4.2.2. Analytical Multi-Scale Methods

Despite the better accuracy of fully deterministic methods, the computational efforts required in these cases are often prohibitive in practical applications due to the fine meshes needed to properly capture the local features. This is especially true for multi-scale surface textures, for which very fine meshes would be necessary to discretize the lubrication domain to entirely represent the geometric details of all scales. Therefore, different analytical multi-scale methods, especially averaging and homogenization methods, have been proposed to avoid dense discretization grids. In these methods, the overall influence of the surface texture features, along with the roughness in mixed lubrication analysis, are represented in terms of averaging parameters (flow factors and homogenization factors) introduced in the governing equations (e.g., averaged Reynolds equation) defined over the entire macro-scale lubrication domain. Thus, only the overall macroscopic geometry of the contacting surfaces is effectively considered in this analysis. The overall calculation process of the flow factors used in the averaged Reynolds equation is schematically illustrated in Figure 15. It is important to emphasize that most of the analytical multi-scale methods discussed in the following paragraphs have been initially proposed to model solely the effect of roughness on lubrication. Nevertheless, they can be extended to deal with the different length scales of multi-scale textures. For a deeper understanding of the principles and fundamentals of multi-scale modeling in science and engineering, the interested reader is referred to the comprehensive textbooks [141–144]. Particularly with respect to tribology, the reader is refereed to [145] for an extensive review on modeling and simulation of various physical, chemical and mechanical phenomena across different scales.

**Figure 15.** Schematic illustrating the calculation process of the flow factors (or homogenization factors) used in the averaged (or homogenized) Reynolds equation which, underlay most analytical multi-scale methods. The flow factors can be calculated either from measured surfaces of actual engineering components or from virtually generated topographies. The flow-factor curve on the left shows the variation of the pressure flow-factor as a function of the dimensionless average interfacial separation. The pressure flow-factor has significant impact on the calculation of the hydrodynamic pressure under mixed lubrication conditions. Similarly, the flow-factor curve on the right represents the variation of the shear flow-factor, which influences the shear stress (and thus friction force) calculations.

#### Averaging Flow Methods

The modeling of the effect of surface roughness patterns on lubrication was first reported by Tzeng and Saibel [146] for one-dimensional transversal roughness and followed by Christensen [147–149] and Chow and Cheng [150] for two-dimensional transverse and longitudinal topographies. All of these pioneering works have been established within the framework of the stochastic process theory, which is based upon the concept of viewing the film thickness as a stochastic process that results in a Reynolds-type equation for the mean or expected fluid pressure.

Patir and Cheng [151,152] proposed an average flow model for general roughness patterns by incorporating "flow factors" coefficients directly in a modified Reynolds equation that is solved in the smooth global domain. The flow factors are determined independently by solving the local deterministic flow problem for a specified rough surface. Unlike the Christensen's methodology that particularly weights the film thickness oscillations according to an expectancy operator defined for a given roughness height distribution, the Patir and Cheng model is derived by "locally averaging the lubricant flows at the microscopic scale" for a representative rough bearing cell. This provides specific flow factors coefficients allowing for the consideration of the roughness (or other length scale components) induced flow perturbation effects directly on the global lubricated domain. The major drawback of this approach is a consequence of its heuristic derivation, which poses limitations on dealing with eventual crossflow produced in the case of surface roughness anisotropy. Such lack of generality was first highlighted by Elrod [153] and subsequently by different authors, including Tripp [154], who proposed a more complete tensor form of the averaged Reynolds equation. In the latter, the effects of roughness anisotropy on lubrication are accounted for in the off-diagonal terms of the diffusion and convective flow tensors, in particular, when the off-diagonal terms are negligible, the Tripp model is essentially identical to the model proposed by Patir and Cheng.

The Patir and Cheng average flow model has proven to be effective to predict the mixed lubrication performance for a wide range of applications, such as thrust and journal bearings, piston and piston-ring cylinder bore systems, mechanical seals, rolling element bearings, gears, and cam-tappet contacts, etc. Regardless of the tribological application, an important aspect for the efficacious use of any average flow model refers to the proper calculation of the flow factor coefficients for a specified topography. Furthermore, the consideration of rough contact mechanics and micro-cavitation effects also significantly affect the accuracy of the flow factor coefficients. In many successful cases, deterministic simulations of representative bearing unit cells at different length scales have been carried out to estimate the flow factors, as reported in several publications involving surface textures [155,156], honing grooves [157,158] and general roughness patterns [159–162].

A comparison between a deterministic hydrodynamic model and the stochastic solution based upon Patir and Cheng approach was undertaken by Dobrica et al. [163] for a partial journal bearing operating under mixed-EHL. The results underlined that the stochastic model correctly predicted the tendencies produced by the different roughness patterns in the fluid pressure distribution and average minimum film thickness, but underestimated the friction torques on both shaft and pad.

A multi-scale method based on the averaging flow concept was proposed by de Kraker et al. [155,156] for surface textures under mixed lubrication including micro-cavitation. In their approach, the local (micro) flow effects for a single micro-scale texture unit cell were evaluated through CFD simulations, and the results were then averaged to flow factors to be used with an averaged Reynolds equation on the macro-scale bearing level. The flow factors are dependent on the ratio between film thickness and texture dimensions, surface velocities and pressure gradient over a texture cell. Additionally, the method presented has no restrictions to the texture dimensions and shape, so that it could be well extended to model multi-scale surface textures.

#### Homogenization Methods

The modeling of the fluid flow problem in mixed lubrication has been addressed by Bayada [164–168], Jai and Bou-Saïd [169–171] and Buscaglia [172–174] within the framework of the

homogenization theory for spatially periodic roughness. This approach is based upon the derivation of a homogenized Reynolds equation, defined at the macroscopic global scale, which captures the overall effects of the surface roughness on the lubricant flow. Besides posing the average flow model on a more rigorous mathematical base that overcomes the pure heuristic induction of the Patir and Cheng model, one additional feature of these techniques is the proper definition of the local (or auxiliary) problem, which has to be solved over a periodic unit cell to compute the average flow tensors [175]. Thus, similarly to the average flow models, the effects occurring at different length scales are incorporated in the homogenized (averaged) Reynolds equation from the solution of well-posed local problems.

A series of works have been published by Almqvist which contributed to the consolidation and widespread use of homogenization methods in lubrication applications [176–181]. The homogenization method proposed by Almqvist has been used to investigate the effect of roughness and surface texture on the tribological performance of different machine elements, such as piston-ring cylinder liner contact and rotating devices [179,182–187].

More recently, Rom and Muller [188] proposed a reduced basis method to accurately solve and speed-up the solution of the homogenized Reynolds equation in a finite element framework. This method replaces the computationally expensive solution of the full texture cell problems (micro-scale) with a reduced basis problem of much smaller dimension, which provides a significantly accelerated solution strategy. After the solution of the texture cell problem for a range of film thicknesses, the homogenized finite element matrix and vector are computed to assemble the homogenized problem. The effectiveness of the combined use of both the homogenization method and reduced basis technique is evaluated for textured journal bearings.

A novel homogenized approach was proposed by Scaraggi et al. [189,190] to study the mixed lubrication behavior of steady sliding contacts of elastically soft solids. The coupled effects of asperity-asperity and asperity-fluid interactions have been considered through a mean field theory based upon a perturbation treatment. The results demonstrated how the asperity flattening induced by the fluid-asperity interactions, as well as the local percolation effects and roughness anisotropic deformation govern the fluid flow at the interface. It was also remarked that the lubrication regime is generally not uniform at the interface due to different local average separations. Furthermore, the potential occurrence of an apparent (elasto)-hydrodynamic regime for those lubrication conditions characterized by values of *h*/*hrms* ≤ 1 at the macroscopic level was discussed, for which the lubricant is expected to have a negligible influence on the frictional stresses. This effective transition from boundary to (elasto)-hydrodynamic regime for a given value of *h*/*hrms* occurs due to the increase of the defined sliding parameter *U*η/(*E*∗ *hrms*), which determines a transition from a constant boundary stress value to a power-law shear stress. It was further noticed that this transition disappears for very small values of *h*/*hrms*, when percolation takes place and the average fluid flow vanishes. Afterwards, Scaraggi [191–193] also presented a homogenized method based upon the application of the Bruggeman effective medium to the Reynolds equation to investigate the average effect of textured surfaces on the macroscopic hydrodynamic characteristics of the interface. The method allows for the assessment of generic texture shape, distribution and area density, and was applied to practical cases involving 1D and 2D thrust bearing geometries.

A heterogeneous multi-scale method has been proposed by Gao and Hewson [194] to analyze micro-EHL with small-scale topographical features. The small-scale problem was solved using full CFD simulations including local elastic deformations and coupled to the global scale via scattered data interpolation method. It has been demonstrated that the proposed multi-scale framework successfully modeled the global pressure and film thickness for a textured bearing while maintaining the small-scale modeling features. Later on, Gao et al. [195] extended this multi-scale framework by incorporating the micro-cavitation and local fluid shear thinning properties.

The homogenized Reynolds equation, which simultaneously considers surface roughness and turbulent flow effects, was proposed by Lahmar et al. [196]. A plain journal bearing with periodic isotropic roughness patterns operating under turbulent conditions was used as a case study. The homogenized results agreed well with results obtained from deterministic simulations, showing that the proposed homogenization approach is suitable to study problems with rough surfaces and turbulent conditions.

A promising computational engineering framework was developed by Waseem et al. [197,198] to support the design process of optimized surface textures for hydrodynamic lubrication. The proposed framework makes use of a combination of a two-scale homogenization method and topology optimization schemes. Another important aspect of the developed multi-scale framework is the consideration of the temporal (squeeze-film effect) and spatial (wedge effect) variations in film thickness in the constitutive tensors, which characterize the homogenized response of the surface texture in terms of fluid pressure generation and load carrying capacity on the macroscopic scale. Although temporal and spatial variations in the film thickness are important for the generation of hydrodynamic pressure within the fluid at the interface, they are not always simultaneously considered in homogenization methods applied to lubrication problems.

More recently, Yildiran et al. [199] investigated the lubrication response of conventional textures (i.e., textures with well-defined, smooth geometries, such as dimples, squares, ellipsoidals, V-shapes, etc.) and representative modern re-entrant textures (i.e., textures with more complex geometries, such as trapezoidal and T-shaped features) based upon the homogenization scheme proposed by Bayada and Chambat [164]. After a comprehensive review of the literature on homogenization techniques applied to lubricated contacts and their limitations, the transition between three microscopic lubrication regimes has been demonstrated for conventional and re-entrant textures. In this work, the difference between Reynolds and Stokes roughness is also discussed. In all the above-mentioned references, except [199], the local roughness slope is always assumed small (Reynolds roughness), so that the flow equations at the microscopic scale are well described by the Reynolds equation (i.e., the local inertia effects can be neglected without significant loss of accuracy). When larger local roughness slope is present (Stokes roughness), the local inertia effects need to be taken into account in the analysis, for instance, through the solution of Stokes equations.

## 4.2.3. Semi-Deterministic Methods

A semi-deterministic modeling strategy was adopted by several authors [30,200–206] to study the combined effect of surface texture and roughness (multi-scale effects) under hydrodynamic and mixed lubrication conditions by solving the averaged Reynolds equation based on the Patir and Cheng model and mass-conservative cavitation. In these works, surface roughness effects (micro-scale) were treated through stochastic models (Patir and Cheng method for the lubricant flow and Greenwood-Williamson based models for asperity contact), while the surface textures (macro-scale) were considered deterministically through proper fine mesh discretization. Figure 16 summarizes the main aspects of different semi-deterministic models proposed in the literature to simulate rough textured surfaces.

Qiu and Khonsari [206] used a mass-conservative cavitation model to investigate the performance of textured dimples in seals and thrust bearings under mixed lubrication conditions. The authors verified the beneficial but minor effect of the surface roughness on the load carrying capacity of dimpled surfaces. It was also concluded that it exists an optimum dimple-to-diameter ratio and dimple density depending on the rotational speed for which the load carrying capacity is maximum. Moreover, it was verified that large dimple depths and increased roughness contribute towards higher seal leakage and that the friction force is decreased due to cavitation over the dimples. Similarly, Brunetière and Tournerie [207] showed that for smooth dimpled surfaces applied to mechanical seals, it is not possible to generate sufficient force to separate the surfaces, whereas a rough dimpled surface can significantly reduce friction.

The semi-deterministic model proposed by Profito et al. [30,203] was validated using experimental results obtained from a reciprocating test with groove surface texture. The same model was used to explain the mechanisms associated with the transient effects induced by moving textures and their influence on the frictional response and film thickness variation in different lubrication regimes. Particularly, with respect to the boundary and mixed lubrication, it was shown that the interplay between inlet suction, asperity contact, cavitation, and fluid squeeze out all contribute to the frictional response and their relative contribution may differ depending on the operating regime. Furthermore, it was also discussed that under certain working conditions in mixed lubrication, as the textures move through the interface, the net effect of inlet suction and the subsequent fluid pressure boosting promoted by the fluid squeeze out tend to increase the film thickness and hence decrease the overall friction.

**Figure 16.** Examples of semi-deterministic models applied to simulate rough textured surfaces. In this modeling strategy, surface textures are considered deterministically, while surface roughness effects are taken into account through averaging (or homogenization) methods for lubricant flow and stochastic models for asperity contact. (**a**) Schematic of the rough contact interface with groove texture used by Profito et al. [30] to investigate the influence of the transient effects induced by moving textures on the frictional response and film thickness in different lubrication regimes. (**b**) Different scales of rough textured surfaces considered by Gu et al. [200] to study mixed lubrication problems in the presence of textures.
