- Email: [email protected]
Hydrodynamic lubrication of micro-textured surfaces: Two dimensional CFD-analysis
Tribology International 88 (2015) 162–169
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Hydrodynamic lubrication of micro-textured surfaces: Two dimensional CFD-analysis Giovanni Caramia n, Giuseppe Carbone, Pietro De Palma Dipartimento di Meccanica, Matematica e Management (DMMM), Politecnico di Bari, Via Re David 200, 70125 Bari, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 6 October 2014 Received in revised form 11 March 2015 Accepted 15 March 2015 Available online 24 March 2015
This paper provides a numerical study of the hydrodynamic lubrication between two parallel surfaces with micro-texturing. The two-dimensional Navier–Stokes equations for an isothermal incompressible steady flow have been considered as a suitable model. A wide variety of geometries characterised by different micro-cavity depth and width, and different gap values have been analysed in order to study the influence of these parameters on the drag force magnitude. A detailed analysis of flow velocity profiles and pressure distributions has been performed to study the forces acting on the textured surface, providing an explanation for the maximum drag reduction achievable with a single-phase lubrication fluid. Furthermore, results indicate that three regions exist, depending on the cavity depth, in which a different flow dynamics occurs and the cavities have a different influence on the drag force. Finally, an “optimal” value of the depth has been found, for which the pressure reaches a minimum value and the probability of cavitation is maximised. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Surface texturing Friction Lubrication Cavitation
1. Introduction Recent technological improvements in surface texturing techniques allow one to exploit the properties of micro-structured surfaces in many engineering applications [1]. Positive effects of artificial surface texturing in lubrication systems concerning load capacity, wear resistance, and friction properties have been investigated starting from the pioneering work of Hamilton et al. [2] and Anno et al. [3]. The present work analyses the influence of surface micro-texturing on the hydrodynamic lubrication properties of two parallel sliding plates. Experimental results in this field [4] show that significant improvements in hydrodynamic performance can be achieved by surface texturing. Recently, with the aim of minimising the friction coefficient, Scaraggi et al. [5,6] experimentally investigated micro-cavity and micro-groove depth influence on friction properties. On the other hand, several numerical techniques have been developed to predict the performance of lubrication systems: from model based on the Reynolds equation to the solution of the more complex Navier–Stokes equations. Fowell et al. [7] highlighted one of the basic mechanisms of textured bearings analysing the contribution of inlet suction to the performance of near-parallel, internally pocketed bearings. Giacopini et al. [8] proposed a mass-conserving formulation of the Reynolds equation to deal with cavitation in lubricated contacts. In order to
n
Corresponding author. E-mail address: [email protected] (G. Caramia).
http://dx.doi.org/10.1016/j.triboint.2015.03.019 0301-679X/& 2015 Elsevier Ltd. All rights reserved.
increase the computational speed with respect to traditional numerical methods, Pei et al. [9] proposed the finite cell method for multi-scale surface texture hydrodynamic lubrication. The mean field approach has been also recently developed to find the optimal geometry of micro-holes which maximises load carrying capacity of the junction or minimises friction [10]. Ma and Zhu [11] solved the Reynolds equation by a finite difference method to find an optimum design model for textured surface with elliptical-shape dimples. De Kraker et al. [12] developed a multi-scale method in which the Navier–Stokes equations’ results were employed to derive suitable flow factors to be used in a “texture averaged Reynolds equation”. Dobrica and Fillon [13] showed the importance of the dimple aspect ratio to establish whether a given configuration could be modelled using the Reynolds equation and to evaluate the expected corresponding accuracy loss. Ausas et al. [14] analysed the impact of the cavitation model on numerical predictions of lubricated journal bearings showing the limits of using Reynolds’ model when simulating flows in the presence of micro-textured surfaces. The finite volume approach was used by Sahlin et al. [15] to solve the incompressible Navier–Stokes equations, providing a twodimensional CFD analysis of micro-patterned surfaces. A three dimensional CFD analysis was then performed by Jing et al. [16] and by Arghir et al. [17] who investigated the net pressure gain as a pure inertia effect due to the combined action of the macroroughness and of Reynolds number. A cavitation phenomenon was considered in the CFD analysis of a low friction pocketed pad bearing by Brjdic-Mitidieri et al. [18] and in the CFD analysis of a journal bearing with surface texturing by Cupillard et al. [19].
G. Caramia et al. / Tribology International 88 (2015) 162–169
0.896 mm
Lubricant bath
inviscid wall
inviscid wall
wall moving wall (disk)
y
16.896 mm x Fig. 2. Computational domain and boundary conditions.
348.125 µ m
348.125 µ m
h
100 µ m
D gap
y
x
Fig. 1. Experimental setup (schematic) employed for friction measurements in [5].
0.6128 m/s
Fig. 3. Local view of the computational domain.
100 µ m
1.5 mm
Holder periodic
The model geometry is inspired to the experimental set-up employed in [5,6], namely, the pin-on-disk tribometer for friction measurements shown in Fig. 1. Scaraggi et al. [5,6] performed several tests with different laser-textured bearing balls assembled in the pin holder. The contact surface was micro-textured with a square lattice of micro-cavities whose diameter, depth, and spacing were controlled during the texturing process. They reported a remarkable drag-reduction effect for some particular combinations of the geometric parameters and flow conditions. Unfortunately, they could not ascertain experimentally the origin of such a phenomenon. One of
wall
2.1. Model geometry
wall
2. CFD-model
the aims of the present work is to provide a theoretical explanation for this behaviour by using a numerical approach. However, a huge computational cost would be needed in order to simulate the real three-dimensional flow. For this reason, we have adopted a simplified two-dimensional geometry which allows us to study as well the fundamental phenomena involved in this particular flow configuration. Therefore, we have considered a two-dimensional section of the ball holder, shown in Fig. 2, with a reduced length, ltot, with respect to the original experimental model. As shown in Fig. 2, a flat wall, moving with velocity U¼0.6128 m/s, represents the rotating disk, whereas the holder supporting the textured surface is superposed to the wall at a distance equal to the film thickness gap (see Fig. 3). The flat-wall moving velocity was calculated on the basis of experimental data as explained in detail in Section 3. The texturing is represented by three equally-spaced rectangular cavities with depth h and width D. Unlike previous studies available in the literature, as far as the authors know, periodic boundary conditions are imposed at the two vertical boundaries of the computational domain, which are located very far from the holder (see Figs. 2 and 3). This allows us to model the isolated pin holder, avoiding spurious fluid interactions, also taking into account the inlet and outlet effects of the fluid at the extrema of the textured surface. For all of the remaining boundaries the no-slip condition is imposed, except for the upper horizontal surface along which the slip flow condition (inviscid wall) is adopted. In order to investigate the effects of the geometric parameters on the drag force on the textured surface, a parametric study based on suitable values of the three parameters gap, D, and h is performed. A comment is in order about the analysis of the results and the comparison with the available experimental data [5,6]. There is a fundamental difference which characterises the experimental and the numerical approach. In the experimental set-up, it is straightforward to assign and control the normal load exerted on the pin holder; therefore the value of the gap is determined by the balance between the fluid forces acting perpendicularly to the wall and the load. On the other hand, in the numerical approach, a great simplification is achieved assigning the value of the gap which allows one to employ a given stationary computational grid. For this reason, we provide the analysis of the drag force for three values of
periodic
The aim of the present paper is to study, by a CFD analysis, the fluid mechanics of the lubrication between two parallel surfaces for an incompressible steady 2D flow. The work of Scaraggi et al. [5,6] inspired the considered geometry model consisting of an upper stationary micro-textured surface with several rectangular cavities and a flat moving surface at the bottom. The influence of several geometric parameters, such as the cavity depth and width, on the drag force has been studied. A thorough analysis of the shear and pressure forces and of the velocity profiles inside the gap is provided, showing how these quantities are influenced by the geometry of the surface micro-texturing. The analysis of the numerical results provides an explanation for the maximum drag reduction achievable with a single phase flow. Furthermore, the study of flow velocity profiles and pressure distributions highlights that three regions exist, depending on the dimple depth, in which a different flow dynamics occurs and the cavities have a different influence on the drag force. Finally, the analysis demonstrates that an “optimal” value of the dimple depth exists for which the pressure achieves a minimum value and the probability of cavitation is maximised. The content of the paper is organised as follows. Section 2 provides the description of the CFD model and of the computational domain with its boundary conditions. A grid refinement study is employed to determine the grid size guaranteeing solution independence of the mesh resolution. Section 3 is devoted to the presentation of the results. Firstly, the influence of geometric parameters on drag force is discussed. Then, the drag forces’ variation with respect to all of the considered geometric parameters is analysed by studying the velocity profiles through the centre of the micro-cavities. Furthermore, the pressure and shearstress distributions along the micro-textured surface are studied in order to explain physical phenomena causing the drag reduction. Concluding remarks are provided in the last section.
163
164
G. Caramia et al. / Tribology International 88 (2015) 162–169
Table 1 Used values for parameter h in μm. gap ¼ 1 μm
gap ¼ 2 μm
gap ¼ 5 μm
0 0.1 0.2 0.335 0.4 0.5 0.6375 0.75 1 1.25 2.5 5 10 20 30 50 100
0 0.2 0.4 0.67 0.8 1 1.275 1.5 2 2.5 5 10 20 40 60 100 ND
0 0.5 1 1.675 2 2.5 3.1875 3.75 5 6.25 12.5 25 50 100 ND ND ND
Fig. 4. Local view of the computational grid near one of the three cavities.
the gap, 1, 2, and 5 μm, which correspond to the range of the gap values measured in the experiments of [5]. In order to consider different percentage values of texture area density (TAD), three values of D are used: 25, 50 and 100 μm. Finally, the employed values of the depth h are listed in Table 1. Notice that values of h greater than 100 μm are not considered because of the experimental difficulty in realising them. Firstly, a set of h values have been defined for gap ¼ 1 μm; the set of h values for the two remaining values of gap has been chosen by scaling the first set of h values proportionally. 2.2. Governing equations Due to large cavity depth, non-negligible pressure gradients in the wall-normal direction may occur; therefore, the Reynolds equation could provide an incorrect prediction of the pressure distribution (see, e.g., [13] and [20]). Accordingly, the flow induced by the motion of the wall is computed by solving the steady twodimensional Navier–Stokes equations for an incompressible fluid (oil). The steady non-dimensional Navier–Stokes equations can be written as ðu U ∇Þu ¼ ∇p þ
1 2 ∇ u; Re
∇ U u ¼ 0; where u is the velocity vector, p is the pressure, ρ is the density, and Re indicates the Reynolds number which is defined as Re ¼
ρU gap ; η
ð1Þ
where η is the dynamic viscosity. Assuming η ¼ 0:048 Pa s and ρ ¼ 960 kg=m3 , one has Re ¼ 0:01; 0:02; 0:05 for gap ¼ 1; 2; 5 μm, respectively. The Navier–Stokes equations are solved by the software FLUENTs employing the SIMPLE [21] algorithm with secondorder-accuracy in space. The residual of the non-dimensional Navier–Stokes equations is reduced to about 10 9. 2.3. Computational grid The grid generation software ICEMCFDs was used to discretise the computational domain by an unstructured triangular grid, as shown in Fig. 4. Using the geometry characterised by gap ¼ 1 μm,
Fig. 5. Non-dimensional drag force and minimal dimension of the computational cell (Mindim) versus the number of cells.
D ¼ 25 μm, and h ¼ 1 μm, a grid refinement study has been performed. Defining Fx as the component along the horizontal x direction (see Figs. 2 and 3) of the force acting on the textured surface, we consider the following non-dimensional parameter: T¼
Fx
ηUL
;
ð2Þ
where L is the unitary length. T has been considered as a suitable parameter to test the grid convergence. The minimal dimension of the computational cell at wall (Mindim), see Fig. 4, is employed to control the size of the grid (global number of cell) automatically generated by the software ICEMCFDs . Varying this reference length it was possible to obtain different grid sizes as shown in Fig. 5. Such a figure also shows the computed corresponding values of T: for grid size grater than about 0.3 million nodes, corresponding to Mindim ¼ 0:08 μm, the value of T converged within a satisfactory tolerance of about 0.01% to a constant value. Therefore, Mindim ¼ 0:08 μm was chosen and employed in all of the simulations.
3. Results and discussion 3.1. Maximum drag reduction The behaviour of the non-dimensional force T versus the depth of the cavities is firstly discussed. Fig. 6(a) shows the values of T versus h for three different cavity widths and gap ¼ 1 μm and Fig. 6(b) shows a local view of the same distribution for low values of h. In fact, in such a region, the curves are characterised by two segments with high slope, separated by a quasi-horizontal plateau. Finally, a second plateau is obtained for high values of h (see Fig. 6(a)). Similar results were presented by Jing et al. [16] only for low h values. Also Scaraggi [22]
G. Caramia et al. / Tribology International 88 (2015) 162–169
165
Fig. 6. Variation of T versus the cavity depth for gap ¼ 1 μm and three values of D: (a) global view, (b) local view.
obtained similar results using a model based on the Bruggeman effective medium approach. The same behaviour is obtained for different values of the gap, i.e., gap ¼ 2 μm and gap ¼ 5 μm (not shown). Therefore, all results indicate that there are two regions in which the force is independent of the cavity depth. Moreover, the force appears to decrease when the width of the cavities, D, increases. A more thorough analysis of the behaviour of these curves will be provided in the next section whereas now the issue of the maximum drag reduction achievable will be addressed with reference to the experimental data obtained by Scaraggi et al. [5]. At this purpose, Fig. 7 provides the reduction (percentage) of the drag force obtained experimentally by Scaraggi et al. [5] varying the cavity depth for a micro-textured surface with D ¼ 100 μm, TAD about equal to 30%, and ηU ¼ 0:02941 Pa m (which corresponds to the values U¼ 0.6128 m/s and dynamic viscosity η ¼ 0:048 Pa s employed in our computations). In Fig. 7 also, the uncertainty range of the depth is provided. In order to explain this remarkable drag-reduction effect, about equal to 80%, we have performed several numerical simulations. Keeping the same textured surface length, ltot (see Fig. 2), and varying the micro-cavity width D leads to TAD variation; three TAD values were considered: 0.07, 0.14 and 0.29. Given a couple of parameter values, gap and D, from the distribution of T versus h (see, e.g., Fig. 6(a)), it is possible to compute the maximum drag reduction achievable, defined as Tðh ¼ 0Þ Tðh ¼ 100Þ : Tðh ¼ 0Þ The results are shown in Fig. 8 where, for the three considered values of the gap, the maximum drag reduction is provided versus TAD. It appears that the maximum reduction is about 26% for a TAD of 0.29 and gap ¼ 1 μm; higher reduction values could be achieved only for higher TAD values. On the other hand, experimental results in Fig. 7 indicate a drag reduction of about 80% for a TAD of about 0.3. It is worth remembering that experiments and computations were performed under different conditions, namely, the load is imposed in experiments, whereas the gap is assigned in the computations. However, since the gap variation during the experiments of Scaraggi et al. [5] was very small, a two-phase flow analysis would be suitable to evaluate the contribution of cavitation or air bubble inclusion to load and friction generation. Finally, Fig. 9 provides the distribution of T versus the gap for h ¼ 100 μm and different values of D: these curves show a diminishing slope as the gap magnitude increases. Moreover, the influence of the cavity width on T reduction decreases as the gap magnitude increases.
Fig. 7. Experimental results [5]: percentage reduction of T versus h.
Fig. 8. Numerical results: percentage reduction of T versus the texture area density.
Fig. 9. Variation of T versus the gap for three values of D and h ¼ 100 μm.
3.2. Analysis of the drag force behaviour In order to study the forces acting on the textured surfaces, we analyse pressure forces and shear-stress forces, separately. Firstly,
166
G. Caramia et al. / Tribology International 88 (2015) 162–169
the zones inside the cavities will be considered. The topology of the flow is very similar in the three cavities considered in our model; therefore, only one cavity will be analysed in this section (the central one). Fig. 10 provides the velocity profiles along the centre of the cavity: the velocity is scaled by the reference slidingwall velocity (see Figs. 2 and 3) and the vertical distance from the wall, y, is scaled by the length hþ g. For clarity reasons, only some profiles corresponding to selected values of h are plotted. In this figure, the h ¼ 0 μm linear profile, which corresponds to a Couette flow, is also presented. For increasing values of h, the profiles initially resemble a Couette–Poiseuille flow in the presence of an adverse pressure gradient, and finally they tend toward a null slope at the cavity bottom (y ¼hþg). From this diagram, one can also notice the presence of a recirculation vortex starting from sufficiently high values of h for which negative velocity values are obtained. To better understand the velocity profile influence on shear stresses, Fig. 11 shows the y-derivative of the velocity profile evaluated at the cavity bottom versus the depth h. Along this curve, three different regions can be identified for increasing h: 1. the region in which the derivative is negative (velocity is positive) with decreasing absolute value; there is no recirculation vortex and the shear stress is decreasing though conserving the same direction of the sliding-wall velocity (see the corresponding velocity contours and streamlines in Fig. 12(a)); 2. the region in which the derivative is positive (velocity is negative) with increasing absolute value; a recirculation vortex is present and the shear stress is increasing with direction opposite to the sliding-wall velocity (see the corresponding velocity contours and streamlines in Fig. 12(b));
Fig. 12. Velocity magnitude contours with streamlines (gap ¼ 1 μm, D ¼ 50 μm): (a) h ¼ 1 μm, (b) h ¼ 2:5 μm, (c) h ¼ 30 μm.
1
0.8
y*
0.6
Fig. 13. Contributions of the shear-stress and pressure to T inside the cavities versus h (gap ¼ 1 μm, D ¼ 50 μm).
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
V* Fig. 10. Velocity profile at the centre of the cavity (gap ¼ 1 μm, D ¼ 50 μm). For plotting reasons the profiles are not in the same scale.
Fig. 11. Slope of the velocity profile at the bottom of the cavities (gap ¼ 1 μm, D ¼ 50 μm).
3. the region in which the derivative is positive (velocity is negative) and its absolute value is decreasing towards zero and the shear stress is decreasing with direction opposite to the sliding-wall velocity. The profile tends to show a very small velocity-magnitude region between the cavity bottom and the recirculation vortex (see the corresponding velocity contours and streamlines in Fig. 12(c)). In the first region, the shear stress applied to the cavity bottom contributes to increase the value of T; whereas, in the second and third regions, it contributes to decrease the value of T. The streamlines reported in Fig. 12(a)–(c) are representative of the flow configurations corresponding to the three regions identified above, respectively. Fig. 12(a) corresponds to h/D¼ 0.02 and demonstrates that also for small values of the aspect ratio, a recirculation region is already found. Fig. 12(b) (h/D¼ 0.05) shows that, for higher values of the ratio h/D, the flow presents a clear two-dimensional configuration with a large vortex that, with its dissipation mechanism, influences the pressure gradient inside the cavity. Finally, Fig. 12(c) (h/D ¼0.6) shows that, for higher values of the aspect ratio, the velocity at the cavity bottom, above the vortex, tends to become negligible. Figs. 13 and 14 show the variation of the shear-stress and pressure contributions to T versus h. Only the surface inside the cavities has been considered to compute these contributions: the cavity bottom contributes with
G. Caramia et al. / Tribology International 88 (2015) 162–169
the shear-stress, the two remaining cavity boundaries contribute with a pressure force (sidewall effects). It appears that the two contributions have mostly opposite sign, the pressure one being larger and always positive (see also [20]). The curve representing the combined effects of shear and pressure forces presents a first zone in which the diminishing shear stress contribution is predominant, as shown for small values of h in Fig. 14. For higher values of h, the curve follows the predominant behaviour of the pressure distribution. The pressure contribution reflects the pressure increase across the cavities as shown in Fig. 15 where a sketch of the textured surface geometry is provided: the corresponding pressure distribution is obtained plotting the pressure value along the textured surface versus x. Lengths are scaled by the textured surface total length, ltot, whereas pressures are scaled by the pressure at the leading edge (left) of the textured surface. From left to right, in the direction of the sliding-wall velocity, there is a
Fig. 14. Local view of Fig. 13 for low values of h.
5 4
P/Pin
3 2 1 0 -1 -2 0
0.2
0.4
0.6
0.8
1
x/ltot Fig. 15. Distribution of the pressure along the textured surface (h ¼ 1 μm, gap ¼ 1 μm, D ¼ 50 μm).
167
pressure drop due to the “blockage” effect of the pin holder. In fact, in the region far from the pin holder, a weak positive pressure gradient is predicted, consistent with the exact solution of the 2D Navier–Stokes equations with the same boundary conditions at top and bottom (neglecting the vertical component of the velocity). This pressure gradient creates a pressure difference between the left and the right face of the pin holder, the left face having an higher pressure with respect to the right face. Most importantly, close to the inlet of the narrow gap a pressure increase is observed, generated by the action of the moving wall on the fluid flowing past the sharp edge of the textured surface. Such a local pressure gradient is necessary to balance the viscous forces, the inertial terms being negligible due to the low Reynolds number. A similar behaviour is observed at the outlet of the narrow gap. Here, the pressure achieves a minimum value just after the corner and then increases to reach the quasi-constant “far field” value. Therefore, flowing through the gap, the fluid must experience a global pressure drop from the inlet to the outlet of the narrow channel. In particular, in the zone between the gap inlet and the first cavity the pressure decreases in agreement with the equations of lubrication. At the inlet of the first cavity, the gap undergoes a step increase, which necessarily leads to an analogous step change in the pressure gradient, as indeed expected on the basis of lubrication theory. The pressure, then, increases until at the outlet of the cavity a step decrease of the pressure gradient is observed caused by the sharp reduction of the gap. The pressure then almost linearly decreases until the second cavity is reached and the process is repeated. Pressure contours are provided in Fig. 16, showing in particular the variation of the pressure around the first corner of the second cavity and the pressure gradient inside the cavity (corresponding to the flow in Fig. 12a)). The pressure distribution is clearly twodimensional, therefore an accurate local analysis of the force distribution, cannot be obtained using the Reynolds equation, especially for higher values the cavity depth. Fig. 17 shows the pressure distribution along the textured surface plotted for several values of the depth h. It is noteworthy that, increasing h, the pressure difference between the right and the left wall of each cavity initially grows, then reaches a maximum, and finally decreases. The maximum pressure jump in the cavity corresponds to the appearance of the recirculation vortex which, with its dissipation, reduces the pressure difference between the vertical walls of the cavity. Such an optimal condition is obtained for h 0:5 μm (h=D 0:01). This behaviour has two important effects. The first one is related to its influence on the flow in the regions outside the cavities. In fact, as a consequence, also the contribution of the shear-stress to the drag in the regions outside the cavities has a non-monotone behaviour, as shown in Figs. 18 and 19. The maximum shear-stress value is reached for the same value of h for which the maximum pressure “jump” is obtained in the cavities. The second important effect is related to the possibility of fluid phase change when the pressure locally reaches the saturation value. Since
Fig. 16. Pressure contours near the first corner of the second cavity (h ¼ 1 μm, gap ¼ 1 μm, D ¼ 50 μm).
168
G. Caramia et al. / Tribology International 88 (2015) 162–169
whereas the latter contribute by a shear force term. In particular, observing Figs. 6(b), 14 and 19 it is possible to note that, for increasing values of h, the initial decreasing inside-cavity shearstress contribution determines the decreasing global drag behaviour. Then, the increasing inside-cavity pressure contribution balances the decreasing global (inside-cavity and outside-cavity) shear-stress contribution creating a quasi-constant plateau for the global drag force. Finally (see Figs. 6(a), 13 and 18), for high values of h, the influence of the inside-cavity shear-stress contribution vanishes and the global drag force behaviour is mainly determined by the outside-cavity contribution.
3
P/Pin
2
1
0
-1
4. Conclusions 0
0.2
0.4
0.6
0.8
1
x/ltot Fig. 17. Distribution of the pressure along the textured surface versus h (gap ¼ 1 μm, D ¼ 50 μm): only eight values of h are shown.
Fig. 18. Contribution of the shear-stress to T outside the cavities versus h (gap ¼ 1 μm, D ¼ 50 μm).
Fig. 19. Local view of Fig. 18 for low values of h.
the pressure gradient inside the cavities has a maximum, a minimum value of the pressure is obtained along the textured surface, as shown in Fig. 17 at x=ltot 0:9. Therefore, in a real flow, it exists a range of intermediate (optimal) values of h, for which cavitation would be possible. This behaviour would agree with experimental measurements provided by [5] which present a minimum of the friction coefficient in correspondence of a particular dimples depth value, with 80% drag reduction. Employing the present numerical single-phase model, in the absence of cavitation, such a drag reduction cannot be obtained. The global drag force (see Fig. 6(b)) is the sum of the contributions due to the surfaces inside (Fig. 14) and outside (Fig. 19) the cavities. The former manly contribute by a pressure force term
The hydrodynamic lubrication between parallel micro-textured surfaces has been studied for isothermal incompressible steady two dimensional flow conditions. A wide variety of geometries characterised by different micro-cavity depth and width, and different gap values have been considered in order to study the influence of these parameters on the non-dimensional drag force magnitude T. The variation of T versus the micro-cavity depth, for several cavity widths and gaps has been studied: the presence of two ranges of h values has been observed in which the drag is independent of the cavity depth. The variation of T versus the gap, with several cavity depths and widths has been analysed as well, showing how the influence of the cavity depth on T reduction decreases as the gap magnitude increases. Furthermore, we have shown that three regions exist, depending on the cavity depth, in which a different flow dynamics occurs and the cavities have a different influence on the drag force. A detailed analysis of flow velocity profiles and pressure distributions has been performed to study the forces acting on the textured surfaces: the pressure–force contribution has been separated from the shear-stress contribution as well as the insidecavity surface part has been separated from the outside-cavity part. Concerning the inside-cavity surface part, the velocity profiles through the cavity geometric center, with several cavity depths, has been analysed. The distributions of shear-stress and pressure contributions to T versus h have been discussed, showing how the two contributions have mostly opposite behaviour, the pressure contribution presenting a higher magnitude. Concerning the outside-cavity surface part, the pressure variation along the textured surface versus h and the shear-stress contribution to T versus h have been discussed. We have observed that an “optimal” value of the depth h exists for which the pressure reaches a minimum and the probability of cavitation is maximised. Finally, the comparison between numerical and experimental results shows a substantial difference in the maximum drag reduction achievable for a given texture area density. In fact, the present analysis demonstrates that a maximum drag reduction of about 20% can be achieved with a single-phase lubrication fluid, whereas the experimental data provide an 80% drag reduction. In order to investigate in more detail this aspect, future work will be dedicated to a two-phase analysis evaluating the influence of cavitation or air bubble inclusion on the load and friction generation.
References [1] Etsion I. State of the art in laser surface texturing. J Tribol 2005;127(1):248–53. [2] Hamilton DB, Walowit JA, Allen CM. A theory of lubrication by microirregularities. J Basic Eng 1966;88(1):177. [3] Anno JN, Walowit J, Allen C. Microasperity lubrication. J Lubr Technol 1968;90 (1):351. [4] Marian V, Gabriel D, Knoll G, Filippone S. Theoretical and experimental analysis of a laser textured thrust bearing. Tribol Lett 2011;44(3):335–43.
G. Caramia et al. / Tribology International 88 (2015) 162–169
[5] Scaraggi M, Mezzapesa FP, Carbone G, Ancona A, Sorgente D, Lugarà PM. Minimize friction of lubricated laser-microtextured-surfaces by tuning microholes depth. Tribol Int 2014;75(7):123–7. [6] Scaraggi M, Mezzapesa F, Carbone G, Ancona A, Tricarico L. Friction properties of lubricated laser-microtextured-surfaces: an experimental study from boundary to hydrodynamic-lubrication. Tribol Lett 2013;49(1):117–25. [7] Fowell M, Olver AV, Gosman AD, Spikes HA, Pegg I. Entrainment and inlet suction: two mechanisms of hydrodynamic lubrication in textured bearings. J Tribol 2007;129(2):336–47. [8] Giacopini M, Fowell M, Dini D, Strozzi A. A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation. J Tribol 2010;132(4) 041702–1–12. [9] Pei S, Ma S, Xu H, Wang F, Zhang Y. A multiscale method of modeling surface texture in hydrodynamic regime. Tribol Int 2011;44(12):1810–8. [10] Scaraggi M. Lubrication of textured surfaces: a general theory for flow and shear stress factors. Phys Rev E: Stat Nonlinear Soft Matter Phys 2012;86(2) 026314–1–12. [11] Ma C, Zhu H. An optimum design model for textured surface with ellipticalshape dimples under hydrodynamic lubrication. Tribol Int 2011;44(9):987–95. http://dx.doi.org/10.1016/j.triboint.2011.04.005. [12] de Kraker A, van Ostayen RAJ, Rixen DJ. Development of a texture averaged Reynolds equation. Tribol Int 2010;43(11):2100–9. [13] Dobrica MB, Fillon M. About the validity of Reynolds equation and inertia effects in textured sliders of infinite width. Proc Inst Mech Eng Part J: J Eng Tribol 2009;223(1):69–78.
169
[14] Ausas R, Ragot P, Leiva J, Jai M, Bayada G, Buscaglia GC. The impact of the cavitation model in the analysis of microtextured lubricated journal bearings. J Tribol 2007;129(4):868–75. [15] Sahlin F, Glavatskih S, Almqvist T, Larsson R. Two-dimensional CFD-analysis of micro-patterned surfaces in hydrodynamic lubrication. J Tribol 2005;127 (1):96–102. [16] Jing H, Fang L, Jiapeng S, Shirong G. Hydrodynamic lubrication of microdimple textured surface using three-dimensional CFD. Tribol Trans 2010;53 (6):860–70. [17] Arghir M, Roucou N, Helene M, Frene I. Theoretical analysis of the incompressible laminar flow in a macro-roughness cell. J Tribol 2003;125(2):309–18. [18] Brajdic-Mitidieri P, Gosman AD, Ioannides S, Spikes HA. CFD analysis of a low friction pocketed PAD bearing. In: Proceedings of the ASME/STLE international joint tribology conference, IJTC; 2004. p. 737–47. [19] Cupillard S, Glavatskih S, Cervantes MJ. Computational fluid dynamics analysis of a journal bearing with surface texturing. Proc Inst Mech Eng Part J: J Eng Tribol 2008;222(2):97–107. [20] Li J, Chen H. Evaluation on applicability of Reynolds equation for squared transverse roughness compared to CFD. J Tribol 2007;129(4):963–7. [21] Patankar S, Spalding D. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 1972;15 (10):1787–806. [22] Scaraggi M. Textured surface hydrodynamic lubrication: discussion. Tribol Lett 2012;48(3):375–91.
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Hydrodynamic lubrication of micro-textured surfaces: Two dimensional CFD-analysis Giovanni Caramia n, Giuseppe Carbone, Pietro De Palma Dipartimento di Meccanica, Matematica e Management (DMMM), Politecnico di Bari, Via Re David 200, 70125 Bari, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 6 October 2014 Received in revised form 11 March 2015 Accepted 15 March 2015 Available online 24 March 2015
This paper provides a numerical study of the hydrodynamic lubrication between two parallel surfaces with micro-texturing. The two-dimensional Navier–Stokes equations for an isothermal incompressible steady flow have been considered as a suitable model. A wide variety of geometries characterised by different micro-cavity depth and width, and different gap values have been analysed in order to study the influence of these parameters on the drag force magnitude. A detailed analysis of flow velocity profiles and pressure distributions has been performed to study the forces acting on the textured surface, providing an explanation for the maximum drag reduction achievable with a single-phase lubrication fluid. Furthermore, results indicate that three regions exist, depending on the cavity depth, in which a different flow dynamics occurs and the cavities have a different influence on the drag force. Finally, an “optimal” value of the depth has been found, for which the pressure reaches a minimum value and the probability of cavitation is maximised. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Surface texturing Friction Lubrication Cavitation
1. Introduction Recent technological improvements in surface texturing techniques allow one to exploit the properties of micro-structured surfaces in many engineering applications [1]. Positive effects of artificial surface texturing in lubrication systems concerning load capacity, wear resistance, and friction properties have been investigated starting from the pioneering work of Hamilton et al. [2] and Anno et al. [3]. The present work analyses the influence of surface micro-texturing on the hydrodynamic lubrication properties of two parallel sliding plates. Experimental results in this field [4] show that significant improvements in hydrodynamic performance can be achieved by surface texturing. Recently, with the aim of minimising the friction coefficient, Scaraggi et al. [5,6] experimentally investigated micro-cavity and micro-groove depth influence on friction properties. On the other hand, several numerical techniques have been developed to predict the performance of lubrication systems: from model based on the Reynolds equation to the solution of the more complex Navier–Stokes equations. Fowell et al. [7] highlighted one of the basic mechanisms of textured bearings analysing the contribution of inlet suction to the performance of near-parallel, internally pocketed bearings. Giacopini et al. [8] proposed a mass-conserving formulation of the Reynolds equation to deal with cavitation in lubricated contacts. In order to
n
Corresponding author. E-mail address: [email protected] (G. Caramia).
http://dx.doi.org/10.1016/j.triboint.2015.03.019 0301-679X/& 2015 Elsevier Ltd. All rights reserved.
increase the computational speed with respect to traditional numerical methods, Pei et al. [9] proposed the finite cell method for multi-scale surface texture hydrodynamic lubrication. The mean field approach has been also recently developed to find the optimal geometry of micro-holes which maximises load carrying capacity of the junction or minimises friction [10]. Ma and Zhu [11] solved the Reynolds equation by a finite difference method to find an optimum design model for textured surface with elliptical-shape dimples. De Kraker et al. [12] developed a multi-scale method in which the Navier–Stokes equations’ results were employed to derive suitable flow factors to be used in a “texture averaged Reynolds equation”. Dobrica and Fillon [13] showed the importance of the dimple aspect ratio to establish whether a given configuration could be modelled using the Reynolds equation and to evaluate the expected corresponding accuracy loss. Ausas et al. [14] analysed the impact of the cavitation model on numerical predictions of lubricated journal bearings showing the limits of using Reynolds’ model when simulating flows in the presence of micro-textured surfaces. The finite volume approach was used by Sahlin et al. [15] to solve the incompressible Navier–Stokes equations, providing a twodimensional CFD analysis of micro-patterned surfaces. A three dimensional CFD analysis was then performed by Jing et al. [16] and by Arghir et al. [17] who investigated the net pressure gain as a pure inertia effect due to the combined action of the macroroughness and of Reynolds number. A cavitation phenomenon was considered in the CFD analysis of a low friction pocketed pad bearing by Brjdic-Mitidieri et al. [18] and in the CFD analysis of a journal bearing with surface texturing by Cupillard et al. [19].
G. Caramia et al. / Tribology International 88 (2015) 162–169
0.896 mm
Lubricant bath
inviscid wall
inviscid wall
wall moving wall (disk)
y
16.896 mm x Fig. 2. Computational domain and boundary conditions.
348.125 µ m
348.125 µ m
h
100 µ m
D gap
y
x
Fig. 1. Experimental setup (schematic) employed for friction measurements in [5].
0.6128 m/s
Fig. 3. Local view of the computational domain.
100 µ m
1.5 mm
Holder periodic
The model geometry is inspired to the experimental set-up employed in [5,6], namely, the pin-on-disk tribometer for friction measurements shown in Fig. 1. Scaraggi et al. [5,6] performed several tests with different laser-textured bearing balls assembled in the pin holder. The contact surface was micro-textured with a square lattice of micro-cavities whose diameter, depth, and spacing were controlled during the texturing process. They reported a remarkable drag-reduction effect for some particular combinations of the geometric parameters and flow conditions. Unfortunately, they could not ascertain experimentally the origin of such a phenomenon. One of
wall
2.1. Model geometry
wall
2. CFD-model
the aims of the present work is to provide a theoretical explanation for this behaviour by using a numerical approach. However, a huge computational cost would be needed in order to simulate the real three-dimensional flow. For this reason, we have adopted a simplified two-dimensional geometry which allows us to study as well the fundamental phenomena involved in this particular flow configuration. Therefore, we have considered a two-dimensional section of the ball holder, shown in Fig. 2, with a reduced length, ltot, with respect to the original experimental model. As shown in Fig. 2, a flat wall, moving with velocity U¼0.6128 m/s, represents the rotating disk, whereas the holder supporting the textured surface is superposed to the wall at a distance equal to the film thickness gap (see Fig. 3). The flat-wall moving velocity was calculated on the basis of experimental data as explained in detail in Section 3. The texturing is represented by three equally-spaced rectangular cavities with depth h and width D. Unlike previous studies available in the literature, as far as the authors know, periodic boundary conditions are imposed at the two vertical boundaries of the computational domain, which are located very far from the holder (see Figs. 2 and 3). This allows us to model the isolated pin holder, avoiding spurious fluid interactions, also taking into account the inlet and outlet effects of the fluid at the extrema of the textured surface. For all of the remaining boundaries the no-slip condition is imposed, except for the upper horizontal surface along which the slip flow condition (inviscid wall) is adopted. In order to investigate the effects of the geometric parameters on the drag force on the textured surface, a parametric study based on suitable values of the three parameters gap, D, and h is performed. A comment is in order about the analysis of the results and the comparison with the available experimental data [5,6]. There is a fundamental difference which characterises the experimental and the numerical approach. In the experimental set-up, it is straightforward to assign and control the normal load exerted on the pin holder; therefore the value of the gap is determined by the balance between the fluid forces acting perpendicularly to the wall and the load. On the other hand, in the numerical approach, a great simplification is achieved assigning the value of the gap which allows one to employ a given stationary computational grid. For this reason, we provide the analysis of the drag force for three values of
periodic
The aim of the present paper is to study, by a CFD analysis, the fluid mechanics of the lubrication between two parallel surfaces for an incompressible steady 2D flow. The work of Scaraggi et al. [5,6] inspired the considered geometry model consisting of an upper stationary micro-textured surface with several rectangular cavities and a flat moving surface at the bottom. The influence of several geometric parameters, such as the cavity depth and width, on the drag force has been studied. A thorough analysis of the shear and pressure forces and of the velocity profiles inside the gap is provided, showing how these quantities are influenced by the geometry of the surface micro-texturing. The analysis of the numerical results provides an explanation for the maximum drag reduction achievable with a single phase flow. Furthermore, the study of flow velocity profiles and pressure distributions highlights that three regions exist, depending on the dimple depth, in which a different flow dynamics occurs and the cavities have a different influence on the drag force. Finally, the analysis demonstrates that an “optimal” value of the dimple depth exists for which the pressure achieves a minimum value and the probability of cavitation is maximised. The content of the paper is organised as follows. Section 2 provides the description of the CFD model and of the computational domain with its boundary conditions. A grid refinement study is employed to determine the grid size guaranteeing solution independence of the mesh resolution. Section 3 is devoted to the presentation of the results. Firstly, the influence of geometric parameters on drag force is discussed. Then, the drag forces’ variation with respect to all of the considered geometric parameters is analysed by studying the velocity profiles through the centre of the micro-cavities. Furthermore, the pressure and shearstress distributions along the micro-textured surface are studied in order to explain physical phenomena causing the drag reduction. Concluding remarks are provided in the last section.
163
164
G. Caramia et al. / Tribology International 88 (2015) 162–169
Table 1 Used values for parameter h in μm. gap ¼ 1 μm
gap ¼ 2 μm
gap ¼ 5 μm
0 0.1 0.2 0.335 0.4 0.5 0.6375 0.75 1 1.25 2.5 5 10 20 30 50 100
0 0.2 0.4 0.67 0.8 1 1.275 1.5 2 2.5 5 10 20 40 60 100 ND
0 0.5 1 1.675 2 2.5 3.1875 3.75 5 6.25 12.5 25 50 100 ND ND ND
Fig. 4. Local view of the computational grid near one of the three cavities.
the gap, 1, 2, and 5 μm, which correspond to the range of the gap values measured in the experiments of [5]. In order to consider different percentage values of texture area density (TAD), three values of D are used: 25, 50 and 100 μm. Finally, the employed values of the depth h are listed in Table 1. Notice that values of h greater than 100 μm are not considered because of the experimental difficulty in realising them. Firstly, a set of h values have been defined for gap ¼ 1 μm; the set of h values for the two remaining values of gap has been chosen by scaling the first set of h values proportionally. 2.2. Governing equations Due to large cavity depth, non-negligible pressure gradients in the wall-normal direction may occur; therefore, the Reynolds equation could provide an incorrect prediction of the pressure distribution (see, e.g., [13] and [20]). Accordingly, the flow induced by the motion of the wall is computed by solving the steady twodimensional Navier–Stokes equations for an incompressible fluid (oil). The steady non-dimensional Navier–Stokes equations can be written as ðu U ∇Þu ¼ ∇p þ
1 2 ∇ u; Re
∇ U u ¼ 0; where u is the velocity vector, p is the pressure, ρ is the density, and Re indicates the Reynolds number which is defined as Re ¼
ρU gap ; η
ð1Þ
where η is the dynamic viscosity. Assuming η ¼ 0:048 Pa s and ρ ¼ 960 kg=m3 , one has Re ¼ 0:01; 0:02; 0:05 for gap ¼ 1; 2; 5 μm, respectively. The Navier–Stokes equations are solved by the software FLUENTs employing the SIMPLE [21] algorithm with secondorder-accuracy in space. The residual of the non-dimensional Navier–Stokes equations is reduced to about 10 9. 2.3. Computational grid The grid generation software ICEMCFDs was used to discretise the computational domain by an unstructured triangular grid, as shown in Fig. 4. Using the geometry characterised by gap ¼ 1 μm,
Fig. 5. Non-dimensional drag force and minimal dimension of the computational cell (Mindim) versus the number of cells.
D ¼ 25 μm, and h ¼ 1 μm, a grid refinement study has been performed. Defining Fx as the component along the horizontal x direction (see Figs. 2 and 3) of the force acting on the textured surface, we consider the following non-dimensional parameter: T¼
Fx
ηUL
;
ð2Þ
where L is the unitary length. T has been considered as a suitable parameter to test the grid convergence. The minimal dimension of the computational cell at wall (Mindim), see Fig. 4, is employed to control the size of the grid (global number of cell) automatically generated by the software ICEMCFDs . Varying this reference length it was possible to obtain different grid sizes as shown in Fig. 5. Such a figure also shows the computed corresponding values of T: for grid size grater than about 0.3 million nodes, corresponding to Mindim ¼ 0:08 μm, the value of T converged within a satisfactory tolerance of about 0.01% to a constant value. Therefore, Mindim ¼ 0:08 μm was chosen and employed in all of the simulations.
3. Results and discussion 3.1. Maximum drag reduction The behaviour of the non-dimensional force T versus the depth of the cavities is firstly discussed. Fig. 6(a) shows the values of T versus h for three different cavity widths and gap ¼ 1 μm and Fig. 6(b) shows a local view of the same distribution for low values of h. In fact, in such a region, the curves are characterised by two segments with high slope, separated by a quasi-horizontal plateau. Finally, a second plateau is obtained for high values of h (see Fig. 6(a)). Similar results were presented by Jing et al. [16] only for low h values. Also Scaraggi [22]
G. Caramia et al. / Tribology International 88 (2015) 162–169
165
Fig. 6. Variation of T versus the cavity depth for gap ¼ 1 μm and three values of D: (a) global view, (b) local view.
obtained similar results using a model based on the Bruggeman effective medium approach. The same behaviour is obtained for different values of the gap, i.e., gap ¼ 2 μm and gap ¼ 5 μm (not shown). Therefore, all results indicate that there are two regions in which the force is independent of the cavity depth. Moreover, the force appears to decrease when the width of the cavities, D, increases. A more thorough analysis of the behaviour of these curves will be provided in the next section whereas now the issue of the maximum drag reduction achievable will be addressed with reference to the experimental data obtained by Scaraggi et al. [5]. At this purpose, Fig. 7 provides the reduction (percentage) of the drag force obtained experimentally by Scaraggi et al. [5] varying the cavity depth for a micro-textured surface with D ¼ 100 μm, TAD about equal to 30%, and ηU ¼ 0:02941 Pa m (which corresponds to the values U¼ 0.6128 m/s and dynamic viscosity η ¼ 0:048 Pa s employed in our computations). In Fig. 7 also, the uncertainty range of the depth is provided. In order to explain this remarkable drag-reduction effect, about equal to 80%, we have performed several numerical simulations. Keeping the same textured surface length, ltot (see Fig. 2), and varying the micro-cavity width D leads to TAD variation; three TAD values were considered: 0.07, 0.14 and 0.29. Given a couple of parameter values, gap and D, from the distribution of T versus h (see, e.g., Fig. 6(a)), it is possible to compute the maximum drag reduction achievable, defined as Tðh ¼ 0Þ Tðh ¼ 100Þ : Tðh ¼ 0Þ The results are shown in Fig. 8 where, for the three considered values of the gap, the maximum drag reduction is provided versus TAD. It appears that the maximum reduction is about 26% for a TAD of 0.29 and gap ¼ 1 μm; higher reduction values could be achieved only for higher TAD values. On the other hand, experimental results in Fig. 7 indicate a drag reduction of about 80% for a TAD of about 0.3. It is worth remembering that experiments and computations were performed under different conditions, namely, the load is imposed in experiments, whereas the gap is assigned in the computations. However, since the gap variation during the experiments of Scaraggi et al. [5] was very small, a two-phase flow analysis would be suitable to evaluate the contribution of cavitation or air bubble inclusion to load and friction generation. Finally, Fig. 9 provides the distribution of T versus the gap for h ¼ 100 μm and different values of D: these curves show a diminishing slope as the gap magnitude increases. Moreover, the influence of the cavity width on T reduction decreases as the gap magnitude increases.
Fig. 7. Experimental results [5]: percentage reduction of T versus h.
Fig. 8. Numerical results: percentage reduction of T versus the texture area density.
Fig. 9. Variation of T versus the gap for three values of D and h ¼ 100 μm.
3.2. Analysis of the drag force behaviour In order to study the forces acting on the textured surfaces, we analyse pressure forces and shear-stress forces, separately. Firstly,
166
G. Caramia et al. / Tribology International 88 (2015) 162–169
the zones inside the cavities will be considered. The topology of the flow is very similar in the three cavities considered in our model; therefore, only one cavity will be analysed in this section (the central one). Fig. 10 provides the velocity profiles along the centre of the cavity: the velocity is scaled by the reference slidingwall velocity (see Figs. 2 and 3) and the vertical distance from the wall, y, is scaled by the length hþ g. For clarity reasons, only some profiles corresponding to selected values of h are plotted. In this figure, the h ¼ 0 μm linear profile, which corresponds to a Couette flow, is also presented. For increasing values of h, the profiles initially resemble a Couette–Poiseuille flow in the presence of an adverse pressure gradient, and finally they tend toward a null slope at the cavity bottom (y ¼hþg). From this diagram, one can also notice the presence of a recirculation vortex starting from sufficiently high values of h for which negative velocity values are obtained. To better understand the velocity profile influence on shear stresses, Fig. 11 shows the y-derivative of the velocity profile evaluated at the cavity bottom versus the depth h. Along this curve, three different regions can be identified for increasing h: 1. the region in which the derivative is negative (velocity is positive) with decreasing absolute value; there is no recirculation vortex and the shear stress is decreasing though conserving the same direction of the sliding-wall velocity (see the corresponding velocity contours and streamlines in Fig. 12(a)); 2. the region in which the derivative is positive (velocity is negative) with increasing absolute value; a recirculation vortex is present and the shear stress is increasing with direction opposite to the sliding-wall velocity (see the corresponding velocity contours and streamlines in Fig. 12(b));
Fig. 12. Velocity magnitude contours with streamlines (gap ¼ 1 μm, D ¼ 50 μm): (a) h ¼ 1 μm, (b) h ¼ 2:5 μm, (c) h ¼ 30 μm.
1
0.8
y*
0.6
Fig. 13. Contributions of the shear-stress and pressure to T inside the cavities versus h (gap ¼ 1 μm, D ¼ 50 μm).
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
V* Fig. 10. Velocity profile at the centre of the cavity (gap ¼ 1 μm, D ¼ 50 μm). For plotting reasons the profiles are not in the same scale.
Fig. 11. Slope of the velocity profile at the bottom of the cavities (gap ¼ 1 μm, D ¼ 50 μm).
3. the region in which the derivative is positive (velocity is negative) and its absolute value is decreasing towards zero and the shear stress is decreasing with direction opposite to the sliding-wall velocity. The profile tends to show a very small velocity-magnitude region between the cavity bottom and the recirculation vortex (see the corresponding velocity contours and streamlines in Fig. 12(c)). In the first region, the shear stress applied to the cavity bottom contributes to increase the value of T; whereas, in the second and third regions, it contributes to decrease the value of T. The streamlines reported in Fig. 12(a)–(c) are representative of the flow configurations corresponding to the three regions identified above, respectively. Fig. 12(a) corresponds to h/D¼ 0.02 and demonstrates that also for small values of the aspect ratio, a recirculation region is already found. Fig. 12(b) (h/D¼ 0.05) shows that, for higher values of the ratio h/D, the flow presents a clear two-dimensional configuration with a large vortex that, with its dissipation mechanism, influences the pressure gradient inside the cavity. Finally, Fig. 12(c) (h/D ¼0.6) shows that, for higher values of the aspect ratio, the velocity at the cavity bottom, above the vortex, tends to become negligible. Figs. 13 and 14 show the variation of the shear-stress and pressure contributions to T versus h. Only the surface inside the cavities has been considered to compute these contributions: the cavity bottom contributes with
G. Caramia et al. / Tribology International 88 (2015) 162–169
the shear-stress, the two remaining cavity boundaries contribute with a pressure force (sidewall effects). It appears that the two contributions have mostly opposite sign, the pressure one being larger and always positive (see also [20]). The curve representing the combined effects of shear and pressure forces presents a first zone in which the diminishing shear stress contribution is predominant, as shown for small values of h in Fig. 14. For higher values of h, the curve follows the predominant behaviour of the pressure distribution. The pressure contribution reflects the pressure increase across the cavities as shown in Fig. 15 where a sketch of the textured surface geometry is provided: the corresponding pressure distribution is obtained plotting the pressure value along the textured surface versus x. Lengths are scaled by the textured surface total length, ltot, whereas pressures are scaled by the pressure at the leading edge (left) of the textured surface. From left to right, in the direction of the sliding-wall velocity, there is a
Fig. 14. Local view of Fig. 13 for low values of h.
5 4
P/Pin
3 2 1 0 -1 -2 0
0.2
0.4
0.6
0.8
1
x/ltot Fig. 15. Distribution of the pressure along the textured surface (h ¼ 1 μm, gap ¼ 1 μm, D ¼ 50 μm).
167
pressure drop due to the “blockage” effect of the pin holder. In fact, in the region far from the pin holder, a weak positive pressure gradient is predicted, consistent with the exact solution of the 2D Navier–Stokes equations with the same boundary conditions at top and bottom (neglecting the vertical component of the velocity). This pressure gradient creates a pressure difference between the left and the right face of the pin holder, the left face having an higher pressure with respect to the right face. Most importantly, close to the inlet of the narrow gap a pressure increase is observed, generated by the action of the moving wall on the fluid flowing past the sharp edge of the textured surface. Such a local pressure gradient is necessary to balance the viscous forces, the inertial terms being negligible due to the low Reynolds number. A similar behaviour is observed at the outlet of the narrow gap. Here, the pressure achieves a minimum value just after the corner and then increases to reach the quasi-constant “far field” value. Therefore, flowing through the gap, the fluid must experience a global pressure drop from the inlet to the outlet of the narrow channel. In particular, in the zone between the gap inlet and the first cavity the pressure decreases in agreement with the equations of lubrication. At the inlet of the first cavity, the gap undergoes a step increase, which necessarily leads to an analogous step change in the pressure gradient, as indeed expected on the basis of lubrication theory. The pressure, then, increases until at the outlet of the cavity a step decrease of the pressure gradient is observed caused by the sharp reduction of the gap. The pressure then almost linearly decreases until the second cavity is reached and the process is repeated. Pressure contours are provided in Fig. 16, showing in particular the variation of the pressure around the first corner of the second cavity and the pressure gradient inside the cavity (corresponding to the flow in Fig. 12a)). The pressure distribution is clearly twodimensional, therefore an accurate local analysis of the force distribution, cannot be obtained using the Reynolds equation, especially for higher values the cavity depth. Fig. 17 shows the pressure distribution along the textured surface plotted for several values of the depth h. It is noteworthy that, increasing h, the pressure difference between the right and the left wall of each cavity initially grows, then reaches a maximum, and finally decreases. The maximum pressure jump in the cavity corresponds to the appearance of the recirculation vortex which, with its dissipation, reduces the pressure difference between the vertical walls of the cavity. Such an optimal condition is obtained for h 0:5 μm (h=D 0:01). This behaviour has two important effects. The first one is related to its influence on the flow in the regions outside the cavities. In fact, as a consequence, also the contribution of the shear-stress to the drag in the regions outside the cavities has a non-monotone behaviour, as shown in Figs. 18 and 19. The maximum shear-stress value is reached for the same value of h for which the maximum pressure “jump” is obtained in the cavities. The second important effect is related to the possibility of fluid phase change when the pressure locally reaches the saturation value. Since
Fig. 16. Pressure contours near the first corner of the second cavity (h ¼ 1 μm, gap ¼ 1 μm, D ¼ 50 μm).
168
G. Caramia et al. / Tribology International 88 (2015) 162–169
whereas the latter contribute by a shear force term. In particular, observing Figs. 6(b), 14 and 19 it is possible to note that, for increasing values of h, the initial decreasing inside-cavity shearstress contribution determines the decreasing global drag behaviour. Then, the increasing inside-cavity pressure contribution balances the decreasing global (inside-cavity and outside-cavity) shear-stress contribution creating a quasi-constant plateau for the global drag force. Finally (see Figs. 6(a), 13 and 18), for high values of h, the influence of the inside-cavity shear-stress contribution vanishes and the global drag force behaviour is mainly determined by the outside-cavity contribution.
3
P/Pin
2
1
0
-1
4. Conclusions 0
0.2
0.4
0.6
0.8
1
x/ltot Fig. 17. Distribution of the pressure along the textured surface versus h (gap ¼ 1 μm, D ¼ 50 μm): only eight values of h are shown.
Fig. 18. Contribution of the shear-stress to T outside the cavities versus h (gap ¼ 1 μm, D ¼ 50 μm).
Fig. 19. Local view of Fig. 18 for low values of h.
the pressure gradient inside the cavities has a maximum, a minimum value of the pressure is obtained along the textured surface, as shown in Fig. 17 at x=ltot 0:9. Therefore, in a real flow, it exists a range of intermediate (optimal) values of h, for which cavitation would be possible. This behaviour would agree with experimental measurements provided by [5] which present a minimum of the friction coefficient in correspondence of a particular dimples depth value, with 80% drag reduction. Employing the present numerical single-phase model, in the absence of cavitation, such a drag reduction cannot be obtained. The global drag force (see Fig. 6(b)) is the sum of the contributions due to the surfaces inside (Fig. 14) and outside (Fig. 19) the cavities. The former manly contribute by a pressure force term
The hydrodynamic lubrication between parallel micro-textured surfaces has been studied for isothermal incompressible steady two dimensional flow conditions. A wide variety of geometries characterised by different micro-cavity depth and width, and different gap values have been considered in order to study the influence of these parameters on the non-dimensional drag force magnitude T. The variation of T versus the micro-cavity depth, for several cavity widths and gaps has been studied: the presence of two ranges of h values has been observed in which the drag is independent of the cavity depth. The variation of T versus the gap, with several cavity depths and widths has been analysed as well, showing how the influence of the cavity depth on T reduction decreases as the gap magnitude increases. Furthermore, we have shown that three regions exist, depending on the cavity depth, in which a different flow dynamics occurs and the cavities have a different influence on the drag force. A detailed analysis of flow velocity profiles and pressure distributions has been performed to study the forces acting on the textured surfaces: the pressure–force contribution has been separated from the shear-stress contribution as well as the insidecavity surface part has been separated from the outside-cavity part. Concerning the inside-cavity surface part, the velocity profiles through the cavity geometric center, with several cavity depths, has been analysed. The distributions of shear-stress and pressure contributions to T versus h have been discussed, showing how the two contributions have mostly opposite behaviour, the pressure contribution presenting a higher magnitude. Concerning the outside-cavity surface part, the pressure variation along the textured surface versus h and the shear-stress contribution to T versus h have been discussed. We have observed that an “optimal” value of the depth h exists for which the pressure reaches a minimum and the probability of cavitation is maximised. Finally, the comparison between numerical and experimental results shows a substantial difference in the maximum drag reduction achievable for a given texture area density. In fact, the present analysis demonstrates that a maximum drag reduction of about 20% can be achieved with a single-phase lubrication fluid, whereas the experimental data provide an 80% drag reduction. In order to investigate in more detail this aspect, future work will be dedicated to a two-phase analysis evaluating the influence of cavitation or air bubble inclusion on the load and friction generation.
References [1] Etsion I. State of the art in laser surface texturing. J Tribol 2005;127(1):248–53. [2] Hamilton DB, Walowit JA, Allen CM. A theory of lubrication by microirregularities. J Basic Eng 1966;88(1):177. [3] Anno JN, Walowit J, Allen C. Microasperity lubrication. J Lubr Technol 1968;90 (1):351. [4] Marian V, Gabriel D, Knoll G, Filippone S. Theoretical and experimental analysis of a laser textured thrust bearing. Tribol Lett 2011;44(3):335–43.
G. Caramia et al. / Tribology International 88 (2015) 162–169
[5] Scaraggi M, Mezzapesa FP, Carbone G, Ancona A, Sorgente D, Lugarà PM. Minimize friction of lubricated laser-microtextured-surfaces by tuning microholes depth. Tribol Int 2014;75(7):123–7. [6] Scaraggi M, Mezzapesa F, Carbone G, Ancona A, Tricarico L. Friction properties of lubricated laser-microtextured-surfaces: an experimental study from boundary to hydrodynamic-lubrication. Tribol Lett 2013;49(1):117–25. [7] Fowell M, Olver AV, Gosman AD, Spikes HA, Pegg I. Entrainment and inlet suction: two mechanisms of hydrodynamic lubrication in textured bearings. J Tribol 2007;129(2):336–47. [8] Giacopini M, Fowell M, Dini D, Strozzi A. A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation. J Tribol 2010;132(4) 041702–1–12. [9] Pei S, Ma S, Xu H, Wang F, Zhang Y. A multiscale method of modeling surface texture in hydrodynamic regime. Tribol Int 2011;44(12):1810–8. [10] Scaraggi M. Lubrication of textured surfaces: a general theory for flow and shear stress factors. Phys Rev E: Stat Nonlinear Soft Matter Phys 2012;86(2) 026314–1–12. [11] Ma C, Zhu H. An optimum design model for textured surface with ellipticalshape dimples under hydrodynamic lubrication. Tribol Int 2011;44(9):987–95. http://dx.doi.org/10.1016/j.triboint.2011.04.005. [12] de Kraker A, van Ostayen RAJ, Rixen DJ. Development of a texture averaged Reynolds equation. Tribol Int 2010;43(11):2100–9. [13] Dobrica MB, Fillon M. About the validity of Reynolds equation and inertia effects in textured sliders of infinite width. Proc Inst Mech Eng Part J: J Eng Tribol 2009;223(1):69–78.
169
[14] Ausas R, Ragot P, Leiva J, Jai M, Bayada G, Buscaglia GC. The impact of the cavitation model in the analysis of microtextured lubricated journal bearings. J Tribol 2007;129(4):868–75. [15] Sahlin F, Glavatskih S, Almqvist T, Larsson R. Two-dimensional CFD-analysis of micro-patterned surfaces in hydrodynamic lubrication. J Tribol 2005;127 (1):96–102. [16] Jing H, Fang L, Jiapeng S, Shirong G. Hydrodynamic lubrication of microdimple textured surface using three-dimensional CFD. Tribol Trans 2010;53 (6):860–70. [17] Arghir M, Roucou N, Helene M, Frene I. Theoretical analysis of the incompressible laminar flow in a macro-roughness cell. J Tribol 2003;125(2):309–18. [18] Brajdic-Mitidieri P, Gosman AD, Ioannides S, Spikes HA. CFD analysis of a low friction pocketed PAD bearing. In: Proceedings of the ASME/STLE international joint tribology conference, IJTC; 2004. p. 737–47. [19] Cupillard S, Glavatskih S, Cervantes MJ. Computational fluid dynamics analysis of a journal bearing with surface texturing. Proc Inst Mech Eng Part J: J Eng Tribol 2008;222(2):97–107. [20] Li J, Chen H. Evaluation on applicability of Reynolds equation for squared transverse roughness compared to CFD. J Tribol 2007;129(4):963–7. [21] Patankar S, Spalding D. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 1972;15 (10):1787–806. [22] Scaraggi M. Textured surface hydrodynamic lubrication: discussion. Tribol Lett 2012;48(3):375–91.