*4.1. CFD Modeling*

The first modeling approach makes use of CFD simulations based upon the full solution of the Navier–Stokes (N–S) equations. The main advantage of CFD is the possibility of considering advanced mathematical models and complex flow phenomena, such as inertia and thermal effects, turbulence, cavitation, fluid compressibility and rheology, wall slip, fluid-structure interaction, among others. The major drawback of CFD simulations resides in the huge, and often prohibitive, computational efforts necessary to simulate problems involving textured surfaces due to the inherent fine meshes needed to properly discretize local geometric features. Particularly considering multi-scale textures, several works investigated the influence of hierarchical structures of bio-inspired shark-skin surfaces on friction drag reduction under turbulent conditions. Choi et al. explored the behavior of the turbulent micro flow field on bio-inspired micro-grooved surfaces using direct numerical simulation (DNS) [95,96]. The distribution of the micro flow field over the real shark-skin surface and its effect on the drag reduction were also analyzed through CFD simulations by Zhang et al. [97] and Luo et al. [98,99]. Figure 10 shows the morphology of scales of shark-skin surfaces studied by Luo et al. [98,99], including a schematic diagram illustrating one of the factors responsible for the drag reduction (Figure 10c) and numerical results obtained from CFD simulations (Figure 10d). The authors concluded that the drag-reduction mechanism of shark-skin is a combination of four factors: (i) a decrease of the wall viscous friction due to reduced turbulence next to the walls induced by the micro-groove tips, which stick out the viscous sub-layer, (ii) a decrease of the turbulence intensity near the wall due to the back-flowing phenomenon associated with the micro-droplets with opposite direction to the main flow (see Figure 10c), (iii) a super-hydrophobic effect produced by the boundary slipping phenomenon on the fluid-solid interface, which significantly decreases the velocity gradient and the local viscous resistance on the surface, and (iv) the presence of a nano-chain of mucus covering the wall, which increases the thickness of the viscous sublayer thus producing the aforementioned slipping phenomenon.

More recently, Martin and Bhushan conducted large-scale CFD simulations to optimize shark-inspired riblet geometries and dimensions for low drags. In that work, it was highlighted that the underlying mechanism responsible for the drag reduction was also associated with vortices lifted away from the surface and hence formed over the riblets (see Figure 11a) under turbulent flow conditions thus decreasing the overall shear stress. Furthermore, it has been identified that the optimum size of riblet design features for low-drag and anti-fouling surfaces can range from nano- to micro-scale depending on the size of the physical components for specific applications [100]. A numerical and experimental investigation of different marine drag reduction technologies based upon shark-skin inspired riblet surfaces was carried out by Fu et al. Illustrative examples of CFD analyses conducted by the authors for herringbone riblets are shown in Figure 11b. They demonstrated that triangular-shaped riblets presented a better trade-off between manufacturing and drag reduction [101].

CFD simulations were also applied by Belhadjamor et al. to study the effect of texturing on the anti-fingerprint and self-cleaning performance. It was verified that multi-scale textures are capable to decrease the finger contact area and promote hydrophobicity thus reducing the surface affinity to skin oil [102]. An example of finite element simulation and contact angle and wettability analysis of a hierarchical textured surface for anti-fingerprint and self-cleaning applications is illustrated in Figure 12. Regarding tribological applications, Brajdic-Mitidieri et al. used a CFD model with cavitation to analyze the lubricant flow behavior, load support and friction of linear, convergent pad bearings having a closed pocket. Depending on the bearing's convergence ratio and the pocket's location, the authors identified two different mechanisms responsible for friction reduction: (i) at moderate to high convergence ratios, the reduction of the shear stress is more pronounced than the pressure build-up within the pocket when the textures are suitably positioned in the high pressure region of the bearing, thus reducing the COF, and (ii) at low convergence ratios, the boost in hydrodynamic pressure within the pocket due to its convergent geometry generates higher load support (and lower friction) compared to the non-textured case [103]. A CFD-based thermo-hydrodynamic study was

carried out by Vakilian et al. to explore the characteristics of Rayleigh step bearings under different steady conditions [104].

Many other works have been published in the literature involving the use of CFD analysis with different model complexities to investigate the lubrication performance of single scale textured thrust and journal bearings. The reader is referred to [21] for a thorough review of recent works based on CFD analysis to investigate the lubrication performance of single scale textured bearings. Furthermore, despite the studies of [103,104] were not directly associated with multi-scale textures, the methodologies adopted can be used as a reference for more advanced CFD analysis involving multi-scale textures.

**Figure 10.** Hierarchical structures of bio-inspired shark-skin surfaces for friction drag reduction. (**a**) Morphology of scales on different locations of the sharkskin. (**b**) 3D image of a biological single shark-skin scale and the corresponding cross-section profile. (**c**) Schematic diagram of the back-lowing phenomenon responsible for the drag reduction due to the attenuation of the turbulence intensity. (**d**) Computational fluid dynamics (CFD) simulation of the resulting shear stress over a real shark-skin surface as well as detailed distribution over a single scale surface. Adapted from [99].

**Figure 11.** Shark-inspired riblet geometries for low drag applications. (**a**) Top: micrographs of samples acquired by scanning electron microscopy. Middle: schematic of the streamwise vortices lifted mechanism responsible for drag reduction on riblet surfaces. Bottom: velocity fields obtained by CFD simulations showing the vortices lifted on blade, sawtooth and scalloped riblet geometries. (**b**) Top: 3D surface topography measurement of micro-riblets applied to marine drag reduction technologies. Shear stress distribution (middle) and velocity field (bottom) of herringbone riblets obtained by CFD simulations. Adapted from [100,101].

**Figure 12.** Hierarchical textured surface for anti-fingerprint and self-cleaning applications. (**a**) Finite element mesh. (**b**) Deformed shape and strain distribution. (**c**) Contact angle and wetting state for hydrophobicity analysis. Adapted from [102].

#### *4.2. Reynolds-Type Equation Modeling*

The second modeling approach is based upon the solution of the Reynolds equation derived from the thin fluid film lubrication theory. The advantage of this approach is the lower computational cost required to simulate textured contacts compared to full CFD methods. Furthermore, many modified versions of the Reynolds equation exist, in which specific physical aspects such as thermal effects, cavitation, turbulence, lubricant rheology, as well as the influence of surface roughness on the lubricant flow (important for mixed lubrication analysis) have been incorporated. Particularly, some methods commonly used to account for the influence of surface roughness on lubrication can be extended to predict the lubrication performance of multi-scale surface textures. In this sense, a brief explanation of the most important models used for mixed lubrication analysis shall be presented here. Special emphasis has been put on works in which such models have been applied to investigate the lubrication behavior of multi-scale surface textures. In this regard, two classes of methods based upon the Reynolds equation approach, namely deterministic and multi-scale, need to be distinguished. Each class is defined according to the way the components of the surface topography are considered for the mathematical representation of the fluid film gap geometry.
