nodal values p c Pa contact pressure r residual of equations (Newton-Raphson process) Δs solution correction (Newton-Raphson process) Torsors Ipressure Iapplied load pressure actions exerted on the housing 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 [C i th ] m Pa -1 thermoelastic compliance matrix for solid S i [C i T ] thermal compliance matrix for solid S i [K] N m -1 stiffness matrix [P] projection matrix [S] m N -1 matrix of elementary solutions [S G ] m N -1 matrix of elementary solutions for the refinement block with the original mesh [S R ] m N -1 matrix of elementary solutions for the refinement block with the refined mesh [A i ] 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 film F film or lubricant S shaft, solid supply lubricant supply amb ambient medium Acronyms ATM CPU EHD FE, FEM FPS GT, GTM PTM R&D TEHD THD averaged temperature method central processing unit Elastohydrodynamic finite element method film partition search global thermal (method) parabolic temperature profile method research and development thermoelastohydrodynamic thermo-hydrodynamic Other notations various various various values a priori known (boundary conditions) values at the time step preceding the current time step values at two time steps before the current time step