A simulation algorithm is required to identify the optimal texture geometry for specific conditions. A suitable simulation algorithm has already been published by Scharf et al. (2024a), Scharf et al. (2024b), and Scharf et al. (2025). Within this work, the performance of textures in full-film lubrication should be analyzed. Consequently, the multiphase Stokes flow, which considers cavitation and is presented through Equation 1, is used. In that equation, u → represents the unknown velocity, and p is the unknown hydrodynamic pressure field. ρ stands for density, ν for kinematic viscosity, and ζ for the second viscosity of the fluid. I is the identity matrix, and f → is the body force. The second viscosity ζ and ν are defined as the same value, which also confirms the inequality presented by Shibata et al. (2018). Additionally, the continuity equation (Equation 2) is necessary. If cavitation does not occur, solving the Stokes flow (cf. Equation 3) is sufficient. − ∇ · ν ∇ u → + ∇ u → T − 2 3 ∇ · u → I + ζ ∇ · u → I + ∇ p = ρ   f , → (1) ∇ · u → = 0 , (2) − ν   Δ u → + ∇ p = ρ   f . → (3)

To consider cavitation, which occurs when the pressure p is less than the vapor pressure p v , the Merkle algorithm (Merkle et al., 1998) is used, as presented in Equations 4–9. This cavitation model has been applied by Concli (2020) to simulate a hydrodynamic journal bearing and by Scharf et al. (2024a) and Scharf et al. (2024b) to simulate a convergent oil film gap. In this model, the vapor volume fraction α v is calculated to determine the density ρ - and kinematic viscosity field ν , which are dependent on the density and viscosity of both the liquid ρ l ,   ν l and the vapor ρ v ,   ν v . The vapor volume fraction α v represents the ratio of the vapor volume to the total volume V v + V l , where the sum of the liquid volume fraction α l and vapor volume fraction α v equals 1. The terms m ˙ + and m ˙ − describe the condensation and vaporization processes and are influenced by the condensation constant C c , the evaporation constant C v , the reference time t ∞ , and velocity U ∞ . ∇ · u →   α v = m ˙ + − m ˙ − , (4) m ˙ + = C c ρ ρ l ρ v 1 − α l 0.5 U ∞ 2 t ∞ max ⁡ p − p v , 0 , (5) m ˙ − = C v ρ ρ v 2 α l 0.5 U ∞ 2 t ∞ max ⁡ p v − p , 0 , (6) α v = V v V v + V l , (7) ρ = ρ v   α v + ρ l   1 − α v , (8) ν = ν v   α v + ν l   1 − α v . (9)

The schematic geometry of the oil film gap is depicted in Figures 2, 3. The finite element method (FEM) implemented in the Python package Netgen/NGSolve (NGSolve, 2025) is applied to solve the previously described partial differential equations. The Sparse Cholesky decomposition is used to solve the matrices. The geometry is discretized by tetrahedral-shaped elements with second-order shape functions. The maximal mesh size was determined through a grid study, resulting in a maximal mesh size of 100 µm being selected. Regarding the geometry of the oil film gap, Figures 2, 3 show a horizontal region on the left side. This surface is necessary to adjust the alignment of the pins (cf. Subsection 2.2) for the novel test methodology. This region is followed by a converging oil film gap. The boundary conditions for this numerical approach are depicted in Figure 2. The top surface is moving, the bottom surface is stationary, and ambient pressure is applied on the lateral surfaces. Consequently, the liquid volume α l is equal to 1 on these surfaces. As a circular disc is used, the velocity field of the moving surface consists of horizontal and vertical components, as shown in Figure 2.

The simulation results include the pressure p and velocity u → fields. To evaluate the performance of a specific oil film gap, the lifting force F → l i f t and drag force F → d r a g are computed (cf. Equations 10, 11). The lifting force F → l i f t is the integral of the hydrodynamic pressure p integrated over the top surface. The drag force is the integral of the shear stresses τ x over the top surface. F → l i f t . = ∫ A p x , y , z = max · n → d A , (10) F → d r a g . = ∫ A τ x x , y , z = max d A . (11)

The required simulation parameters are presented in Table 1. HLP22 oil is used for lubrication, and the viscosity at 23 °C is calculated utilizing the Walther (1931) equation. The density ρ l at different temperatures is estimated using the equation presented by GOST (2025). All other oil- and cavitation parameters are based on the work of Hong et al. (2018) and Savio et al. (2021).

Figure 3 illustrates the geometry of the single- and multi-dimple textures. To ensure manufacturability, the dimple geometry must be easily reproducible and cost-effective. Previous research has shown that rectangular textures are more effective than other simple dimple shapes (Scharf et al., 2025). Scharf et al. (2025) showed that textures that can cover a larger region, such as rectangular textures, perform better than other texture shapes (e.g., rectangular dimples). To ensure manufacturability, the edges are rounded with a radius R , which is defined by the minimum of a t e x . / 4 and b t e x . / 4 .

An optimization algorithm is used to identify the optimal dimple parameters for specific conditions.

The negative performance enhancement ratio (PER), presented by Sharma et al. (2019) (cf. Equation 12), is used as the cost function for this optimization algorithm. Because such algorithms are designed to find a minimum, the negative PER value is defined as the cost function. This methodology was previously published by Scharf et al. (2025), and a Tree-structured Parzen Estimator (TPE) (Bergstra et al., 2025) is employed to determine the optimal parameters. A TPE is a variant of the Bayesian optimization algorithm (Watanabe, 2025), which categorizes parameters into trials that perform better and those that do not (Wang L. et al., 2025). Initially, the parameters are defined randomly, and then the algorithm attempts to maximize the ratio between better-performing trials and the others (Wang L. et al., 2025). This process should guide parameter selection toward better-performing regions. The TPE sampler has the advantage of being able to be used for different data types (categorical, float, integer), which is essential for creating manufacturable textures. P E R = F → l i f t , t e x t u r e d F → l i f t , u n t e x t u r e d F → d r a g , t e x u r e d F → d r a g , u n t e x t u r e d   . (12)

In Table 2, the intervals for the optimization algorithm parameters are introduced. A categorized list of values for the possible optimal parameters is defined for the texture length a t e x . , width b t e x . , and height h t e x . The spacing interval for the multi-dimple texture depends on a t e x . and b t e x . , causing the limits for d h o r . and y s t a r t to change, depending on a t e x . and b t e x . . Therefore, a floating parameter interval is defined for d h o r . and y s t a r t . The limits are set to ensure that the dimples do not overlap each other and do not protrude beyond the global oil film gap. Three parameters are varied for the single-dimple textures, and five parameters are varied for the multi-dimple texture.

The stopping criteria for both geometries were based on the number of trials. Specifically, 1000 simulations were conducted for the single-dimple texture, and 5000 simulations were conducted for the multi-dimple texture. Note that such a large number of simulations is not necessary to identify the optimal dimple parameters, as these were found within a few hundred numerical trials. However, the extensive dataset does allow for verification of whether the identified optimal parameters are indeed the best ones.

With the aid of the simulation algorithm described, the optimal dimple parameters were found and are depicted in Table 3. It is evident that the hyper-parameter toolbox aims to reduce the spacing between the dimples, as d h o r . reaches the lower limit of the interval, and y s t a r t reaches the upper limit of y s t a r t . These results suggest that a single-dimple texture leads to a higher PER.

Figure 4 depicts the test methodology using a modular compact rheometer MCR702e from Anton Paar. A specific oil film gap height h min ⁡ . is adjusted to conduct the experiments. The generated normal force, which corresponds to the lifting force F → l i f t in the simulation approach, is measured. The basic principle of this experimental methodology is a three-pin-on-disc test rig. The disc rotates while the pins remain stationary. The pin specimen consists of three single pins arranged to avoid tilting effects and ensure that the MCR702e’s air bearing is not loaded through shear forces. The resolution of the vertical displacement is approximately 0.6 µm, and the maximum tolerable normal load is 50 N. The load cell and sensor to measure the displacement are located in the upper part of the rheometer. The test cell itself is flooded with oil, and the splashback avoids oil contamination of the surrounding area. Additionally, this component serves as a type of oil storage, ensuring that the oil does not form a parabolic shape due to centrifugal force. In addition to the lifting force F → l i f t , the drag moment M → d r a g , the rotational speed n and the minimum oil film gap height h min are measured.

The specimens must be accurately adjusted before each test. The axial runout of the disc and the vertical position of each pin are arranged. The axial runout is measured using a dial indicating gauge with a resolution of 2 µm. To quantify the vertical position of each pin, a feeler gauge stock is placed on the alignment surface of a pin, and a normal force of 1 N is applied to measure the vertical displacement for each pin. Feeler gauge stocks are added under the socket to adjust any displacement differences, achieving a height difference of less than 2 µm.

To determine the zero gap, a normal load of 1 N is applied when the oil bath is flooded and heated. This allows consideration of the thermal expansion.

As previously mentioned in this study, a three-axle milling machine is used to manufacture each texture presented in Table 3. A ball nose cutter (with a diameter of 0.5 mm) is installed in the milling machine (cf. Figure 5d). The manufacturing times for a complete specimen, consisting of three pins, are 4 h for a single-dimple and 6.5 h for a multi-dimple textured pin. It appears that the milling process is not as time efficient as estimated, but the milling machine used is not designed for high cutting speeds. The maximum spindle speed is limited to 8000 rpm, and the tool does not have an internal cooling system. Sommer et al. (2024), for example, used a high-speed milling machine with a maximum spindle speed of 45,000 rpm, which would significantly decrease the manufacturing time.

Figure 5 shows the surface of a single pin (untextured, single-dimple, and multi-dimple texture). The machining grooves are oriented in the direction of movement. The parallel alignment surface is located on the left side, followed by the converging region.

A contour measurement device, MarSurf VD140, is used to quantitatively evaluate the pin geometry. In Figure 6a, the methodology for evaluating the pin’s surface is detailed. Using a tactile probe, the surface contour is measured over different horizontal lines with a displacement of 0.25 mm between each measurement. Figures 6b–d display the evaluated surface profiles. It can be observed that the surface is parallel up to 2 mm, followed by an inclination. Theoretically, the height difference for an angle of 0.16° and a length of 10 mm is 28 µm. This height difference is evident in Figure 6b–d. As the surface contour measurement device is used to evaluate the geometry of the texture, the y coordinates do not cover the entire area of 10 mm. Figures 6c–d show that the optimal texture parameters, for example, the textured depth h t e x . of 20 µm, can be achieved with this common manufacturing technique. The irregularities in the height profile at different y coordinates can be attributed to the different tool tracks.

These different tool tracks also influence the roughness of the pins, as presented in Table 4. For the untextured specimens, the surface roughness is measured three times for each pin, perpendicular to the lay direction. The roughness parameters for the single-dimple texture are determined inside the dimple (two tracks per pin) and outside the dimple (one track per pin). Due to the limited space inside the dimples for the multi-dimple textures, the roughness parameters are measured outside the dimples, twice for each pin. The disc roughness is analyzed in the radial direction four times.

Standard aluminum discs are used for the disc material in the pin-on-disc test rig. The pins are made from stainless steel, and hydraulic oil HLP22 (LIQUI MOLY GmbH, 2025) is used as the lubricant (cf. Table 5).

Each experiment consists of three different stages (cf. Figure 7):

Heating

During the heating period, the oil film gap height h min ⁡ . is set at 1 mm, and the disc rotates at 300 rpm to establish a uniform temperature field. This is followed by the adjustment of the oil film gap height, where the minimum film thickness h min ⁡ . is defined. During this process, the disc is stationary. The final stage is the testing phase, where the rotational speed is increased from 100 rpm to 1000 rpm, which corresponds to a magnitude of the velocity between 0.23 m/s and 2.3 m/s. Two different temperature levels (23 °C and 40 °C) were tested, along with seven different minimum oil film gap heights h min ⁡ . . The minimum oil film gap height h min ⁡ . is controlled and measured during the experiment to prevent the bodies from coming into contact. To evaluate churning losses, a test with a minimum oil film gap height h min ⁡ . of 1 mm is conducted. The drag moment M → d r a g is subtracted from the results obtained to identify the friction in the tribological contact.

Figure 8 shows specific test graphs for different pin geometries. Generally, it can be observed that with increasing rotational velocity, the lifting force F → l i f t and drag force F → d r a g increase. Furthermore, the temperature measured at the bottom of the oil bath is constant at 40 °C during the testing period. Although the control system to keep the minimum oil film gap height constant was enabled, these graphs illustrate that the minimum oil film gap height h min ⁡ . increases with increasing rotational speed. Theoretically, a minimum oil film gap height h min ⁡ . = 25 µm is defined in the graphs of Figure 8. This defined value is only maintained at the beginning of the experiment. With increasing normal load, h min ⁡ . also increases stepwise. Further experiments have shown that this behavior is more dominant at lower minimum oil film gap heights h min ⁡ . (cf. excerpt of the experimental results in the appendix). Therefore, the increasing lifting force F → l i f t with decreasing h min ⁡ . could potentially explain this phenomenon. Note that h min ⁡ . is measured as a displacement of the air bearing.

Regarding the performance of different texture geometries, the single-dimple texture leads to the highest lifting force F → l i f t , followed by the multi-dimple texture and the untextured oil film gap (cf. Figure 8). This trend was also seen at a temperature of 23 °C and with other minimum oil film gap heights h min ⁡ .

Referring to the drag force F → d r a g , Figure 8 shows that the F → d r a g of both textures is quite similar, and it is greater than the F → d r a g of the untextured oil film gap. Additional experiments revealed a controversial result at 23 °C. At this temperature level, the F → d r a g of both textures is also similar, but less than the F → d r a g of the untextured pin (cf. Figure 9). However, note that the drag force F → d r a g is, in general, low, up to approximately 0.3 N at room temperature, depending on the rotational speed n, the oil film gap height h min ⁡ . , and the surface of the pin.

Figure 9 presents the experimental and numerical results for specific conditions. Similar to Figure 8, h min ⁡ . increases with increasing F → l i f t . For the simulated points, the effective h min ⁡ . is determined according to the experimental results. Consequently, there appear to be some fluctuations in the numerical results, which are a result of varying two parameters: rotational speed n and minimum oil film gap height h min ⁡ . . The trends and rankings between the numerical and experimental results are similar, but there is an offset between them. This difference could be explained by a higher temperature in the oil film gap, which reduces the viscosity, or through the observation that, according to DIN 51524-2 (2017), the kinematic viscosity of an HLP22 can vary between 19.8 mm²/s and 24.2 mm²/s, or a combination of both factors. Henry et al. (2015) found that the temperature inside the oil film gap is significantly higher. They measured the temperature at various positions in a hydrodynamic thrust bearing and concluded that the temperature can increase by approximately 10 K (Henry et al., 2015). As the temperature increases, the oil viscosity decreases, resulting in a decrease in the lifting force F → l i f t . This effect is especially significant at low temperatures. At a temperature of 23 °C, a temperature increase of 10 K leads to a 40% decrease in viscosity for the oil used in this study. Consequently, the numerical results overestimate the lifting force F → l i f t . It can be observed that the relative difference between the numerical and experimental results decreases as the temperature increases. This also supports the potential mechanism, as at higher temperatures, there are fewer viscosity deviations due to temperature deviations. Regarding the drag force F → d r a g , it should be noted that these values are generally low, making them difficult to evaluate. Although the churning losses are determined at a minimum oil film gap height h min ⁡ . of 1 mm, it is possible that the churning losses vary, leading to overestimation of the drag force in the experiments. The difficulty of achieving the same numerical and experimental results was also mentioned by Wang et al. (2021), where a laser-textured surface pin-on-disc contact is analyzed. They compared the simulated and measured coefficient of friction COF and found a significant deviation between the values (Wang et al., 2021). Nevertheless, this test methodology can be used to identify similar trends as the simulation would predict.

The simulation algorithm showed that a single-dimple texture leads to a higher PER than the multi-dimple texture (cf. Table 3). This observation was also mentioned by Codrignani et al. (2020), although they simulated a pin-on-disc contact with a minimum oil film gap height h min ⁡ . of 1 µm and a velocity of 0.1 m/s. In this work, a significantly higher oil film gap and greater velocity were analyzed. This single-dimple texture resembles a Rayleigh step, which is already known in the literature. Rahmani et al. (2009) found that the optimal length of such a step is 70% of the oil film gap length for a parallel slider. In this study, the length of the texture a t e x . is 83% of the slider’s length l , but it should be noted that this is the upper limit of the examined interval, which was defined to guarantee a manufacturable texture geometry. Additionally, note that in this study, the performance of a convergent oil film gap is analyzed, which means that two converging regions overlap, as presented in Figure 1. Regarding the texture width b t e x . Bei et al. (Rahmani et al., 2009) concluded that for rectangular textures on a convergent oil film gap, the optimal width b t e x . is approximately 70% of the oil film gap width b . This observation can also be found in Table 3, where the optimal texture width b t e x . is 71% of the complete width b .

Liu et al. (2019) investigated the performance of micro dimples for a parallel oil film gap and found that the optimal texture height h t e x . is 30 µm to increase the pressure the most for a minimum oil film gap height h min ⁡ . of 30 µm. In this study, the optimal texture height h t e x . is 20 µm for a minimum oil film gap height h min ⁡ . equivalent to 25 µm. Although Liu et al. (2019) simulated a parallel oil film gap, their work and this study present similar optimal texture heights h t e x . .

As already mentioned, the optimization algorithm tries to reduce the spacing by searching for parameters that reduce the spacing between the dimples. Xing et al. (2021) focused on the performance of multi-dimple textures on a parallel oil film and showed that with increasing spacing perpendicular to the movement direction, the load-carrying capacity decreases. This phenomenon matches the results obtained in this work. Xing et al. (2021) explored the influence of the spacing in the movement direction and showed that with increasing spacing in the movement direction, the load-carrying capacity decreases starting with a spacing length of 10 µm. Below this 10 µm, the load-carrying capacity increases (Xing et al., 2021). It should be noted that Xing et al. (2021) changed the global geometry of the oil film gap depending on the spacing distance and evaluated the average fluid pressure. So, in the case of increasing spacing at constant dimple geometries, the average hydrodynamic pressure is lower than that of smaller spacing because the relative area of the converging region, related to the area of the oil film gap, becomes lower. Nevertheless, similar trends in this work and in Xing et al. (2021) could be observed because the minimum spacing was defined as 100 µm.

As mentioned in the introduction, most publications use a test methodology for parallel oil film gaps. Pusterhofer et al. (2025), Rosenkranz et al. (2019), and Guo et al. (2018) used a test rig to evaluate the performance of textures in full-film lubrication by applying an external force and measuring the minimum oil film gap height h min ⁡ . . This differs from the methodology of this article, where the oil film gap height h min ⁡ . is applied, and the resulting lifting force F → l i f t is measured. This principle can prevent mixed lubrication, but Figure 8 shows that h min ⁡ . is not kept constant with increasing lifting force F → l i f t , as previously mentioned.

Guo et al. (2018) conducted their experiments at low velocities, less than 0.02 m/s, compared to Rosenkranz et al. (2019) (1 m/s) and Pusterhofer et al. (2025) (5 m/s). The test methodology used in this work varies the velocity between 0.23 m/s and 2.3 m/s. Additionally, glass discs are required for the methodology described by Rosenkranz et al. (2019) and Guo et al. (2018) to measure the oil film thickness with optical devices. Pusterhofer et al. (2025) developed a test methodology using an eddy current sensor to measure the minimum oil film gap height h min ⁡ . This sensor specification is also calibrated for a specific disc material. The test methodology presented in this study has the advantage of measuring the minimum oil film gap height h min ⁡ . in the air bearing, eliminating the influence of specimen material on the measurement, in contrast to Pusterhofer et al. (2025), Rosenkranz et al. (2019), and Guo et al. (2018).

Because the viscosity of the oil is temperature dependent, the control of temperature is crucial. Rosenkranz et al. (2019) and Pusterhofer et al. (2025) published that the oil is pumped to the contact. In Rosenkranz et al. (2019), the oil temperature at the inlet ranges from 23.5 °C to 26 °C, while Pusterhofer et al. (2025) did not control the oil temperature. No information regarding temperature is provided Guo et al. (2018). This study uses a fully flooded oil bath, allowing precise temperature adjustment. Nevertheless, it should be noted in the tribological contact itself that the temperature potential is higher. For instance, Henry et al. (2015) measured a temperature difference of over 10 K in the tribological contact.

Friction force measurement is complex due to the small values. Rosenkranz et al. (2019) determined the friction force by small displacements of the load arm, while Pusterhofer et al. (2025) measured the friction force by an integrated mounted force cell. Guo et al. (2018) do not consider drag force. In the test methodology of the module compact rheometer MCR702e, the friction moment is measured by the current in the engine. The installed air bearing reduces friction losses significantly. Therefore, this solution does not depend on any displacements.

Although the presented test methodology has specific advantages, the manufacturing process of the pin specimens is complex, as the convergent region must be manufactured and cannot be adjusted, unlike Pusterhofer et al. (2025), Rosenkranz et al. (2019), and Guo et al. (2018).

In terms of the lubrication system, this test methodology has an advantage, as the oil bath is fully flooded. This allows for easy consideration of thermal expansion when heating the complete specimens. In Pusterhofer et al. (2025), Rosenkranz et al. (2019), and Guo et al. (2018), the oil is transported through a pipe and supplied to the contact, which may pose challenges when conducting experiments at higher temperatures, if hot oil is only added to the surface.

Codrignani et al. (2020) and Fouflias et al. (2015) show numerically that single-dimple textures can increase the lifting F → l i f t and drag force F → d r a g more than multi-dimple textures. This behavior is observed in this study (cf. Figures 8, 9). Theoretically, h min ⁡ . should be kept constant, but with an increasing lifting force F → l i f t , h min ⁡ . increases. As h min ⁡ . of both textures increases more than the untextured oil film gap, this could be an indicator of the beneficial behavior of surface textures, as presented in Figure 8.

Zhang et al. (2015), Fouflias et al. (2015), Babu et al. (2021), and Ullah et al. (2021) investigated the performance of different dimple geometries in different velocity intervals numerically. They show that the ranking of the performance, including lifting F → l i f t and drag force F → d r a g , over their analyzed velocity interval does not change. This behavior also is observed in this study and is depicted in Figures 8, 9.

The influence of different temperatures, corresponding to different viscosities, is examined by Gao and Chen (2020). They conducted experiments on parallel grooved textured oil film gaps. They found that the reduction of the load-carrying capacity with increasing temperature is not constant for all analyzed grooves (Gao and Chen, 2020). Because their reference is a parallel oil film gap, the comparison between this study and their work is not useful, because their reference experiments result in a lifting force F → l i f t of approximately 0 N. However, Figure 9 shows that with increasing temperature, the beneficial effect of textures, which decreases the drag force F → d r a g compared to an untextured oil film gap, can no longer be observed. Nevertheless, the surface modifications increase the lifting force, compared to the untextured oil film, even at higher temperatures.