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1 This article was downloaded by: [A. Gherca] On: 28 November 2013, At: 01:17 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Tribology Transactions Publication details, including instructions for authors and subscription information: Influence of Texture Geometry on the Hydrodynamic Performances of Parallel Bearings A. R. Gherca a, P. Maspeyrot a, M. Hajjam a & A. Fatu a a Institut Pprime, UPR 3346 Département Génie Mécanique et Systèmes Complexes CNRS, Université de Poitiers ENSMA SP2MI, Teleport 2, Boulevard Pierre et Marie Curie, 86962, Futuroscope Chasseneuil Cedex, France Accepted author version posted online: 20 Dec 2012.Published online: 14 Mar To cite this article: A. R. Gherca, P. Maspeyrot, M. Hajjam & A. Fatu (2013) Influence of Texture Geometry on the Hydrodynamic Performances of Parallel Bearings, Tribology Transactions, 56:3, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Tribology Transactions, 56: , 2013 Copyright C Society of Tribologists and Lubrication Engineers ISSN: print / X online DOI: / Influence of Texture Geometry on the Hydrodynamic Performances of Parallel Bearings A. R. GHERCA, P. MASPEYROT, M. HAJJAM, and A. FATU Institut Pprime, UPR 3346 Département Génie Mécanique et Systèmes Complexes CNRS Université depoitiers ENSMA SP2MI, Teleport 2, Boulevard Pierre et Marie Curie 86962, Futuroscope Chasseneuil Cedex, France It has been proven experimentally that surface texturing represents a viable solution for increasing the load-carrying capacity of parallel fluid bearings. Along with several loadsupporting mechanisms that have been identified in the literature, the texture geometry remains an important feature. With the main objective of evaluating the effects of the texture geometry, a mass-conserving model is employed. While avoiding the use of the bulk modulus β, the algorithm also deals with the cavitation phenomenon and provides rapid and accurate results. For given operating conditions (supply pressure, surface speed, or lubricant viscosity), essential geometrical features such as size, density, and shape are analyzed in detail. In terms of load support and friction, the results reveal a strong dependence between certain parameters such as the texture cell number and dimple depth, and an increase in the texture density has beneficial effects in most cases. With regard to shape, the influence of this feature proves to be more significant in the case of single-grooved bearings than in the case of textures. KEY WORDS Hydrodynamic Lubrication; Cavitation; Load-Carrying Capacity; Texture INTRODUCTION Improving the hydrodynamic performance of lubricated devices has always represented one of the focus points of research in the domain of tribology. Among the most popular solutions to this issue, surface texturing has gradually become the basis of numerous optimization techniques (Etsion (1)). In addition to enhancing the load-carrying capacity, this procedure has also been proven to reduce friction and increase the wear resistance of machine elements. The concept of surface texturing was initially proposed in the 1960s, when Hamilton, et al. (2) advanced an innovative lubrication approach based on the effect of antisymmetric pressure distribution. It was stated that the cavitation phenomenon induced by the so-called surface irregularities would allow high film pres- Manuscript received June 15, 2012 Manuscript accepted November 15, 2012 Review led by Ted Keith 321 sures to overbalance low film pressures and thus generate liftoff effects between parallel sliders. In order to assess the capacity of textured surfaces to induce beneficial effects on the hydrodynamic behavior of lubricated devices, numerous experimental studies were performed. Etsion (1) proved that laser surface texturing (LST) can induce significant improvements in various lubricated applications. For instance, applying textures to mechanical seals (Etsion and Kligerman (3)) has led to a reduction in friction of up to 40% in comparison to untextured seals, and textured piston rings provided a decrease in fuel consumption of up to 4% (Etsion and Sher (4)). The efficiency of textured thrust bearings was also demonstrated, with results showing that microgrooves can induce a significant loadsupporting effect (Brizmer, et al. (5)). Moreover, a twofold effect of LST was reported: on the one hand, partial LST relies on a collective effect of the texture consisting in a gradual increase in pressure, whereas full LST is based on the local cavitation effect that occurs in each individual dimple. With the aim of explaining the experimental success of surface texturing, several lubrication theories were developed. Tønder (6) (8) promoted the idea that textures provide the equivalent virtual effect of an inlet step, subsequently allowing a net increase in load support. Furthermore, he stated that microgrooves act as small reservoirs that entrap the fluid within the contact region, thus providing better lubrication. The works of Fowell, et al. (9) also made a significant contribution to this subject by identifying the inlet suction effect as an essential load-supporting mechanism in the hydrodynamic lubrication of textured bearings. Although different from the classical entrainment mechanism, which is present in the case of convergent bearings, inlet suction was reported to induce the same effect of drawing fluid into the contact region. The whole process was explained by the fact that when the lubricant reaches the divergent region of the texture, the subambient pressure that is generated has the effect of sucking lubricant into the bearing. It is well worth noting that in this type of approach, taking the cavitation phenomenon into account is essential. Dealing with the issue of film breakdown and re-formation, commonly called cavitation, can pose several problems, because the effects of a fluid that reaches subambient pressure are not yet fully understood. Most cavitation models are based on the Jakobsson-Floberg-Olsson (JFO) theory (Jakobsson and Floberg (10); Olsson (11)), which deals with separating the fluid film into

3 322 A. R. GHERCA ET AL. NOMENCLATURE a = Inlet land breadth (mm) B = Bearing breadth (mm) b = Pocket breadth (mm) c = Exit land breadth (mm) D = Universal function F = Switch function F f = Friction force (N/mm) h = Film thickness (μm) h d = Pocket depth (μm) h p = Pocket film thickness (h 0 + h d )(μm) h 0 = Land film thickness (μm) l c = Cell length (mm) = Dimple length (mm) l d N = Cell number p atm = Atmospheric pressure (MPa) p cav = Cavitation pressure (MPa) Re = Reynolds number r = Filling factor U = Speed of lower surface (mm/s) W = Load (N/mm) x = Coordinate axis in the flow direction (mm) x = Nodal distance in the x direction (mm) β = Lubricant bulk modulus (Pa) η = Dynamic viscosity (MPa.s) λ = Dimple aspect ratio (l d /h d ) ρ = Density of a mixture of gas and fluid (kg/mm 3 ) ρ 0 = Lubricant density (kg/mm 3 ) ρ t = Texture density (l d /l c ) two regions: a pressurized zone corresponding to a complete fluid film and a nonpressurized region corresponding to the film breakdown. Although its results are in good agreement with experimental data, the JFO model is complex and its implementation is time consuming. Consequently, Elrod (12) developed a more adaptable model, based on a single and independent equation that integrates JFO theory. By using a simple switch function, the Elrod cavitation model automatically predicts cavitation regions and easily eliminates the pressure term in the cavitated region. A recent work by Ausas, et al. (13) has highlighted the importance of mass flow conservation in numerical models that treat the sensible issue of cavitation. They noted that the absence of mass conservation points to misleading predictions of the pressure distribution. A particular series of cavitation models found in the literature is based on the utilization of the bulk modulus β (e.g., Wang, et al. (14)). This coefficient expresses the resistance of fluid to compression and can vary with pressure, temperature, and molecular structure. Though using this method is not necessarily incorrect, it was observed that β can have an important influence on the results and can pose convergence problems. In addition, choosing a realistic value for this coefficient can often prove difficult, considering the fact that the literature provides values for β starting from Pa, as used by Vijayaraghavan and Keith (15), upto Pa, as used by Elrod (12). Consequently, the authors of the present article have purposely tried to avoid this formulation. With regard to the precise geometrical parameters that influence the hydrodynamic behavior of textured devices, recent literature has provided several surveys that examine the issue. In a 2009 article, Pascovici, et al. (16) performed an analytical investigation in order to study the effects of various geometrical parameters. They found that parameters such as the number of dimples, dimensionless textured length, or dimensionless dimple depth have an important influence on the load-carrying capacity and the friction coefficient of a partially textured slider. However, it should be noted that in order to avoid the treatment of cavitation, Pascovici, et al. (16) have chosen a texture configuration that starts with a dimple. A series of reviews authored by Rahmani,et al. (17), (18) provided an optimization technique for textured slider bearings that would deliver favorable values for loadcarrying capacity and friction force. In this case, the respective authors applied the so-called Reynolds boundary condition and set all negative pressures to zero. Similarly, Papadopoulos, et al. (19) extended an optimization technique for a three-dimensional study of a microthrust bearing, and Fu, et al. (20) used an analytical model to investigate a partially textured slider of infinite width with orientated parabolic grooves. While considering this scientific background, the authors propose a different approach to the investigation of textured bearings. The main objective of the current study is to perform a more appropriate parametrical investigation of a textured slider, in which the effects induced by cavitation are fully taken into account while ensuring mass flow conservation. The first part of the article focuses on presenting the algorithm used by the authors. The analytical formulation, which is based on JFO theory of separating the fluid film into two distinct regions, is presented in Eq. [1]. Its discretized form (using the finite volume method) is given in Eq. [9]. Next, in order to achieve model validation, several comparisons are made between the results provided by the algorithm and those found in the literature. It is proved that the model ensures mass conservation throughout the domain and provides accurate results even in discontinuous film profiles, thus constituting an appropriate instrument for the study of textures. The main part of this work is dedicated to a one-dimensional study of the geometrical features that influence the hydrodynamic performance of a textured parallel bearing. The most important parameters such as cell number, texture density, groove depth, and groove shape are examined in detail. The choice of a onedimensional investigation instead of a two-dimensional one was based on several foundations. First, there is the obvious gain in computation time, which offers the opportunity to expand the number of cases investigated. With regard to textures, this is an important advantage because of the numerous geometrical parameters that come into play. In addition, one-dimensional analyses allow a finer mesh grid to be employed, with a direct effect on the precision of the results. Once again, this is an important advantage, especially when treating discontinuous film profiles, as in the case of rectangular-shaped textures. MODELING APPROACH A finite element version of the algorithm employed in this article was published in 2007 by Hajjam and Bonneau (21), who

4 Hydrodynamic Lubrication of Textured Parallel Bearings 323 primarily employed it to optimize the design of radial lip seals. In 2011, Fatu, et al. (22) adapted and used the model to study the influence of wall slip on the load-carrying capacity and power loss in hydrodynamic journal bearings. The presented formulation is equally appropriate, as will be demonstrated in the current study, for the treatment of complex problems such as the lubrication of textured devices. The algorithm used in this investigation can be easily adapted to different discretization methods, such as the finite differences method, the finite volume method, or the finite element method. It is worth mentioning that all three methods lead to similar results. Due to its more adaptable implementation, the finite volume formulation is presented in this study. Validity of the Reynolds Equation The Reynolds equation is commonly used to analyze the hydrodynamic performance of surface textures. However, Dobrica and Fillon (23) proved that when analyzing the hydrodynamic behavior of textured surfaces, certain elements such as the effects of convective inertia may influence and limit the use of the Reynolds equation. Distinct reference was made to the Reynolds number, Re, and the texture aspect ratio, λ, which has been defined as the ratio between the dimple length, l d, and the dimple depth, h d. Their conclusion stated that the Reynolds equation should only be applied in the analysis of textured sliders as long as λ is sufficiently large and Re is sufficiently small, values for which the inertia effects are negligible. Throughout the entire investigation, the evolution of the Reynolds number and the texture aspect ratio was checked systematically. In all analyses, Re did not surpass 0.1, and the minimum value of λ was 400. Analytical Formulation The analytical formulation employed in this study was introduced by Hajjam and Bonneau (21) and consists of the use of a modified version of the Reynolds equation, which can be applied throughout the entire domain, both in the active and the nonactive regions: { ( d h 3 dx 12η )} dd F = U dh dx 2 dx + { U 2 } dd (1 F) [1] dx The equation utilizes a universal function D and a so-called index function F, both of which satisfy the following conditions: { D = p p cav Intheactiveregion,, F = 1 where p cav represents the cavitation pressure, which for simplification reasons is considered constant. For the case where F = 1, Eq. [1] degenerates into a classical form of the Reynolds equation for incompressible fluids: ( d h 3 ) d(p p cav ) = U dh [2] dx 12η dx 2 dx In the nonactive region, { D = r h,where r = ρ h F = 0 ρ 0 In this situation, for F = 0, Eq. [1] becomes an expression of mass conservation: U dh 2 dx + U d(r h) = 0 [3] 2 dx Fluid pressure is superior to the cavitation pressure at all times, and the so-called filling factor r (describing the ratio between the density of the lubricant, ρ 0, and the density of the mixture of fluid and gas, ρ) is always inferior to h, thus leading to a negative value of D. Therefore, a complementarity approach was developed based on the following inequalities: { D 0 in the active region D < 0 in the nonactive region These inequalities are particularly important when establishing the transition boundaries between the active and nonactive regions. Essentially, the D = 0 condition is present in both situations, even if the respective boundary marks the breakdown or re-formation of the lubricating film. Furthermore, the same condition is automatically taken into account by integrating the discretized expressions (which are presented in the next section) into the cavitation algorithm. Boundary Conditions The boundary conditions used for the resolution of the modified Reynolds equation employed in this study are based on the JFO theory of separating the fluid film into two distinct regions, according to the pressurization regime of the lubricant. Therefore, on the outside boundaries of the domain, for x = 0andx = B the lubricant pressure p is set to the atmospheric pressure p atm and subsequently: D x=0 = D x=b = p atm [4] The so-called Reynolds conditions are applied on the boundaries of the nonactive region. Therefore, on the film breakdown/cavitation boundary, pressure is set to the cavitation pressure, and the pressure gradient is null: p = p cav ; dp dx = 0 [5] On the re-formation boundary, the only necessary condition applies to the pressure p, which is also set to the cavitation pressure p cav : p = p cav [6] In addition, mass conservation must be ensured on the boundaries of the nonactive region: h 3 dp + (r h) = 0 [7] 6ηU dx On the breakdown/cavitation boundary, because the pressure gradient is null, Eq. [7] degenerates into the following expression: r h = 0 [8]

5 324 A. R. GHERCA ET AL. Fig. 1 Single-grooved parallel bearing, as defined by Fowell, et al. (9). Finite Volume Solution By using the finite volume method, Eq. [1] can be rewritten in discretized form as follows: A P D P + A E D E + A W D W + S c = 0, [9] where ( ) A P = 1 h 3 i 1/2 + h3 i+1/2 F(i) U [1 F(i)], 12η x 2 2 x A E = 1 h 3 i+1/2 F(i + 1) 12η x2 A W = 1 h 3 i 1/2 U F(i 1) + [1 F(i)], 12η x2 2 x S c = U (h i h i 1 ) 2 x This results in a tridiagonal system, which can be solved by different iterative methods, such as Gauss-Seidel, Jacobi, or the successive overrelaxation method. For this investigation, the tridiagonal matrix algorithm, also known as the Thomas algorithm, is employed. This is a direct resolution method, based on a simplified form of Gaussian elimination. In association with the complementarity approach, the algorithm produces a fast and precise resolution of the problem. Model Validation In order to validate the model, specific comparisons to the analytical study by Fowell, et al. (9) were made. Figure 1 illustrates the geometry of a single-grooved bearing, and the input data used TABLE 1 MAIN BEARING PARAMETERS Bearing breadth B 20 mm Inlet land breadth a 4mm Pocket breadth b 6mm Exit land breadth c 10 mm Speed of lower surface U 1,000 mm/s Lubricant viscosity η 10 8 MPa s Atmospheric pressure p atm 0.1 MPa Cavitation pressure p cav 0MPa Minimum film thickness h 0 1 μm Pocket depth h d 5 μm Pocket film thickness (h 0 + h d ) h p 6 μm for the analysis are presented in Table 1. The geometry proposed by Fowell, et al. (9) is a challenge for any numerical model for several reasons. Primarily, the divergence of the film thickness profile produces a cavitation problem. In this case, the difficulty rests in locating the boundaries of the cavitation region while ensuring mass conservation. Secondly, the rectangular shape of the groove results in a discontinuous film profile, which poses an important problem from the numerical standpoint. It is known that in the case of such an abrupt variation in the film thickness, discretization and resolution of the Reynolds equation have to be performed carefully (Vijayaraghavan and Keith (15); Cioc and Keith (24)). The discretization technique employed in this investigation was described in detail by Arghir, et al. (25). As a starting point, an analysis of the pressure distribution corresponding to the single-grooved bearing (Fig. 1) is presented in Fig. 2a. The diagram shows a comparison between the pressure profiles produced by the numerical algorithm employed in this article (for 50 and 400 nodes) and the one obtained using the analytical model of Fowell, et al. (9). In addition to the influence of grid size on the precision of the model, Table 2 presents the difference between the results in terms of maximum pressure, load, and friction force. Due to the small gap between the values, the differences are better highlighted if expressed in terms of percentages, with the analytical results taken as a reference. To better explain how the proposed model works, Fig. 2b illustrates the evolution of the ρ/ρ 0 ratio (describing the ratio between the density of the lubricant, ρ 0, and the density of the mixture of fluid and gas, ρ) corresponding to the single-grooved bearing presented in Fig. 1. This shows that in the active region, the lubricant maintains its density at the default value (the case in which ρ/ρ 0 = 1), whereas in the nonactive region, the lubricant transforms into a mixture of fluid and gas, with a density that in this case is approximately five times lower than in the active region. Overall, the pressure profiles (Fig. 2a) corresponding to the single-grooved bearing (Fig. 1) show small discrepancies. The differences in terms of maximum pressure and load presented in Table 2 are also small, even for a coarse grid. The load-carrying capacity is computed through the integration of the pressure field using: W = B 0 [p (p atm p cav )] dx [10] This expression is valid even for negative values of p cav and p atm. It should also be noted that if the p atm = p cav condition were applied, the suction effect would not be present and no load support would be generated in the bearing. This was explained in detail by Wang, et al. (14). In this study, the values associated with the TABLE 2 RELATIVE DIFFERENCE BETWEEN THE NUMERICAL SOLUTION AND THE ANALYTICAL RESULTS OF FOWELL, ET AL. (9) Grid Size ,000 4,000 10,000 Maximum pressure (%) Load-carrying capacity (%) Friction force (%)

6 Hydrodynamic Lubrication of Textured Parallel Bearings 325 Fig. 2 (a) Pressure fields for a single-grooved bearing (comparison between Fowell, et al. s (9) analytical solution and the finite volume solution for 50 and 400 nodes) and (b) corresponding ρ/ρ 0 ratio (color figure available online). atmospheric pressure, p atm, and cavitation pressure, p cav, were set to constant values of 0.1 and 0 MPa, respectively. In the case of the friction force analysis, the results provided by the numerical model were slightly higher than those proposed by Fowell, et al. (9), even for a more refined mesh grid. This was due to the fact that in the nonactive region, where Fowell, et al. (9) considered the friction force to be null, a more accurate formulation of the friction force is employed. From a technical point of view, although considerably reduced, friction still exists in the cavitation area due to the presence of a mixture of liquid and gas. Though the Poiseuille term has a minimal influence in the cavitation region, the Couette term should be regulated by the ρ/ρ 0 ratio. Consequently, in the current investigation, the following expression is applied throughout the entire domain: F f = B 0 [ ( Uη F h + h 2 ) ( dp r Uη (1 F) dx h h + h 2 )] dp dx [11] dx where F represents the same index function used in Eq. [1]. Thus, for F = 1 corresponding to the active region, the resulting formulation is similar to the one ordinarily used to compute friction in

7 326 A. R. GHERCA ET AL. Fig. 3 (a) Elementary texture cell: 1, cell inlet land; 2, dimple; 3, cell exit land; and (b) partially textured bearing geometry. a pressurized fluid film: F f = B 0 [ ( Uη h + h )] dp dx [12] 2 dx In the nonactive region, where F = 0, Eq. [11] degenerates into F f = B 0 [ ( r Uη h h + h )] dp dx [13] 2 dx where the term r h transposes into ρ ρ 0. This formulation results in a more precise expression of the friction force in the cavitation zone. RESULTS AND DISCUSSION This section presents a detailed examination of the influence of several geometrical characteristics on the hydrodynamic performances (pressure, load-carrying capacity, and friction force) of a textured parallel bearing. The most important parameters analyzed are the texture cell number, dimple depth, texture density, and dimple shape. In the current investigation, it is considered that a texture represents a symmetrical set of geometrical features, as illustrated in Fig. 3b. Consequently, the texture is divided into elementary components. Figure 3a presents the main component of a texture, which is the texture cell. During the entire investigation, a certain number of parameters such as the bearing breadth, B; land film thickness, h 0 ;and cell length, l c, are kept constant. At the same time, in an attempt to understand the influence that they exert on the hydrodynamic parameters, the cell number, N, and texture density, ρ t, along with the groove shape; depth, h d ; and length, l c, are all modified according to the data presented in Table 3. Dependence between Geometry and Grid Size It should be noted that a more refined mesh grid naturally allows a more accurate definition of the texture geometry. For a given grid size, an increase in cell number may complicate the texture geometry, especially by the introduction of film profile Fig. 4 Impact of grid size on the precision of the algorithm (relative error for load) for N = 1, 2, 5, 10 (color figure available online).

8 Hydrodynamic Lubrication of Textured Parallel Bearings 327 TABLE 3 MAIN GEOMETRICAL PARAMETERS Bearing features Breadth B 20 mm Land film thickness h 0 1 μm Cell features Length l c 2mm Cell number N 1 10 Density ρ t = l d / l c 0.25, 0.5, 0.75 Dimple features Shape,, Depth h d 2.5, 5.0, 7.5 μm Length l d 0.5, 1.0, 1.5 mm Fig. 5 a) Maximum pressure, (b) load, and (c) friction force versus cell number for various dimple depths (color figure available online). discontinuities, thus affecting the precision of the results. In order to assess this dependence, four patterns were analyzed. The grid size was gradually increased from 200 up to 10,000 nodes and the load-carrying capacity was evaluated. The 10,000-point results were used as a reference and the relative error was calculated. Figure 4 shows the evolution of this error with respect to the mesh grid for different cell numbers. The results indicate that after reaching a certain grid size, the relative error decreased significantly and the number of cells became irrelevant. During the texture investigation, the cell number was frequently varied, so in order to ensure an acceptable precision of the results, a sufficiently fine mesh grid consisting of 4,000 nodes was used at all times. Influence of Cell Number and Dimple Depth With regard to cell number, it was found that there are two possible techniques of distributing the texture cells on the surface of the bearing. The first method was presented by Fowell, et al. (9) and consists of replacing a single groove by multiple dimples of smaller width while maintaining the same dimensions for the Fig. 6 Pressure profiles for a five-cell texture with different dimple depths (color figure available online).

9 328 A. R. GHERCA ET AL. Fig. 7 (a) Pressure fields for different texture configurations and (b) ρ/ρ 0 ratio for two cells and (c) eight cells (color figure available online).

10 Hydrodynamic Lubrication of Textured Parallel Bearings 329 Fig. 8 Dependence of the Poiseuille flow on the dimple depth for a two-cell texture. inlet and exit lands. This method was shown to provide the same maximum pressure regardless of the cell number and had minimal impact on load support. The other method, which is analyzed in this article, entails a successive addition of symmetric texture cells on the surface of the bearing. The inlet and outlet lands of the bearing are therefore directly modified, with immediate impact on friction and load. During the investigation, it was found that the influence of cell number was highly dependent on the depth of the grooves and vice versa. Therefore, these two geometrical parameters are analyzed in the same section. For a bearing breadth, B, of 20 mm and a cell length, l c,of 2 mm, the bearing becomes completely textured for a maximum cell number, N, of 10. For all possible values of N, three different dimple depths, h d, were analyzed: 2.5, 5.0, and 7.5 μm. The maximum pressure, load-carrying capacity, and friction force were calculated for each case and are plotted in Fig. 5. This permits a parallel observation of these three hydrodynamic parameters with regard to cell number and dimple depth. Figure 5a illustrates the maximum pressure values obtained for each case and with reference to the cell number, N. The results Fig. 9 Influence of texture density on the pressure profile for N = 5andh d = 5 μm (color figure available online).

11 330 A. R. GHERCA ET AL. Fig. 10 (a) Geometry of a single-grooved bearing for various shapes and (b) the influence of groove shape on load and (c) friction force for various inlet lengths a (color figure available online). indicate that when using a lower number of cells, lower dimple depths provided higher maximum pressures. However, when N was increased, the peak pressure decreased linearly and the dimple depth became less significant. Although different texture configurations of equal depth can provide the same maximum pressure, the pressure profiles can differ greatly. This is illustrated in Fig. 6, which offers the pressure distribution corresponding to a five-cell texture for three values of groove depth. The load-carrying capacity is plotted against the cell number in Fig. 5b. The results show that for a lower number of cells, shallower dimples provided a higher load. However, when the number of cells was increased, the deeper dimples provided enhanced load support. This apparent contradiction can be explained by the data presented in Fig. 7. For the two-cell configuration, Fig. 7a shows that the shallower texture (h d = 2.5 μm) produced an increased pressure buildup compared to the deeper dimples (h d = 7.5 μm). In contrast, for the eight-cell situation, the improved load support was produced by the deeper dimples. This effect may be explained by the influence of cavitation. Figure 7b presents the evolution of the ρ/ρ 0 ratio for the two-cell textures. On the one hand, for the shallower dimples, cavitation was present in the first pocket and only partially in the second one. On the other hand, for the deeper texture, cavitation did not occur at all and the ρ/ρ 0 ratio remained constant along the entire breadth of the bearing. This suggests that in this particular case, cavitation helped draw fluid into the bearing, with beneficial effects on load support. Due to the influence of the pressure gradient, the suction effect was more clearly displayed in the case of the Poiseuille component of the flow rate (Fig. 8). The results show that for the two-cell textures, increasing the dimple depth significantly reduced the Poiseuille term. Thus, for a shallow texture (h d = 2.5 μm), the Poiseuille flow was approximately two times greater than in the case of a deeper one (h d = 7.5 μm). In the case of the eight-cell texture, the effect was reversed and a superior pressure buildup was obtained in the case of deeper dimples (Fig. 7a). Figure 7c illustrates the ρ/ρ 0 ratio for this case, and again it is shown that the shallower dimples cavitated more easily. This is better observed on the seventh and eighth cell of the textures. In this case, cavitation was present along the entire length of the bearing and lost its suction effect, thus becoming detrimental with regard to load support. In comparison, deeper cells better restricted the flow and provided a slightly superior pressure buildup. Figure 5c shows a plot of the friction force versus cell number for several dimple depths. Contrary to the previous diagrams, this case presents a clearer tendency, as the friction force was diminished with an increase in cell number and dimple depth. This was predictable, because increasing N or h d has the same effect of increasing the mean gap between the two surfaces. In terms of cell number and dimple depth, the diagrams presented in Fig. 5 can be used to choose an optimum texture configuration with regard to the hydrodynamic performance of the bearing. For instance, if the objective were to increase the loadcarrying capacity, regardless of the friction force, then the onecell configuration, with a 2.5-μm depth, would be very effective. However, if a minimal friction force were preferred, then the 10- cell configuration, with a 7.5-μm depth, should be applied to the bearing. The 5-cell texture could also be employed if one were searching for an optimum ratio between the friction force and the load-carrying capacity that is generated in the bearing. The following sections of the article offer similar possibilities for identifying the optimum texture configuration in terms of texture density and dimple shape. Texture Density The term texture density, ρ t, was previously defined by Dobrica and Fillon (23) as the ratio between the dimple length, l d,

12 Hydrodynamic Lubrication of Textured Parallel Bearings 331 Fig. 11 Influence of texture shape on (a) load and (b) friction force for N = 1,..., 10 (color figure available online). and the texture cell length, l c. A common value of the texture density is 50%, and this value was used in all of the analyses presented thus far. In this section, two additional values of density are analyzed: 25 and 75%. It should be noted that for a density of 100%, the texture becomes a Rayleigh step, which is not the subject of this investigation. Figure 9 illustrates the effect of density modification on the pressure profile for N = 5andh d = 5 μm. Overall, a single tendency was observed in all of the analyses of texture density, regardless of the dimple depth or cell number: increasing the texture density enhanced the load-supporting effect and reduced the friction force. A possible explanation is that when modifying the density of a texture, the inlet and exit lands of the bearing also change, with a direct impact on the pressure profile. The analytical study by Fowell, et al. (9) showed that the inlet and exit lands of single-grooved bearings are important geometrical features with a great influence on load and friction. The same statement is equally accurate for textured bearings, where the inlet of the first cell and the outlet of the last cell become the inlet and the exit lands of the bearing. Texture Shape In this section, three texture shapes are analyzed: rectangular, triangular, and parabolic (Fig. 10a). It should be noted that in these geometrical conditions it is difficult to apply a circular shape, because the dimple depth, h d, is much smaller than the dimple depth, l d. First, the influence of shape is studied for a simpler geometry, that of the single-grooved bearing. Overall, it was found that the influence of shape is strongly dependent on the inlet length. In terms of load (Fig. 10b), the results show that for a small inlet length a, the rectangular shape of the groove proved to

13 332 A. R. GHERCA ET AL. be much more efficient. However, after a certain limit, the shape ceased to have an impact. This limit most certainly corresponds to a geometrical configuration in which cavitation occurs in the pocket. With regard to the friction force (Fig. 10b), the rectangular grooves offered slightly higher values of friction, but this was only true for small values of inlet length. Because cavitation occurs after increasing the inlet length, the behavior is reversed. Thus, for higher values of a, the rectangular groove offers lower values of friction force. Next, the same three shapes were applied to textures. As seen in the case of the single-groove bearing, cavitation plays an important role in the multigrooved configuration. In this case, the dimple shape was also dependent on the cell number. In terms of load support (Fig. 11a), when N = 1, the triangular texture provided better results, but this tendency was reversed when N was increased. With regard to the friction force, Fig. 11b shows that the influence of texture shape was less significant. CONCLUSIONS The first part of this article focused on presenting the algorithm used for analyzing textured surfaces. In order to validate the model, several comparisons to an analytical model were made. It was shown that while assuring mass conservation, the model successfully handles important problems such as cavitation or film thickness discontinuities. The main part of the article was dedicated to a geometrical investigation of surface textures. The following conclusions summarize the results of this investigation: 1. Depending on the number of cells, the dimple depth can yield contradictory effects on the resulting pressure profile. This tendency was most clearly observed in the load support analysis, which showed that for a reduced number of cells, higher loads were provided by the most superficial dimples, whereas for greater cell numbers, the improvement was noted for deeper dimples. With regard to friction, the results showed a general trend: the friction force diminished with an increase in cell number and an increase in dimple depth. 2. Overall, the results showed that an increase in texture density enhanced the load support and successfully diminished the friction force, regardless of the cell number or groove depth. 3. Three pocket shapes were analyzed: rectangular, triangular, and parabolic. It was revealed that the effect of the groove shape was strongly related to those of other geometrical parameters such as the inlet length in the case of a single-grooved bearing or the cell number in the case of textures. In the case of single-grooved bearings, the rectangular shape yielded remarkable improvements in terms of load, whereas in the case of textures, the impact of dimple shape proved to be less important. REFERENCES (1) Etsion, I. (2005), State of the Art in Laser Surface Texturing, Journal of Tribology, 127, pp (2) Hamilton, D. B., Walowit, J. A., and Allen, C. M. (1966), A Theory of Lubrication by Microirregularities, Journal of Basic Engineering, 88, pp (3) Etsion, I. and Kligerman, Y. (1999), Analytical and Experimental Investigation of Laser Textured Mechanical Seal Faces, Tribology Transactions, 42, pp (4) Etsion, I. and Sher, E. (2009), Improving Fuel Efficiency with Laser Surface Textured Piston Rings, Tribology International, 42, pp (5) Brizmer, V., Kligerman, Y., and Etsion, I. (2003), A Laser Surface Textured Parallel Thrust Bearing, Tribology Transactions, 46(3), pp (6) Tønder, K. (1996), Dynamics of Rough Slider Bearings: Effects of One- Sided Roughness/Waviness, Tribology International, 29, pp (7) Tønder, K. (2001), Inlet Roughness Tribodevices: Dynamic Coefficients and Leakage, Tribology International, 34, pp (8) Tønder, K. (2004), Hydrodynamic Effects of Tailored Inlet Roughness: Extended Theory, Tribology International, 37, pp (9) Fowell, M., Olver, A. V., Gosman, A. D., Spikes, H. A., and Pegg, I. (2007), Entrainment and Inlet Suction: Two Mechanisms of Hydrodynamic Lubrication in Textured Bearings, Journal of Tribology, 129, pp (10) Jakobsson, B. and Floberg, L. (1957), The Finite Journal Bearing Considering Vaporization, Chalmers Tekniska Hoegskola och Goteborgs Universitet, Geologiska Institutionen, 190, pp (11) Olsson, K. O. (1965), Cavitation in Dynamically Loaded Bearing, Chalmers Tekniska Hoegskola och Goteborgs Universitet, Geologiska Institutionen, 308, pp (12) Elrod, H. G. (1981), A Cavitation Algorithm, Journal of Lubrication Technology, 103, pp (13) Ausas, R., Ragot, P., Leiva, J., Jai, M., Bayada, G., and Buscaglia, G. C. (2007), The Impact of the Cavitation Model in the Analysis of Microtextured Lubricated Journal Bearings, Journal of Tribology, 129, pp (14) Wang, H., Yang, S., and Guo, F. (2011), Modeling of a Grooved Parallel Bearing with a Mass-Conserving Cavitation Algorithm, Tribology Transactions, 54, pp (15) Vijayaraghavan, D. and Keith, T. G. (1989), Development and Evaluation of a Cavitation Algorithm, Tribology Transactions, 32(2), pp (16) Pascovici, M. D., Cicone, T., Fillon, M., and Dobrica, M. (2009), Analytical Investigation of a Partially Textured Parallel Slider, Proceedings of the Institution of Mechanical Engineers - Part J: Journal of Engineering Tribology, 223(2), pp (17) Rahmani, R., Shirvani, A., and Shirvani, H. (2007), Optimization of Partially Textured Parallel Thrust Bearings with Square-Shaped Micro- Dimples, Tribology Transactions, 50, pp (18) Rahmani, R., Mirzaee, I., Shirvani, A., and Shirvani, H. (2010), An Analytical Approach for Analysis and Optimisation of Slider Bearings with Infinite Width Parallel Textures, Tribology International, 43, pp (19) Papadopoulos, C. I., Efstathiou, E. E., Nikolakopoulos, P. G., and Kaiktsis, L. (2011), Geometry Optimization of Textured Three- Dimensional Micro-Thrust Bearings, Journal of Tribology, 133, pp (20) Fu, Y., Ji, J., and Bi, Q. (2012), The Influence of Partially Textured Slider with Oriented Parabolic Grooves on the Behavior of Hydrodynamic Lubrication, Tribology Transactions, 55, pp (21) Hajjam, M. and Bonneau, D. (2007), A Transient Finite Element Cavitation Algorithm with Application to Radial Lip Seals, Tribology International, 40, pp (22) Fatu, A., Maspeyrot, P., and Hajjam, M. (2011), Wall Slip Effects in (Elasto) Hydrodynamic Journal Bearings, Tribology International, 44,pp (23) Dobrica, M. B. and Fillon, M. (2009), About the Validity of Reynolds Equation and Inertia Effects in Textured Slider of Infinite Width, Proceedings of the Institution of Mechanical Engineers - Part J: Journal of Engineering Tribology, 223, pp (24) Cioc, S. and Keith, T. G. (2002), Application of the CE/SE Method to One-Dimensional Flow in Fluid Film Bearings, Tribology Transactions, 45(2), pp (25) Arghir, M., Alsayed, A., and Nicolas, D. (2002), The Finite Volume Solution of the Reynolds Equation of Lubrication with Film Discontinuities, International Journal of Mechanical Sciences, 44, pp