4.2.1. Full Deterministic Methods

In deterministic methods, full-scale representation of the surface topography, including both micro- and macro-scale features of the surface texture, alongside roughness, are considered in the definition of the lubricant film thickness and solution of the Reynolds equation. In the last few decades, quickly advancing computational power and the development of improved numerical techniques (e.g., multi-grid methods and parallel computing) permitted the effective digitalization of engineering surfaces, as well as the efficient solution of the coupled contact mechanics and lubrication problems. Therefore, increasing attention has been devoted to deterministic simulations. Early deterministic solutions of full-film EHL under limited rough contact were proposed for artificial topographies with simple irregularities, such as dimples and sinusoidal waviness [105–109]. Actual 2D roughness profiles were then employed by [110,111] and later 3D measured topographies were considered by [112,113]. More recently, Hu and Zhu [114–125] proposed a fully coupled mixed-TEHL model assuming a continuous lubricant film in the non-contacting regions as well as asperity contact wherever the local fluid film is sufficiently thin. In this model, the film thickness is computed from the deformed average gap, while the lubricant flow and asperity interactions are accounted for in a unified solution framework. Using this formulation, different types of multi-scale surface textures can be used as input data to deterministically simulate the entire transitions from full-film and mixed-TEHL as well as boundary lubrication under more severe conditions. However, since no averaging technique is considered in this formulation, it can only be applied to relatively small regions such as point contacts. The Hu and Zhu deterministic model was applied by [121,126] to propose a virtual surface texturing simulation tool being able to provide comparative information and directions for innovative texture design and optimization, including the relationship between textured surfaces and mixed lubrication characteristics of non-conformal contacts. Figure 13 illustrates examples of different groove textures evaluated with the deterministic mixed-TEHL model proposed by Hu and Zhu. The full-scale mixed-TEHL model developed by Hu and Zhu was also coupled to different multi-scale surface texture decomposition models to investigate the influence of surface roughness [127] and groove texture patterns placed on cylinder surfaces of internal combustion engines [128,129] on the COF.

Li and Chen proposed a deterministic mixed-lubrication model applied to the simulation of the piston ring cylinder liner contact of internal combustion engines. The model is based upon the calculation of the oil transport and the hydrodynamic pressure generation for the contact between a parallel and flat rigid plane sliding against a rough surface [131–135]. Similarly, Profito et al. presented a deterministic mixed-lubrication model based upon the simultaneous solution of the asperity contact and fluid flow problems at the roughness scale considering inter-asperity mass-conservative cavitation. The influence of the cylinder liner wear on the lubrication performance of a Twin Land Oil Control Ring (TLOCR) was analyzed using measured surface topographies of a honed cylinder liner prior to and after 100 h engine tests (see Figure 14a). The results showed that under mixed lubrication, the worn liner surface yielded to an increase of the average hydrodynamic load capacity and a decrease of the asperity contact pressures compared to the unworn liner surface. As illustrated in Figure 14b,c, this was traced back to the smoothing of the plateau regions caused by the wear-out of the highest asperities and the general decrease in the summits curvature, which contributed to facilitate the pressure-driven lubricant flow throughout the inter-grooves zones thus intensifying the role of the honing grooves in the hydrodynamic pressure generation [130]. Afterwards, Tomanik et al. applied the deterministic mixed-lubrication model proposed by Profito et al. to investigate the effect of waviness and roughness of two measured mirror-like coated bore topographies on the hydrodynamic and asperity pressure distributions. The simulation results revealed that most of the fluid pressure was generated by the honing grooves rather than by the localized pores on the coated bore surfaces [136]. Moreover, Biboulet et al. and Noutary et al. proposed multi-grid techniques to solve deterministic hydrodynamic lubrication models with non-mass-conservative cavitation for the piston ring cylinder liner contact with measured textured surfaces. It was shown that the groove depth and density are

important factors determining the load carrying capacity, whereas the groove shape has only a minor influence [137–139].

**Figure 13.** Deterministic simulations obtained with the fully coupled mixed-TEHL model proposed by Hu and Zhu. Sample cases with textured surfaces: rectangular (left), fishbone (middle) and sinusoidal (right) grooves with corresponding film thickness countors, centerline pressures, and normalized film thicknesses. Adapted from [118].

**Figure 14.** Influence of cylinder liner wear on the lubrication performance of a Twin Land Oil Control Ring (TLOCR) investigated by deterministic simulations. (**a**) Schematic of the TLOCR system and cylinder liner topography. (**b**) Field results for an instantaneous position of the TLOCR land on the liner prior to the engine test. (**c**) Field results for the same instantaneous position of the TLOCR land on the liner after 100 h engine test. Dashed circles highlight regions with most significant changes after wear. Adapted from [130].

A deterministic mixed lubrication model was also proposed by Minet et al. for mechanical seals applications. The model is based on the simultaneous solution of the Reynolds equation with mass-conservative cavitation and asperity contact considered through the Hertzian contact model. The results reproduced numerically the hydrodynamic load carrying capacity between nominally parallel surfaces promoted by the surface roughness, as well as the transition from mixed to full hydrodynamic lubrication regimes in face seals [140].
