Anstassios Charitopoulos
email: anastasios.charitopoulos@univ-poitiers.fr

Michel Fillon

Christos I Papadopoulos

Numerical investigation of parallel and quasi-parallel slider bearings operating under ThermoElastoHydroDynamic (TEHD) Regime

Keywords: Thermoelastohydrodynamic (TEHD) regime, Slider bearing, Parallel surface, Thermal deformation. _____________________________________________________________________________________________

In the present work, a CFD numerical investigation of the tribological characteristics of parallel and quasi-parallel slider bearings has been performed. The work is initiated following experimental observations of load carrying capacity on parallel thrust bearings. The computational results conclude that the principal mechanism of pressure build-up is the hydrodynamic wedge generated due to thermal deformation of the active pad surface. The calculated final geometry of the pad consists of a converging region, extending from the oil entrance to the maximum pressure region, and a diverging region extending from the maximum pressure region to the oil outflow. Finally, initial surface defects of the pad active surface, contribute moderately to building-up the final load carrying capacity of the slider.

INTRODUCTION

The parallel slider exhibits an anomalous behaviour in terms of the classic lubrication theory, which states that a parallel slider cannot support the observed, by the experiments, loads [START_REF] Fogg | Fluid Film Lubrication of Parallel Thrust Surfaces[END_REF][START_REF] Robinson | Studies in Hydrodynamic Thrust Bearings. The parallel surfaced bearing[END_REF][START_REF] Henry | An experimental analysis of the hydrodynamic contribution of textured thrust bearings during steady-state operation: A comparison with the untextured parallel surface configuration[END_REF][START_REF] Taniguchi | A Thermo-Elastic Analysis of the Parallel Surface Thrust Washer[END_REF]. Many different explanations have been proposed in the literature. Those can be categorised in three different groups: (a) geometrical imperfections of the pad and the runner surfaces that generate pressure build-up [START_REF] Taniguchi | A Thermo-Elastic Analysis of the Parallel Surface Thrust Washer[END_REF][START_REF] Burton | Effects of Two-Dimensional, Sinusoidal Roughness on the Load Support Characteristics of a Lubricant Film[END_REF][START_REF] Hamilton | A Theory of Lubrication by Microirregularities[END_REF][START_REF] Lebeck | Parallel Sliding Load Support in the Mixed Friction Regime. Part 2-Evaluation of the Mechanisms[END_REF][START_REF] Anno | Microasperity Lubrication[END_REF][START_REF] Salama | The Effect of Macro-Roughness on the Performance of Parallel Thrust Bearings[END_REF][START_REF] Hargreaves | Surface waviness effects on the load-carrying capacity of rectangular slider bearings[END_REF][START_REF] Charitopoulos | Effects of manufacturing errors on tribological characteristics of 3-D textured[END_REF], (b) variations of the lubricant properties (mainly lubricant viscosity) in the film domain [START_REF] Ettles | The Action of the Parallel-Surface Thrust Bearing[END_REF][START_REF] Currie | Thermal Wedge Lubrication of Parallel Surface Thrust Bearings[END_REF][START_REF] Cameron | The Viscosity Wedge[END_REF][START_REF] Tieu | A Numerical Simulation of Finite-Width Thrust Bearings, Taking into Account Viscosity Variation with Temperature and Pressure[END_REF], and (c) distortion of the pad geometry due to thermal or mechanical deformations [START_REF] Tan | The Flow Factors Considering the Elastic Deformation for the Rough Surface with a Non-Gaussian Height Distribution[END_REF][START_REF] Hahn | The Effects of Thermal Expansion in Infinitely Wide Slider Bearings-Free Thermal Expansion[END_REF][START_REF] Xu | A Thermal Elastohydrodynamic Lubricated Thrust Bearing Contact Model[END_REF][START_REF] Zhai | 3D Two-way Coupled TEHD Analysis on the Lubricating Characteristics of Thrust Bearings in Pump-turbine Units by Combining CFD and FEA[END_REF][START_REF] Huang | Thrust Bearings In the 2-D Thermal Elastohydrodynamic Lubrication: On the Slider Velocity Corresponding to the Maximum Load[END_REF][START_REF] Greenwood | Elasto-hydrodynamic lubrication of centrally pivoted thrust bearings[END_REF]. Regarding geometrical imperfections, it is well known that it is impossible to manufacture sliders or bearings with perfect parallel surfaces, or no leading-edge imperfections, thus the tolerances of what is considered as parallel surface needs to be investigated. Moreover, in experiments conducted with rough and polished parallel thrust bearings, differences of the results have been observed, but those differences were not large enough to drive to the conclusion that the small geometrical imperfections of the polished thrust bearing are the only reason of load carrying capacity of a parallel surface thrust bearing or slider [START_REF] Henry | An experimental analysis of the hydrodynamic contribution of textured thrust bearings during steady-state operation: A comparison with the untextured parallel surface configuration[END_REF]. On a different path, the leadingedge imperfection has been considered to account for pressure build-up, operating as a small tapered-land geometry. However, this geometry wedge is limited to a very small part of the bearing or slider, therefore offering only a small contribution to pressure build-up. The other major category of publications on the present subject is the existence of a so-called "viscosity wedge". The variation of pressure and temperature in the fluid domain, generate a viscosity variation, leading to a wedge effect capable of generating oil pressure. Finally, other researchers [START_REF] Chen | The Contribution of Fluid Film Inertia to the Thermohydrodynamic Lubrication of Sector-Pad Thrust Bearings[END_REF] have suggested that the inertia forces and the fluid expansion play also a significant role in the parallel slider load carrying capacity. The last major group of publications on this phenomenon, is the thermal deformation of the pad geometry, due to temperature gradients at the bearing-oil interface. This would alter the geometry between the two initially parallel surfaces, leading to generation of a non-parallel fluid film shape. This shape is considered to be a converging wedge at the entrance of the lubricant, therefore generating the load carrying capacity of the parallel bearing or slider. This last assumption is consistent with the results of experimental observations [START_REF] Henry | An experimental analysis of the hydrodynamic contribution of textured thrust bearings during steady-state operation: A comparison with the untextured parallel surface configuration[END_REF], since the load carrying capacity of such bearings is less pronounced during the start-up of the experimental procedure, however, as the experimental tests progress, the generated heat alters substantially the solid temperature values, causing thermal deformation, which leads to a rapid increase of the observed load carrying capacity. In the present work, a CFD-based thermoelastohydrodynamic (TEHD) computational approach is used to study the tribological performance of quasi-parallel as well as parallel slider bearings. The paper is organized as follows: The problem definition and computational approach are first presented. Simulation results are then presented and discussed, and, finally, the main findings are summarized.

NUMERICAL MODEL

Problem description

In the present study, a three-dimensional CFD-based TEHD model of a slider bearing has been generated. A CFD model has been utilized for the calculation of flow, pressure build-up and heat dissipation in the fluid domain, as well as for the conjugate heat transfer through the solid parts of the bearing. A Finite Element (FE) model of the pad geometry has been used to calculate the thermal and mechanical deformations of the bearing pad. Further, a two-way Fluid Surface Interaction (FSI) approach has been used in order to transfer the interface pressure and thermal loads between the fluid and the solid solvers.

The numerical procedure of the present study comprises the following steps:

• First, the FE solver calculates the initial thermal expansion of the pad geometry, from an initial temperature of 20 o C to the ambient temperature of 40 o C, and transfers the displaced geometry to the CFD solver. • The Navier-Stokes and the energy equations are solved in the lubricant and slider domains, respectively, calculating the corresponding flow and temperature distributions. • The pressure and heat flux at the fluid/pad interface are transferred to the FE solver, which calculates the thermal and mechanical deformations of the pad. • The displaced geometry of the pad is transferred to the CFD solver, and an updated lubricant geometry is computed. • This described procedure iterates until final convergence, which is reached if the difference of the geometry change (measured in terms of maximum nodal mesh displacement between two consecutive steps) is less than 3x10 -8 . • The performance indices of the bearing are calculated for the final converged film geometry. The above procedure is followed for different values of the initial distance between the slider and the pad. The computational calculation algorithm is depicted in Fig. 1.

Fig. 1-Thermoelastohydrodynamic analysis of parallel slider bearings: Calculation algorithm.

In the present model, the conservation equations are solved with the CFD code ANSYS CFX and the FE code ANSYS MECHANICAL, for steady, incompressible flow, with no gravitational forces and no cavitation modeling. The equations are presented below:

Mass conservation equation:

𝛻𝛻 ⋅ 𝑉𝑉 = 0 [1]
Momentum equations:

𝜌𝜌(𝑉𝑉𝑉𝑉)𝑉𝑉 = -𝛻𝛻𝛻𝛻 + 𝛻𝛻 ⋅ (𝜇𝜇 ⋅ 𝛻𝛻𝑉𝑉) [2]
Energy equation, fluid domain:

𝜌𝜌 ⋅ 𝑐𝑐 𝑝𝑝𝑝𝑝 ⋅ 𝑣𝑣 ⋅ 𝛻𝛻𝛻𝛻 = 𝛻𝛻�𝜆𝜆 𝑝𝑝 ⋅ 𝛻𝛻𝛻𝛻� + 𝜇𝜇𝜇𝜇 [3]
Energy equation, solid domains:

𝛻𝛻(𝜆𝜆 𝑠𝑠 ⋅ 𝛻𝛻𝛻𝛻) = 0 [4]
Where V is the velocity vector (m/s), p the static pressure (Pa), Τ the temperature (K), τ the viscous stress tensor, ρ the oil density (kg/m 3 ), μ the oil dynamic viscosity (kg/m.s), cpf the oil specific heat capacity (J/kg.K), λf the oil thermal conductivity (W/m.K), λs the runner thermal conductivity (W/m.K).

The dissipation term 𝜇𝜇 describes the heat generated by internal fluid friction.

The pad undergoes elastic deformation due to (a) pressure distribution on the fluid/pad interface, and (b) the thermal expansion of the solid. The stress is related to strain by:

{σ}=[D][{ε el } [5]
where {ε el }={ε}-{ε th }, and {ε th }= ΔΤ[α se α se α se 0 0 0] T [START_REF] Hamilton | A Theory of Lubrication by Microirregularities[END_REF] where α se ,is the coefficients of thermal expansion.

The load carrying capacity (LCC) of the slider is calculated as the integral of positive pressure over the fluid/pad interface:

𝐿𝐿𝐿𝐿𝐿𝐿 = ∬ 𝛻𝛻(𝑥𝑥, 𝑦𝑦) 𝑑𝑑𝑥𝑥 𝑑𝑑𝑦𝑦 𝐹𝐹𝐹𝐹𝐹𝐹 [7]
Where:

[8]

The friction force is calculated as the integral of the shear force component over the fluid/pad interface.

𝐹𝐹𝐹𝐹 = ∬ 𝐹𝐹 𝑦𝑦 (𝑥𝑥, 𝑦𝑦) 𝑑𝑑𝑥𝑥 𝑑𝑑𝑦𝑦 𝐹𝐹𝐹𝐹𝐹𝐹 [9]

Geometry

The present computational model consists of three domains: the pad, the fluid, and the runner. The geometrical dimensions and operating conditions have been selected as close as possible to those of the thrust bearing of the experiment of Henry et al. [START_REF] Henry | An experimental analysis of the hydrodynamic contribution of textured thrust bearings during steady-state operation: A comparison with the untextured parallel surface configuration[END_REF]. The fluid domain is defined by the space between the runner and pad, which contains the thin film region and half groove fore and half groove aft. This configuration has been selected in order to account for the oil mixing and the hot oil carry-over phenomena. In the present study, the fluid/pad interface has not been considered as a "perfect" parallel surface. In particular, a sinusoidal surface defect has been introduced (Fig. 3). From industrial applications, the typical specific load for such geometries is between 0.13 MPa and 0.7 MPa, but larger specific pressures can be acquired, in the order of magnitude of 1.5 MPa. The minimum film thickness of parallel thrust slider bearings operating at a specific load of approximately 1.5 MPa is of the order of few microns, as that of the typical surface defect amplitudes. Therefore, it can easily be assumed that those defects will be affecting the hydrodynamic lubrication characteristics of the parallel thrust slider bearing. In that direction, an initial surface defect was chosen for the following calculations. The reference magnitude of the defect is 1μm, as the smallest defect amplitude observed in the experimental study of Henry [START_REF] Henry | An experimental analysis of the hydrodynamic contribution of textured thrust bearings during steady-state operation: A comparison with the untextured parallel surface configuration[END_REF].

Table 1-Dimensions of the slider bearing model of the present study.

Dimension

Boundary Conditions

The slider is consider fully flooded, with feeding pressure at the fluid inner surface, as denoted in Fig. 4. The thermal boundary conditions of the slider bearing system have been considered following the work in [START_REF] Papadopoulos | Computational Fluid Dynamics Thermohydrodynamic Analysis of Three-Dimensional Sector-Pad Thrust Bearings With Rectangular Dimples[END_REF], see Fig. 4 and Table 2. The runner is considered rigid. The pad is considered fixed on the bottom surface (no displacements or rotations allowed). The thermophysical properties of the pad and runner materials are presented in Table 3. The thermophysical properties of the oil are presented in Table 4. The density of the lubrication oil utilized (ISO VG 46) is 870 kg/m3, while a temperature dependent viscosity is considered, modeled according to the McCoull and Walther relation [START_REF] Frene | Hydrodynamic lubrication: Bearings and thrust bearings[END_REF]:

log(log(v+a) )=b-nlog(T)
where, ν is the kinematic viscosity (cSt), and a = 0.6, b = 9.02865 and n = 3.52681.

Meshing procedure

The final mesh parameters of the present model have been selected following a detailed mesh study, presented in Appendix A. Regarding the pad domain, due to the thermal and mechanical deformations, a dense hexahedral mesh has been utilised, composed of 127,310 elements. The fluid domain is discretized with hexahedral elements, utilizing 18 layers of elements in the cross-flow (film thickness) direction, whereas in the longitudinal and transverse directions, 156 and 70 elements have been used, respectively. The total number of elements in the fluid and runner domains is 268,240. The runner domain is assumed rigid, therefore, only the heat transfer equations are solved. Simple tetrahedral elements have been selected for simplifying the meshing procedure. The number of elements used for discretizing the pad geometry is approximately 600,000 elements. Mesh details about the computational domain are presented in Fig. 5.

THERMOELASTOHYDRODYNAMIC ANALYSIS OF BEARING OPERATION

Comparison between TEHD, THD with mechanical deformations and THD analyses

As reported in the introduction of the present work, different physical mechanisms have been proposed to explain the capability of parallel thrust bearings to support thrust load. In the present section, in order to quantify the effect of each pressure build-up mechanism, three different numerical models have been generated, with the same initial sinusoidal defect geometry (defect amplitude of 1 μm, see also Fig. 3), and their results have been assessed. In particular, (a) a TEHD model, taking into consideration the thermal and mechanical deformations of the bearing pad, (b) a THD model taking also into consideration only the mechanical deformations, assuming a very small value of thermal expansion coefficient of the bearing pad (order of magnitude 2x10 -45 ), and (c) a THD model, assuming that the bearing pad is non-deformable, have been generated . The first model (TEHD) accounts for almost all pressure build-up mechanisms suggested in the literature, except for micro-asperity lubrication. The second model (THD with mechanical deformations) accounts for elastic deformations of the bearing pad, but neglects thermal deformations due to temperature gradients. In the third model (THD), the bearing pad and runner are assumed rigid, therefore the film geometry remains constant throughout the simulations. In all three models, a steady-state simulation scheme is selected, whereas cavitation is not taken into consideration.

The results of the three different models are presented in Fig. 6 and7. The presented results have been acquired for constant runner velocity of 22 m/s and constant thrust load of 226 N ± 3 N. The feeding temperature is assumed to be constant and equal to 40 °C (imposed boundary condition at "Fluid Inner surface"). Large differences are observed between the calculation results of each model. In particular, maximum lubricant temperature attains a lower value for the TEHD analysis, whereas minimum film thickness and maximum pressure are substantially higher. The differences of the calculated bearing performance indices across the three different analyses vary from 5% to 30%, leading to the conclusion that the thermal deformation of the bearing pad, account only in the TEHD analysis, plays a significant role in the pressure build-up behavior of the system, affecting substantially the performance indices of the bearing.

Fig. 8 -Thrust load: 226 N, Runner velocity: 22 m/s: Calculated bearing pad profiles at the longitudinal mid-section, corresponding to (a) TEHD analysis, (b) THD with mechanical deformation analysis, and (c) THD analysis. Fig. 9 -Thrust load: 226 N, Runner velocity: 22 m/s: Calculated pressure profiles along the longitudinal mid-section of the bearing pad, corresponding to (a) TEHD analysis, (b) THD with mechanical deformation analysis, and (c) THD analysis.

In Fig. 8, the three final film shapes are depicted for each of the three different analyses. The final minimum film thickness predicted by the THD model is the lowest amongst the three different models, and thus resulting to the largest values of oil temperature. The THD model with mechanical deformation (not accounting for thermal deformation) yields a bearing pad geometry in close resemblance to the initial geometry, but with small alterations which conclude to larger minimum film thickness. The pressure profiles for the three different models are depicted in the Fig. 9, and they present different regions of maximum pressure for the different modeling techniques. In the TEHD model, the high pressure region is close to the leading edge, the exact opposite of the THD with mechanical deformation. The THD model presents a maximum pressure region near the middle of the pad, with no large negative pressure region. The different behaviour of the pressure distribution on the three models is due to the final film geometry differences. Here, it should be noted that the high pressure zone was also measured in the leading edge zone of the pads by Henry et al. [START_REF] Henry | An experimental analysis of the hydrodynamic contribution of textured thrust bearings during steady-state operation: A comparison with the untextured parallel surface configuration[END_REF] for the configuration of the parallel surfaces thrust bearings, as it is calculated by the TEHD model. Moreover, in Fig. 10, the pressure temperature contours are depicted. The temperatures at the fluid pad interface are lower for the TEHD model, whereas those of the THD model are considerably higher.

Fig. 10 -Pressure and temperature contours of the fluid/pad interface for constant load for the three different models

Effect of Initial Defect Amplitude on Bearing Performance

A parametric study has been performed on the upper pad surface to investigate how a defect, in the present case of a macroscopic sinusoidal geometry, can contribute to the supporting load of the parallel slider. In Fig. 11, the initial upper surface of the pad is depicted, for different defect amplitude values, in particular 2 μm, 1 μm, 0.5 μm, and 0.2 μm. All other bearing parameters are kept constant; runner velocity is assumed 22 m/s.

In Fig. 12, pressure, temperature, and mesh displacement profiles are presented. All the resulting fields are similar with the cases of higher defect amplitudes exhibiting higher values of pressure and decreased values of pad surface elastic deformation. In Fig. 13, the differences of the initial geometry can be observed. In the pressure profiles, as the initial defect amplitude increases the pressure positive area increases and the absolute value of the minimum pressure decreases, resulting in a higher load carrying capacity for the same H min , or, equivalently, in a higher H min for the same load carrying capacity. Moreover, whatever the amplitude of the defect, the positive pressure zone is always located in the first half of the pad, which is in good agreement with the pressure measured by Henry et al. [START_REF] Henry | An experimental analysis of the hydrodynamic contribution of textured thrust bearings during steady-state operation: A comparison with the untextured parallel surface configuration[END_REF]. Also, larger initial defect amplitude result in slightly lower temperatures. The final fluid profiles differ not only due to the initial defect amplitude, but also due to the different thermal deformation of the pad. For large amplitude defects, the thermal deformation of the bearing pad is at least three times the value of the initial defect amplitude, whereas at small defect amplitudes, this figure attains a value of 20. Therefore, for the bearing exhibiting a 2 μm defect, the final film wedge shape has a larger converging region than that of the bearings with smaller defect amplitudes, driving to the conclusion that the initial surface of macroscopic sinusoidal defect improves the performance as the defect amplitude increases. Those results do not conclude that all initial macroscopic defects improve the behaviour of a parallel thrust slider, but there are demonstrative for a large percentage of the observed ones. Usually, the machining processes utilised in thrust slider manufacturing leads to pad profiles that resemble a sinusoidal or a convex profile, thus, the results are representative for a vast majority of the of initial surface defects present in bearing manufacturing.

The load carrying capacity of the quasi-parallel thrust slider is partially affected by the surface defect, but this is not the main pressure build-up mechanism, as it can been seen in Fig. 14. The defect amplitude of 0 μm line represents a perfect parallel thrust slider, indicating that the fact that initial surface defect just contributes to the load carrying capacity, and that the main mechanism of pressure build-up must be the thermal deformation of the pad geometry. This is also supported by the experimental study of Henry et al. [START_REF] Henry | Experimental analysis of the hydrodynamic effect during startup of fixed geometry thrust bearings[END_REF], who have analyzed the behavior during the startup period of polished and rough pad surfaces (the later could be considered as a surface with small geometrical local defects) of two parallel surface thrust bearings. The main difference observed for the bearing with a rough surface, is the faster initiation of the hydrodynamic pressure build-up.

Study of Bearing Performance at Different Thrust Loads

In order to explain how the thermal deformation affects the fluid film geometry, a parametric study for a constant initial geometry and velocity has been performed. The varying parameter for the current study is the load carrying capacity. The longitudinal points of maximum and minimum pressure are almost identical in all studied configurations, as it can be observed in pressure profile of Fig. 15. Moreover, the final pad deformation is of the same shape, but as the load carrying capacity increases, the maximum deformation increases, but not proportionally. The resulting thermally generated fluid wedge angle increases as the load increases, producing shapes resembling more and more those of a tapered land slider bearing.

CONCLUSIONS

In the present work, three different studies have been conducted in order to quantify the significance of different phenomena occurring in the parallel thrust slider bearings. First, three different models have been generated, namely a THD model, a TEHD model, and a THD model with mechanical deformation not including the effect of thermal deformation of the bearing pad.

From the results of those three models, it can be concluded that the THD model and the THD model with mechanical deformations both underestimate the value of H min for given load carrying capacity, thus resulting in a smaller maximum load carrying capacity for given values of Hmin. As observed in experiments, parallel thrust bearings can withstand specific loads up to 1.5MPa, which cannot be computed by the THD models of the present study (either with or without mechanical deformations), since minimum film thickness is very low, leading to high temperature rise and extremely low oil viscosity, which cannot support the high specific loads that are expected. The TEHD model computes a modified lubricant geometry with more pronounced pressure build-up regions, leading to higher values of minimum film thickness and lower oil temperatures, thus being able to support higher thrust loads. Further, the maximum pressure zone calculated by the TEHD model is in accordance with the experimental observations made by Henry et al. [START_REF] Henry | Experimental analysis of the hydrodynamic effect during startup of fixed geometry thrust bearings[END_REF]. Continuing on the parametric analysis for the sinusoidal macroscopic surface defect, the initial surface does not play a significant role in the load carrying capacity of the slider, but it helps the geometrical shape of resulting wedge generated by the thermal deformation on the pad. As the defect amplitude increases, the load carrying capacity increases. The sinusoidal defect moves the Hmin longitudinal position closer to the trailing edge, therefore, giving a larger converging region and a smaller diverging. The macroscopic defect studied here is not the only observed defect in the manufacturing of such sliders, but it represents a large part of them. Different macroscopics defect shapes need to be analysed in order to be able to caracterise better the manufacturing tolerances for the thrust slider bearings. As it can be concluded from the results, the main reason for load carrying capacity for thrust slider bearing is the thermal deformation of the fluid/pad interface, but initial surface defects and the viscosity wedge can contribute substantially to the overall performance of the slider bearing. In order to be able to acquire better results, and more relatable with the experiments conducted in other works, the presented models need to take into account cavitation. Further, the present results demonstrate that introduction of initial defects on the bearing surface may affect substantially bearing performance, especially at high values of defect amplitude. Therefore, for the characterization of the tolerances of quasi-parallel thrust slider bearings, additional types of initial defects, such as convex, concave, diverging and converging shapes, at different amplitudes, need also to be studied.

Fig. 2 -

 2 Fig. 2 -3D representation of the solid and fluid domains of the slider bearing. Origin point.

Fig. 3 -

 3 Fig. 3 -(a) Initial Pad quasi-parallel surface. (b) Plot of reference sinusoidal defect (1μm).

Fig. 4 -

 4 Fig. 4 -Computational model: designation domains surfaces.

Fig. 5 -

 5 Fig. 5 -Mesh details of the computational domain: (a) Pad mesh details. (b) Fluid mesh details. (c) Runner mesh details.

Fig. 6 -Fig. 7 -

 67 Fig. 6 -Thrust load: 226 N, Runner velocity: 22 m/s: Calculated maximum temperature at the fluid/pad interface and friction force, corresponding to (a) TEHD analysis, (b) THD with mechanical deformation analysis, and (c) THD analysis.

Fig. 11 -Fig. 12 -Fig. 13 -Fig. 14 -

 11121314 Fig. 11 -Initial pad quasi-parallel surface.

Fig. 15 -

 15 Fig. 15 -Longitudinal and transverse section profiles for varying load carrying capacity at the 50% of the width and the length, (a) regarding pressure, and (b) regarding the final interface profile.

Table 3 -Thermophysical properties of the pad and runner materials.

 3

	Property	Runner(Steel) Pad (Copper)	Units
	Specific Heat Capacity	434	377	J/(kg.K)
	Thermal Conductivity	60.5	385	W/(m.K)
	Molar Mass	55.85	63.55	kg/kmol
	Density	7854	8933	kg/m 3
	Thermal Expansion Coef.	-	1.8x10 -5	C -1
	Young's Modulus	-	1.17x10 11	Pa

Table 4 -Thermophysical oil properties.

 4

		ISO VG 46 Oil Parameters
	Symbol Value	Units	Comments
	ρ	870	kg m 3	Oil density
	c	2.1	kJ (kg.K)	Specific heat capacity
	k	0.13	W/(m.K)	Thermal conductivity

NOMENCLATURE H min :

Minimum film thickness (μm) V:

velocity vector (m/s) p:

the static pressure (Pa) T:

the temperature (K) τ:

the viscous stress tensor ρ:

the oil density (kg/m 3 ) μ:

the oil dynamic viscosity (kg/(m*s)) cp:

oil specific heat capacity (J/(kg*K)) λf:

oil thermal conductivity (W/(m*K)) λsp:

pad thermal conductivity (W/(m*K)) λsr:

slider thermal conductivity (W/(m*K)) [D]: elasticity matrix {ε el }: elastic strain vector {ε}:

total strain vector {ε th }: thermal strain vector α se : secant coefficient of thermal expansion

APPENDIX MESH STUDY

Two seperate mesh studies have been conducted (a) for varying number of elements in the longitudinal direction, and (b) for varying the element number in the cross-flow direction (film thickness). In all presented cases, the number of elements in the transverse direction is the half of the elements in the longitudinal direction.

Fig. A.1 -Calculated values of (a) load carrying capacity and maximum temperature, (b) Maximum pressure and minimum film thickness, and (c) Specific load and friction force, for varying number of elements in the longitudinal direction.

CONVERGENCE STUDY

In order to calculate the TEHD model, an iterative procedure is needed, alternating the calculations between the FE and the CFD solvers. The amount of iterations needed to fully converge to the solution was not known. In order to investigate the required number of iterations needed, a convergence study has been conducted.

Regarding the maximum number of iterations, Fig. A.3 presents calculations of the main bearing performance indices, namely thrust load, maximum lubricant temperature, maximum lubricant pressure, minimum film thickness, specific load and friction force, as a function of iteration number. Based on the results, an iteration number value of 25 has been selected. This value yields a relative error of 2.7%, 0.037%, 2.9%, 0.8%, 2.7% and 0.22% of the six performance indices of the bearing, in comparison to the values corresponding to iteration number 52 (fully converged model). Calculation up to an iteration number of 52 leads to extreme computational times (25 iterations: required time is 3 days and 10 hours in 24 CPUs, 52 iterations: required time is approximately 6 days). In all cases, the maximum value of mesh displacement between two consecutive steps, after the 25th iteration was always less than 3e10-8, well below the minimum defect amplitude.


Shuhui Cui

Chuanwei Zhang

Michel Fillon

Le Gu

Optimization performance of plain journal bearings with partial wall slip

Keywords: heavily loaded journal bearing, partial wall slip, mixed lubrication, simulation model

In present work, in order to optimize partial slip zone for heavily loaded journal bearings, an isothermal hydrodynamic model considering wall slip occurring at oil-bush interface is built. Two-component slip model is utilized to calculate slip velocity. The performances of the journal bearing in different slip zone shapes and eccentricity ratios are evaluated. The optimal slip parameters in a range of eccentricity ratios are selected by considering their effects on hydrodynamic behavior, such as hydrodynamic pressure, load carrying capacity and friction coefficient/torque. In addition, the application of optimal partial slip zone is presented. Results show that the optimal range of slip zone in high eccentricity ratio is quite restricted. The slip zone should be located at the convergent region of the film. The circumferential end of slip zone should be close to the position of minimum film thickness. It is better to widen the axial length of slip zone, i.e. rectangular shape could better improve bearing performance.

Introduction

In hydrodynamic bearings, coatings have been widely used due to their helpfulness to improve the tribological performance and prolong the serving life of bearings. Many kinds of plastic coatings, such as polytetrafluoroethylene (PTFE) and polyetheretherketone (PEEK), have lower surface energy and non-stick property, which are very different from the metal ones. For heavily loaded hydrodynamic journal bearings under high speed, wall slip is easier to occur, and the traditional nonslip boundary condition is not appropriate in the condition of high-speed sliding bearing with small gap [START_REF] Senatore | Partial Slip Texture Slider and Journal Bearing Lubricated With Newtonian Fluids: A Review[END_REF]. Therefore, the influence of wall slip at fluid-solid interface is necessary to be considered.

In many literatures, experimental studies have been conducted to investigate the occurrence and influence of wall slip at the fluid-coating interface. Yu et al. [START_REF] Yu | Measurement of lubricant viscosity and detection of boundary slip at high shear rates[END_REF] developed a rheometer and tested the perfluoropolyethers (PFPE) coated sample with the slip length in the order of 10 μm. They indicated that the critical shear rate of water was higher than that of PC and 3# mineral oil. Jin et al. [START_REF]Theoretical and experimental research of Elasto-Metal-Plastic(EMP) journal bearings[END_REF] tested that the critical shear stress at the oil-PTFE interface of a plastic journal bearing was about 1126.94 Pa. It was proved that, in hydrodynamic lubrication, the wall slip could decrease friction coefficient [START_REF] Choo | A low friction bearing based on liquid slip at the wall[END_REF], but this reduction became smaller in mixed lubrication [START_REF] Kalin | The Stribeck curve and lubrication design for non-fully wetted surfaces[END_REF]. Ryan et al. [START_REF] Ryan | Traction of Lubricated Rolling Contacts between Thin-Film Coatings and Steel[END_REF] studied the Stribeck curve of the coated/uncoated surface and steel with a ball-on-flat test. They observed that when the film thickness was small, the micro-texture and coatings would affect the traction coefficient. With the lubricant of PAO oil, the CrxN coatings could decrease the traction coefficient more than DLC coatings. Some slip models have been applied in the studies of hydrodynamic bearings. The modified Reynolds equation is deduced based on the Navier-Stokes equation [START_REF] Rao | Analysis of slider and journal bearing using partially textured slip surface[END_REF][START_REF] Bailey | Dynamics of a high speed coned thrust bearing with a Navier slip boundary condition[END_REF], critical shear stress slip model [START_REF] Tauviqirrahman | Combined effect of texturing and boundary slippage in lubricated sliding contacts[END_REF][START_REF] Fatu | Wall slip effects in (elasto) hydrodynamic journal bearings[END_REF][START_REF] Ma | Influence of wall slip on the dynamic properties of a rotor-bearing system[END_REF][START_REF] Lv | Mixed-lubrication analysis of thin polymer film overplayed metallic marine stern bearing considering wall slip and journal misalignment[END_REF] and two-component slip model [START_REF] Spikes | Equation for slip of simple liquids at smooth solid surfaces[END_REF][START_REF] Ismail | Effects of texture shape and fluid-solid interfacial slip on the hydrodynamic lubrication performance of parallel sliding contacts[END_REF][START_REF] Zhang | Boundary slip surface design for high speed water lubricated journal bearings[END_REF]. Cheng et al. [START_REF] Cheng | A velocity-slip model for analysis of the fluid film in the cavitation region of a journal bearing[END_REF] presented a slip model between cavitation air and lubricant oil to solve the problem of slip occurring in the cavitation zone. They deduced the Reynolds equation from Navier-Stokes equation considering both cavitation and wall slip. Jao et al. [START_REF] Jao | A modified average Reynolds equation for rough bearings with anisotropic slip[END_REF] studied the effect of anisotropy slip combined with roughness on the lubrication behavior of journal bearings. They studied the coefficients of the average Reynolds equation and concluded that the wall slip could offset part effect of roughness. Moreover, it was found that wall slip was easier to occur in the zones owning high shear stress gradient and small film thickness. The special oil wedge and high eccentricity ratio could decrease the influence of wall slip. In addition, the rupture zone and the maximum pressure are also affected by wall slip [START_REF] Ma | Influence of wall slip on the dynamic properties of a rotor-bearing system[END_REF].

For journal bearings, the optimal designs of wall slip parameters have already been investigated, because the performance of the slip zone is significantly affected by working conditions. Ma et al. [START_REF] Ma | Wall slip and hydrodynamics of two-dimensional journal bearing[END_REF] systematically investigated and obtained the optimal parameters of the slip zone for the eccentricity ratios from 0 to 0.8. They also indicated that if the wall slip occurred on the oil-shaft interface, there were almost no extra bearing load. Zhang et al. [START_REF] Zhang | Boundary slip surface design for high speed water lubricated journal bearings[END_REF] studied the effects of water-bush boundary slip on the high-speed water lubricated journal bearings under low eccentricity ratios. They found it would decrease the load carrying capacity for full slip configuration on the bush, but an appropriate partial slip zone could improve the performance of the bearing. Rao [START_REF] Rao | Analysis of single-grooved slider and journal bearing with partial slip surface[END_REF] analyzed the effect of partial slip immediately followed by a single grooved on the journal bearing and found that the maximum hydrodynamic pressure was larger in partial slip condition. However, the shear stress was large before the slip/nonslip interface and obviously decreased in the nonslip zone. Fortier et al. [START_REF] Fortier | Numerical analysis of a journal bearing with a heterogeneous slip/no-slip surface[END_REF] evaluated the performance of the load carrying capacity, friction force, and the leakage rate. They concluded that for low and moderate eccentricity ratios, slip boundary condition had an advantage for the improvement of load carrying capacity and friction coefficient. However, for the eccentricity ratios larger than 0.92, the effect of slip became much smaller. Lv et al. [START_REF] Lv | Mixed-lubrication analysis of thin polymer film overplayed metallic marine stern bearing considering wall slip and journal misalignment[END_REF] demonstrated that, in mixed lubrication regime, wall slip occurring at thin polymer film coated on metallic bush could decrease the load bearing capacity and increase the friction coefficient. Wang et al. [START_REF] Wang | Study on the influence of critical shear stress on wall slip of spiral oil wedge journal bearing[END_REF] found that with an increasing eccentricity ratio, the slip was more difficult to occur, leading to an increase in load capacity in a parallel sliding gap.

In the current work, further optimization of the partial wall slip zone for high eccentricity ratios is studied by considering the surface roughness. The average Reynolds equation considering wall slip is applied and solved by finite element method (FEM). The optimal slip zone parameters for moderate shaft eccentricity ratios are first obtained. Then, the optimal parameters are imposed on the working conditions at high eccentricity ratios. Thirdly, the optimal slip zone parameters for high eccentricity ratio are chosen to investigate the performance in moderate eccentricity ratio. Finally, the best slip zone parameters for a range of eccentricity ratios are determined.

Model and formulation

Geometry model

In order to shape the slip zone at the coordinate of θ, we defined three coefficients of α=Bs/B1 (0≤ α ≤2), λ=L 1 /L s1 (0≤ λ), and γ=L s1 /L (0≤ γ ≤1) as shown in Fig. 1. α<1 and α>1 indicate that the slip zone locates at the convergent region and divergent region of the film respectively. λ>1 means the end width L 1 is larger than the front width L s1 . B 1 is the length from attitude angle φ to the minimum film thickness position. L is bearing width, ω is shaft velocity, e is eccentric distance and R is shaft radius. Fig. 1. The slip zone at the oil-bush interface

Governing equation

The lubricant is assumed to be incompressible and Newtonian and, the flow regime is laminar. The modified Reynolds ) ( )

3 3 0 0 6 6 s s s s z c s c u u u w h p h p h u u h p x x z z x x x z θ Φ Φ Φ Φ σ Φ ξ η η - ∂ +       ∂ ∂ ∂ ∂ ∂ ∂ ∂     + = + + + + +       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂       (1)
where p is hydrodynamic pressure. h is film thickness. η is the dynamic viscosity of lubricant. u 0 is the velocity of shaft surface. u s and w s are the slip velocities in circumferential and axial directions. ξP -is the penalty item which could make flow mass conservation [START_REF] Habchi | Stabilized fully-coupled finite elements for elastohydrodynamic lubrication problems[END_REF]. ξ is an infinitely positive number and p -represents the negative pressure in cavitation zone. Φ c is the contact coefficient, Φ θ , Φ z, and Φ s are the flow rate coefficients that can be determined in Ref. [START_REF] Wu | An average Reynolds equation for partial film lubrication with a contact factor[END_REF] and [START_REF] Patir | Application of average flow model to lubrication between rough sliding surfaces[END_REF]. σ is the standard deviation of roughness asperity height, which is set to be 1 μm.

Taking the dimensionless parameters as z =z/L, H=h/C, ψ=C/R, P=p/(ηω/ψ 2 ), U=u 0 /(Rω), U s =u s /(Rω), and W s =w s /(Rω),

where C is the radial clearance of the bearing, the dimensionless average Reynolds equation is obtained as

( ) ( )
2 0 3 3 6 6 s s s s z c s c U U U W P R P H R H H U U H P L z z C L z θ Φ σ Φ Φ Φ Φ ξ θ θ θ θ θ -   ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂           + = + + + + +         ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂         (2)
In order to obtain the wall slip velocity, two-component slip model [START_REF] Spikes | Equation for slip of simple liquids at smooth solid surfaces[END_REF] is used. The shear stress is expressed by

c co s u b η τ τ = + (3)
where τ co is the critical shear stress at which wall slip occurs. b is slip length. Wall slip begins when the fluid-solid shear stress is larger than the critical shear stress of τ co . The slip velocity u s increases linearly with the increasing shear stress at a constant slip length of b.

The wall slip velocity in circumferential direction is ( )

( ) 2 0 1 sgn 2 1 co s s h p h u u x u h b τ η η ∂ - - ∂ = + (4)
where sgn(u s ) is the sign function for sgn(u s >0)=1, sgn(u s <0)=-1 and sgn(u s =0)=0.

As shown in Eq.( 4), slip velocity is directly influenced by the gradient of hydrodynamic pressure d / d p x , the film thickness h, the critical shear stress τ co, and the slip length b, which are discussed in section 3.

The dimensionless parameters of shear stress and slip length are

0 R C η ω τ τ τ τ = = , b Cb =
The dimensionless wall slip velocity in circumferential direction is related to the HD pressure gradient

( ) ( ) ( ) ( ) 2 0 0 2 2 0 0 2 0 2 2 2 ,
P H U H P U if u U H H H b P H U H P U if u U H H H b P U if u U H τ θ τ θ τ θ τ θ θ ∂ - - ∂ ∂ = < - ∂ + ∂ - + ∂ ∂ = > + ∂ + ∂ = = = ∂ >0 <0 (5)
Similarly, the slip velocity w s in axial direction can be expressed in the same way.

Solution procedure

The modified Reynolds equation of Eq.( 2) is solved by finite element method (FEM) with a four-node element. The detailed solution is presented in Ref [START_REF] Cui | Numerical analysis on the dynamic contact behavior of hydrodynamic journal bearings during startup[END_REF].

The corresponding matrix equation for the four-node element is

[ ]{ } 4 4 4 1 4 1 K P R δ × × × =
The pressure is expressed by P=P 0 +δP and 0 P P P δ ∇ = ∇ + ∇ . P 0 and δP are the previous pressure and pressure increment.

Applying them to the above matrix equation, the coefficient matrixes are deduced by the Galerkin method as follows:

( )

( ) 2 3 3 2 3 0 0 3 0 0 d 6 d 6 e e j j e i i ij z i j e i i z s i e i e i i i c s s s s c N N N N R K H H N N L z z P N P U U U W R N R H H H U U N L z z R N HN C L N P z θ θ Φ Φ θ θ Φ Φ θ θ ξ ξ θ θ Φ Φ σ Φ θ - - Ω Ω --    ∂   ∂ ∂ ∂ ∂     = + + Ω  + ∂ ∂     + +      ∂ ∂ ∂ ∂         ∂ ∂ ∂ ∂ ∂   + + +     ∂ ∂ ∂ ∂ ∂     = - Ω     + + ∂ ∂ ∂       ∫∫ ∫∫           (6)
where N i and N j are the shape function of the four-node element with i, j=1, 2, 3, 4. And the shape function can be expressed as

1 2 3 4 1 ( , ) (1 )
(1 ) 4 1 ( , ) (1 )
(1 ) 4 1 ( , ) (1 )
(1 ) 4 1 ( , ) (1 )(1 ) 4
N - - N + - N + + N - + ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η  =    =    =    =  % % % % % % % % % % % % % % % % (7)
where ξ % andη %are the normalized coordinate of the element, 0 , 1 ξ η ≤ ≤ % % .

Friction torque and friction coefficient

The oil shear force is

2 0 0 d dz L f z F R π θ τ θ = ∫ ∫ (8)
where z θ τ is the fluid shear stress, and ( ) ( )

0 2 s z f fs fp u u h p h R θ η τ φ φ φ θ + ∂ = + ± ∂ . ± is used for the oil-shaft friction torque (+)
and oil-bush friction torque (-). f fs fp φ φ φ 、 、 are the coefficients considering surface roughness, which can be determined in Ref [START_REF] Patir | Application of average flow model to lubrication between rough sliding surfaces[END_REF].

The friction torque T f is

2 0 0 = d dz L f f z T RF R R π θ τ θ = ∫ ∫ (9)
The hydrodynamic bearing forces in x and y directions are

( ) ( ) 2 0 0 2 0 0 , cos dzd , sin d d L hx L hy F R p z F R p z z θ θ θ θ θ θ π π  =    =  ∫ ∫ ∫ ∫ (10)
The load carrying capacity is

2 2 h hx hy F F F = +
. The friction coefficient f can be expressed by

f h f F F = (11)
The load carrying capacity gain is calculated by F G =(F h_slip -F h_nonslip )/F h_nonslip ×100%. The reduction in friction coefficient is calculated by f D =-(f slip -f nonslip )/f nonslip ×100%. The decrease in friction torque is calculated by T D =-(T f_slip -T f_noslip )/T f_noslip ×100%. The subscripts slip and nonslip mean the parameters calculated with and without the wall slip.

Result and Discussion

Verification of the model

The present model is validated by the data in Ref. [START_REF] Lv | Mixed-lubrication analysis of thin polymer film overplayed metallic marine stern bearing considering wall slip and journal misalignment[END_REF] as shown in Fig. 2. The dimensionless parameters are used. Eccentricity ratio is ε=0.8, length to diameter ratio is L/D=1, dimensionless critical shear stress is c τ =2. In the present work, the slip dimensionless length is set to be 0.0345. The dimensionless hydrodynamic pressure is noted as P=pψ 2 /2μω. The effect of wall slip in axial direction in different slip zone shapes is shown in Fig. 3. The maximum slip velocity W s in axial direction occurs at both axial ends. With an increase in γ, the maximum W s increases. Meanwhile, the circumferential slip velocity U s occurs in a larger area. The negative effect of W s can be offset by the positive effect of U s , and the effect of W s is still small. Therefore, the influence of slip velocity in axial direction is not considered in this analysis. Fig. 3 The shape of the slip zone and the dimensionless slip velocity in (a) circumferential direction U s and (b) axial direction W s at ε=0.9

Optimization of slip zone in moderate eccentricity ratio

Without any special notification, the common parameters used in this simulation are listed in Table 1. For different configurations of slip zone, the effect of slip length is not regular. Here we set the slip length to be b=10 μm. The critical shear stress is 1126.94 Pa (for dimensionless 0.28 τ =

) based on the oil-PTFE interface measured by Ref [START_REF]Theoretical and experimental research of Elasto-Metal-Plastic(EMP) journal bearings[END_REF].

Effect of slip zone location (coefficient α)

Fig. 4 shows the effect of coefficient α on the hydrodynamic behavior with the constant parameters of λ=0.5 (β=3.64°) and γ=0.8. The pressure distribution is obviously affected by the slip zone coefficient α in both maximum value and pressure distribution (Fig. 4a). The dimensionless circumferential slip velocity U s in the middle plane variates with the slip zone coefficient α (Fig. 4b). For α>1, U s increases first and then decreases in circumferential direction, reaching the maximum value at the position of minimum film thickness. According to Eq.( 5), this is caused by the film thickness that changes inversely with U s . Moreover, the hydrodynamic (HD) pressure is zero in the cavitation zone, but the slip velocity is not zero for α>1, which is caused by shaft velocity. Since the shaft velocity is constant, the slip velocity in the cavitation zone is almost the same for different coefficient α.

In Fig. 4 (c) and (d), the slip zones at α<1.0 lead to a positive effect on hydrodynamic behavior by improving the load carrying capacity, decreasing the friction coefficient, and reducing slightly the friction torque in both oil-shaft and oil-bush interfaces. An optimal value of α=0.85 is observed, which generates an increase of 3.4% in the load carrying capacity, a reduction of 4.1% in the friction coefficient. However, for α>1.0, the slip zone generates a decrease in the load capacity and a rise in both the friction coefficient and friction torque. Therefore, it is better to set the slip zone at the convergent region of the film, instead of the divergent region (in Fig. 1).

Effect of slip zone shape (coefficients λ and γ)

Fig. 5 shows the effect of coefficient λ on the hydrodynamic behavior with the constant parameters of γ=0.8 by setting the optimal slip zone coefficient α=0.85 for ε=0.5. It demonstrates that the performance of journal bearings is better improved with large coefficient λ (in Fig. 5a). For λ=1.2, the load carrying capacity gain climbs up to 4.8%, meanwhile the friction coefficient drops by 5.9% with 1.5% reduction in friction torque. In addition, the coefficient λ has little influence on the hydrodynamic (HD) pressure and slip velocity U s in the middle plane (Fig. 5b andc), but it significantly affect the distribution of slip velocity U s (Fig. 5d), leading to the improvement of the bearings. Therefore, in order to better improve the performance of journal bearings, it is necessary to choose large coefficient λ (or large L 1 ) to enlarge the action range of slip zone.

Dimensionless slip velocity

Circumferential angle (°) coefficient of α=0.85 for ε=0.5. λ is set to be 1/γ, so the end width L 1 of the slip zone is fixed to be the widest value. The distribution of slip velocities U s change with slip zone shapes based on the coefficient γ (Fig. 6a). Fig. 6 (b) shows the increasing of γ can slightly increase the load carrying capacity gain (with maximum F G =1.5%) and the reduction of friction coefficient (with maximum f D =2%), as well as the reduction of friction torque (less than 1%). The coefficient γ mainly affects the front part of the slip zone where slip velocity is much smaller than the end part, so its effect is not so significant. However, it is better to widen the slip zone in the inlet region, i.e. enlarge L s1 , to augment the inlet flow rate.

Effect of eccentricity ratio ε

Fig. 7 shows the effect of eccentricity ratio on the hydrodynamic behavior. The slip zone coefficients are chosen as α=0.85, λ=0.5 and γ=0.8. Slip length is b=10 μm. The partial slip zone behaviors much better in low eccentricity ratio than that in high eccentricity ratio (Fig. 7a). For ε=0.1~0.8, the partial slip zone significantly augment the load carrying capacity and reduced the friction coefficient with a slight change in friction torques. However, when ε rises up to 0.9, the performance of partial slip zone is negative by diminishing the load carrying capacity and increasing the friction coefficient. It can be seen in Fig. 7 (b), the slip velocity is large in low eccentricity ratio at the front part of slip zone. In high eccentricity of ε=0.9, negative slip velocity appears, i.e. the slip velocity goes into the opposite direction of the shaft speed, resulting in the decrease of the load carrying capacity. In order to improve the performance of the bearings, it is necessary to further optimize the slip zone coefficients in high eccentricity ratio.

Optimization of slip zone in high eccentricity ratio

This part mainly focuses on the optimization of journal bearings in high eccentricity ratio. The common parameters used are eccentricity ratio ε=0.9, slip zone coefficients λ=0.5 and γ=0.8, slip length b=10 μm, dimensionless critical shear stress c τ =0.28.

Effect of slip zone location (coefficient α)

Fig. 8 shows the effect of slip zone coefficient α on hydrodynamic behavior at the high eccentricity of ε=0.9. For α>1.0, the load carrying capacity and friction coefficient are much worse than that of α<1.0 (Fig. 8a), so it is better to set α<1.0. In the positive region of α=0.91~1.0, slip zone could improve the load carrying capacity with the maximum gain of 4.5%, while the friction coefficient decrease f D rises up to the peak value of 2.4% during α=0.95~0.98 (Fig. 8b). Fig. 8 (c) indicates that the oil-bush friction torque does not always decrease in the positive region, but the increment of friction torque is smaller than 2%, so in high eccentricity ratio, the load carrying capacity and friction coefficient are used to obtain the optimal partial slip zone.

Effect of slip zone shape (coefficient λ and γ)

Fig. 9 shows the effect of coefficient λ on the performance of journal bearings with the eccentricity ratio of ε=0.9, the slip zone coefficient of α=1.0. It can be seen that the load carrying capacity gain F G and friction coefficient reduction f D increase with coefficient λ (Fig. 9a). F G and f D reach their maximum values of 4.8% and 11% respectively at λ=1.2. This means that the slip zone with a wider length L 1 in axial direction works better. For constant α, large coefficient λ causes a little increase in maximum hydrodynamic pressure (Fig. 9b). In Fig. 9 (c), before θ=150° slip velocity is negative, while it becomes positive after θ=150°. dimensionless circumferential slip velocity U s in middle plane Fig. 10 shows the effect of slip zone coefficient γ on the hydrodynamic behavior of journal bearings with coefficients of α=1 and λ=1/γ. For more obvious effect, larger eccentricity ratio of ε=0.92 is used. It is clear that the increasing coefficient γ can improve the load carrying capacity slightly with F G <1%, and lower friction coefficient and friction torque with f D <1.23% and T D <1.4% respectively. Moreover, coefficient γ has little influence on hydrodynamic pressure and slip velocity U s in the middle plane (Fig. 10b andc). This is because the shape change caused by coefficient γ mainly influences the slip zone where the slip velocity is close to zero and make little change in the pressure distribution, and the variation of γ could not offset the negative zone of slip velocity U s (Fig. 10d). Despite the subtle influence of γ, it is still beneficial to widen the front size of the slip zone, i.e. enlarge L s1 . The influence of slip zone shape, i.e. the coefficients of λ and γ at high eccentricity ratio is similar to that in moderate eccentricity ratio. It is helpful to increase both the coefficients of λ and γ. That is to say setting λ=1 and γ=1 and the optimal shape of the slip zone is rectangular.

Effect of high eccentricity ratio ε

Fig. 11 shows the effect of high eccentricity ratio on the hydrodynamic behavior with the optimized slip zone parameters of α=1, λ=1, and γ=1, slip length b=5 μm. As shown in Fig. 11(a), the load carrying capacity gain F G increases with the rise of ε. The friction coefficient decrease f D experiences rise and meets the largest value of 12.5% at ε=0.96. However, there is a special point of ε=0.93, whose performance is better than the two adjacent points of ε=0.92 and ε=0.94. This is because the present slip zone parameters are not optimal for all the eccentricity ratios. In this case, these parameters can best improve the performance of ε=0.93 and it is marked as a special point. For ε>0.96, the F G and f D go lower, but it can be concluded that the rectangular slip zone can still improve the performance of journal bearings for ε=0.9~0.97. As shown in Fig. 11 (b) and (c) and Eq.( 5), in slip zone 1 the pressure gradient dp/dθ is close to zero and the film thickness is high, so the slip velocity is almost zero. In slip zone 2, though film thickness becomes smaller, the slip velocity is still negative because of the increasing dp/dθ. In slip zone 3, dp/dθ is quite large, but the film thickness also becomes small enough to offset the negative effect of dp/dθ, so there is a sharp increase in slip velocity.

Considering operating conditions

In this part, the optimized slip zone coefficients of λ=1 and γ=1 are selected. The influence of attitude angle in the high eccentricity ratio is considered. The fixed coordinate system of Θ is applied (Fig. 1b), and its origin of Θ=0 is the vertical direction of Y=0. The slip zone starts from the system origin of Θ=0, instead of θ=0. In the fixed coordinate system Θ, attitude angle φ is not zero. As the partial slip zone of Θ slip =0~φ locates at the end of the divergent region of the film, its influence can be neglected. Therefore, in high eccentricity ratio the optimal α still located at 0.9~1.0. The slip zone is better to end at the position of Θ end =π+κφ (

[0,1] κ ∈ ), i.e. B s =π+κφ, α=(B s -φ)/B 1 .
The optimized parameters of ε=0.94 are selected for further exploring the performance of journal bearing with eccentricity ratios ranging from 0.4 to 0.97 as shown in Fig. 12. The slip length is b=5 μm. The slip zones endings are set to be Θ end =π, Θ end =π+0.25φ, Θ end =π+0.5φ, Θ end =π+0.75φ, and Θ end =π+1.0φ, corresponding to κ=0, κ=0.25, κ=0.50, κ=0.75 and κ=1.00 respectively. The nonslip attitude angle of ε=0.94 is φ=20.3°, so the corresponding values of α are 0.889, 0.915, 0.943, 0.971 and 0.999. For ε<0.94, the selected slip zone can raise the load carrying capacity and lower the friction coefficient (Fig. 12a andb). For κ=0.75 and κ=1.00, the partial slip zone better improves the bearing performance in different eccentricity ratios with the maximum load gain of 4.5% and the maximum friction coefficient decrease of 10%. However, for ε>0.94, except κ=0.5, partial slip zone diminish the load carrying capacity in any case.

Usually with the increase in eccentricity ratio, the attitude angle drops, and the slip zone coefficient α rises. In Fig. 12(c), it can be seen for ε<0.94 and κ=0.5~1 the attitude angle goes down, so the exact value of coefficient α becomes higher. If the journal bearing always works in a constant heavy load, in order to get the optimal performance it is better to set the partial slip zone with the coefficient κ ranging from 0.5 to 0.9.

Conclusions

In the present work, the improved performance of journal bearings with partial slip zone is obtained by optimizing slip zone shape at both moderate and high eccentricity ratios. The key conclusions are as follows:

(1) It is better to enlarge the wall slip zone in axial direction in the oil-wedge convergent zone, i.e. setting the slip zone in rectangular. For high eccentricity ratio, in circumferential direction, the optimal slip zone is located at the convergent region of film and the optimal coefficient α is approximately 0.9~1.0 which is limited much more strictly than that in moderate eccentricity ratio.

(2) For the working condition of heavy load, it is better to locate partial slip zone during Θ=0~π+κφ with the coefficient of κ=0.5~0.9 in circumferential direction, where φ is the attitude angle of the highest eccentricity ratio.

(3) In high eccentricity ratio, there exists a part of slip zone where the slip velocity goes in the opposite direction of the shaft speed, which weakens the performance of journal bearings.

As the influence of wall slip on journal bearings are very complex, the present model makes some hypotheses for simplification. Therefore, further studies would be needed considering the thermal and elastic deformation. Also, experiments need to be conducted for the application of the optimal slip zone.

  Loadequation considering both slip and surface roughness[START_REF] Wu | An average Reynolds equation for partial film lubrication with a contact factor[END_REF] is used as follows

(

Fig. 2

 2 Fig. 2 Verification of the hydrodynamic pressure distribution in the middle plane

Fig. 4

 4 Fig.[START_REF] Choo | A low friction bearing based on liquid slip at the wall[END_REF] The effect of slip zone coefficient α on the hydrodynamic (HD) behavior of journal bearings in moderate eccentricity ratio, (a) HD pressure P in middle plane, (b) dimensionless slip velocity in circumferential direction U s , (c) load carrying capacity gain F G and friction decrease f D , and (d) friction torque decrease at oil-shaft (T sD ) and oil-bush interfaces (T bD )

Fig. 5

 5 Fig. 5 Effect of slip zone coefficient λ on the hydrodynamic (HD) behavior of journal bearings, (a) the load carrying capacity gain F G , friction coefficient decrease f D and friction torque decrease T D , (b) the HD pressure P in middle plane, (c) the circumferential slip velocity U s in middle plane, (d) the distribution of slip velocity U s in different shapes of slip zone

Fig. 6

 6 Fig. 6 The effect of slip zone coefficient γ on the hydrodynamic behavior of journal bearings, (a) the distribution of slip velocity U s in different slip zone shapes, and (b) the load carrying capacity gain F G , friction coefficient decrease f D and decrease of friction torque T D

Fig. 7

 7 Fig. 7 Effect of eccentricity ratio on the hydrodynamic (HD) behavior, (a) the load-carrying load gain F G , friction coefficient decrease f D and friction torque decrease T D , and (b) the dimensionless circumferential slip velocity U s in the middle plane

Fig. 8

 8 Fig. 8 Effect of slip zone coefficient α on the hydrodynamic behavior under high eccentricity ratio of ε=0.9, (a) the load carrying capacity gain F G and friction coefficient decrease f D , (b) the load carrying capacity gain F G and friction coefficient decrease f D in positive region, and (c) friction torque T D decrease in positive region

Fig. 9

 9 Fig. 9 Effect of slip zone coefficient λ on the hydrodynamic (HD) behavior of journal bearings, (a) the load carrying capacity gain F G , friction coefficient decrease f D and friction torque decrease T D , (b) the HD pressure P in middle plane, and (c) the

Fig. 10

 10 Fig. 10 Effect of slip zone coefficient γ on the hydrodynamic behavior of journal bearings in high eccentricity ratio, (a) the load carrying capacity gain F G , friction coefficient decrease f D and friction torque decrease T D , (b) the HD pressure P in middle plane, (c) the slip velocity U s in middle plane, (d) the distribution of slip velocity U s

Fig. 11

 11 Fig. 11 Effect of high eccentricity ratio on hydrodynamic behavior of journal bearings, (a) effect on load carrying capacity gain F G , friction coefficient decrease f D and friction torque decrease T D , (b) effect on hydrodynamic (HD) pressure in the middle plane, and (c) effect on circumferential dimensionless slip velocity U s in the middle plane

Fig. 12

 12 Fig. 12 The performance of journal bearing in partial slip zone in different eccentricity ratio, (a) the load carrying capacity gain F G , (b) the friction coefficient decrease f D , (c) the attitude angle, and (d) the decrease percentage of attitude angle

Table 1

 1 Parameters of simulation

	Parameters	Values
	Shaft radius R, mm	50
	Bearing length L, mm	80
	Length to diameter ratio, L/D	0.8
	Radial clearance C, μm	145
	Eccentricity ratio ε	0.5
	Rotational speed N, rpm	4000
	Viscosity at 40℃ η, Pa•s	0.0277
	Dimensionless critical shear stress τ	0.28
	Wall slip length b, μm	10
	Supply pressure ps, MPa	Ps=0

Acknowledgment

This research is supported by the National Natural Science Foundation of China (No.51675120 and No.U1537214) and Self-Planned Task (SKLRS201814B). The first author also wants to thank China Scholarship Council (No.201806120132) for the financial support of studying aboard.