Numerical analysis and optimization of surface textures for a tilting pad thrust bearing

Abstract A thermo-hydrodynamic model previously developed by the authors is applied in this paper to study the influence of surface texturing on the performance of a tilting pad thrust bearing with offset line pivots. Utilizing an interior-point algorithm, texture depth, circumferential extent and radial extent are numerically optimized to improve three bearing performance parameters: minimum film thickness, friction torque and maximum temperature. Results are presented for various operating conditions and texture densities. It is found that, for most cases, optimum texturing parameters depend significantly on the operating conditions, optimization objective and texture density. Whereas minimum film thickness values can be increased by up to 12%, only minor improvements are achievable in terms of friction torque and maximum temperature.

texture design vector ! relative texture extent in circumferential direction ! pitch angle (rad) " relative texture extent in radial direction # difference in load carrying capacity and applied load (N) #$ circumferential distance from centre of pressure to pivot (rad) % , % , % fractional residuals for pressure, equilibrium and temperature M A N U S C R I P T A C C E P T E D

Introduction
Although first papers on the application of artificial surface textures for enhancing the performance of tribological contacts were published decades ago, a large quantity of research is still being conducted. This is mainly due to the lack of universal texture design recommendations, as it has become clear that, despite the fact that surface textures can be optimized, the optimum design depends significantly on the application, the lubrication regime and even the operating conditions [1]. One of the most popular applications of surface texturing are hydrodynamic bearings, where the main aim is a performance improvement in terms of minimum film thickness, friction torque, temperature and wear.
This paper is concerned with textured tilting pad thrust bearings, an application of surface texturing that has hardly been given any attention. One of the only studies was published recently by Zouzoulas and Papadopoulos [2]. They compared the hydrodynamic performance of a conventional point-pivoted tilting pad thrust bearing with a pocketed, grooved and dimpled design using commercial CFD software. The results showed that minimum film thickness, friction torque and maximum temperature can be improved considerably. Best performance improvements were achieved by the pocketed bearing, followed by the one having circumferential grooves. A parametric study revealed the dependency of optimum texturing parameters on the operating condition and optimization objective. For the investigated cases, a texture depth of 30 µm and a radial texture extent of around 70 % were recommended.
Considerably more studies are available concerning fixed geometry thrust pad bearings.
Marian et al. [3], for example, presented a parametric numerical study on a parallel thrust pad bearing with square dimples and recommended circumferential and radial texture extents of 50 % and from 90 to 100 % respectively in terms of load carrying capacity. Henry Gropper 5 [7], they numerically optimized pocket dimensions and shape utilizing Genetic algorithms.
Another optimization study was published by Papadopoulos et al. [8] for convergent thrust pad bearings. They showed that the optimum texture extent in circumferential direction and the optimum texture depth depend on the convergence ratio and that the highest possible texture density should be chosen. Papadopoulos et al. [9] presented a parametric CFD study and concluded that 67 % and 75 % of the pad should be textured in circumferential and radial direction respectively in terms of load carrying capacity. Moreover, a texture depth close to the encountered minimum film thickness was recommended.
Although these findings are to a certain extent applicable to pivoted pad bearings, more research is needed for this application; in particular regarding the influence of surface texturing on the equilibrium position of the pads. Moreover, previous research is generally limited to only a few operating conditions and restricted parametric studies due to the complexity of the flow and associated high computation times. No optimization of surface patterns has been performed previously for tilting pad bearings.
The present paper is focussed on optimizing texture patterns for tilting pad thrust bearings and investigate their dependency on the optimization objective and operating condition. To allow a mathematical optimization for various conditions in reasonable time, a fast numerical model previously developed by the authors finds application [10]. The model utilizes an adaptive and non-uniform finite volume discretization, where discontinuities are directly incorporated in the discrete system. Moreover, the computational speed is improved by taking advantage of multicore processing, using results from the equivalent untextured bearing and evaluating the bearing equilibrium through a combination of the Newton-Raphson method and Broyden's algorithm. In this paper, the previously developed model is applied in combination with an interior-point method to mathematically optimize texture designs in terms of circumferential extent, radial extent and texture depth for three optimization objectives: a maximization of the encountered minimum film thickness, a minimization of frictional torque and a minimization of the maximum temperature. An in-depth analysis regarding the dependency of the optimum texture design on the operating conditions and optimization objective as well as achievable performance improvements under hydrodynamic conditions are presented. Furthermore, texture design recommendations are given for a wide range of conditions. The thermo-hydrodynamic model applied in this paper is described in detail in Ref. [10] and is therefore only given briefly in the following.

Bearing and texture geometry
The bearing considered is a partially textured tilting pad thrust bearing with offset line pivots (see Fig. 1). configuration as recommended in literature [1]. The texture pattern is fully defined by four parameters: The texture depth (ℎ ), the circumferential extent of the textured area (!), the radial extent of the textured area (") and the texture density (, ). The film thickness over the pad area can be expressed as where ℎ is the film at the level of the pivot, $ the angular location of the pivot and ! the pitch angle. The texture depth is simply added wherever textures are located during the numerical preparation of the film thickness distribution.

Fluid mechanics
The where is the local pressure, , the lubricant density,the rotational speed, & the dynamic viscosity and Θ the fractional film content as described in [11,12] to incorporate JFO boundary conditions. A modified adaptive non-uniform finite volume method is applied to discretize the equations, where discontinuities are directly incorporated in the discrete system to reduce discretization errors and computation time [13]. Although the approach allows the consideration of concentrated inertia effects, they are disregarded in the present study for improved computational performance. This is justified by the low Reynolds numbers and high texture aspect ratios encountered. Dobrica and Fillon [14] showed that the validity of the Reynolds equation for textured surfaces depends on these two parameters.
Here, the Reynolds number is defined as . , ℎ D /& and the texture aspect ratio as The mesh is adapted to the investigated texture pattern by aligning control volume boundaries with discontinuities. The mesh is defined by the number of control volumes inside individual textures, in-between adjacent textures as well as the number of control volumes for the untextured portions of the pad (see Fig. 2).

Fig. 2 Computational mesh with 4 x 4 control volumes inside individual textures, 2 x 2 control volumes in-between adjacent textures, 8 control volumes for the untextured pad area in circumferential direction and 4 control volumes each for the untextured pad areas in radial direction.
The utilized mass-conserving cavitation algorithm requires an iterative solution of the discrete Reynolds equation [12], therefore, a Gauss-Seidel method with pointwise relaxation is applied for both pressure and film content. A constant pressure at the pad boundaries (10 FG Pa) and no-slip conditions are imposed. The iteration stops when the fractional residual reaches values smaller than or equal to a pre-defined tolerance value (for this work . 10 FH ).

Bearing equilibrium
The bearing equilibrium is found by a combination of the Newton-Raphson method and Broyden's method with Sherman-Morrison formula, where the Jacobian matrix is evaluated using a perturbation approach and finite difference formulae. The Newton-Raphson method is applied only for the first temperature iteration. The equilibrium for following temperature iterations is evaluated with Broyden's method for enhanced computational speed due to the Gropper 9 time saved in the determination of the Jacobian matrix. For details about the applied methods and formulae the reader is referred to Ref. [10]. The convergence of the equilibrium solver is checked by the following criterion: where a tolerance value of . 10 FS is used.

Thermal effects
The lubricant temperature is evaluated using an effective temperature method, where a convection parameter of . 0.75 is used [15,16], i.e. it is assumed that 75 % of the heat caused by viscous shearing is removed by convection: where the tolerance value used in this work is . 10 FG .

Texture optimization
The optimization of a given texture design involves finding the texturing parameters that minimize or maximize a certain bearing performance parameter. In the present study, three optimization objectives are investigated: (i) the maximization of the minimum film thickness

Numerical procedure
After importing the input file, the numerical procedure starts by preparing the mesh and initializing the film thickness distribution (see Fig. 3). After this, the discrete Reynolds equation is assembled and solved using the Gauss-Seidel method until pressure convergence is reached (% Q ). The film thickness distribution is then updated by the Newton-Raphson and Broyden's method until the pad is in equilibrium (% Q ). The effective temperature method is subsequently applied to update inlet temperature, effective temperature and effective lubricant viscosity until thermal equilibrium is reached (% Q ). This series of steps is initially executed for the equivalent untextured pad, where a coarse uniform mesh is used, and then repeated for the textured pad using all results from the untextured pad. This methodology leads to a significant reduction in computation time [10]. What follows is the computation of the objective functions as well as other important bearing performance parameters, such as tilt angle, power loss and flow rates.
To evaluate the gradients of the objective functions, the above computations are performed simultaneously on the available processor cores for different texture design vectors . The interior-point algorithm then updates the texture design to minimize or maximize the objective function until the optimality tolerance reaches the specified tolerance (% Q ).

M A N U S C R I P T
A C C E P T E D  Fig. 4 in terms of the difference in predicted load carrying capacity with respect to the load carrying capacity computed with the finest mesh. area in circumferential direction and 4 control volumes each for the untextured pad areas in radial direction (this mesh is shown in Fig. 2).

Validation
The accuracy of the applied numerical model in terms of ℎ D , and DW was investigated in detail in Ref. [  its optimum is different for the three optimization objectives. Whereas the minimum film thickness is very sensitive to a change in texturing parameters around the optimum, reasonably good friction and temperature characteristics can be achieved with a texture design that is a bit further away from the optimum. This means that, for the considered cases, the chosen texture design would most likely be based on the design resulting in the highest minimum film thickness.

Optimization results
The texture design optimization is performed for a total of twelve operating conditions (1000, 2000, 3000 rpm at 0.5, 1.0, 1.5, 2.0 MPa), three optimization objectives (ℎ D , , DW ) and three texture densities (40, 50, 60 %), resulting in a total of 108 simulations. The required computation time for each optimization was around 40 minutes. Note that a cubic spline interpolation is used to smoothen the graphs presented in the following.

Optimum texture depth
Results of the optimization in terms of the optimum texture depth are shown in Fig. 6. To further analyse the relation between the optimum texture depth and the minimum film thickness, results are plotted again in terms of the relative texture depth defined as . ℎ /ℎ D , where ℎ D is taken from the equivalent untextured bearing rather than the texture bearing to allow a selection of the texture depth based on easily available data from the conventional bearing (see Fig. 7). changes are marginal and a texture depth deviating only slightly from the optimum will still result in almost optimal performance as presented in Fig. 5, justify the simplification of the data by averaging over the 12 investigated operating conditions. This results in a recommended relative texture depth of 0.87, 0.93 and 1.01 for densities of 40, 50 and 60 % respectively if a high minimum film thickness is desired. To optimize the bearing in terms of friction and maximum temperature, the relative texture depth should be slightly lower at 0.8, 0.84 and 0.9 for the three investigated densities. As a rule of thumb, a relative texture depth of just under 1 seems to be preferable. In fact, it is well known that a relative texture depth of approximately 1 results in best performance for partially textured fixed geometry parallel and near-parallel hydrodynamic contacts [1]. The found values for the relative texture depth confirm that this is also true for the case of pivoted pad bearings. Furthermore, slightly deeper textures seem be the better for increasing the minimum film thickness whereas lower texture depths are preferred for a reduction of friction and temperature. This was also concluded in the parametric CFD study for point-pivoted pads by Zouzoulas and Papadopoulos [2]. The high dependency of the optimum texture depth on bearing speed and specific load makes it clear that the texture depth has to be selected for the expected operating condition or will be a compromise for the expected range of conditions.

Optimum circumferential texture extent
Results for the optimum extent of the textured region in circumferential direction are given in Fig. 8.

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Regarding the optimum extent of the textured region in radial extent, values range from 56 to 91 % (see Fig. 9).

Fig. 9 Optimum radial texture extent for different operating conditions and optimization objectives for a texture density of (a) 40 %, (b) 50 % and (c) 60 %.
" is mostly independent of the rotational speed and only shows a relevant dependency on the specific load when optimized in terms of the friction torque, where " gets smaller with an increase in load, ranging from 70 to 56 %. It is also evident that a higher texture extent is required for higher texture densities, although differences are small. Similarly to the optimum extent in circumferential direction, the most influential parameter is the optimization objective. If optimized in terms of minimum film thickness, the radial extent should be between 66 and 72 %. Much higher values are predicted when the bearing is optimized for minimizing the maximum temperature. In this case, " should be between 84 and 91 %. The reason behind this is that the higher the texture extent in radial direction, the higher the lubricant inflow, which promotes cooling and results in lower temperatures. In fact, the lubricant inflow is increased by 9 % on average with respect to the untextured bearing. The results show that as a rule of thumb, it can be concluded that a relatively high extent in radial direction of approximately 87 % is recommended when the objective is a minimization of the maximum temperature. About 2/3 of the pad should be textured for optimum performance regarding the minimum film thickness and good frictional behaviour.

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Performance of the untextured bearing
To evaluate the influence of texturing on the bearing performance, the key characteristics of the conventional untextured bearing are evaluated. For these computations, a uniform mesh with 101 x 101 control volumes is used. Results are presented in Fig. 10. For the considered bearing, values range from 0.39 to 1.03 Nm and from 36 to 58 °C for and DW respectively.

Performance of the textured bearing
The relative performance of the textured bearing with respect to the untextured bearing is evaluated for all 108 optimized texture patterns (see Fig. 11).

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A C C E P T E D ACCEPTED MANUSCRIPT bearing. This highlights the importance of a thorough texture design for a specific application.
The results also show that the beneficial influence of texturing increases with an increase in texture density for all considered cases. However, this is expected, as it is known for other partially textured contacts that the density cannot be optimized as a higher density will always improve bearing performance [1]. Therefore, the beneficial impact of the considered texture designs could most likely be further enhanced by using higher texture densities.
However, this was not done as the density is limited by stress concentration and mixed lubrication considerations.
Interestingly It is also noteworthy that the convergence ratio of the pads in equilibrium is no longer determined solely by the pivot position, which is the case for conventional tilting pad thrust bearings. The high influence of the texture extent in circumferential direction on the centre of pressure also entail a dependency of the convergence ratio on the circumferential texture extent, as investigated also by Yagi and Sugimura [19]. Interestingly, the studied texture patterns always decrease the convergence ratio (see Fig. 12).

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A C C E P T E D ACCEPTED MANUSCRIPT Research highlights of the paper with title:

Numerical analysis and optimization of surface textures for a tilting pad thrust bearing
• Texture patterns are optimized in terms of texture depth, circumferential extent and radial extent. • Three optimization objectives are considered: minimum film thickness, friction torque and maximum temperature.
• For most cases, optimum texture parameters depend on the operating conditions, optimization objective and texture density.
• The minimum film thickness can be improved by up to 12 %.
• Only marginal improvements are possible with regards to friction torque and maximum temperature.