1. Introduction
In a lubricated contact, when the fluid enters a converging zone, the pressure increases and reaches a maximum near the minimum of the film thickness. Assuming that a diverging area follows the converging entrance, the fluid is stretched and the pressure decreases. The Reynolds equation as originally expressed does not account for the cavitation area in a realistic way and abundant literature has followed. Among the methods that guarantee the Reynolds conditions, the penalty method and the well-known Elrod’s algorithm are quite easy to implement. The penalty method promoted by Wu [
1] treats both the film area and the cavitation area by introducing a penalty term into the Reynolds equation. When the pressure falls below the cavitation pressure
, its value is forced to
. Whilst providing an accurate pressure boundary, the method fails in predicting the film reformation boundary. The seminal Elrod and Adams algorithm [
2,
3] splits the Reynolds equation by the means of a switch function to ensure the equation validity in both the gaseous and liquid areas. Mass conserving is guaranteed in both areas; however, the location of the boundaries is mesh dependent.
Let us describe the cavitation phenomenon a little more. If the pressure falls below the ambient pressure, three situations can be considered [
4]: (a) the dissolved gas is released and forms bubbles, (b) the bubbles already present expand, and (c) the fluid, which contains no gas, evaporates. The cases are referred to as a (a) gaseous cavitation, (b) pseudocavitation, and (c) vaporous cavitation. In a more recent study, Bai et al. [
5] propose a review on the cavitation in a thin liquid layer, in which a section is dedicated to the hydrodynamic cavitation. However, the authors do not distinguish the different cavitation phenomena as reported by Braun and Hannon [
4]. The simplified taxonomy (a)-(b)-(c) is that the latter present does not include adsorption effects, as studied by Belova et al. [
6], on the heterogeneous cavitation. Although rough surfaces are likely to adsorb much more gas than smooth surfaces because of their fractal nature, the related phenomenon significantly complicates the lubrication modeling of rough surfaces and therefore is not addressed in the present work.
Regarding the gaseous cavitation in oil, Li et al. [
7] and Song and Gu [
8] consider the lubricant as a mixture of pure liquid and air, as the result of the dissolution and the release of air in an instantaneous equilibrium saturation state. The model satisfies the classical Jakobson–Floberg–Olsson (JFO) conditions, but it does not account for the gaseous cavitation rate, as performed by Hao and Gu [
9] and more recently by Ding et al. [
10].
Ransegnola et al. [
11] model both the vaporous and gaseous cavitation in an oil-lubricated bearing, predicting not only the cavitation area but also the distribution of the gas, vapor, and liquid states. However, with respect to water as a lubricant, Magaletti et al. [
12] provide a graph that shows the cavitation pressure of ultra-pure water as a function of temperature. The remarkable data are the cavitation pressure of the water at ambient conditions, about −120 MPa, because of its high tensile stress. This very low value makes one think that behind the common cavitation in water, there is essentially a gaseous cavitation or even a pseudocavitation.
In the present work, the fluid is assumed to be a biphasic mixture: air bubbles are present in an incompressible liquid, which leads to the pseudocavitation in the diverging zones. Considering the mixture as a homogeneous media allows for the use of the Reynolds equation in the whole contact zone: the fluid rheology is modified in the depressurized zones and is constant in the full-film zones. Three advantages are brought with this model presented by Brunetière [
13] and referred to as the “Lubricant General Model” (LGM). (1) It is a handy model because once the fluid density and the fluid viscosity are defined, the Reynolds equation is solved in the same way, whatever the zone. (2) The cavitation area transition is smooth, making it particularly suitable for multiscale meshes. (3) The varying density model is a mass conserving one; as such, it accurately accounts for the film rupture and reformation.
Grützmacher et al. [
14] propose a review of the multiscale approaches about texturing in tribology. The classification that is proposed is well suited to distinguish the different strategies related to the lubrication of rough contacts. In particular, two classes are of importance here: the analytical multiscale methods and the numerical multiscale modeling.
Lubricated contacts may require multiscale approaches for various reasons. Heavily loaded contacts can lead to high gradient pressures, such as in the outlet spike in point contacts. The problem can be addressed with fine meshes, but without speedup convergence algorithms, such as multigrid methods, the computing time becomes prohibitive. The multiscale approach is then a means to accelerate the convergence of the Reynolds equation resolution. The mesh can also exhibit different element scales based on the pressure gradient: the steeper the pressure field, the finer the mesh. In both situations, the Reynolds equation is not modified. The above common techniques belong to the numerical multiscale modeling class.
The analytical multiscale methods involve treatments applied to the Reynolds equation. When roughness influences the fluid flow because of the small film thicknesses, instead of a deterministic resolution, the equation can be enriched with flow factor modifiers [
15,
16]. The flow factors are determined with a few of the statistical properties of the rough surface, which limit the accuracy of the results. However, once the flow factors are determined, the surfaces being considered smooth, coarse meshes can be used, decreasing the computing effort. This stochastic approach has since been improved to cope with the micro-cavitation to handle a broader variety of surface roughnesses [
17], but it remains global: the local effects of the roughness on the pressure cannot be captured. In order to get rid of the stochastic approximation of the roughness, the Reynolds equation can be viewed as a set of equations solving different wavelength pressure problems [
18,
19,
20]. Indeed, an asymptotic expansion of the pressure is written with respect to a scalar related to the roughness wavelength, leading to a modified Reynolds equation. The process, based on rigorous mathematical developments, is called homogenization and is compatible with any periodic roughness. Homogenization techniques have received more attention in recent years than flow factor modifiers with the works of, among others, de Boer et al. [
21,
22], de Boer and Almqvist [
23], and Han et al. [
24]. In Rom et al. [
25], the reader can find the homogenization advantages that explain its greater development. However, flow factor modifiers seem to be more widely used to date, such as in [
26,
27,
28,
29,
30].
Computing flow factors and homogenization suffer from a common drawback: for flat parallel rough surfaces, no pressure build-up exists, although it is experimentally observed [
31,
32,
33,
34]. A compromise between the deterministic and the stochastic methods has been proposed by Brunetière and Wang [
35]. The Reynolds equation is filtered: above an
roughness frequency, averaging is used; and below
, a deterministic solution is computed on a coarse grid. For this analytical multiscale method, the computational effort is much less than for the deterministic case, but the micro-cavitation is not taken into account. Pei et al. [
36,
37] use a Guyan reduction to condensate finite element cells. In doing so, the general linear system is smaller than the one obtained with the usual finite element method. A further reduction is obtained while defining the master/slave nodes on the cell boundary. Even if the bandwidth is larger, the computing time is up to five times smaller than for the conventional finite element method (FEM). The major drawback of this numerical multiscale modeling is, however, that the cavitation is not taken into account.
The present paper, belonging to the numerical multiscale approaches, is a step beyond the work of Brunetière and Francisco [
38]: the domain is divided into macro-cells inside which a deterministic FEM is used to solve the Reynolds equation; then, the macro-cells are linked together using a mass-conserving principle to cover the whole domain. In addition, when the average film thickness over a macro-element is sufficient to ignore the roughness, the macro-element is not submeshed but rather replaced by a linear finite element: it is the hybrid version of the algorithm.
4. Conclusions
In the incompressible case, the FMS model provides for a very fast tool to solve the Reynolds equation. However, strong departures from linearity affect the FMS/HMS model efficiency for compressible fluids. Indeed, with the cavitation phenomenon, the computing process iterates much more: iterations are not only needed at the top-scale level but also at the bottom-scale level. In addition, some relaxation coefficients must be tuned on both levels, and particularly when the ratio exceeds 1.
Another a priori negative point is that reducing the mesh density allows for the same computing time reduction along with the same result accuracy. And yet, the slider is flat, meaning that the pressure build-up is only generated by the roughness (). Despite the preceding points, it is worth noting that the computing times—when the HMS is used with a few BS elements—remain attractive, compared to the deterministic model, whether it is reduced or not.
The benefits of the FMS/HMS approach are much more expected in the following cases:
- 1.
Shorter wavelengths— and mm;
- 2.
Rough texturized surfaces—square dimples being modeled with simple FEM macro-elements, and rough contacting parts discretized with BS elements;
- 3.
As many threads as there are macro-elements—which leads to a GPU implementation of the numerical code.
As for the numerical aspects, additional computing time can be saved.
- 1.
The TS element boundaries are updated once the whole TS element batch is processed. However, some TS elements converge slowly—mainly because of a local narrow slider gap. Therefore, a new criterion has to be set up to locally guarantee the best compromise accuracy/iteration number.
- 2.
The TS element boundary update must be monitored because important changes in the TS pressure field affect the four other connected macro-elements—in particular, oscillations are undesirable.
- 3.
The heights of the element boundaries are the result of the domain division, and this can lead to rough relief for some of them, with convergence problems. In a future work, a sensitivity analysis will study the effect of numerically smoothed boundaries on the slider lubrication. We are confident that the results will not change much and that the Reynolds equation will be solved faster on TS elements.
To conclude the present work, the FMS/HMS approach is accurate and fast due to a highly parallelizable structure, and promising keys to improvement make this technique a good candidate for the lubrication of rough parallel surfaces.