Virtual texturing of lightweight crankshafts : Texturização virtual de mancais de virabrequins leves

Faculdade de Engenharia Mecânica

JONATHA OLIVEIRA DE MATOS REIS

Virtual Texturing of Lightweight Crankshafts

Bearings

Texturização Virtual de Mancais de Virabrequins

Leves

CAMPINAS

2017

Virtual Texturing of Lightweight Crankshafts

Bearings

Texturização Virtual de Mancais de Virabrequins

Leves

Dissertation presented to the School of Me-chanical Engineering of the University of Campinas in partial fulfillment of the require-ments for the Master’s degree, in the field of Solid Mechanics and Mechanical Design.

Dissertação apresentada à Faculdade de Engenharia Mecânica da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Mestre em Engenharia Mecânica, na Área de Mecânica dos Sólidos e Projeto Mecânico.

Orientador: Prof. Dr. Marco Lucio Bittencourt

ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA DIS-SERTAÇÃO DEFENDIDA PELO ALUNO JONATHA OLIVEIRA DE MATOS REIS, E ORIENTADA PELO PROF. DR. MARCO LUCIO BITTENCOURT.

CAMPINAS 2017

Ficha catalográfica

Universidade Estadual de Campinas Biblioteca da Área de Engenharia e Arquitetura

Luciana Pietrosanto Milla - CRB 8/8129

Reis, Jonatha Oliveira de Matos,

R277v ReiVirtual texturing of lightweight crankshafts / Jonatha Oliveira de Matos Reis. – Campinas, SP : [s.n.], 2017.

ReiOrientador: Marco Lucio Bittencourt.

ReiDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica.

Rei1. Lubrificação. 2. Rolamentos. 3. Tratamento de superfícies. 4. Rugosidade de superfície. 5. Efeito da temperatura. I. Bittencourt, Marco Lucio,1964-. II. Universidade Estadual de Campinas. Faculdade de Engenharia Mecânica. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Texturização virtual de mancais de virabrequins leves Palavras-chave em inglês: Lubrication Bearings Surface treatment Surface roughness Temperature effect

Área de concentração: Mecânica dos Sólidos e Projeto Mecânico Titulação: Mestre em Engenharia Mecânica

Banca examinadora:

Marco Lucio Bittencourt [Orientador] Gregory Bregion Daniel

Francisco José Profito

Data de defesa: 04-08-2017

Programa de Pós-Graduação: Engenharia Mecânica

FACULDADE DE ENGENHARIA MECÂNICA

COMISSÃO DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA

DEPARTAMENTO DE SISTEMAS INTEGRADOS

DISSERTAÇÃO DE MESTRADO ACADÊMICO

Virtual Texturing of Lightweight

Crankshafts Bearings

Texturização Virtual de Mancais de

Virabrequins Leves

Autor: Jonatha Oliveira de Matos Reis

Orientador: Prof. Dr. Marco Lucio Bittencourt

A Banca Examinadora composta pelos membros abaixo aprovou esta Dissertação:

Prof. Dr. Marco Lucio Bittencourt FEM/UNICAMP

Prof. Dr. Gregory Bregion Daniel FEM/UNICAMP

Prof. Dr. Francisco José Profito PME/EPUSP

A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vida acadêmica do aluno.

Gostaria de agradecer, primeiramente, aos meus pais, Ana e Valter, que me apoia-ram em todos os momentos e que conhecem os caminhos que me levaapoia-ram até aqui. Foi a estrada que vocês trilharam que me trouxe tão longe e onde ela começa é o lugar que chamarei sempre de lar. Obrigado também à minha irmã Jocasta e à minha prima Thais, por fazerem parte dos ótimos momentos que uso de recordação e inspiração.

Agradeço imensamente ao professor Marco Lúcio Bittencourt, pela chance de fazer parte desse projeto. O aprendizado proporcionado por esse mestrado estará comigo sempre e se hoje dou mais um passo em direção à pesquisa, é porque ele me deu a oportunidade e me inspirou a continuar buscando o conhecimento sempre.

Deixo aqui também um agradecimento especial ao professor Francisco José Profito. Sem a sua ajuda, eu não teria chegado tão longe nessa pesquisa. Muito obrigado pelas conversas e pelas dúvidas respondidas.

Muito obrigado a todos do lab, por tornarem a experiência do mestrado muito mais tranquila e divertida. Alfredo, obrigado pelo template em LATEX, mas também por dividir os problemas e alegrias do mestrado comigo. Guilherme, muito obrigado por toda a ajuda no início da minha pesquisa, em especial pelo suporte na utilização dos softwares comerciais de simulação. Darla e Mari, muito obrigado pelo convívio e pela dose de humor e loucura quase diária. Eduardo, obrigado pela ajuda a cada novo documento que precisava ser entregue e pela companhia nas reuniões de projeto. Agradeço imensamente a todos e espero o melhor para vocês.

Agradeço eternamente aos meus conterrâneos e amigos, Alessandro, Breno e Lívia, que passaram pelas mesmas dificuldades e saudades que passei para virem até aqui e que esti-veram comigo durante toda essa jornada. Vocês foram essenciais para tornar essa jornada mais fácil e saibam que, aonde quer que estejamos, poderão sempre contar comigo.

Meu muito obrigado àquela que esteve comigo durante todos os momentos. Cintia, meu amor, obrigado por fazer parte dessa história e dividir comigo os melhores momentos. Você foi e vai ser sempre minha referência do melhor que posso ser. Seu apoio foi vital para minha sanidade durante esse mestrado. Obrigado por estar comigo na alegria e na tristeza, na saúde e na doença, na pobreza e na menos pobreza, por todos os dias.

Por fim, meus agradecimentos ao professor Waldyr Luiz Ribeiro Gallo, pela gerên-cia do projeto temático, e à Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) pela concessão do auxílio à pesquisa e bolsa de mestrado, sob o processo número 2015/16470-1 .

Em 2009, 56 milhões de litro de combustível foram gastos com o atrito em mo-tores de combustão, sendo os mancais responsáveis por cerca de um quarto dessa perda. Essa pesquisa tem o objetivo de modelar numericamente os mancais de um virabrequim com alívio de peso, estudando os efeitos que a texturização superficial desses mancais pode ter no seu desempenho e focando em reduzir o atrito deste componente. Uma revisão bibliográfica foi realizada, tratando da texturização superficial, dos regimes de lubrificação e aspectos de rugosi-dade, contato de superfícies e temperatura. Um programa de computador foi desenvolvido para estudar o comportamento de um mancal carregado dinamicamente, levando em conta efeitos de rugosidade, contato de asperezas, efeitos térmicos e texturização superficial. Simulações foram realizados no mancal de um virabrequim com alívio de peso, aplicando diferentes designs de textura à sua superfície. Os resultados mostraram que é possível reduzir o coeficiente de atrito hidrodinâmico médio e máximo destes componentes, mas alguns designs de texturas tendem a aumentar a pressão no fluido lubrificante, especialmente se forem levados em conta os efeitos térmicos.

Palavras-Chave: Texturização Superficial; Equação de Reynolds; Regime de Lubrificação Mista; Mancal de Deslizamento;

In 2009, over 56 million liters of fuel were used to overcome friction in combustion engines, being the bearings alone accountable for nearly a quarter of this loss. This research aims to numerically model a lightweight crankshaft bearing and to study the effects that surface tex-tures can have on its performance, aiming to reduce friction in this component. A literature review was performed, covering the surface texture treatments, lubrication regimes and aspects of rough contact and temperature. A computer program was developed to study the behavior of a dynamically loaded bearing, considering roughness effects on lubrication, asperity contact, global thermal effects and surface texturing. Simulations were performed for the bearing of a lightweight crankshaft, considering different texture designs on its surface. The results showed that it is possible to reduce the mean and maximum hydrodynamic friction coefficient of these bearings, but some texture designs may lead to an increase in the fluid pressure, specially when the thermal effects are take into account.

Keywords: Surface Texture; Reynolds Equation; Mixed Lubrication;Crankshaft Journal Bear-ings; Roughness Effects.

Figure 2.1 – Textured mechanical seal model, as present by Etsion (ETSION; HALPERIN,

2002) . . . 27

Figure 2.2 – Parallel bearing with a surface pattern, as shown by Fesanghary (FESANG-HARY; KHONSARI, 2013) . . . 29

Figure 3.1 – Coordinate system and velocity components for the Navier-Stokes equation (adapted from (SANTOS, 1997)). . . 32

Figure 3.2 – Laminar flow between the two lubricated surfaces (adapted from (SANTOS, 1997)). . . 35

Figure 3.3 – Journal bearing geometry. . . 37

Figure 3.4 – Fluid film cavitation on a lubricated domain. The inside region is a non cavitated domain, where the fluid film is not broken and θ = 1. Outside that region is the cavitated domain, where the thin blue lines represent the gas/vapor mixture and where 0 ≤ θ < 1 (adapted from (PROFITO et al., 2015)). . . 39

Figure 3.5 – Pressure distributions along the center line of a dimple cell, comparing the three cavitation models, as found by Khonsari (QIU; KHONSARI, 2009). . 41

Figure 3.6 – Total film thickness hT between two surfaces with roughness δ1and δ2, with a nominal film thickness h (PROFITO et al., 2015). . . 45

Figure 3.7 – The main surface variables on the Greenwood-Tripp model (PROFITO et al., 2015). . . 48

Figure 3.8 – Relation between dynamic viscosity and temperature for different viscosity models. . . 52

Figure 3.9 – Cartesian system of a journal bearing, with the isometric view (left) and the front view (right). . . 54

Figure 3.10–Textured journal bearing (a), with detail for the dimple geometry (b) and the textured surface (c) (KANGO et al., 2014). . . 55

Figure 3.11–Isometric view of a single dimple on the surface of a mechanical seal (top) and the center line view from the dimple (bottom). . . 56

Figure 3.12–Arrangement of dimples in the circumferential and axial directions of a jour-nal bearing surface (KANGO et al., 2014). . . 57

Figure 3.13–Fluid film thickness of a textured journal bearing. . . 57

Figure 3.14–Computational domain, . . . 58

Figure 3.15–Computation grid. . . 58

Figure 3.16–Computational procedure flow chart. . . 63

Figure 4.1 – Cross-section of the journal bearing geometry with two grooves, as used by Elrod and Adams (adapted from (FESANGHARY; KHONSARI, 2011)). . . 65

as achieved by Elrod (top) and by this study (bottom) (ELROD; ADAMS, 1974). . . 66 Figure 4.3 – Results for the pressure and film content for different mesh sizes . . . 66 Figure 4.4 – Textured mechanical seal with the detail for the surface roughness of the

texture (adapted from (QIU; KHONSARI, 2011b)). . . 67 Figure 4.5 – Load-carrying capacity for a single cell of a texture mechanical seal as found

by Khonsari (left) and by us (righ) (QIU; KHONSARI, 2011b). . . 68 Figure 4.6 – Pressure profile for a journal bearing considering temperature variation

(TALA-IGHIL; FILLON, 2015). . . 70 Figure 4.7 – Fluid film: 3D view (left) and cross-section (right) for a smooth journal

bear-ing. . . 71 Figure 4.8 – Fluid film: 3D view (left) and cross-section (right) for a textured journal

bearing. . . 71 Figure 4.9 – Resulting pressure profile for the textured journal bearing (TALA-IGHIL;

FILLON, 2015). . . 72 Figure 4.10–Journal bearing schematics and computational domain, as presented by Ausas

(AUSAS et al., 2009) . . . 73 Figure 4.11–Loads evolution with time. (Adapted from (AUSAS et al., 2009)) . . . 74 Figure 4.12–The maximum pressure over time for a dynamically loaded journal bearing. 74 Figure 5.1 – Crankshaft without (top) and with (bottom) weight reduction (RODRIGUES,

2013). . . 75 Figure 5.2 – Load chart for a dynamically load journal bearing for a complete engine

cycle, from 0 to 720 degrees. . . 76 Figure 5.3 – Numerical results of fluid film maximum pressure over time in journal

bear-ing usbear-ing different mesh sizes. . . 77 Figure 5.4 – Convergence curve for the fluid film peak pressure. . . 78 Figure 5.5 – Fluid film maximum hydrodynamic pressure on a dynamically load journal

bearing. . . 78 Figure 5.6 – Lubricant thickness on the journal bearing. . . 79 Figure 5.7 – Journal center position in both polar (left) and Cartesian (right) coordinate

systems. . . 79 Figure 5.8 – Journal eccentricity over time on the dynamically load bearing. . . 79 Figure 5.9 – Power loss during the journal bearing operation. . . 80 Figure 5.10–Hydrodynamic friction coefficient for the journal bearing during its operation. 80 Figure 5.11–Results of fluid film maximum pressure over time for different operation

speeds with detail for the maximum pressure peaks. . . 81 Figure 5.12–Lubricant thickness on the journal bearing, highlighting the minimum film

loaded journal bearing. . . 83 Figure 5.14–Relative minimum fluid film thickness for a rough and a smooth dynamically

loaded journal bearing. . . 83 Figure 5.15–The hydrodynamic coefficient of friction for a rough and a smooth

dynami-cally loaded journal bearing. . . 84 Figure 5.16–Computational domain . . . 84 Figure 5.17–Temperature distribution over time for a dynamically loaded journal bearing

considering thermal, roughness and asperities effects. . . 85 Figure 5.18–Lubricant maximum pressure values for a dynamically loaded journal

bear-ing with and without thermal effects. . . 86 Figure 5.19–Fluid film thickness for a dynamically loaded journal bearing with and

with-out thermal effects . . . 86 Figure 5.20–Asperity pressure for a dynamically loaded journal bearing with and without

thermal effects. . . 87 Figure 5.21–Hydrodynamic coefficient of friction for a dynamically loaded journal

bear-ing with and without thermal effects. . . 87 Figure 5.22–Lubricant film thickness for a journal bearing with dimples and different

depths. . . 89 Figure 5.23–Fluid maximum pressure for a journal bearing with the presence of dimples

with different depths. . . 89 Figure 5.24–Power loss for a journal bearing with the presence of dimples with different

depths. . . 90 Figure 5.25–Average power loss for a journal bearing with the presence of dimples with

different depths. . . 91 Figure 5.26–Hydrodynamic friction coefficient over time for a journal bearing with the

presence of dimples with different depths . . . 91 Figure 5.27–The maximum and mean friction coefficient on the dynamically load journal

bearing for different values of dimple depths. . . 92 Figure 5.28–Comparison between the hydrodynamic friction coefficient and the power

loss of a journal bearing for different values of dimple depths. . . 92 Figure 5.29–Fluid maximum pressure for a journal bearing dimples and different depths,

considering thermal effects. . . 93 Figure 5.30–Fluid minimum thickness for a journal bearing with the presence of dimples

with different depths, considering thermal effects. . . 93 Figure 5.31–Power loss for a journal bearing with the presence of dimples with different

depths, considering thermal effects. . . 94 Figure 5.32–Average power loss for a journal bearing with the presence of dimples with

presence of dimples with different depths and considering thermal effects . . 95 Figure 5.34–The maximum and mean friction coefficient on dynamically load journal

bearing for different values of dimple depth, considering thermal effects. . . 96 Figure 5.35–Comparison between the hydrodynamic friction coefficient and the power

loss of a journal bearing for different values of dimple depths,considering thermal effects. . . 97 Figure 5.36–Lubricant film thickness for a journal bearing with dimples and different

radius. . . 98 Figure 5.37–Fluid maximum pressure for a journal bearing with dimples and different

radius. . . 98 Figure 5.38–Average power loss for a journal bearing with the presence of dimples with

different radius. . . 99 Figure 5.39–The maximum and mean friction coefficient on dynamically load journal

bearing for different values of dimple radius. . . 99 Figure 5.40–Comparison between the hydrodynamic friction coefficient and the power

loss of a journal bearing for different values of dimple depths. . . 100 Figure 5.41–Fluid maximum pressure for a journal bearing dimples and different radius,

considering thermal effects. . . 100 Figure 5.42–Average power loss for a journal bearing with the presence of dimples with

different radius, considering thermal effects. . . 101 Figure 5.43–The maximum and mean friction coefficient on dynamically load journal

bearing for different values of dimple radius, considering thermal effects. . . 101 Figure 5.44–Comparison between the hydrodynamic friction coefficient and the power

loss of a journal bearing for different values of dimple depths,considering thermal effects. . . 102 Figure 5.45–Lubricant film thickness top view for a journal bearing with the presence of

different dimples distribution . . . 103 Figure 5.46–Fluid maximum pressure for a journal bearing with different dimples

distri-bution. . . 103 Figure 5.47–Pressure profile in a journal bearing with the presence of different dimples

distribution and considering thermal effects. . . 104 Figure 5.48–Fluid maximum pressure for a journal bearing with the presence of different

Table 3.1 – Dynamic viscosity-temperature equations: . . . 51

Table 3.2 – Constants for the dynamic viscosity models. . . 51

Table 4.1 – Input parameters for the Elrod’s hydrodynamic lubrication problem . . . 65

Table 4.2 – Input parameters for Khonsari’s mechanical seal problem . . . 67

Table 4.3 – Tala-Ighil input parameters for the journal bearing under thermal effects . . . 69

Table 4.4 – Geometrical parameters for the dimples on the textured journal bearing . . . 71

Table 4.5 – Ausas dimensionless parameters for the dynamically load journal bearing . . 73

Table 5.1 – Geometrical parameters for the dynamically load journal bearing. . . 76

Table 5.2 – Surface properties values . . . 82

Table 5.3 – Parameters for the thermal model . . . 85

Table 5.4 – Geometrical properties of the dimples, considering different depths. . . 88

Table 5.5 – Geometrical properties of the dimples, considering different radius. . . 97

Table 5.6 – Geometrical properties of the dimples, considering different distributions. . . 102

Table 5.7 – Hydrodynamic friction coefficient values for the different texture distribu-tions. . . 104

Table 5.8 – Hydrodynamic friction coefficient values for the different texture distribu-tions, considering thermal effects. . . 105

a, b, c Constants for the dynamic viscosity thermal models x, y, z Cartesian axis

u_x, u_y, u_z Local velocity components r_x, r_y, r_z Elliptical dimple radius x_c, yc, zc Elliptical dimple origin center

h Fluid film thickness h_o Radial clearance

h′ Mean surface separation

h_T Film thickness between two rough surfaces

t Time

e Eccentricity

k Successive over relaxation iteration step qx, qz Lubricant film flow

nx, nz Number of dimples

P Hydrodynamic pressure

P_cav Cavitation pressure P_amb Ambient pressure

R Bearing radius

R_j Journal radius

U,W Rotational speeds on the x and y axes.

T Temperature in Kelvin

T_∘C Temperature in Celsius Cp Lubricant specific heat

N Number of asperities

N_x, Nz Number of integration points Lx Bearing thickness

L_z Bearing length

L_tx, Ltz Length of dimple cell

Wx,Wy Length of dimple cell

F_x, F_y Length of dimple cell

Z_s Mean asperity height of combined rough surfaces

µ Dynamic viscosity

ρ Lubricant density

ρc Lubricant density on cavitation pressure

τ Shear stress

ε Eccentricity ratio

β Bulk modulus

βs Mean asperity radius of combined rough surfaces

σr Standard deviation of combine roughness amplitude

σs Standard deviation of asperity heights

δr Combined roughness amplitude

ηs Asperity density

ϑ Dissipated power

ζ Friction torque

ωp Over relaxation parameter for the pressure

ωp Over relaxation parameter for the film fraction

ξp Pressure tolerance

ξy Tolerance for the support load on the y axis

ξT Temperature tolerance

Λ Ratio between fluid film thickness and surface roughness φp(x,z) Patir and Cheng pressure flow factors

φs(x) Patir and Cheng shear flow factors

φf p(x,z) Patir and Cheng friction pressure flow factors φf sx Patir and Cheng friction shear flow factors φf(x,z) Friction contact factor

1 Introduction . . . 18 1.1 Objective . . . 19 1.2 Dissertation Outline . . . 19 2 Literature Review . . . 20 2.1 Lubrication . . . 20 2.2 Surface Texturing . . . 26 2.3 Solution Schemes . . . 30 3 Methodology . . . 32

3.1 Hydrodynamic Lubrication Modeling and Solution . . . 32

3.2 Mixed Lubrication Modeling Solution . . . 44

3.3 Global Thermal Effects . . . 51

3.4 Fluid Film for Spherical Geometry Dimples . . . 54

3.5 Solution Scheme . . . 58

3.6 Computational Procedure . . . 63

4 Program Validation . . . 64

4.1 Case 1: Cavitation Model . . . 65

4.2 Case 2: Roughness Effects . . . 67

4.3 Case 3: Temperature Model . . . 69

4.4 Case 4: Texture Model . . . 71

4.5 Case 5: Dynamically Loaded Journal Bearing . . . 73

5 Results and Discussion . . . 75

5.1 Input Parameters . . . 76

5.2 Convergence Study . . . 77

5.3 Operation Speeds . . . 80

5.4 Roughness and Asperity Effects . . . 82

5.5 Thermal Effects . . . 85

5.6 Surface Texture . . . 88

6 Conclusion . . . 106

6.1 Future Works . . . 109

1 INTRODUCTION

This work aims to develop procedures for virtual texturing of crankshaft bearings to improve their performance in terms of friction and load carrying capacity. There is an ever-increasing demand for automobiles worldwide and enhancing the tribological properties of en-gine components is important since nearly one third of the fuel energy is used to overcome friction (HOLMBERG et al., ). According to the author, only in 2009, 208,000 millions of liters of fuel were used to overcome friction, and, as gasoline is a nonrenewable power source and, therefore, has a limited life cycle, the amount of money wasted due to friction is only increasing. If would be possible to reduce the amount of friction in crankshaft bearings, it is possible to reduce the amount of gasoline required to provide the same power to the engine and, as a result, increase the vehicle efficiency, allowing it to run with lower costs.

In a regular four-stroke engine, gasoline works in an intake-compression-combustion-exhaust cycle, performed using mechanical components. About 28% of the energy fuel, without considering braking friction, is lost due to friction. Lubrication is important to reduce friction on these mechanical parts, but for a bearing, even when lubricated, the friction still represents about 25% of the power loss in engines (LIGIER; NOEL, 2015).

Researches were performed in mechanical seals and concluded that it is possible to reduce friction in seals by modifying their surfaces. In this study, performed by Etsion and Burstein, they added dimples to the seal surface and improved the component’s performance, for an optimum set up. With those modifications, there was a friction reduction up to 50% (ETSION; BURSTEIN, 1996). This research led to questions about whether it is possible to reproduce this results for different engine components. If so, the amount of energy lost due to friction in engine components would fall drastically, improving the vehicles’ power efficiency.

1.1 Objective

The objective of this dissertation is to evaluate the surface texturing applied to crankshaft’s bearings in order to decrease friction in engines aiming to improve the vehicle’s performance.

The study will implement numerical procedures to simulate the bearing behavior during its operation following the steps:

∙ Solve the Reynolds equation for a hydrodynamic lubricated journal bearing;

∙ Solve the modified Patir and Cheng Reynolds equation to deal with the roughness effects in the fluid;

∙ Use the Greenwood-Trip rough contact model to deal with the asperities interaction; ∙ Use an average temperature model to deal with the variation of lubricant viscosity with

the temperature during the bearing operation;

∙ Numerically apply textures to the journal bearing’s surface.

1.2 Dissertation Outline

This dissertation is divided into the following chapters:

∙ Literature review: where the main lubrication regimes are presented along with the sur-face texturing and the solution methods used to solve the approximate Reynolds equation; ∙ Methodology: with a brief explanation of the equations used to model the journal

bear-ing’s behavior;

∙ Program validation: with the comparison between this dissertation models for the cavita-tion, roughness, temperature, texture and dynamical with papers from the literature;

∙ Results and discussion: with application of the program to investigate lightweight crankshaft textured bearings.

∙ Conclusion: with a summary of what was done and ideas for future works.

This project is part of São Paulo Research Foundation (FAPESP) Center of Research in Engineering Prof. Urbano Ernesto Stumpf, under project number 2013/50238-3 and grant number 2015/16470-1.

2 LITERATURE REVIEW

The increasing need of the automobile industry for more efficient cars has lead researchers to push the boundaries of science when trying to find ways to make engine com-ponents smaller and lighter. These features are desirable without losing the engine ability to sustain high loads. One of the components that require high capability of sustain these loads is the engine bearings. The bearings are responsible, among other things, to uphold the shaft movement in different velocities and loads, with the least wear and friction. There has been an increasing number of papers related with the enhancement of tribological properties of engine bearings, specially through the manipulation of their surface (MARIAN et al., 2007; MA; ZHU, 2011; ARSLAN et al., 2015).

This review aims to present the current worldwide progress when dealing with the modeling of textured surfaces in order to enhance their tribological properties, particularly when applied to journal bearings. In order to better understand the behavior of bearings, we are going to present aspects of lubrication and how it influences the bearings’ operation. The most relevant papers related to the subject are presented here, including common designs for textures and the most common used modeling techniques for the different lubrication regimes.

2.1 Lubrication

The main purpose of a lubricant is to reduce friction and wear on machine elements, making sure that these elements run smoothly during their life cycle. The most used type of lubricant in the automobile industry is liquid, although it can also be presented in solid and gas forms. There are four main lubricant regimes, each one dealing differently with the physical and chemical interactions between the lubricant and the contacted surface (HAMROCK et al., 2004). This section is going to discuss each of these regimes and present the most well known works on the field, scaling to their applications in car engine components, specific on bearings. 2.1.1 Hydrodynamic Lubrication

Hydrodynamic lubrication is characterized by a lubricant film thick enough to keep the lubricated surfaces separated from each other. This lubrication regime is considered ideal because of the low friction and wear associated with the positive pressure created by the converging surfaces, the relative movement and fluid viscosity, occurring specially on confor-mal surfaces such of journal and thrust bearings, although it can also happen for non-conforconfor-mal contacts (HAMROCK et al., 2004).

2.1.1.1 Reynolds Equation

The Reynolds equation is a partial differential equation that governs the fluid film pressure distribution for lubricated contacts. It was first derived in 1886, by Osborne Reynolds, who assumed the condition of dominance of the pressure and viscosity laminar flow terms of the Navier-Stokes equations. Reynolds isothermal and isoviscous equation can be written as (REYNOLDS, 1886; GROPPER et al., 2016):

∂ ∂ x  h3 12µ ∂ p ∂ x  + ∂ ∂ z  h3 12µ ∂ p ∂ z  =U 2 ∂ h ∂ x+ ∂ h ∂ t (2.1) where: h: film thickness; p: hydrodynamic pressure; µ : dynamic viscosity; U: velocity in x-direction.

Later on, Dowson extended the lubrication theory, writing generalized equations, that included the thermohydrodynamic effects and the lubricant properties variation across the film thickness (DOWSON, 1962). Berther and Godet took into account the existence of local-ized perturbations and defects on both surfaces, creating the today known as Generallocal-ized Equa-tions of the Mechanics of Viscous Thin Films (BERTHE; GODET, 1973). Afterwards, Patir and Cheng proposed a modification to the Reynolds equation, this time including the roughness effects on the pressure distribution by introducing an average flow model (PATIR; CHENG, 1978; GROPPER et al., 2016).

2.1.1.2 Fluid Film Cavitation

Cavitation is the phenomenon where the hydrodynamic pressure reaches values lower than the gas saturation or vapor pressure, resulting on a rupture of the lubricant film. The created cavity is filled with a mixture of liquid and gases or vapor, modifying the lubrication system behavior (PROFITO et al., 2015).

The first researcher to study lubricant film rupture was Gumbel, in 1914. He pro-posed that only the pressures with values equal or higher than a defined cavitation pressure limit should be considered. All the pressures bellow this values were admitted to be equal to the as-sumed cavitation pressure. The Gumbel model is the simplest model to deal with cavitation on the Reynolds equation and is also known as the half-Sommerfeld cavitation model (GUMBEL, 1914).

A few years later, Swift and Steiber proposed a new formulation to improve the half-Sommerfeld model. The new formulation considered that the cavitation boundaries had a null pressure gradient and within the cavitated zones, the pressure were constant and equal

to the cavitation limit pressure. This formulation is known as the Reynolds cavitation model (SWIFT, 1932; STIEBER, 1933).

The half-Sommerfeld and the Reynolds cavitation models both have the drawback that there is no enforcement of mass-conservation on the reformation boundaries. It was Jakobs-son, Floberg and Olsson who proposed complementary boundary conditions that ensured mass-conservation of all the lubricant flow domain (JAKOBSSON; FLOBERG, 1957; OLSSON, 1965).

The first algorithm considering mass conservation, was developed by Elrod and Adams, who implemented the boundary conditions proposed by Jakobsson, Floberg and Ols-son (JFO model). They divided the cavitation region using a switch function, eliminating the need for multiple equations and making sure that there was a pure Couette flow present in the cavitated areas (ELROD; ADAMS, 1974; ELROD, 1981).

Recently, the Elrod algorithm was compared with Reynolds model by Ausas and co-workers, in 2007, when studying the cavitation influence on micro-textured journal bear-ings. The conclusion they reached was that, for textured journal bearings, models without mass conservation underestimate the cavitation area, resulting in higher values for the friction torque. It is important that the cavitation models used when modeling journal bearings consider mass-conservation, in order to avoid calculation errors. Ausas also made his algorithm available, making it easier for new studies (AUSAS et al., 2007; AUSAS et al., 2009).

Qiu and Khonsari used a mass conserving algorithm to predict the cavitation behav-ior on flat dimpled surfaces. They ran experimental tests to validate their model and concluded that it’s possible to predict the cavitation effect using the JFO boundary conditions. The study also showed that using half Sommerfeld and Reynolds models leads to distinct solutions when different dimple sizes are used (QIU; KHONSARI, 2009).

The stability problem of the Elrod algorithm was later studied by Fesanghary and Khonsari. They focused their effort on developing a variation for the switch function in the Elrod algorithm, that would result in less instability. The solution they found was to substitute the binary switch proposed in the JFO method by a progressive regressive exponential equation. The results showed an improvement in the solution stability and also in the computation time (FESANGHARY; KHONSARI, 2011).

2.1.2 Mixed-Lubrication

The mixed-lubrication, or partial lubrication, is the regime where both the fluid film and contact of asperities take place during a certain operation condition. It can occur, for example, when lubricated components operate under high pressure loads (or too low velocities) (HAMROCK et al., 2004; ADJEMOUT et al., 2014).

operate under mixed-elastohydrodinamic lubrication regime with very thin film thickness. He used a fast Fourier transform (FFT) technique to solve the forward (surface displacement) and inverse problem (asperity contact pressure), developing a code that analyzed asperity of contact on 3D surfaces with a good computational time (JIANG et al., 1999).

A deterministic model was developed later by Shi and Salant to treat the behavior of two surfaces operating in a mixed soft elastohydrodynamic lubrication regime, a flat and an elastic rough. The model took into account the effects of inter-asperity cavitation, using the JFO boundary conditions through the Elrod method. They showed that it is necessary to consider inter-asperity when modeling mixed EHL, for better understatement of the soft EHL problems (SHI; SALANT, 2000).

Later on, in 2006, Dobrica and other researchers proposed a deterministic model and compared it with the Patir and Cheng stochastic model. They concluded that roughness had an influence on all the parameters they observed (film thickness, attitude angle and friction torque) and that the stochastic model proposed by Patir and Cheng was able to predict accurately the effect of different roughness types, giving good predictions for average minimum film thickness, but underestimating friction torque (DOBRICA et al., 2006).

Experimental tests were performed by Braun and co-workers to analyze the effi-ciency that laser surface texturing had in the reduction of friction in steel sliding pairs, con-sidering a mixed lubrication regime. The group used different sizes of dimples and performed tests using two different lubricants temperatures, 50 and 100∘C They concluded that there were no linear correlation between dimple size and friction coefficient, operation speed or temper-ature. Despite that, they managed to reduce the friction by 80% for an optimum diameter and operation speed, being this reduction highly dependent on the oil temperature (BRAUN et al., 2014).

Profito modelled the different lubrication regimes on journal and sliding bearings, using the isothermal generalized equation of the mechanics of viscous thin films and the mass-conserving p-θ Elrod-Adams cavitation model. He proposed a new way to numerically calculate the modified mass-conserving Reynolds equation based on the element based finite volume method. This technique gathers the finite elements conservation flexibility to work with complex geometries and the finite volume to work with the flow, making it ideal to work with irregular meshes, such as textured surfaces (PROFITO et al., 2015).

2.1.2.1 Roughness Effects

The interaction between two surfaces is limited by the roughness and can have its location defined by an influence area, being this area of contact the responsible for carrying all the external load. Asperity can be understood as the highest roughness peaks, with the actual contact area being the one generated by the deformation of these asperities (PROFITO, 2010).

The milestone for contact mechanics was the paper published by Hertz, in 1881, where he considered two surfaces with no lubrication between them. After that, many other studies were published, trying to define models for the contact area between surfaces (dry con-tact). For instance, Greenwood and Williamson presented one of the first models where the ac-tual random nature of roughness asperity peaks was considered (GREENWOOD; WILLIAMSON, 1966).

Harp and co-workers’ paper presented a variation of the Reynolds equation, includ-ing the effects of cavitation induced by inter-asperity. The equation is based on the JFO model, implemented in the Elrod algorithm, and the Patir and Cheng method. They later published another paper with the application of this equation for the analyzes of a mechanical seal, show-ing that the microscopic cavitations are meanshow-ingful in seals with small film/roughness ratio. Including inter-asperity changed the predictions of previous models that neglected it, reaching higher load supports numbers and very different leakage rates, even for hydrodynamic lubrica-tion regimes (HARP; SALANT, 2001; HARP; SALANT, 2002).

Almost a decade later, an algorithm was created by Qiu and Khonsari to analyze the performance of seals and thrust bearings with surface texture. The algorithm considers mass-conservation, using JFO boundary conditions and applies the Patir and Cheng formulation to take into account roughness effects. Among other things, they studied the film thickness, depth-diameter ratio and dimple density necessary to improve the surface performance. The main outcome were that the cavitation is only relevant for thin fluid films, the roughness influence is small when compared with the cavitation effect and there is an optimum depth and density of dimples that improves the performance, but it depends on the operation conditions (QIU; KHONSARI, 2011b).

2.1.3 Elastohydrodynamic Lubrication

The elastohydrodynamic lubrication can be understood as a variation of the hy-drodynamic lubrication, where the elastic deformation of the contact surfaces are considered (HAMROCK et al., 2004).

Zhu and Cheng wrote a FORTRAN code to calculate point and line contacts lubrica-tion, using inlet heating and thermal reduction factor, evaluating their effects on film thickness. They also adopted a viscoelastic fluid model to calculate shear stress. The code and results were useful to determine the efficiency of mechanical components and predict failure on elastohy-drodynamic contacts. A year later, they improved their code in order to predict both mixed and elastohydrodynamic lubrication parameters, including friction. This new code used real 3D sur-face roughness under critical loads to predict lubrication characteristics, although it still needed improvement in the numerical approach and the computational time (ZHU; CHENG, 1989; ZHU; HU, 2001).

and his colleagues: direct summation, multilevel multi-integration (MLMI) and discrete con-volution fast Fourier transform (DC-FFT). These methods are used for contact problems and results obtained by Wang showed that all three methods are capable of reaching the same pre-cision, but with different calculation times. Out of the three methods, the DC-FFT had the best computational efficiency, specially when dealing with dense meshes. It was able to solve sur-face deformation and also temperature rise problems, being three times faster than the MLMI (WANG et al., 2003).

2.1.4 Thermohydrodynamic and Thermoelastohydrodynamic Lubrication

Thermohydrodynamic (THD) and thermoelastohydrodynamic (TEHD) lubrica-tions regimes are important when the temperature has an influence on the hydrodynamic and elastohydrodynamic behavior of the lubricants, respectively (HAMROCK et al., 2004).

Shi and Wang developed a thermoelastohydrodynamic model for journal bearings conformal contacts where they considered the roughness, asperity contact, thermal and ther-moelastic effects, working with large eccentricity ratios. They concluded that roughness effects should be considered when working with thermal-tribological analysis and, afterwards, wrote another paper where they put this model into practice. This paper simulated the performance of journal bearings for different arrangements of convective heat-transfer conditions. The conclu-sion was that, when considering thermal effects, the contact asperity does not decrease linearly with the increase of operational speed, because the speed increases the hydrodynamic effect and also the amount of heat generated. They also reported a transition of the contact asperity areas, from the edge to a global central area. This phenomenon was explained by the transition process of lubrication regimes or by the thermal expansion difference between the bearing inside and outside surfaces (SHI; WANG, 1998; SHI et al., 1998).

A transient studied was done by Zhang considering non-Newtonian dynamically loaded journal bearings in mixed lubrication to analyze their TEHD behavior. He solved the problem using the Reynolds equation, a 2D conduction equation and heat balance, demon-strating the effects, among other things, of the texture roughness and the thermal and elastic deformations performance of bearings under dynamic load. His results showed that the bear-ing system is very sensitive to the contact conditions and temperature distribution varies with the degree of contact asperity. He also concluded that, for better results, roughness, degree of contact and thermal and elastic deformation must be coupled (ZHANG, 2002).

Kucinschi and colleagues analyzed the performance of radially groove thrust washer with the presence of thermal deformations, coupling the flow effects of the lubrication with heat transfer of both solid and fluid. They created a model to study elastic deformation and temperature influence on this mechanical component and found higher minimum film thickness values than the ones presented by rigid stators. Also, when comparing THD and TEHD regimes, they found only a small difference between results, when a small misalignment was present

(KUCINSCHI et al., 2004).

A new model for the thermoelastohydrodynamic lubrication was proposed by Fatu and his team to analyze a dynamically loaded journal bearing. The model considered mass conservation cavitation and the effect of temperature in the viscosity, proposing a new heat flux algorithm to take into account the lubricant film temperature. They used a big-end connecting-rod bearing to validate their algorithm, verifying its efficiency and also concluding, from the results, that temperature for the oil film varies significantly over space and time (FATU et al., 2006).

A 3D thermohydrodynamic analysis was performed by Cupillard and his team on a textured slider, considering laminar and steady flow. They used hot and cold lubricants in the grooves and studied their performance on different operation conditions. They concluded that thermal effects must be considered for better results, having a maximum performance with long grooves and achieving an increase in the loading carry capacity of about 16% for crit-ical conditions. They also discovered that, when compared with smooth sliders, the textured ones performed better for a large range of operation conditions, specially under high speeds (CUPILLARD et al., 2009).

A micro-textured journal bearing was thermally analyzed using JFO boundary con-ditions and non-Newtonian rheology by Kango and co-workers, comparing with a model using Reynolds boundary conditions. They showed that the loading carrying capacity was similar for flat and full textured surfaces, but the JFO provided more realistic results for the textured bearings. Additionally, they investigated the partial texturing and found that, when located in the convergence area, it improves the loading carrying capacity, decreasing the temperature, specially for small eccentricity ratios (KANGO et al., 2014).

2.2 Surface Texturing

The first group to associate surface texturing with the improvement of a surface tribological properties was Hamilton and his co-workers, in 1966 (HAMILTON et al., 1966). They pointed out that the classical theory of lubrication did not predict the existence of an additional pressure that separate two planar parallel surfaces, so there should exist a mechanism that would explain this effect. The proposed mechanism was the cavitation generated in the small irregularities of the surface. Therefore, the cavitation in the micro cavities was responsible for creating a hydrodynamic pressure that would balance the asymmetric pressure distribution of the fluid film, producing a high carrying load capacity.

Very little attention was paid for this discovery at that time and only few papers were published related to the subject (ANNO et al., 1968; ANNO et al., 1969; WILLIS, 1986). It was only a few years later, with the publication of Etsion and Burstein and the positive results they obtained with a surface treatment on mechanical seals, that surface texturing became a

subject of interest in academia. They found that with a proper selection of the size and ratio of the surface pores, it was possible to improve the mechanical seal’s performance (ETSION; BURSTEIN, 1996). After their work, there were a huge number of publications, most of them theoretical, exploring the surface texturing in different components, such as thrust and journal bearings, piston rings, piston pins, etc.

Etsion published many other theoretical and experimental papers about surface tex-turing. He investigated the effects of laser surface texturing on partially textured mechanical seals in order to find an optimum texture design to reduce friction and heat generation on me-chanical components, discovering that this type of treatment enable an impressive improvement in the tribological properties. Figure 2.1 shows the textured mechanical seal model used by Etsion (ETSION; HALPERIN, 2002). Later, Brizmer and Etsion published a paper about the advantages of using laser surface texture in parallel thrust bearings and, furthermore, the rela-tion between the porous density and influence on the pressure (BRIZMER et al., 2003; ETSION et al., 2004). Finally, Etsion published a review showing the efforts worldwide related to laser surface texturing, describing different models for particular machine elements (ETSION, 2005).

Fig. 2.1 – Textured mechanical seal model, as present by Etsion (ETSION; HALPERIN, 2002)

Other researchers investigated the effects of surface texturing using different ap-proaches. Kucinschi and his team published a paper in 2004 describing a numerical model con-sidering the thermoelastohydrodynamics effects of a grooved thrust washer. He used Reynolds equation to solve the fluid flux problem, the energy equation for the temperature and the equa-tion of thermo-elasticity for the deformaequa-tion (KUCINSCHI et al., 2004). A year later, Ko-valchenko and co-workers performed an experimental study on laser surface texturing effect on transitions and found out that, for a hydrodynamic lubrication regime, the loading carrying per-formance was improved, specially for high loads, velocities and viscosities (KOVALCHENKO et al., 2005).

It was Wang and Zhu, in 2005, who published the first known paper using the term "virtual texturing" when referencing to the act of modeling surfaces with different treatments. In their paper, they related the texture designs with the component tribological properties and illustrated the idea of what was virtual texturing. They listed these concepts as: 1. Virtually generation of the surface, based on the operation requirements; 2. Analysis of the contact and the lubrication to evaluate the performance; 3. Prediction of the efficiency, life and the evolution of the surface; 4. Modification and optimization of the surface; 5. Validation of the model through experimental analysis (WANG; ZHU, 2005).

More recently, in 2013, Kango and his team made a comparison with textured and grooved journal bearings, studying the performance of components with dimples, paral-lel grooves and transverse grooves. They conclude that both dimples and groove can improve the average temperature, as long as they are optimally located and operating under certain ec-centricities ratios (KANGO et al., 2013).

2.2.1 Definition and Purpose

The act of creating features on a surface, like micro dimples or grooves, with the intention of improving its mechanical properties, is denominated surface texturing. There are three main mechanism responsible for the improvement of the tribological parameters of tex-tured surfaces. Firstly, the micro-dimples are believed to function as micro reservoirs, providing lubricant for the mechanical components in case of mixed or boundary lubrication. Secondly, the dimples function as micro-traps for wear particles. Finally, the generation of lift, also called the micro-bearing effect (mechanism only captured by theoretical models). Even though these effects are addressed repeatedly in the literature, there are no researches that corroborate with the first two assumptions (ARAUJO et al., 2004; LU; KHONSARI, 2007; QIU; KHONSARI, 2011a; GROPPER et al., 2016; PROFITO, 2010).

2.2.2 Texturing Design

Texturing design depends on many parameters, but it is possible to limit the most important ones to texture density, dimple aspect ratio and dimple depth. There are many studies focused solely on optimizing these parameters, aiming to achieve better tribological performance, either for full or partial textured surfaces.

Tönder studied the influence of dimples on a full textured plane bearing, using the Reynolds equation to run numerical simulations. He found that the loading carrying capacity can be increased under specific operation conditions, even though the friction increases as well. In addition, for certain operation conditions, it is possible to improve the film stiffness and damping, with the cost of higher values for the coefficient of friction (TONDER, 2010).

Ma and Zhu considered the optimum design for surfaces textured with elliptical-shaped dimples. They used many dimple depths and found that an increase in the optimum

diameter is followed by an increase in the optimum corresponding depth. Furthermore, they discovered that the ideal area ratio does not depend on texture or operating parameters, validat-ing these results with experimental tests (MA; ZHU, 2011).

Podgornik and his colleagues analyzed the best texture designs for different lubrica-tion regimes, using experimental tests, fluid dynamic models and 2D finite element simulalubrica-tions. They discovered that, under starved lubrication, friction was increased and only low dimple density resulted in the micro oil reservoir effect. The best results for friction were obtained when the full film lubrication was reached, but a CFD simulation was needed to define the ideal densities and shapes (PODGORNIK et al., 2012).

Fesanghary and Khonsari used the Sequential Quadratic Programming optimization algorithm (SQP) to find the optimum groove shapes for loading carrying capacity on parallel bearings. They ran a great amount of theoretical and experimental analyzes, validating their re-sults with a good correlation and finding that the aspect ratio has a major influence in the loading carrying capacity, reaching an improvement of 36% when comparing with parallel bearings nor-mal spiral grooves. Figure 2.2 presents one of the texture designs used by Fesanghary on the parallel bearing (FESANGHARY; KHONSARI, 2013).

Fig. 2.2 – Parallel bearing with a surface pattern, as shown by Fesanghary (FESANGHARY; KHONSARI, 2013)

Qiu and co-workers investigated the effects of many different textured shapes on the friction coefficients and stiffness of parallel slider bearings. They concluded that there is an optimum textured shape that minimizes the friction coefficient, but that one is not neces-sarily the one that gives the higher values of loading carrying capacity. Also, they found that curved shaped dimples produce the lowest friction coefficient, specially ellipsoid shapes, but for isotropic performance, an spherical shape dimple is the best option (QIU et al., 2013).

Changing the focus of the studies for a more practical sense, Adjemout and col-leagues showed the effects of real dimple shapes on the performance of textured mechanical seals. They analyzed the shapes obtained after the fabrication process and implemented them in a hydrodynamic lubrication model, studying the influence of different dimple defects. Results

showed that there is a limit where the fabrication imperfection ceases the positive effects of the texturing, being the depth and orientation of roughness inside the dimples the most rele-vant parameters for the leakage. They concluded that an increase in the curvature angle of the edge of the dimples increases the leakage, affecting the pumping and increasing the friction (ADJEMOUT et al., 2014).

2.2.3 Journal Bearings

As a result of the circumferential converging-diverging gap of the journal bear-ings film thickness, the surface texturing on journal bearbear-ings are the most challenge ones and there are not many publications in the field (GROPPER et al., 2016).

Tala-Ighil and co-workers studied the effects of surface texture and the influence of texturing location on the performance of journal bearings. They considered the inertia ef-fects and found that there is a negative influence of these efef-fects on the bearing’s performance, concluding that the optimum texture depends on the geometry and operation conditions of the bearing (TALA-IGHIL et al., 2011).

A paper published by Cupillard, in 2008, showed how he used computational fluid dynamics analysis on a textured journal bearing, trying to determinate the texture effect on the loading carrying capacity. The paper considered hydrodynamic lubrication and used a multi-phase flow cavitation model for the analyzes. Cupillard and his team tested the textured design in a wide range of eccentricity ratios and discovered that it is possible to improve the loading carrying capacity for a specific range of eccentricity ratios. They also figured that the friction coefficient is reduced for deep dimples located in the regions of maximum pressure when the eccentricity ratio is high, but these effects depend on the operation conditions (CUPILLARD et al., 2008).

A theoretical research was conducted by Brizmer and Kligerman in 2012 to evalu-ate the potential of laser surface texturing on journal bearings. Optimum parameters were found for the dimples when seeking for high values of loading carrying capacity, analyzing both par-tial and fully textured bearings. Their results showed that, for fully textured bearings, there is a detrimental in the tribological properties of the bearings and, for the partial texturing, the load-ing carryload-ing capacity increases for low eccentricities, but no real improvements were found for high eccentricities (BRIZMER; KLIGERMAN, 2012).

2.3 Solution Schemes

There are many types of solution for the lubrication models commented so far, each one with its benefits, trying to solve the equations using the least amount of time. Re-searchers try to avoid too many simplifications, in such way that the model becomes too simple or to little (or too heavy with prohibited time consuming). The most common methods to solve

the Reynolds equations are:

∙ Finite Element Method (FEM): A numerical method for the solution of boundary value problems involving ordinary and partial differential equations. This method was first de-veloped for civil engineering applications, but, with the work of mathematicians, a solid theoretical base was created and it was further used in many different applications (POD-GORNIK et al., 2012; JACKSON; GREEN, 2008; FATU et al., 2006; KUCINSCHI et al., 2004; ZHANG, 2002; BITTENCOURT, 2015);

∙ Finite Difference Method (FDM): Numerical method to obtain the approximate solu-tion of a partial differential equasolu-tion. This method uses the Taylor’s series to expand the derivated function (ADJEMOUT et al., 2014; CUPILLARD et al., 2009; DOBRICA et al., 2006);

∙ Finite Volume Method (FVM): Consists of approximating the integral form of a dif-ferential equation into discrete regions called finite volumes. The basic assumption of this method is taking a physical representation of the problem through a differential equation (KANGO et al., 2014; FESANGHARY; KHONSARI, 2013; QIU et al., 2013; BRIZMER; KLIGERMAN, 2012; MA; ZHU, 2011; TALA-IGHIL et al., 2011; PROF-ITO et al., 2015).

Each of these methods have a specific characteristic that makes them preferable for solving lubrication problems and they were analyzed by Woloszynski and colleagues, consid-ering the difference between analytical and numerical results, size of the mesh, load carrying capacity and friction coefficient. Woloskynski and his team came up with the following con-clusions, when analyzing the methods for hydrodynamics bearing with and without surface texturing (WOLOSZYNSKI et al., 2013):

∙ The finite element method gives good results, having the second best result when dealing with textured surfaces;

∙ The finite volume method has the best stability when mesh size parameters were changed, maintaining small difference in the results for pressure and load carrying capacity, for example. Unfortunately, it was the method that presented the least accurate results; ∙ The FDM showed fairly precise results for pressure and other parameters only when used

3 METHODOLOGY

The methodology used in this work was the gathering and replication of papers from the literature review, coming up with a new computational program capable of accurately describe the behavior of dynamically loaded journal bearing, considering effects of roughness, temperature and surface textures. The articles were selected to improve the learning process, beginning with a hydrodynamic lubrication solution of the Reynolds equation and increasing its complexity, scaling up to consider surface roughness, temperature and texture effects.

3.1 Hydrodynamic Lubrication Modeling and Solution

The Reynolds equation is a second order partial differential equation that deals with the pressure distribution in the fluid film. The equation is named after Osborne Reynolds, who was the first person to describe the basic differential equation of fluid film, comparing his the-oretical predictions with experimental results (REYNOLDS, 1886). The Reynolds equation is obtained from the Navier-Stokes and mass conservation equations. Consider two lubricated sur-faces with the Cartesian system described in Figure 3.1.

Fig. 3.1 – Coordinate system and velocity components for the Navier-Stokes equation (adapted from (SANTOS, 1997)).

We can write the Navier-Stokes equation for this system as (SANTOS, 1997): ρ∂ ux ∂ t = − ∂ p ∂ x+ µ  2∂ 2_u x ∂ x2 + ∂2ux ∂ y2 + ∂2ux ∂ z2  + 2∂ µ ∂ x ∂ ux ∂ x + ∂ µ ∂ y  ∂ uy ∂ x + ∂ ux ∂ y  +∂ µ ∂ z  ∂ uz ∂ x + ∂ ux ∂ z  ρ∂ uy ∂ t = − ∂ p ∂ y+ µ  2∂ 2_u y ∂ x2 + ∂2uy ∂ y2 + ∂2uy ∂ z2  + 2∂ µ ∂ y ∂ uy ∂ y + ∂ µ ∂ x  ∂ uy ∂ x + ∂ ux ∂ y  +∂ µ ∂ z  ∂ uz ∂ y + ∂ uy ∂ z  ρ∂ uz ∂ t = − ∂ p ∂ z+ µ  2∂ 2_u z ∂ x2 + ∂2uz ∂ y2 + ∂2uz ∂ z2  + 2∂ µ ∂ z ∂ uz ∂ z + ∂ µ ∂ x  ∂ uz ∂ x + ∂ ux ∂ z  +∂ µ ∂ y  ∂ uz ∂ y + ∂ uy ∂ z  (3.1) where:

x, y and z: Cartesian coordinates ux, uyand uz: velocity components

ρ : fluid density

µ : fluid dynamic viscosity p: pressure of the fluid film t: time variable

Now consider the following hypotheses:

∙ The lubricant oil flow is laminar and therefore, the fluid inertia effects can be neglected and ρ∂ ux ∂ t = 0, ρ ∂ uy ∂ t = 0, ρ ∂ uz ∂ t = 0. (3.2)

∙ The velocity variations in the x and z directions are null, as illustrated in Figure 3.1. Therefore, ∂2ux ∂ x2 ≪ ∂2ux ∂ y2 ≫ ∂2ux ∂ z2 ; ∂2uy ∂ x2 ≪ ∂2uy ∂ y2 ≫ ∂2uy ∂ z2 ; ∂2uz ∂ x2 ≪ ∂2uz ∂ y2 ≫ ∂2uz ∂ z2 . (3.3)

∙ The pressure gradient and lubricant properties are negligible across the film thickness dimension, since its too small when compared to the other dimensions (10−6). Therefore,

∂ p ∂ y = 0,

∂ µ

∙ The product between the derivative terms of viscosity and velocity are small and negligi-ble. Consequently, ∂ µ ∂ x  ∂ uz ∂ x + ∂ ux ∂ z  = 0, ∂ µ ∂ x  ∂ uy ∂ x + ∂ ux ∂ y  = 0; ∂ µ ∂ z  ∂ uz ∂ x + ∂ ux ∂ z  = 0, ∂ µ ∂ z  ∂ uz ∂ y + ∂ uy ∂ z  = 0. (3.5)

Using the described assumptions in Equation (3.1), we have:

∂ p ∂ x = µ ∂2ux ∂ y2 ; ∂ p ∂ y = µ ∂2uy ∂ y2 ; ∂ p ∂ z = µ ∂2uz ∂ y2. (3.6)

We can obtain the velocity profiles by integrating Equation (3.6) twice. Performing the integrations for the x, y and z coordinates, we have:

u_x= y 2 2µ ∂ p ∂ x + c1y+ c2; u_y= y 2 2µ ∂ p ∂ y + c3y+ c4; u_z= y 2 2µ ∂ p ∂ z + c5y+ c6; (3.7)

where c1through c6are the integration constants. In order to determine these constants, consider

Fig. 3.2 – Laminar flow between the two lubricated surfaces (adapted from (SANTOS, 1997)).

From Figure 3.2, we can define the components of velocity for y = 0

u_x= U1, uy= 0, uz= 0. (3.8) And for y = h u_x= U₂cosα −Vysinα; u_y= U2sinα +Vycosα; uz= 0; (3.9) where: U₁: speed of surface 1 U₂: speed of surface 2 Vy= ∂ h_{∂ t}

As the curvature of the solid components is much greater, when compared with the fluid film thickness, the angle α can be considered very small, thus:

cos α = 1, sin α ≈ tan α = ∂ h ∂ x, Vy

∂ h

∂ x = 0. (3.10)

Equations (3.8) and (3.9) in Equation (3.7), we have: u_x= 1 2µ ∂ p ∂ x(y 2_{− yh) +}h− y h U1+ y hU2; u_y= y hVy; u_z= 1 2µ ∂ p ∂ z(y 2_{− yh).} (3.11)

The fluid is considered a continuum medium and the continuity equation is define by

Dρ

Dt + ρ div~u = 0. (3.12)

Considering that density is constant and substituting Equation (3.11) on Equation (3.12), we have ∂ ∂ x  1 2µ ∂ p ∂ x(y 2_{− yh) +} h − y h U1+ y hU2  + ∂ ∂ y y hVy  + ∂ ∂ z  1 2µ ∂ p ∂ z(y 2_{− yh)}  = 0. (3.13)

Integrating Equation (3.13) on the y direction for a domain [0, h], we have

∂ ∂ x " 1 2µ ∂ p ∂ x  h3 3 − h3 2  +h− h2 2 h U1+ h2 2hU2 # +Vy+ ∂ ∂ z  1 2µ ∂ p ∂ z  h3 3 − h3 2  = 0. (3.14)

Considering surface 1 fixed, we have velocity U1null. Rearranging and simplifying

Equation (3.14), we then have:

∂ ∂ x  h3 12µ ∂ p ∂ x  + ∂ ∂ z  h3 12µ ∂ p ∂ z  | {z } I =U 2 ∂ h ∂ x | {z } II + ∂ h ∂ t |{z} III . (3.15)

where U is the velocity of surface 2. Equation (3.15) is known as the Reynolds equation. The first term (I) is the Poiseuille, or the pressure flow term, and deals with the lubricant flow due to the pressure gradients. The second term (II) is the Couette term, also known as the wedge flow term, and is responsible for the relative motion between the surfaces in contact and variations of the fluid density and surface velocities. Finally, the third term (III) is the normal squeeze term and deals with the transient effects of the film thickness.

We can now use the same assumptions to define the lubricant flows and shear stress. The lubricant flows for the x and z components can be defined as:

q_x= −ρ h 3 12µ ∂ p ∂ x+ ρhU; q_z= −ρ h 3 12µ ∂ p ∂ z. (3.16)

The shear stresses in the lubricant are defined as: τ_x1,2= −₊ 1 2 ∂ p ∂ x(2y − h) + µU h ; τ_z1,2= −₊ 1 2 ∂ p ∂ z(2y − h). (3.17) 3.1.1 Film Thickness

The film thickness geometry of a journal bearing is a function of the nominal clear-ance (ho) and the axis eccentricity (e). The equation for the film thickness can be obtained

geometrically from Figure 3.3, where R and Rjare the bearing and journal radius respectively.

Fig. 3.3 – Journal bearing geometry.

We know that for a typical journal bearing, the nominal clearance hois much smaller

than R and Rjand

Hence,

OB≃ CF. (3.19)

We can now say that the film thickness h is

h(ϕ) = EF = AB. (3.20) Thus, h= OB− DE = OB − (CE −CD) = OB −CE + e cos(ϕ) = R − Rj+ e cos(ϕ) = ho+ e cos(ϕ). (3.21)

We can use the following eccentricity relation:

ε = e/ho. (3.22)

Substituting Equation (3.22) in Equation (3.21) we have the film thickness equation for a journal bearing i.e.,

h= ho(1 + εcosϕ). (3.23)

3.1.2 Film Cavitation

The Reynolds equation can provide the pressure profile on a journal bearing, but depending on the operational conditions, these pressures can reach negative values in some regions of the fluid film. These negative values mean that the fluid film is under tension and is likely to be dissolved in that region. This cavity is filled with a biphasic mixture of liquid and gas/vapor (PROFITO et al., 2015). This process is known as cavitation or film rupture and affects directly the performance of lubricated journal bearings. There are many models that try to predict this phenomenon numerically, but this dissertation is going to focus in only three of them: Gümbel, Swift-Steiber and JFO models. Figure 3.4 represents a lubricated domain under the cavitation effect, according to JFO approach, where the boundary between the full film and the broken film is the rupture boundary (left boundary) and the boundary between the broken film and the reconstructed film is the reformation boundary (right boundary).

Fig. 3.4 – Fluid film cavitation on a lubricated domain. The inside region is a non cavitated domain, where the fluid film is not broken and θ = 1. Outside that region is the cavitated domain, where the thin blue lines represent the gas/vapor mixture and where 0 ≤ θ < 1 (adapted from (PROFITO et al., 2015)).

3.1.2.1 Gümbel

The Gümbel model was the first proposed procedure to deal with the cavitation in hydrodynamic pressure problems. The proposed model, also known as half-Sommerfeld model, consists of excluding all pressures values lower than a specific cavitation pressure, setting them to be equal to the cavitation pressure, as

p= (

p_cav i f p< pcav

p i f p≥ pcav

(3.24)

The Gümbel model is probably the simplest way to deal with the cavitation problem, but it fails to enforce the mass conservation principle in the cavitation boundaries. Because of the abruptly break of the pressure film, the Reynolds conservative nature is not maintained. 3.1.2.2 Swift-Steiber

The Swift-Steiber model is an extension of the Gümbel model, assuming that the pressure gradient on the cavitation boundaries are null. Therefore,

(

p= pcav (cavitation region) ∂ p

∂~n = 0 (cavitation boundary)

(3.25)

The Swift-Steiber model, also known as the Reynolds cavitation model, predicts accurately the film rupture, since it enforces the mass conservation principle in that region. The main issue with the Swift-Steiber cavitation model is that it is not capable of predicting the

re-formulation boundary with precision, due to the lack of mass conservation on the rere-formulation boundary of the fluid film.

3.1.2.3 JFO

The cavitation model proposed by Floberg and Jakobsson and later on improved by Olsson is a more accurate model to predict cavitation, specially for practical approaches, since it deals with the mass conservation on both rupture and reformulation boundaries. The main assumptions proposed by them are (PROFITO et al., 2015):

∙ The cavitation pressure inside the broken film region is constant and equal to the limit pressure (pcav);

∙ Inside the cavitated region, the mixture liquid-gas/vapor flows in thin lines that are com-pletely separated by the vapor/gas phase;

∙ On the reformulation and rupture boundaries, the mass conservation flow is enforced using the following complementary conditions:

ρ h(θ − 1) Un 2 −Wn  +ρ h 3 12µ ∂ p ∂~n = 0 (3.26)

where θ is the film fraction, ~n is the unit vector normal to the cavitation boundaries and U_nand Wn are the components of the sliding and moving boundary velocities for a local

direction ~n, respectively.

Figure 3.5 shows the difference between the cavitation zones for the three different models, as presented by Khonsari (QIU; KHONSARI, 2009). There are other types of cavitation models, but the ones described here are the most common used. This dissertation is going to use the JFO complementary boundary conditions to solve the cavitation problem on journal bearings. The JFO boundary conditions are not simple to implement due to the fact that the rupture and reformulation boundaries are not fixed and their location are not known beforehand. In order to implement the JFO boundary conditions, we are going to use the Elrod and Adams algorithm, proposing two ways to implement it, the gfunction and the p − θ approaches.

Fig. 3.5 – Pressure distributions along the center line of a dimple cell, comparing the three cav-itation models, as found by Khonsari (QIU; KHONSARI, 2009).

3.1.2.3.1 gFunction Model

Vijayaraghavan and Keith modified the Elrod algorithm in a way that it automat-ically changed the form of differentiation (central or upwind) of the shear flow terms in both regions. Their method avoids the trial and error step used during the development of the El-rod algorithm. Consider the following form of the Reynolds equation (VIJAYARAGHAVAN; KEITH, 1989): ∂ ∂ x  ρ h3 12µ ∂ p ∂ x  + ∂ ∂ z  ρ h3 12µ ∂ p ∂ z  =U 2 ∂ (ρ h) ∂ x + ∂ (ρ h) ∂ t (3.27)

We can relate the fluid density with the film pressure using the definition of the bulk modulus. The bulk modulus can be defined as:

β = ρ∂ p

∂ ρ (3.28)

Elrod proposed a variable called fraction film content that made it possible to im-plement an universal partial differential equation that covers both the cavitated and full-film regions. The film fraction variable proposed by Elrod is:

θ = ρ ρc

where:

θ : fraction film content

ρc: fluid density at the cavitation pressure

The fraction film removes the need to distinguish the boundaries of the cavitated region, working as an auxiliary variable that represents the proportion of lubricant at every point of the solution domain. However, it is necessary to implement a switch function, capable of making the partial differential equation consistent with the uniform pressure assumption within the cavitated region. Elrod’s switch function is defined as (ELROD; ADAMS, 1974; ELROD, 1981; FESANGHARY; KHONSARI, 2011):

g= (

0 θ < 1 1 θ ≥ 1

(3.30)

Using Equations (3.28), (3.29) and (3.30) in Equation (3.27), we have:

∂ ∂ x  β h3g(θ ) 12µ ∂ θ ∂ x  + ∂ ∂ z  β h3g(θ ) 12µ ∂ θ ∂ z  =U 2 ∂ θ h ∂ x + ∂ θ h ∂ t (3.31)

Now solving Equation (3.31) for θ , we can obtain the pressure profile in the fluid film using the following equation:

p= pc+ β lnθ (3.32)

with pcis the cavitation pressure. Elrod’s algorithm is highly nonlinear, making it susceptible to

numerical instabilities and often causes results to diverge. Fesanghary and Khonsari proposed a modification to the Elrod’s switch function, aiming to improve the stability and convergence speed of the Elrod algorithm (FESANGHARY; KHONSARI, 2011). Instead of using the binary switch function, they proposed the a continuous function implement in Algorithm 3.1: