CFD Modeling of the Effect of Different Surface Texturing Geometries on the Frictional Behavior

Transcription

1 lubricants Article CFD Modeling the Effect Different Surface Texturing Geometries on the Frictional Behavior Luís Vilhena 1, * ID, Marko Sedlaček 2, Bojan Podgornik 2, Zlatko Rek 3 and Iztok Žun 1 CEMMPRE, Centre for Mechanical Engineering, Materials and Processes, University Coimbra, Rua Luís Reis Santos, Coimbra, Portugal 2 IMT, Institute Metals and Technology, Lepi pot 11, SI-1000 Ljubljana, Slovenia; Marko.Sedlacek@imt.si (M.S.); bojan.podgornik@imt.si (B.P.) 3 Faculty Mechanical Engineering, University Ljubljana, SLO 1000 Ljubljana, Slovenia; zlatko.rek@fs.uni-lj.si * Correspondence: luisvilhena@gmail.com; Tel.: This author has been passed away. Received: 5 December 2017; Accepted: 22 January 2018; Published: 26 January 2018 Abstract: In order to understand the effect surface texturing parameters on the frictional behavior textured surfaces and to correlate results different lubrication regimes, Computational Fluid Dynamics (CFD) numerical analysis the fluid flow was performed for four different textured surface geometries. The aim the present research paper is to get theoretical background for the frictional behavior textured surfaces under hydrodynamic lubrication. Since it is unrealistic to make a direct analysis a real problem that can possess more than several thousand micro-dimples, the purpose is then to investigate the flow in single cells periodical micro-dimple patterns and to extract useful conclusions for the lubrication s framework. Among all geometries studied, optimum geometry shapes in terms hydrodynamic performance were reported. It was found that the best hydrodynamic performance was achieved with the rectangular geometry (lowest shear force). Keywords: modeling; CFD; surface texturing; lubrication mode; Reynolds equation 1. Introduction 1.1. Lubrication Theory The purpose lubrication is to generate a separating force between two closely spaced surfaces separated by a fluid in relative motion in order to reduce friction and wear. The separation and load-carrying capacity are achieved by generating a pressure in the fluid film between the surfaces [1]. For a good understanding lubrication theory, three different regimes must be considered: boundary lubrication, mixed lubrication and hydrodynamic lubrication. If a full lubricating film is separating the surfaces and there are no asperities in contact, this regime is denoted as hydrodynamic or full-film. When hydrodynamic lubrication occurs, full film lubrication problems can be analyzed using the Reynolds equation to model the lubricant pressure and fluid flows, and the interaction between the lubricant and surfaces. However, due to an increasing availability user-friendly, commercial CFD codes, there has been a growing application the full Navier-Stokes equations to solve some lubrication problems [2]. If the amount asperities in contact increases, the lubrication regime is denoted as mixed. In a mixed lubrication regime, modeling is more complicated since a full lubricating film does not exist and at the same time, only a part the surface is in contact. Finally, when the load is carried by the asperities in contact, this is called a boundary lubrication regime. Thus, surface contact becomes the major part the load-supporting mechanisms, and both plastic and elastic deformations occur. Lubricants 2018, 6, 15; doi: /lubricants

2 Lubricants 2018, 6, 15 x FOR PEER REVIEW models on lubricant chemistry and contact mechanics do exist and can help to understand this Some lubrication models regime. on lubricant Therefore, chemistry contact and under contact starved mechanics lubrication do exist can and be simulated can help to using understand a Finite this Element lubrication Model regime. (FEM). Therefore, results contact obtained under are starved valid for lubrication a dry sliding can becontact, simulated but using they amay Finite be Element extended Model to boundary-lubricated (FEM). The resultscontacts obtainedwith are a valid small for film a dry thickness sliding [3]. contact, For large but film they thickness, may be extended FEM analysis to boundary-lubricated is not valid since hydrodynamic witheffects a small would film be thickness dominant. [3]. For large film thickness, FEM The analysis three ismain not valid lubrication since hydrodynamic regimes are illustrated effects would schematically be dominant. Figure 1 in a typical Stribeck curve, The that three shows main the lubrication evolution regimes the coefficient are illustrated friction schematically μ with the in Figure non-dimensional 1 in a typical parameter Stribeck curve, H that that is shows defined the as evolution ηu/p, in which the η coefficient is the lubricant friction dynamic µ with viscosity, the non-dimensional u is the velocity parameter the H contacting that is defined surfaces as and ηu/p, p in is which the mean contact is the lubricant pressure. dynamic H is strongly viscosity, related u to isthe thefilm velocity thickness. the contacting surfaces and p is the mean contact pressure. H is strongly related to the film thickness. Figure 1. Typical Stribeck curve. Figure 1. Typical Stribeck curve. As As can can be be seen seen in in Figure Figure 1, 1, at at the the right right part part the the Stribeck Stribeck curve curve for for high high speeds, speeds, and/or and/or lower lower loads, loads, and/or and/or consistent consistent film film viscosity viscosity conditions, conditions, the region the could region be associated could be with associated a hydrodynamic with a regime hydrodynamic where friction regime increases where friction almost linearly increases with almost H. It is linearly thus realistic with H. toit assume is thus that realistic lubricant to assume film is thick that lubricant enough tilm fully is thick support enough normal to fully pressure support during normal sliding. pressure With diminishing during sliding. H, friction With diminishing coefficient reaches H, friction a minimum. coefficient The reaches lubricant a minimum. film thickness The lubricant is reduced, film revealing thickness the is reduced, limit the revealing hydrodynamic the limit lubrication the hydrodynamic regime, andlubrication thus the beginning regime, and thethus mixed the lubrication beginning regime. the mixed Consequently, lubrication for lower regime. H, the Consequently, surfaces in contact for lower are supported H, the surfaces on a combination contact are asperities supported and/or on a fluid. combination The left-hand asperities section and/or the Stribeck fluid. The curve left-hand is associated section with the the Stribeck boundarycurve lubrication is associated regimewith where the friction boundary coefficient lubrication rises up regime to thewhere quasi-static friction values coefficient in correspondence rises up to the critical quasi-static pressure values and/or in correspondence vanishing sliding speed. critical pressure and/or vanishing sliding speed Modelling Textured Surfaces 1.2. Modelling Recent publications Textured Surfaces [1,4 13] have shown that the pressure-generating effect textured surfaces that operate in full film lubrication conditions may be the result three different mechanisms: Recent publications [1,4 13] have shown that the pressure-generating effect textured surfaces micro-cavitation [5 8,12,13] that happens when the pressure in a fluid suddenly drops due to an that operate in full film lubrication conditions may be the result three different mechanisms: microcavitation [5 8,12,13] that happens when the pressure in a fluid suddenly drops due to an enlargement the flow section, convective inertia [1,9] that consists the inertial force due to the convective acceleration in a flow, and piezo viscosity [10,11], a pressure dependence viscosity. enlargement the flow section, convective inertia [1,9] that consists the inertial force due to the However, from all these mechanisms that happen in textured surfaces, it seems that micro-cavitation convective acceleration in a flow, and piezo viscosity [10,11], a pressure dependence viscosity. can make a decisive contribution in order to understand the load-carrying capacity effect. However, from all these mechanisms that happen in textured surfaces, it seems that micro-cavitation Micro-cavitation can occur when the amplitude the pressure variation, induced by the textured can make a decisive contribution in order to understand the load-carrying capacity effect. surface, is larger than the nominal pressure level at the particular location. Numerical simulations can Micro-cavitation can occur when the amplitude the pressure variation, induced by the account for micro-cavitation phenomena and thus help elucidate this issue. By using the Reynolds textured surface, is larger than the nominal pressure level at the particular location. Numerical equation for an incompressible flow, the pressure variation in the surrounding area the dimple simulations can account for micro-cavitation phenomena and thus help elucidate this issue. By using should result in antisymmetry about a point located in a mid-plane (x = 0 and y = 0), since the the Reynolds equation for an incompressible flow, the pressure variation in the surrounding area pressure first decreases due to the channel height enlargement then increases under the influence the dimple should result in antisymmetry about a point located in a mid-plane (x = 0 and y = 0), since the diminishing transversal flow section at the right corner, as demonstrated by Sahlin et al. [1] where the pressure first decreases due to the channel height enlargement then increases under the influence they showed a comparison between Navier-Stokes and Stokes solutions for the pressure distribution the diminishing transversal flow section at the right corner, as demonstrated by Sahlin et al. [1] where they showed a comparison between Navier-Stokes and Stokes solutions for the pressure

3 Lubricants 2018, 6, x FOR PEER REVIEW 3 26 Lubricants 2018, 6, distribution on the upper smooth wall for a cylindrical geometry. The usual practice is then to consider that the incompressible fluid will cavitate and pressures lower than the cavitation threshold are on the discarded upper smooth in the calculation wall for a cylindrical the load-carrying geometry. capacity. The usual One practice thus obtains is then to a consider nonsymmetrical that the pressure incompressible distribution fluid will that cavitate leads and to a pressures lift effect lower that than is exemplified the cavitation in threshold Figure 2. are A discarded second model in the proposed calculationby Elrod the load-carrying and Adams capacity. [8] has as One main thus advantage obtains athat nonsymmetrical it enforces mass pressure conservation, distribution which that is leads not satisfied to a lift effect in the that Reynolds is exemplified model. inhowever Figure 2. the A second Reynolds model base proposed method by should Elrodbe and sufficient Adams to [8] illustrate has as main the advantage problem. In that order it enforces to include mass inertia conservation, effects, the which full is Navier-Stokes not satisfied inequations the Reynolds have model. to be used However instead the Reynolds the Reynolds base method equations should since be the sufficient Reynolds toequation illustratedoes the problem. not incorporate In order convective to include inertia forces. effects, the full Navier-Stokes equations have to be used instead the Reynolds equations since the Reynolds equation does not incorporate convective inertia forces. Figure2. 2. The The hydrodynamic effect idealized micro asperity [14] Simplificationsto to the Reynolds Equation The Reynolds equationis is a differential equation governing pressure distributionin in a thin fluid film. This equationis is frequently usedin fluid film lubrication andin three dimensions can be written in the form [15]: ( 3 3 h h 3 ) p p h p h h = 6 U + V 12( wh w0 ) x x + + ( h 3 ) ( p y y = 6 U h x y + (1) x η x y η y x + V h ) + 12(w y h w 0 ) (1) The The termson onthe left-handside side the Reynolds equation representthe the flowdue dueto to pressure gradients where p/ x p/ x and and p/ y p/ y are are the the pressure pressure gradients gradients acting acting respectively respectively along along the x the axis x and axis along and along the y the axis. y The axis. right-hand The right-hand side shows side shows the shear the shear induced induced flow by flow the by (top) the surface (top) surface sliding sliding with velocity with velocity components components U and U V and in Vthe in x the x and and y y directions, directions, respectively. The The second secondterm termon onthe right-hand side showsthe flows inducedby by normal (squeeze) motions the bounding surface where w0 0 is isthe velocityat at which the bottom the column moves up and wh h is isthe velocityat at whichthe top top the column moves moves up. up. The The film film thickness thickness is represented is represented by h by and h η and is the η lubricant is the lubricant viscosity. viscosity. Some simplifications to the Reynolds equation are required before it can be used in practical engineering Some simplifications applications [15]. to the Reynolds equation are required before it can be used in practical engineering Unidirectional applications Velocity [15]. Approximation (It is always possible to choose axes in such a way that one Unidirectional the velocitiesvelocity is equalapproximation to zero, i.e., V = (It 0) [15]. is always possible to choose axes in such a way that one Assuming the velocities thatis V equal = 0, Equation to zero, i.e., (1) can V = be 0) [15]. rewritten in a more simplified form: Assuming that V = 0, Equation ( (1) can be rewritten in a more simplified form: h 3 ) p 3 + ( h 3 ) p 3 = 6U h x h η p x y h η p y h x + 12(w h w 0 ) (2) + = 6U 12( wh w0 ) x x y y x + (2) Steady Film Thickness η Approximation η (Movement surfaces normal to the sliding velocity, known Steady as a Film squeeze Thickness film effect, Approximation is not considered) (Movement [15]. surfaces normal to the sliding velocity, known as a squeeze film effect, is not considered) [15].

4 Lubricants 2018, 6, It is always possible to assume that there is no vertical flow across the film, i.e., w h w 0 = 0. This assumption requires that the distance between the two surfaces remains constant during the operation. Assuming, however, that there is no vertical flow and w h w 0 = 0, Equation (2) can be written in the form: ( h 3 x η ) p + ( h 3 x y η ) p = 6U h y x Isoviscous approximation (This approach is known in the literature as the isoviscous model where the thermal effects in hydrodynamic films are neglected. However lubricant viscosity varies with thermal effects in hydrodynamic films and must be considered in a more elaborate and accurate analysis) [15]. For many engineering applications it is assumed that the lubricant viscosity is constant over the film, i.e., η = constant. Assuming that η = constant Equation (3) can further be simplified [15]: ( h 3 p ) + ( h 3 p ) = 6Uη h x x y y x where the terms on the left-hand side the Reynolds equation represent the flow due to pressure gradients and the right-hand side shows the shear induced flow by the (top) surface sliding with velocity U. Or in dimensionless coordinates: ( h 3 p ) + ( h 3 p ) = h (5) x x y y x This is the most common form the Reynolds equation used for solving engineering problems. In the area outside the dimples the Laplace equation is applied: 2 p x p y 2 = 0 (6) The Laplace equation is also applied on the discontinuity nodes because the film thickness is considered to be equal with the film thickness the nodes situated on nearby. The above equations are solved using the finite difference method Reynolds vs. Navier-Stokes Equation One recurrently used approximation to the governing equations for fluid flow in tribology is the Reynolds equation, which is used in thin fluid films [16]. For a film thickness that is the order or larger than the dimple depth (h m /h c 1), a large difference in pressure solution was found [4] between the Navier-Stokes and Reynolds equation. This is due to convective inertia. The magnitude the pressure variation is relatively small and will not lead to micro-cavitation. For a film thickness that is small compared to the dimple depth (h m /h c < 1) the contribution convective inertia becomes negligible. The amplitude the pressure is large and micro-cavitation is almost certain to occur. In this last case the Reynolds equation can lead to a good approximation. However if the walls contain some geometrical perturbations, the inertial term can significantly influence the flow and the Reynolds approximations are not obvious as showed by Sahlin et al. [1] Load-Carrying Capacity and Coefficient Friction Equations When the pressure distribution is integrated over the area the cell surface the corresponding load-carrying capacity the lubricating film can be calculated. Since including a cavitation model would increase the complexity and time consuming, a simple model (where pressures smaller than zero are forced to zero inside each iteration) was used. The real pressure occurring inside the contact is certainly higher than the pressure obtained without taking cavitation into account, but the difference is (3) (4)

5 Lubricants 2018, 6, extremely small [3]. The mean pressure would be practically the same using the model with or without cavitation [3]. As a consequence the load-carrying capacity is calculated using only the positive values the pressure: F = L L 0 0 p dx dy (7) By definition the coefficient friction is expressed as a ratio the friction and normal forces acting on the surface: µ = F f F where the friction force F f generated due to the shearing the lubricant is obtained by integrating the shear stress τ over the area the cell surface. F f = L L 0 0 (8) τ dx dy (9) As was performed with load, coefficient friction can also be expressed in a so-called normalized form. Therefore µ is defined entirely in terms other non-dimensional parameters [17]: µ = F f F = µ L h m (10) The aim the present research paper is to get theoretical background for the frictional behavior textured surfaces under hydrodynamic lubrication. 2. Materials and Methods 2.1. Computational Fluid Dynamic (CFD) Analysis Modeling using simplified the Reynolds equation showed that texturing provides pressure increase at the trailing edge the dimples, which then creates lift force and consequently leads to reduced friction. However, in order to investigate in detail the effect different texturing parameters and to get correlation with experimental results, computational fluid dynamic analysis (CFD) was carried out [18,19]. The present work was carried out by numerically integrating the full system Navier-Stokes equations for a laminar, incompressible and isothermal flow in macro-roughness cells. Contact under full film lubrication was modeled through computational fluid dynamics (CFD) using FLUENT v6.3. Simulation a fully wetted smooth plate sliding (u = m/s) over a stationary groove separated by 5 µm lubricating film was performed for 2D steady-state flow viscous incompressible fluid (density 940 kg/m 3, dynamic viscosity 1.06 kg/ms). In simulation, where periodicity in the direction movement was taken into account (Figures 3 and 4), a Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) pressure velocity coupling was used. The SIMPLE algorithm, used to solve a steady-state problem iteratively, consists the following steps [18,19]: (1) An approximation the velocity field was obtained by solving the momentum equation and then the pressure gradient term was calculated using the pressure distribution from the previous iteration or an initial guess; (2) The pressure equation was formulated and solved in order to obtain the new pressure distribution; (3) Velocities were corrected and a new set conservative fluxes was calculated. The convergence criteria was set to ε = 10 4.

6 (Table 1) and in y-direction set to 5 μm. The film thickness the fluid field was set to 5 μm as predicted by Hamrock and Dowson equation [7] for given contact conditions. In the present research work, four different textured geometries have been studied. The first geometry is the cylindrical geometry and can be seen in Figure 3. The other three geometries are: rectangular, triangular and wedge and are shown in Table 2. Lubricants 2018, 6, Lubricants 2018, 6, x FOR PEER Figure Figure REVIEW 3. Geometrical parameters for the cylindrical geometry. 3. Geometrical parameters for the cylindrical geometry Varied Parameters A standard fluid model (cylindrical geometry, depth = 22.8 μm, width = 128 μm, spacing = 500 μm and u = m/s; light grey color in Table 1) was used as a reference and then certain parameters have been changed in a systematic order to investigate their effect on pressure distribution, velocity field and fluid flow. The influence the cavity geometry (cylindrical, rectangular, triangular and wedge) in the flow characteristics was studied first, whereas the other reference conditions remain unchanged. A parametric study was performed by considering the cylindrical geometry and three different cavity depths, namely 8, 16, and 22.8 μm. The cavity width, spacing and sliding speed were kept constant within the reference conditions. The cavity width was also varied from 40 up to 128 μm and spacing from 150 up to 500 μm, whereas the other reference conditions remain unchanged. Finally, the effect different Figure 4. sliding Boundary speeds conditions (0.01, 0.05, on the fluid and domain. 0.2 m/s) was also studied. The cylindrical geometry was Figure used with 4. Boundary cavity depth conditions 22.8 on the μm, fluid cavity domain. width 128 μm and spacing 500 The equations are steady and solved in the x- and y-direction only. The Navier-Stokes and the continuity 2.5. Computational equations Grid can be written as follows: In Computational the x-direction, area was ( discretized with quadrilateral cells a maximum length 0.2 μm as shown in Figure 5. The discretization ρ u u 2nd order pressure and 2nd order with upwinding x + v u ) = p ( y x + 2 ) η u momentum as well as under-relaxation (pressure = 0.3, momentum x u y = 0.7) 2 (11) were applied. and in the y-direction, The continuity equation becomes: Figure 3 represents the elementary textured cell considered in this study, and its geometrical parameters. In the case cylindrical geometry the groove was described by a circular arc defined depth and width, which are given Figure in Table 5. Grid 1. with Thequadrilateral length in the cells. x-direction was defined by spacing (Table 1) and in y-direction set to 5 µm. The film thickness the fluid field was set to 5 µm as predicted by3. Hamrock Results anddiscussion Dowson equation [7] for given contact conditions Influence Cavity Geometry ( ρ u v x + v v ) = p ( y y + 2 ) η v x v y 2 Figure 6 shows streamlines and velocity vectors for the smooth surface. It can be seen that if the walls are parallel without a cavity, a simple Couette flow is to be found with an overall laminar flow the viscous fluid in the space between the two parallel surfaces. Streamlines for the four different groove geometries are shown in Figures 7a, 8a and 9a. It can be seen that the streamlines present the expected pattern with a recirculation zone located in the cavity. In the case square-shaped cavity, the disturbance in streamlines is the most obvious and the largest followed by wedge shape, where asymmetry in streamlines can also be observed due to asymmetry in geometry. The smallest effect on fluid flow and the smallest vortex can be observed for triangular and semicircular type cavities. Figures 7b, 8b and 9b show the velocity field and illustrate how the vortex formation influences the inward fluid with a consequent development a frontier (12) u x + v y = 0 (13) where u and v are the components the velocity vector, x and y are the coordinate directions, p is the pressure, ρ the mass density, and η the dynamic viscosity the fluid Geometrical Model

7 Lubricants 2018, 6, Table 1. Individually varied parameters (with the standard fluid model indicated in Bold ). Lubricants 2018, 6, x FOR PEER REVIEW 7 26 μm. μm. A Lubricants summary summary 2018, 6, x the the FOR PEER parameters parameters REVIEW that that are are being being varied, varied, entirely entirely independently independently from from one one 7 another, another, 26 are are listed listed in in Tables Tables 1 and and μm. A summary the parameters that are being varied, entirely independently from one another, are Table Table listed in Individually Individually Tables 1 and varied varied 2. parameters parameters (with (with the the standard standard fluid fluid model model indicated indicated in in Bold ). Bold ). Table 1. Individually varied parameters Cavity (with Depth the standard Cavity fluid Width model indicated Spacing Bold ). Sliding Speed Varied Parameters Geometry (μm) (μm) (μm) (μm) (μm) (μm) (m/s) (m/s) Cavity Depth Cavity Width Spacing Sliding Speed Varied Parameters Cylindrical Cylindrical Geometry (μm) (μm) (μm) (m/s) Study Study the the influence influence Rectangular Rectangular Cylindrical Study geometry the influence Triangular Rectangular geometry Wedge Triangular Study Study the the influence influence Cylindrical Cylindrical Wedge cavity cavity Study depth depth the influence Cylindrical Cylindrical Study Study the the cavity influence influence depth Cylindrical Cylindrical cavity cavity Study width width the influence Cylindrical Cylindrical cavity width Study the influence Cylindrical In the present research Study the work, influence four different Cylindrical textured 22.8 geometries have been 150 studied. The first spacing Cylindrical spacing Cylindrical geometry is the cylindrical geometry and can be seen in Figure 3. The other three geometries are: Cylindrical Cylindrical Study rectangular, triangular and Study the influence wedge the influence and are shown Cylindrical in Table Cylindrical sliding sliding speed speed Cylindrical Cylindrical Table 2. Schematic representation textured geometries. Table Table Table Schematic Schematic 2. representation representation textured textured geometries. geometries. Cylindrical Cylindrical Cylindrical Rectangular Rectangular Rectangular Rectangular Triangular Triangular Triangular Triangular Wedge Wedge Wedge Wedge Although the form the cavity can be described by a different number parameters depending on whether it is rectangular, triangular, cylindrical, or wedge shaped, at least the peak to bottom distance (cavity depth) can be always defined and used for comparison all shapes. Although Although the the form form the the cavity cavity can can be be described described by by a different different number number parameters parameters depending depending 2.3. Varied Parameters on on whether whether 2.4. Boundary it it is is rectangular, rectangular, Conditions triangular, triangular, cylindrical, cylindrical, or or wedge wedge shaped, shaped, at at least least the the peak peak to to bottom bottom A standard distance distance fluid model (cavity (cavity (cylindrical depth) depth) can can geometry, be be always always defined defined depth = and and 22.8 used used µm, for for width comparison comparison = 128 µm, all all spacing shapes. shapes. = 500 µm In Figure 4, the boundary conditions for the fluid domain are presented. The fluid is and u = m/s; light experiencing grey color no-slip in Table conditions 1) wasat used the walls. as a reference The upper wall andis then moving certain with a parameters velocity equal have to u = been changed in 2.4. a systematic Boundary Conditions m/s while order the lower to investigate wall is stationary their(u effect = 0). Periodic on pressure boundary distribution, conditions are velocity imposed field on the and fluid flow. The influence In In left Figure Figure and on the 4, 4, the the the right cavity boundary boundary boundaries geometry conditions conditions (edges (cylindrical, at high for for and the the low rectangular, fluid fluid x-coordinates) domain domain triangular and are are a presented. presented. symmetry and wedge) condition The The in fluid fluid is is is applied in the z-plane. Periodic boundary conditions indicate that full periodicity in terms field the flow characteristics experiencing wasno-slip studiedconditions first, whereas at the walls. the other The reference upper wall conditions is moving with remain a velocity unchanged. equal to u = m/s m/s values while while and the the fluxes lower lower are wall wall ensured. is is stationary stationary (u (u = 0). 0). Periodic Periodic boundary boundary conditions conditions are are imposed imposed on on the the A parametric study was performed by considering the cylindrical geometry and three different cavity left left and and on on the the right right boundaries boundaries (edges (edges at at high high and and low low x-coordinates) x-coordinates) and and a symmetry symmetry condition condition is is depths, namely applied applied 8, 16, and in in the the 22.8 z-plane. z-plane. µm. The Periodic Periodic cavity boundary boundary width, spacing conditions conditions and indicate indicate sliding that that speed full full periodicity periodicity were kept in in constant terms terms field field within the reference values values conditions. and and fluxes fluxes are are The ensured. ensured. cavity width was also varied from 40 up to 128 µm and spacing from 150 up to 500 µm, whereas the other reference conditions remain unchanged. Finally, the effect

8 Lubricants 2018, 6, Lubricants 2018, 6, x FOR PEER REVIEW 8 26 different sliding speeds (0.01, 0.05, and 0.2 m/s) was also studied. The cylindrical geometry was used with cavity depth 22.8 µm, cavity width 128 µm and spacing 500 µm. A summary the parameters that are being varied, entirely independently from one another, are listed in Tables 1 and 2. Although the form the cavity can be described by a different number parameters depending on whether it is rectangular, triangular, cylindrical, or wedge shaped, at least the peak to bottom distance (cavity depth) can be always defined and used for comparison all shapes Boundary Conditions In Figure 4, the boundary conditions for the fluid domain are presented. The fluid is experiencing no-slip conditions at the walls. The upper wall is moving with a velocity equal to u = m/s while the lower wall is stationary (u = 0). Periodic boundary conditions are imposed on the left and on the right boundaries (edges at high and low x-coordinates) and a symmetry condition is applied in the z-plane. Periodic boundary conditions indicate that full periodicity in terms field values and fluxes are ensured. Figure 4. Boundary conditions on the fluid domain Computational Grid Computational area was discretized with quadrilateral cells a maximum length 0.2 µm μm as shown in Figure 5. The discretization 2nd order pressure and 2nd order with upwinding momentum as well as under-relaxation (pressure = 0.3, momentum = 0.7) were applied. 3. Results and Discussion 3.1. Influence Cavity Geometry Figure 5. Grid with quadrilateral cells. Figure 6 shows streamlines and velocity vectors for the smooth surface. It can be seen that if the walls are parallel without a cavity, aa simple Couette flow is is to to be be found with with an an overall laminar flow flow the the viscous fluid fluid in the in the space space between the the two two parallel parallel surfaces. Streamlines for the four different groove geometries are shown in Figures 7a, 8a and 9a. It can be seen that the streamlines present the expected pattern with a recirculation zone located in the cavity. In the case square-shaped cavity, the disturbance in streamlines is the most obvious and the largest followed by wedge shape, where asymmetry in streamlines can also be observed due to asymmetry in geometry. The smallest effect on fluid flow and the smallest vortex can be observed for triangular and semicircular type cavities. Figures 7b, 8b and 9b show the velocity field and illustrate how the vortex formation influences the inward fluid with a consequent development a frontier layer next to the upper surface. This boundary layer does not expandto to coverthe the dimples, so sothat that a a high-velocity gradient is is maintained close close to the to the moving moving surface surface even even with with deep deep dimples dimples for the forfour the different geometries. Figures 7c, 8c and 9c present the static pressure for the different geometries and show that pressure first diminishes at the left corner the cavity and then increases when encountering the right corner. The pressure decrease and increase corresponds to an enlargement

9 Lubricants 2018, 6, four different geometries. Figures 7c, 8c and 9c present the static pressure for the different geometries and show that pressure first diminishes at the left corner the cavity and then increases when encountering Lubricants the 2018, right 6, x FOR corner. PEER REVIEW The pressure decrease and increase corresponds to an enlargement 9 26 and to a diminishing transversal flow section. This is the well-known ram effect that occurs at rapidly and to a diminishing transversal flow section. This is the well-known ram effect that occurs at converging rapidly constrictions. converging constrictions. It can be It seen can be that seen thethat static the static pressure pressure is nearly is nearly symmetrical about about a a point locatedpoint on the located mid-section the mid-section for the triangular, for the triangular, rectangular rectangular and and cylindrical geometries. However for the wedge for geometry the wedge the geometry pressure the rise pressure is higher rise is than higher the than pressure the pressure drop drop as as shown in in Figure 10c. (a) Streamlines (m/s) (b) Velocity vectors (m/s) Figure 6. (a) Streamlines, (b) velocity vectors (colored by velocity magnitude [m/s]) in a smooth surface (u = m/s). Figure 6. (a) Streamlines, (b) velocity vectors (colored by velocity magnitude [m/s]) in a smooth surface (u = m/s).

10 Lubricants 2018, 6, Lubricants 2018, 6, x FOR PEER REVIEW (a) Streamlines (Pa) (b) Velocity Magnitude (m/s) (c) Static Pressure (Pa) Figure 7. (a) Streamlines (colored by static pressure [Pa]), (b) velocity magnitude and, (c) static Figure 7. pressure (a) Streamlines a single (colored pocket for bythe static cylindrical pressure geometry [Pa]), (cavity (b) velocity width magnitude = 128 μm, cavity and, depth (c) static = 22.8 pressure in a single μm, pocket spacing for = 500 theμm, cylindrical u = m/s). geometry (cavity width = 128 µm, cavity depth = 22.8 µm, spacing = 500 µm, u = m/s).

11 Lubricants 2018, 6, Lubricants 2018, 6, x FOR PEER REVIEW (a) Streamlines (Pa) (b) Velocity Magnitude (m/s) (c) Static Pressure (Pa) Figure 8. (a) Streamlines (colored by static pressure [Pa]), (b) velocity magnitude and, (c) static Figure 8. pressure (a) Streamlines a single (colored pocket for by the static rectangular pressure geometry [Pa]), (b) (cavity velocity width magnitude = 128 μm, cavity and, depth (c) static = 22.8 pressure in a single μm, pocket spacing for = 500 theμm, rectangular u = m/s). geometry (cavity width = 128 µm, cavity depth = 22.8 µm, spacing = 500 µm, u = m/s).

12 Lubricants 2018, 6, Lubricants 2018, 6, x FOR PEER REVIEW (a) Streamlines (Pa) (b) Velocity Magnitude (m/s) (c) Static Pressure (Pa) Figure 9. (a) Streamlines (colored by static pressure [Pa]), (b) velocity magnitude and, (c) static Figure 9. pressure (a) Streamlines a single (colored pocket for bythe static triangular pressure geometry [Pa]), (cavity (b) velocity width magnitude = 128 μm, cavity and, depth (c) static = 22.8 pressure in a single μm, pocket spacing for = 500 theμm, triangular u = m/s). geometry (cavity width = 128 µm, cavity depth = 22.8 µm, spacing = 500 µm, u = m/s).

13 Lubricants 2018, 6, Lubricants 2018, 6, x FOR PEER REVIEW (a) Streamlines (Pa) (b) Velocity Magnitude (m/s) (c) Static Pressure (Pa) Figure 10. (a) Streamlines (colored by static pressure [Pa]), (b) velocity magnitude and, (c) static Figure 10. pressure (a) Streamlines a single pocket (colored for the by wedge static geometry pressure (cavity [Pa]), width (b) = 128 velocity μm, cavity magnitude depth = 22.8 and, μm, (c) static pressure inspacing a single = 500 pocket μm, u = for m/s). the wedge geometry (cavity width = 128 µm, cavity depth = 22.8 µm, spacing = 500 µm, u = m/s). Pressure variations on the upper wall are presented in Figure 11 for different geometries. The pressure first decreases due to the channel height enlargement at the left corner then increases when encountering the right corner due to the diminishing transversal flow section. The rectangular geometry produces lower amplitude pressure distribution than the other three geometries. This was already seen in Figures 7c, 8c and 9c where the static pressure for the different geometries inside the cavity showed lower amplitude values for the rectangular geometry ( 75,079 to 74,971 Pa), followed by the cylindrical geometry ( 89,271 to 89,261 Pa) and triangular geometry ( 104,807 to 105,040 Pa). For the wedge geometry, due to asymmetry in geometry, the pressure rise was higher than the pressure drop ( 85,229 to 139,817 Pa).

14 than the pressure drop ( 85,229 to 139,817 Pa). Following the pressure curves in Figure 11, the curves are entirely anti-symmetric about the point located at x = 0 and y = 0 for the rectangular, cylindrical and triangular geometry and the amplitude the pressure distribution increases in the mentioned order with the curves being more sharp, whereas for the wedge geometry shows a net pressure rise in the domain. For the wedge Lubricants 2018, geometry, 6, 15 the pressure distribution becomes asymmetric; namely, the pressure rise closer to the right corner being more important than the pressure drop. Figure 11. Pressure distribution on the flat moving wall for different geometries (cavity width = 128 Figure 11. Pressure distribution on the flat moving wall for different geometries (cavity width = 128 µm, μm, cavity depth = 22.8 μm, spacing = 500 μm, u = m/s). cavity depth = 22.8 µm, spacing = 500 µm, u = m/s). In Figure 12 the shear force is plotted for the four different geometries (cylindrical, rectangular triangular and wedge). The rectangular geometry produces the lowest shear force between all the geometries investigated. This is in part explained by the fact that the rectangular groove inherits a greater area than the other three, using the same width and depth, which in turn gives rise to a lower friction force. Also, greater disturbances in fluid flow and greater vortex formation can be observed for rectangular geometry. As will be seen in next subsection (influence cavity depth), the vorticity is clearly dependent on groove depth, increasing with its depth and giving rise to lower shear force. Following the pressure curves in Figure 11, the curves are entirely anti-symmetric about the point located at x = 0 and y = 0 for the rectangular, cylindrical and triangular geometry and the amplitude the pressure distribution increases in the mentioned order with the curves being more sharp, whereas for the wedge geometry shows a net pressure rise in the domain. For the wedge geometry, the pressure distribution becomes asymmetric; namely, the pressure rise closer to the right corner being more important than the pressure drop. In Figure 12 the shear force is plotted for the four different geometries (cylindrical, rectangular triangular and wedge). The rectangular geometry produces the lowest shear force between all the geometries investigated. This is in part explained by the fact that the rectangular groove inherits a greater area than the other three, using the same width and depth, which in turn gives rise to a lower friction force. Also, greater disturbances in fluid flow and greater vortex formation can be observed for rectangular geometry. As will be seen in next subsection (influence cavity depth), the vorticity is clearly dependent on groove depth, increasing with its depth and giving rise to lower shear force. Lubricants 2018, 6, x FOR PEER REVIEW Cylindrical Rectangular Triangular Wedge Shear Force, Fx (N) Cylindrical Rectangular Triangular Wedge Textured geometry Figure 12. Shear force (Fx) as a function groove geometry (cavity width = 128 μm, cavity depth = Figure 12. Shear 22.8 force μm, spacing (F x ) as = 500 a μm, function u = m/s). groove geometry (cavity width = 128 µm, cavity depth = 22.8 µm, spacing 3.2. Influence = 500 µm, Cavity u Depth = m/s). Streamlines for different depths semicircular cavity, shown in Figure 13, can give a qualitative idea about how the flow is organized and how it influences the hydrodynamic performance. In Figure 13a the streamlines are ordered and smooth, but as depth increases to a value 16 and 22.8 μm, as can be seen in Figure 13b,c, a recirculation zone is developed in the cavity having closed contours. This shows that the vorticity/recirculation is clearly dependent on groove depth and increases with its depth.

15 0.45 Cylindrical Rectangular Triangular Wedge Textured geometry Lubricants 2018, 6, Figure 12. Shear force (Fx) as a function groove geometry (cavity width = 128 μm, cavity depth = 22.8 μm, spacing = 500 μm, u = m/s) Influence Cavity Depth 3.2. Influence Cavity Depth Streamlines for different depths semicircular cavity, shown in Figure 13, can give a qualitative idea about Streamlines how the for flow different is organized depths semicircular and how cavity, it influences shown in the Figure hydrodynamic 13, can give a performance. qualitative idea about how the flow is organized and how it influences the hydrodynamic performance. In Figure In Figure 13a the streamlines are ordered and smooth, but as depth increases to a value 16 and 13a the streamlines are ordered and smooth, but as depth increases to a value 16 and 22.8 μm, as 22.8 µm, as can be seen in Figure 13b,c, a recirculation zone is developed in the cavity having closed can be seen in Figure 13b,c, a recirculation zone is developed in the cavity having closed contours. contours. This shows that the vorticity/recirculation is clearly dependent on groove depth and This shows that the vorticity/recirculation is clearly dependent on groove depth and increases with increases with its depth. its depth. (a) depth = 8 μm Lubricants 2018, 6, x FOR PEER REVIEW (b) depth = 16 μm (c) depth = 22.8 μm Figure 13. Streamlines (colored by static pressure [Pa]) in the cylindrical geometry for different values Figure 13. Streamlines (colored by static pressure [Pa]) in the cylindrical geometry for different values depth where cavity width = 128 μm, spacing = 500 μm, u = m/s: (a) 8 μm, (b) 16 μm and, (c) 22.8 depth where cavity width = 128 µm, spacing = 500 µm, u = m/s: (a) 8 µm, (b) 16 µm and, (c) 22.8 µm. μm. The The graphics, showing streamlines for for the the cylindrical geometry with cavity depths 8 and 8 and µm (Figure μm (Figure 13a,b, 13a,b, respectively), were were already already shown shown in in a previous a study [20] in orderto to illustrate the the formation the the vortex inside the cavity, that are responsible for the shear force and global friction reduction. These were achieved by comparing fluid dynamic modeling with experimental tribological investigation showing friction results for different dimple depths. Comparisons pressure distribution for the cylindrical geometry are made in Figure 14, where the pressure is plotted as a function the spatial dimension x for different values cavity depth. The other parameters remain constant for all subfigures. Before arriving at the groove edge, located at x = 64 μm, pressure continues decreasing until it reaches the groove edge where the pressure curve suddenly changes to a pressure rise at a cavity depth 8 μm. As already observed before, the

16 Lubricants 2018, 6, reduction. These were achieved by comparing fluid dynamic modeling with experimental tribological investigation showing friction results for different dimple depths. Comparisons pressure distribution for the cylindrical geometry are made in Figure 14, where the pressure is plotted as a function the spatial dimension x for different values cavity depth. The other parameters remain constant for all subfigures. Before arriving at the groove edge, located at x = 64 µm, pressure continues decreasing until it reaches the groove edge where the pressure curve suddenly changes to a pressure rise at a cavity depth 8 µm. As already observed before, the total pressure rise is completely anti-symmetric to the total pressure drop, about a point located at y = 0 on the vertical midsection. As the depth the cavity is increased it is clear that the pressure magnitude in the dimple region is decreased and change in pressure inside the cavity shifts away from linear to more Lubricants exponential, 2018, 6, x FOR aspeer shown REVIEW in Figure Figure 14. Pressure distribution on the flat moving wall for different cavity depths (cylindrical geometry, cavity width = 128 µm, spacing = 500 µm, u = m/s). Figure 14. Pressure distribution on the flat moving wall for different cavity depths (cylindrical geometry, cavity width = 128 μm, spacing = 500 μm, u = m/s). In Figure 15 the shear force in the lubrication film is plotted for three different cavity depths. In Figure 15 the shear force in the lubrication film is plotted for three different cavity depths. As As shown, the shear force decreases when the dimple depth increases from 8 to 22.8 µm. The result is shown, the shear force decreases when the dimple depth increases from 8 to 22.8 μm. The result is lower friction and thus a reduced friction coefficient. Thus, a minimum value F x = 0.46 N is found lower friction and thus a reduced friction coefficient. Thus, a minimum value Fx = 0.46 N is found for a cavity depth 22.8 µm. However to obtain a true estimate the performance benefit the for a cavity depth 22.8 μm. However to obtain a true estimate the performance benefit the cavity, comparison should be made between a grooved surface and a smooth surface, under the same cavity, comparison should be made between a grooved surface and a smooth surface, under the same contact conditions. The shear force, for a smooth surface (cavity depth is set to zero), corresponds to contact conditions. The shear force, for a smooth surface (cavity depth is set to zero), corresponds to a value F x = 0.55 N, with the result that the 22.8 µm deep cavity produces a considerable overall a value Fx = 0.55 N, with the result that the 22.8 μm deep cavity produces a considerable overall reduction 16 % in shear force and gives rise to lower friction. reduction 16 % in shear force and gives rise to lower friction smooth , Shear Forec (N)

17 shown, the shear force decreases when the dimple depth increases from 8 to 22.8 μm. The result is lower friction and thus a reduced friction coefficient. Thus, a minimum value Fx = 0.46 N is found for a cavity depth 22.8 μm. However to obtain a true estimate the performance benefit the cavity, comparison should be made between a grooved surface and a smooth surface, under the same contact conditions. The shear force, for a smooth surface (cavity depth is set to zero), corresponds to Lubricants 2018, 6, a value Fx = 0.55 N, with the result that the 22.8 μm deep cavity produces a considerable overall reduction 16 % in shear force and gives rise to lower friction smooth , Shear Forec (N) smooth ,8 Cavity depth (μm) Figure 15. Shear force (F x ) as a function cavity depth (cylindrical geometry, cavity width = 128 µm, spacing = 500 µm, u = m/s). Hydrodynamic efficiency depends on load-carrying capacity (normal wall force) and shear force (tangential wall force), respectively, in the y- and x-direction. As seen in the previous section, the load-carrying capacity results from the integration the pressure distribution over the area the cell surface. If considered a simplified cavitation model where pressures smaller than zero are forced to zero it can be concluded that the highest load-carrying capacity is achieved by using dimple depth 8 µm and is decreasing as dimple depth increases. At the same time, it is seen that the shear force is also monotonically decreased by increasing values dimple depth, due to a local increase in film thickness. It can be seen that a recirculation zone occurs in the dimple when the depth increases from 8 to 16 µm. Increasing the depth the dimples tends to generate a stretched recirculation zone. According to Shankar et al. [21] this backflow leads to a lower pressure gradient and lower pressure generation, which can explain the obtained results for pressure distribution. Since the main objective is to achieve as high a load-carrying capacity at as low a shear force as possible, the increase dimple depth gives contradictory results for this range dimple depth variation. However, the decrease in shear force with the increase in dimple depth from 8 to 22.8 µm is also in agreement with the experimental results published in previous papers from the same authors [22]. For higher sliding speeds, especially at 0.45 m/s where the contact is under full film lubrication, as the dimple depth increases friction gets reduced as shown in previous studies performed by Vilhena et al. [22], where they showed the variation the coefficient friction with the cavity depth Influence Cavity Width In Figure 16, streamlines are plotted for the cylindrical geometry with constant depths but different widths. At a sufficiently small value width (see Figure 16a) no vortex and no back-flow exists in the groove anymore. As width increases to a value 80 µm, see Figure 16b, a recirculation zone is

18 increases friction gets reduced as shown in previous studies performed by Vilhena et al. [22], where they showed the variation the coefficient friction with the cavity depth Influence Cavity Width LubricantsIn 2018, Figure 6, 15 16, streamlines are plotted for the cylindrical geometry with constant depths 18 but 25 different widths. At a sufficiently small value width (see Figure 16a) no vortex and no back-flow exists in the groove anymore. As width increases to a value 80 μm, see Figure 16b, a recirculation developed zone is developed in the cavity. in the It can cavity. be seen It can that be seen the vortex that the decreases vortex decreases in size as in the size cavity as the width cavity increases width fromincreases 80 to 128 from µm. 80 to 128 μm. (a) cavity width = 40 μm Lubricants 2018, 6, x FOR PEER REVIEW (b) cavity width = 80 μm (c) cavity width = 128 μm Figure 16. Streamlines (colored by static pressure [Pa]) in the cylindrical geometry for different values Figure cavity 16. Streamlines width where (colored cavity depth by static = 22.8 pressure μm, spacing [Pa]) in = 500 theμm, cylindrical u = m/s: geometry (a) cavity forwidth different = 40 values μm, cavity (b) cavity width width where = 80 cavity μm and, depth (c) cavity = 22.8 width µm, spacing = 128 μm. = 500 µm, u = m/s: (a) cavity width = 40 µm, (b) cavity width = 80 µm and, (c) cavity width = 128 µm. Comparisons pressure distribution for the cylindrical geometry are made in Figure 17, where the pressure is plotted as a function the spatial dimension x for different values cavity width. The same observations hold for the pressure distribution presented in Figure 14. The distribution is anti-symmetrical about the x = 0 and y = 0 midsection. However, it is clear that the amplitude the pressure variation in the dimple region and the maximum pressure at the dimple edge increases with increasing width.

19 Lubricants 2018, 6, Comparisons pressure distribution for the cylindrical geometry are made in Figure 17, where the pressure is plotted as a function the spatial dimension x for different values cavity width. The same observations hold for the pressure distribution presented in Figure 14. The distribution is anti-symmetrical about the x = 0 and y = 0 midsection. However, it is clear that the amplitude the pressure variation in the dimple region and the maximum pressure at the dimple edge increases with increasinglubricants width. 2018, 6, x FOR PEER REVIEW Lubricants 2018, 6, x FOR PEER REVIEW Figure 17. Pressure distribution on the flat moving wall for different cavity widths (cylindrical geometry, cavity depth Figure = Pressure µm, spacing distribution = 500 µm, on the u = flat moving m/s). wall for different cavity widths (cylindrical Figure geometry, 17. Pressure cavity depth distribution = 22.8 μm, on spacing the flat = moving 500 μm, wall u = for m/s). different cavity widths (cylindrical geometry, cavity depth = 22.8 μm, spacing = 500 μm, u = m/s). Figure 18 shows how the shear force is affected by different values cavity width for the Figure 18 shows how the shear force is affected by different values cavity width for the cylindrical cylindrical geometry. Figure 18 geometry. shows As the how As cavity the cavity shear width force width increases, is increases, affected by the different shear force, force, values F Fx decreases x cavity decreases width monotonically. monotonically. for the When compared cylindrical When compared to geometry. un-textured to un-textured As the or cavity smooth or smooth width surface, increases, aa drop the shear in shear force, force Fx decreases for for 40 μm 40 monotonically. µm wide wide grooves grooves is is only about When only 5%. compared about However, 5%. to However, un-textured as theas width the or width smooth increases surface, to to a drop and in 128 shear μm µm force wider for 40 groves, μm wide reduction grooves in shear is in shear only force increases force about increases 5%. However, to 10 and to 10 20%, and as respectively. 20%, the respectively. width increases to 80 and 128 μm wider groves, reduction in shear force increases to 10 and 20%, respectively smooth smooth Shear Forec (N) Shear Forec (N) smooth smooth 40 cavity width (μm) cavity width (μm) Figure 18. Shear force (Fx) as a function cavity width (cylindrical geometry, cavity depth = 22.8 μm, Figure spacing 18. = Shear 500 μm, force u (Fx) = as m/s). a function cavity width (cylindrical geometry, cavity depth = 22.8 μm, Figure 18. Shear force (F spacing = 500 μm, u x ) as a function cavity width (cylindrical geometry, cavity depth = 22.8 µm, = m/s). spacing = 500 µm, u = m/s). The best hydrodynamic performance is achieved for cavity width 128 μm since the shear force decreases The best with hydrodynamic increasing values performance cavity is width achieved and for the cavity load-carrying width capacity, 128 μm since which the can shear be related force decreases with increasing values cavity width and the load-carrying capacity, which can be related

20 Lubricants 2018, 6, The best hydrodynamic performance is achieved for cavity width 128 µm since the shear force decreases with increasing values cavity width and the load-carrying capacity, which can be related to the amplitude the pressure distribution on the flat moving wall, increases with cavity width Lubricants 2018, 6, x FOR PEER REVIEW as well. to the amplitude the pressure distribution on the flat moving wall, increases with cavity width as 3.4. Influence well. Distance between Cavities (Spacing/Density) In Figure 3.4. Influence 19, streamlines Distance are between plotted Cavities for (Spacing/Density) the cylindrical geometry with constant depth and width but for different In spacing. Figure 19, streamlines As the spacing are plotted increases for the cylindrical vortex isgeometry shiftedwith closer constant to the depth cavity and surface. width It can also be observed but for different that as spacing. the distance As the goes spacing from increases 150 vortex to 250 is µm shifted andcloser thento to the 500 cavity µm, surface. vortex It can changes its shape becoming also more observed flat that as and the narrow, distance goes withfrom only 150 ato thinner 250 μm and section/layer then to 500 μm, affected vortex changes by theits vortex. shape becoming more flat and narrow, with only a thinner section/layer affected by the vortex. (a) spacing = 150 μm (b) spacing = 250 μm (c) spacing = 500 μm Figure 19. Streamlines (colored by static pressure [Pa]) in the cylindrical geometry for different values spacing where cavity depth = 22.8 µm, cavity width = 128 µm, u = m/s: (a) spacing = 150 µm, (b) spacing = 250 µm and, (c) spacing = 500 µm.

21 Pressure variations on the upper wall are presented in Figure 20 for different distances between cavities (spacing). As for all cases, the pressure first decreases due to the channel height enlargement then increases under to the influence the diminishing transversal flow section at the right corner. However, it can be seen that the distance between cavities influences the amplitude the pressure Lubricants 2018, 6, variation and the pressure curves are convexly shaped across the left half the dimple and have concave shapes at the right half, with the curve for 250 μm spacing, showing extreme convexity and concavity Pressure and variations lower amplitude on the upper variation. wall are presented in Figure 20 for different distances between cavities Since (spacing). a reference As forpressure all cases, must the pressure be imposed first decreases in a point due tothe thefluid channel field, height and enlargement the resulting then pressure increases field under is computed to the influence with respect theto diminishing this imposed transversal value (zero flowin section this case). at the By right imposing corner. However, different spacings, it can be seen we are thatshifting the distance the periodic betweenboundary cavities influences conditions the and amplitude the reference thepressure is variation considered andat the different pressure locations. curves are Consequently, convexly shaped a longer across pressure the left cycle halfis obtained the dimple in case andwhere have concave spacing shapes is 500 μm at the (blue right line half, in Figure with the 20), curve while fora 250 shorter µm spacing, cycle is obtained showingin extreme case where convexity spacing andis concavity 250 or 150 and μm lower (yellow amplitude line in Figure variation. 20). Figure Pressure distribution on the flat moving wall for different values spacing (cylindrical geometry, cavity depth = = 22.8 µm, μm, cavity width = = 128 µm, μm, u u = = m/s). Since In Figure a reference 21, the pressure shear force must is plotted be imposed as a function in a point distance the fluid between field, and cavities the resulting (spacing). pressure There field exists is computed an optimal with value respect spacing to this in imposed terms value shear (zero force in that this case). could By fit imposing in the range different 150 < spacings, spacing < we 500. are shifting the periodic boundary conditions and the reference pressure is considered at different locations. Making Consequently, a comparison a longer between pressure Figures cycle is obtained reveals the in case relationship where spacing between is 500 vortex µm formation, (blue line in amplitude Figure 20), while the pressure a shorter distribution cycle is obtained and shear in case force where in terms spacing different is 250 or 150 values µm (yellow spacing. line This in Figure discussion 20). shows that the minimum shear force and minimum amplitude pressure distribution appears In Figure at a 21, certain the shear value force vorticity is plotted represented as a function by distance more spaced between streamlines, cavities (spacing). which There means exists that an lower optimal velocity value can spacing be found. in terms shear force that could fit in the range 150 < spacing < 500. Making This simulation a comparison is in agreement between Figures with the experimental reveals the results relationship from previous between studies vortex formation, performed amplitude by Podgornik the et al. pressure [20], where distribution they showed and the shear variation force in the terms coefficient different friction values with the spacing. cavity This density. discussion At higher shows sliding that the speeds, minimum as the shear lubricant force fluid and film minimum thickness amplitude increases and pressure the contact distribution enters appears the full at fluid a certain film lubrication value vorticity regime, represented an optimal by value more spaced density streamlines, in terms which coefficient means that friction, lower velocity that could can fit be in found. the range between 3.5 % (300 μm spacing) and 31.2 % (100 μm spacing), was observed. The This minimum simulation coefficient is in agreement friction with was obtained the experimental for 7.8 % results density from (200 previous μm spacing). studies performed by Podgornik et al. [20], where they showed the variation the coefficient friction with the cavity density. At higher sliding speeds, as the lubricant fluid film thickness increases and the contact enters the full fluid film lubrication regime, an optimal value density in terms coefficient friction, that could fit in the range between 3.5 % (300 µm spacing) and 31.2 % (100 µm spacing), was observed. The minimum coefficient friction was obtained for 7.8 % density (200 µm spacing).

22 Lubricants 2018, 6, Lubricants 2018, 6, x FOR PEER REVIEW Shear Force, Fx (N) Spacing (μm) Figure 21. Shear force (Fx) as a function distance between cavities spacing (cylindrical geometry, Figure 21. Shear force (F x ) as a function distance between cavities spacing (cylindrical geometry, cavity width = 128 μm, cavity depth 22.8 μm, m/s). cavity width = 128 µm, cavity depth = 22.8 µm, u = m/s) Influence Sliding Speed 3.5. Influence Sliding Speed Figure 22 shows the variation shear force at different sliding speeds. It is shown that the shear force Figure also 22 monotonically shows the variation increases by shear increasing force atthe different sliding sliding speed, speeds. for a given It is thickness. shown that For thevery shear force small alsosliding monotonically speeds, when increases viscous byeffects increasing are dominant the sliding the speed, shear for force a given is very thickness. small; with Forthe very small augmentation sliding speeds, the when sliding viscous speed and effects the are increased dominant importance the shear inertia force is forces verythe small; shear with force the augmentation increases exponentially. the sliding However, speed in and real the contacts, increased lubrication importance film thickness inertia is not forces constant the shear and is force a increases function exponentially. viscosity, sliding However, speed, in and real applied contacts, load. lubrication This, in turn, film determines thickness the islubrication not constant regime. and is a function As sliding viscosity, speed increases, sliding speed, the film and thickness appliedalso load. increases This, and turn, starts determines to move thefrom lubrication boundary regime. Aswhere slidingthe speed two increases, surfaces mostly the filmare thickness in contact alsowith increases each other and starts and results to move in from very boundary high friction, where to the mixed two surfaces where mostly the two are surfaces in contact are partly withseparated, each other partly andin results contact inand very a sharp high drop friction, in coefficient to mixed where friction the two is surfaces observed, are and, partly finally, separated, to full film partly or hydrodynamic in contact and alubrication sharp drop where in coefficient the two surfaces friction is observed, are separated and, finally, by a fluid to full film filmand or hydrodynamic the coefficient lubrication friction where reach the its two minimum. surfaceshowever are separated in by hydrodynamic a fluid film and lubrication the coefficient the coefficient friction reach friction its minimum. slightly increases However with in sliding hydrodynamic speed. This lubrication is due to the friction produced by the fluid. At higher sliding speeds, the fluid film becomes thicker. the coefficient friction slightly increases with sliding speed. This is due to the friction produced by However, the fluid drag on the moving surfaces also increases, which can explain the results shown the fluid. At higher sliding speeds, the fluid film becomes thicker. However, the fluid drag on the in Figure 22. Therefore, Reynolds and Computational Fluid Dynamics (CFD) analysis fluid flow moving surfaces also increases, which can explain the results shown in Figure 22. Therefore, Reynolds between parallel surfaces are only valid when contact is under full film lubrication. and Computational Fluid Dynamics (CFD) analysis fluid flow between parallel surfaces are only Strong similarities between CFD analysis textured surfaces presented in this study and results valid when contact is under full film lubrication. reported by Sahlin et al. [1] can be further underlined. Thus, Sahlin et al. [1] analyzed the influence micro-patterned surfaces in hydrodynamic lubrication. Their calculations were made for two types parameterized grooves with the same order depth as the film thickness. They show that an introduction a micro groove with a given geometry on one the surfaces affects the fluid flow and pressure pattern and that the shear force decreases with increasing values groove depth and width. It was also noticed that shear force increased with Re number. These results corroborate entirely our CFD simulation (see Figure 22). They also show that load-carrying capacity increases with groove width and a general optimum is achieved when dimensionless groove depth (d + ) is in

23 an increase in groove depth leads to a decrease in load-carrying capacity. This agrees with our experiments since the magnitude pressure distribution decreases with increasing dimple depth in the range 8, 16 and 22.8 μm (dimensionless dimple depth 1.6, 3.2 and 4.6 for a given film thickness 5 μm). Based on our experiments and Sahlin et al. results [1], it can be concluded that if Lubricants the depth 2018, is 6, below 15 some critical value, the increase dimple depth increases the load-carrying capacity. If it is above, contributes to a reduction the load-carrying capacity Shear Force, Fx (N) sliding speed (m/s) Figure 22. Shear force (Fx) as function sliding speed (cylindrical geometry, cavity width 128 μm, Figure 22. Shear force (F x ) as a function sliding speed (cylindrical geometry, cavity width = 128 µm, spacing 500 μm, cavity depth 22.8 μm). spacing = 500 µm, cavity depth = 22.8 µm). 4. Conclusions Strong similarities between CFD analysis textured surfaces presented in this study and results reported A limited by Sahlin number et al. [1] can analytical be further models underlined. to study Thus, surface Sahlin et texturing al. [1] analyzed have been the influence proposed micro-patterned [1,2,4,19,23], mostly surfaces considering in hydrodynamic hydrodynamic lubrication. effects Their micro-texturing, calculations were while made the formajority two types studies parameterized have been grooves experimental with the by trial sameand order error approach depth as the [20 22,24 26]. film thickness. For full They film show lubrication, that an introduction another frequently a micro used groove approximation with a given to the geometry governing on equations one thefor surfaces fluid flow affects is the the fluid Reynolds flow and equation pressure where pattern the inertia and that term the is shear neglected. forcethe decreases current with work increasing is based on values the full groove system depth Navier- and width. Stokes equations It was also and noticed provides that the shear following force increased conclusions: with Re number. These results corroborate entirely For our low CFD Reynolds simulation numbers, (see Figure an analogous 22). Theyresult also show can be that obtained load-carrying using capacity the Reynolds increases and with the groove Navier-Stokes width and a general equations optimum since the is achieved pressure when distribution dimensionless is anti-symmetrical groove depth (d about + ) is inthe between point 0.5< d located + <0.75. at However, the mid-plane for dimensionless the cell, groove as shown depth during higher the than present 0.75 itinvestigation. was shown that However, an increase for in groove large depth Reynolds leadsnumbers to a decrease the pressure in load-carrying distribution capacity. becomes Thisasymmetric, agrees with resulting our experiments in a positive since the magnitude net force, which pressure requires distribution the use decreases the Navier-Stokes with increasing equation. dimple depth in the range 8, 16 and 22.8 µm The (dimensionless CFD simulation dimple performed depthin this 1.6, research 3.2 and work 4.6 fordiscussed a given film only thickness the effect 5 the µm). shear Based stress on our experiments due to the presence and Sahlin et the al. micro-dimple. results [1], it can However, be concluded for the that conditions if the depth simulated is belowin some this critical study value, (high the increase viscosity fluid), dimple the depth Reynolds increases number, the load-carrying Re << 1 shows capacity. that the convective If it is above, inertia contributes forces can to a reduction be neglected. the load-carrying (The Reynolds capacity. number is defined as Re = ρuh/μ where ρ and μ are the density and dynamic viscosity the fluid, h the film thickness and u is the velocity the moving wall.) 4. Conclusions Although these results need further investigation, they clearly indicate that optimization Asurface limited texturing numberfor analytical full-film lubrication models to study can be surface done through texturingcfd. have been proposed [1,2,4,19,23], mostly considering hydrodynamic effects micro-texturing, while the majority studies have been experimental by trial and error approach [20 22,24 26]. For full film lubrication, another frequently used approximation to the governing equations for fluid flow is the Reynolds equation where the inertia term is neglected. The current work is based on the full system Navier-Stokes equations and provides the following conclusions:

24 Lubricants 2018, 6, For low Reynolds numbers, an analogous result can be obtained using the Reynolds and the Navier-Stokes equations since the pressure distribution is anti-symmetrical about the point located at the mid-plane the cell, as shown during the present investigation. However, for large Reynolds numbers the pressure distribution becomes asymmetric, resulting in a positive net force, which requires the use the Navier-Stokes equation. The CFD simulation performed in this research work discussed only the effect the shear stress due to the presence the micro-dimple. However, for the conditions simulated in this study (high viscosity fluid), the Reynolds number, Re << 1 shows that the convective inertia forces can be neglected. (The Reynolds number is defined as Re = ρuh/µ where ρ and µ are the density and dynamic viscosity the fluid, h the film thickness and u is the velocity the moving wall.) Although these results need further investigation, they clearly indicate that optimization surface texturing for full-film lubrication can be done through CFD. An introduction a micro-groove on one the surfaces affects the flow and pressure distribution. For very small Reynolds numbers, when viscous effects are largely dominant and the approximation the Stokes flow regime is acceptable, the pressure distribution is anti-symmetrical about the point located at x = 0 and y = 0. The shear force is also monotonically decreased by increasing values cavity depth, due to a local increase in film thickness. The result is lower friction and thus a reduced friction coefficient. This agrees with our CFD simulation since the magnitude pressure distribution decreases with increasing dimensionless dimple depth from 1.6 to 3.2 and 4.6. CFD simulation for different distances between cavities (densities) is also in agreement with the experimental results obtained under full-film lubrication in previous work [20]. They clearly showed that there exists an optimal value spacing in terms shear force. Furthermore, according to the CFD simulation the best hydrodynamic performance is expected with the rectangular geometry (lowest shear force). Acknowledgments: The author Luís Vilhena gratefully acknowledges the financial support the Portuguese Foundation for Science and Technology (FCT), through the program QREN-POPH, reference: SFRH/BPD/92787/2013. The present work is part the EC WEMESURF project MRTN-CT Characterization WEear MEchanisms and SURFace functionalities with regard to life time prediction and quality criteria-from micro to the nano range. Author Contributions: All authors contributed equally. Conflicts Interest: The authors declare no conflict interest. References 1. Sahlin, F.; Glavatskih, S.B.; Almqvist, T.; Larsson, R. Two-dimensional CFD-analysis micro-patterned surfaces in hydrodynamic lubrication. Trans. ASME 2005, 127, 96. [CrossRef] 2. Brajdic-Mitidieri, P.; Gosman, A. CFD Analysis a Low Friction Pockeed Pad Bearing. ASME J. Tribol. 2005, 127, [CrossRef] 3. Marian, V.G.; Kilian, M.; Scholz, W. Theoretical and experimental analysis a partially textured thrust bearing with square dimples. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 2007, 221, [CrossRef] 4. Kraker, A.; Ostayen, R.; Van Beek, A. A Multiscale Method Modeling Surface Texture Effects. J. Tribol. 2007, 129, [CrossRef] 5. Feldman, Y.; Kligerman, Y.; Etsion, I.; Haber, S. The validity the reynolds equation in modelling hydrostatic effects in gas lubricated textured parallel surfaces. ASME J. Tribol. 2006, 128, [CrossRef] 6. Sahlin, F.; Almqvist, A.; Larsson, R.; Glavatskih, S. A cavitation algorithm for arbitrary lubricant compressibility. Tribol. Int. 2007, 40, [CrossRef] 7. Ausas, R.; Ragot, P.; Leiva, J.; Jai, M.; Bayada, G.; Buscaglia, G.C. The impact the cavitation model in the analysis micro-textured lubricated journal bearings. ASME J. Tribol. 2007, 129, [CrossRef] 8. Adams, H.G.; Elrod, M. A computer program for cavitation. Cavitation relat. Phenom. Lubr. 1974, 103,

25 Lubricants 2018, 6, Arghir, M.; Roucou, N.; Helene, M.; Frene, J. Theoretical Analysis the Incompressible Laminar Flow in a Macro-Roughness Cell. J. Tribol. 2003, 125, [CrossRef] 10. Echávarri Otero, J.; Guerra Ochoa, E.; Bellón Vallinot, I.; Chacón Tanarro, E. Optimising the design textured surfaces for reducing lubricated friction coeficiente. Lubr. Sci. 2017, 29, [CrossRef] 11. Vladescu, S.-C.; Olver, A.V.; Pegg, I.G.; Reddyhf, T. The effects surface texture in reciprocating contacts An experimental study. Tribol. Int. 2015, 82, [CrossRef] 12. Zavos, A.; Nikolakopoulos, P.G. Cavitation effects on textured compression rings in mixed lubrication. Lubr. Sci. 2016, 28, [CrossRef] 13. Morris, N.J.; Rahmani, R.; Rahnejat, H. A hydrodynamic flow analysis for optimal positioning surface textures. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 2017, 231, [CrossRef] 14. Wang, X.; Kato, K.; Adachi, K.; Aizawa, K. Loads carrying capacity map for the surface texture design SiC thrust bearing sliding in water. Tribol. Int. 2003, 36, [CrossRef] 15. Stachowiak, G.W.; Batchelor, A.W. Engineering Tribology, 3rd ed.; Elsevier Science Publishers: New York, NY, USA, Fillon, M.B.; Dobrica, M. About the validity Reynolds equation and inertia effects in textured sliders infinite width. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 2008, 223, Stachowiak, G.W.; Batchelor, A.W. Engineering Tribology, 2nd ed.; Elsevier: New York, NY, USA, Patankar, S.V.; Spalding, D.B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transf. 1972, 15, [CrossRef] 19. Ferziger, J.H.; Peric, M. Computational Methods for Fluid Dynamics; Springer: Berlin, Germany, 2001; ISBN Podgornik, B.; Vilhena, L.M.; Sedlaček, M.; RekI, Z. Effectiveness and design surface texturing for different lubrication regimes. Meccanica 2012, 47, [CrossRef] 21. Deshpande, P.N.; Shankar, M.D. Fluid mechanics in the driven cavity. Ann. Rev. Fluid Mech. 2000, 32, Vilhena, L.M.; Podgornik, B.; Vižintin, J.; Možina, J. Influence texturing parameters and contact conditions on tribological behavior laser textured surfaces. Meccanica 2011, 46, [CrossRef] 23. Hsu, S.M. Surface Texturing: Principles and Design; National Institute Standards & Technology: Gaithersburg, MD, USA, Vilhena, L.M.; Sedlaček, M.; Podgornik, B.; Vižintin, J.; Babnik, A.; Možina, J. Surface texturing by pulsed Nd:YAG laser. Tribol. Int. 2009, 42, [CrossRef] 25. Vilhena, L.M.; Sedlaček, M.; Podgornik, B. Answer to the discussion the paper entitled surface texturing by pulsed Nd:YAG laser (L.M. Vilhena, M. Sedlaček, B. Podgornik, J. Vižintin, A. Babnik, and J. Možina, tribology international 42 (2009) ). Tribol. Int. 2018, 118, [CrossRef] 26. Vilhena, L.M.; Ramalho, A.; Cavaleiro, A. Grooved surface texturing by electrical discharge machining. (EDM) under different lubrication regimes. Lubr. Sci. 2017, 29, [CrossRef] 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions the Creative Commons Attribution (CC BY) license (