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Research Papers: Gas Turbines: Structures and Dynamics

# An Optimum Design Approach for Textured Thrust Bearing With Elliptical-Shape Dimples Using Computational Fluid Dynamics and Design of Experiments Including CavitationOPEN ACCESS

[+] Author and Article Information
Gen Fu

Laboratory for Turbomachinery and Components,
Department of Biomedical
Engineering and Mechanics,
Virginia Tech,
Norris Hall, Room 107,
495 Old Turner Street,
Blacksburg, VA 24061
e-mail: gen8@vt.edu

Alexandrina Untaroiu

Laboratory for Turbomachinery and Components,
Department of Biomedical
Engineering and Mechanics,
Virginia Tech,
Norris Hall, Room 324,
495 Old Turner Street,
Blacksburg, VA 24061
e-mail: alexu@vt.edu

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received December 22, 2016; final manuscript received February 14, 2017; published online April 11, 2017. Editor: David Wisler.

J. Eng. Gas Turbines Power 139(9), 092502 (Apr 11, 2017) (9 pages) Paper No: GTP-16-1594; doi: 10.1115/1.4036188 History: Received December 22, 2016; Revised February 14, 2017

## Abstract

Textured thrust bearings are capable of providing higher load capacity and lower friction torque compared to nontextured bearings. However, most previous optimization efforts for texturing geometry were focused on rectangular dimples and employed Reynolds equation. Limited studies have been done to investigate the effects of partially textured thrust bearings with elliptical dimples. This study proposes a new optimization approach to find the optimal partially texture geometry with elliptical dimples, which maximize the loading capacity and minimize the friction torque. In this study, a 3D computational fluid dynamics (CFD) model for a parallel sector-pad thrust bearing is built using ANSYS cfx. Mass conserving cavitation model is used to simulate the cavitation regions. Energy equation for Newtonian flow is also solved. The results of the model are validated by the experimental data from the literature. Based on this model, the flow pattern and pressure distribution inside the dimples are analyzed. The geometry of elliptical dimple is parameterized and analyzed using design of experiments (DOE). The selected geometry parameters include the length of major and minor axes, dimple depth, radial and circumferential space between two dimples, and the radial and circumferential extend. A multi-objective optimization scheme is used to find the optimal texture structure with the load force and friction torque set as objective functions. The results show that the shape of dimples has a crucial effect on the performance of the textured thrust bearings. Searching the design space for a proper combination among the design variables satisfying the constraints has the advantage of capturing the codependence among design variables and leads to a surface patterning of the bearing, which showed a 42.7% improvement on the load capacity.

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## Introduction

For engineering application, energy efficiency is always a big issue. One of the common factors causing energy loss is friction. To reduce friction torque, an effective way of controlling friction at oil lubricated interface has been shown to be surface texturing. These structures have been widely used in bearings and seals [15]. Applying textured surface on bearing lubrication has been studied for years because of its beneficial effects on the tribological performance [6]. In textured bearings, identical geometries are introduced on the supporting surface to alter the surface topology. With the new surface geometry on the stator surface, the hydrodynamic behavior of fluid film bearings can be improved. In addition, the thin fluid film can still provide loading forces with textured surface on the lubricated interface even though the two matting surfaces are parallel [7].

Over the past 50 years, different research efforts have elaborated the application of texture patterns on the stator surface of thrust bearings. Most of these studies are based on numerical modeling. Among all these numerical methods, Reynolds equation is widely used in the early research [810]. In addition, energy equation is coupled with momentum equations by viscosity terms. Based on the coupled energy equation, most of the thermal analysis used adiabatic boundaries. In the last two decades, CFD has been used to analyze the bearing performance. Besides these simulation methods, a lot of experimental works have been done to study the characteristics of the partially textured thrust bearing. At present, to obtain the best bearing performance, researchers are mostly focused on the optimization of the texture patterns on the stator surface.

The manufacturing methods used to create these dimples on the surface of the pad also affect the lubricating condition. Photolithographic process can produce partially textured bearings having higher fluid film thickness than that produced by laser textured bearings. Based on all the previous studies, it is believed that optimum textured geometry can be achieved. In addition, surface texturing can also be used on microbearings. Papadopoulos et al. [15,16] optimized the geometry shape of the trapezoidal dimples and rectangular dimples on microthrust bearings. At high operating speed, compared to pocketed bearing and tapered-land bearing, textured thrust bearing exhibits better friction characteristics [17].

After reviewing the previous studies, there is no standard approach to optimize the texture pattern and the size of the dimples has not been systematically and quantitatively studied. The goal of this study is to analyze the influence of texture geometry on the performance of parallel thrust bearings and optimize the performance at a typical operating condition. In this study, the basic approach is to combine the optimization algorithm and CFD simulation. The objectives of the optimization process are to get maximum load capacity and minimum friction. Based on previous studies, the fraction of the texture is expected to have significant influences. This process can be considered as a standard approach for future research.

## CFD Model

###### Governing Equations.

In the current study, laminar flow model is used since the Reynolds number in the flow domain is in the laminar regime. To solve the fluid flow in this study, three main governing equations are calculated simultaneously Display Formula

(1)$∂ρ∂t+∇⋅(ρU)=0$

where U is the velocity vector, and $ρ$ is the density Display Formula

(2)$∂(ρU)∂t+∇⋅(ρU×U)=−∇p+∇⋅τ+SM$

where stress tensor , and $SM$ is the external force Display Formula

(3)

where is the total enthalpy, T represents the temperature, and $SM$ is the external heat source.

Incompressible and isoviscous conditions are assumed for the fluid. Rayleigh Plesset model is used in this study to analyze the effect of cavitation on the load capacity and friction of the bearing. The tendency of flow to have cavitation is described by cavitation number in the ANSYS CFX package, defined as Display Formula

(4)$Ca=p−pv12ρU2$

where p is a reference pressure for the flow, and pv is the vapor pressure in the liquid. Rayleigh–Plesset model describes the evolution of the bubble radius in an infinite liquid domain. The stress at the interface between gas and liquid is balanced. The Rayleigh–Plesset equation is as follows: Display Formula

(5)$RBd2RBdt2+32(dRBdt)2+2σρfRB=pv−pρf$

where $RB$ is the bubble radius, $ρf$ is the liquid density, and $σ$ represents the surface tension coefficient between the liquid and vapor. There are no thermal barriers assumed to bubble growth in this equation.

###### Bearing Geometry and CFD Model Validation.

The geometry of the base model is shown in Fig. 1. It is a typical configuration of a 12 pads thrust bearing with partially textured square dimples. All the geometry parameters are derived from the literature [2] to facilitate validation of the initial computational model. The computational model is built in ANSYS, and a mesh refinement with inflation layers along the pad surface is performed to increase mesh quality. The mesh density is decided based on a grid convergence study to determine the trade-off between accuracy of the numerical solution and the computational time required to attain that solution. The resulting computational model has approximately 980,000 grid elements.

The numerical results are compared with the experimental data from the literature [2], as shown in Table 1. From the table, it can be seen that the CFD results agree well with the experimental data. Based on this validated model, the square dimples are revised to elliptical dimples in this study in order to investigate the influence of the new surface patterning. All other bearing geometry parameters are constant. The configuration of the elliptical dimples is shown in Fig. 2. Table 2 lists the complete geometry parameters for the bearing model with elliptical dimples.

###### Boundary Conditions.

A 3D steady-state CFD model is built in this study. The reference pressure used in this simulation is set to be 1 atm. The boundary conditions and operating condition used in this model are listed in Table 3.

The lubricant properties used in this model are similar with the fluid used in the validation experiments from the literature. The properties of the fluid are listed in Table 4.

## Design of Experiments

###### Parameterization.

The geometry of the thrust bearing with elliptical dimples is parameterized to conduct a sensitivity analysis. The key geometry parameters that define the configuration and distribution of the dimples are selected as design variables. These parameters include the length of major axis, the length of minor axis, dimple depth, circumferential space between two dimples, radial space between two dimples, radial extend, and circumferential extend. The set of values for these parameters used to construct the baseline model is provided in Table 5.

In the DOE process, central composite design (CCD) is used to generate the sampling points. Central composite design, also known as Box–Wilson design, is one of the most typical techniques in DOE process, which is suitable for calibrating the quadratic response model. It is basically a five level fractional factorial design method [18,19]. Each design variable is divided into five levels according to the upper and lower limits of the design space. In this study, a face-centered CCD is employed. The ranges of each design parameters are chosen considering both physical limit and bearing design conventions. A total of 85 design points are generated for this study.

###### Optimization.

Based on the response surface model created from the DOE process, a multi-objective optimization method is used to search the optimal design that results in maximum load and minimum friction torque. A genetic algorithm of goal-driven optimization is employed in this study [20,21]. The multi-objective genetic algorithm (MOGA) is a hybrid variant of popular nondominated sorted genetic algorithm. It is based on controlled elitism concepts. It can evolve uniformly sampled versions of trade-off surface regions. Table 6 shows the parameters used in the MOGA optimization process.

## Results and Discussion

###### Response Surface Model.

Regression analysis is one of the most commonly used statistical methods to estimate the relationship between several dependent variables. As mentioned in the Design of Experiments section, a total of 85 design points are used in the DOE process to sample the design space. The load and friction torque for each point are calculated using the CFD model. Based on these design points, two second-order polynomial response surface models are established for both load capacity and friction torque.

The response surface models describe the relationship between the load and friction force and the seven design variables. When developing the surface response model, a forward stepwise method is used to ensure the statistical significance of each term. All the terms in the response surface model are tested by a partial F-test. The terms which cause a significant improvement of the regression results are iteratively added to the regression model. Figure 3 shows the response plots for the load capacity ((a)–(c)) and friction torque ((d)–(f)).

It can be observed from these plots that most of the parameters have a strong interaction effect with each other. All the plots have some levels of twist, which means that a quadratic model can better fit the data.

The local sensitivity of each design variables for both load and friction torque is shown in Fig. 4. The local sensitivity reflects the influence of each variable in the quadratic surface response model. The positive value indicates positive correlation, while the negative value represents reversed correlation. From Fig. 4(a), one can see that the most influential factor for friction torque is rTheta, which is the extension of the textured area in the circumferential direction. Also, the negative value of the sensitivity indicates that the friction torque is reversely related with the circumferential extension of the texture. Other parameters do not show significant impact on friction torque. In the sensitivity plot for bearing load shown in Fig. 4(b), the circumferential axis length of the elliptical dimples shows the most significant impact. Also, the circumferential axis length is positively related with load capacity.

To evaluate the goodness-of-fit of the generated response surface model, the predicted values versus observed values chart are plotted for both load and friction torque. Figure 5 shows the values predicted from the response surface versus the values observed from the design points. It can be seen from Fig. 5 that all the points are close to the diagonal line, which reflects that the response surface model fits the design points. Table 7 shows the statistical parameters of the regression model. As shown in the table, the coefficient of determination of both load and friction torque is very close to unit, which means the regression model explains the response variation about its mean. Also, the adjusted coefficient of determination is very close to the coefficient of determination. Therefore, the two response models created for load and friction force are of good quality and fit the design points. Based on the above statistical results, it is reasonable to conduct an optimization algorithm with these response surface models.

###### Flow Patterns.

The load capacity for the CFD simulation with cavitation model and without cavitation model under different rotating speeds is compared together in Fig. 6. It can be observed from the figure that the effect of the cavitation model strongly depends on the rotational speed. With lower and higher speeds, the cavitation model has little effect on the load capacity of thrust bearing. However, reviewing the overall trend, the inclusion of the cavitation model will result in higher predicted value of the load capacity.

Figure 7 illustrates the pressure distribution on the rotating surface: (a) shows the results for the simulation with cavitation model and (b) shows the case without cavitation model under the same operating condition. In both cases, the high-pressure region is located at the end of the textured area. At this location, the textured area becomes flat surface. Therefore, the variation of the film thickness at this location reaches the maximum value. In the hydrodynamic lubrication theory, the rate change of the film thickness along the flow direction is positively related with the load capacity. As a result, maximum pressure occurs in this region. Also, the pressure profile is symmetrical in radial direction. The difference due to the cavitation model occurs at the beginning of the textured area. The low-pressure area in this region is shown in the case with cavitation model. This is similar to the half-Sommerfeld pressure boundary condition where all the negative pressures are set to be zero.

###### Optimization.

The multi-objective genetic algorithm is selected in this study to attain the optimal design. Figure 8 shows the Pareto front of the optimization. As objective functions, the load capacity and friction torque are selected and the goal is to obtain a design providing the maximum load capacity and minimum friction torque. The points on the lower right boundary in Fig. 8 show the best set of samples (first Pareto front). The best performing design is selected to keep the maximum load capacity in this study.

Table 8 shows the optimal parameters of the texture configuration. The predicted load capacity is 3255.8 N. To verify this, a CFD simulation was conducted using the parameters corresponding to the optimal design. The results for this CFD model provided a load capacity of 3020.8 N. Compared to the base model, the load capacity shows an improvement of 42.7%. The pressure distribution of the optimal design is shown in Fig. 9.

Figure 10 shows the velocity vectors on the cross section of the elliptical dimple at different depths. As shown in the figure, the length d is the distance of the cross section from the top of the dimple. Figures 10(a)10(d) show the flow pattern for the baseline model, while Figs. 10(e)10(h) represent the flow pattern inside the dimples corresponding to the optimal design. It can be seen that the velocity vectors change to the opposite direction as the distance becomes longer for the baseline model. This indicates that the flow inside the dimples contains vertical vortex. On the other hand, for the optimal design, the flow direction remains the same near the bottom. This indicates that there is no obvious vertical vortex inside the dimples of the optimal design. Therefore, based on the simulated results, the vortex inside the dimple can be harmful to the load capacity in such working condition. In addition, the dimple of the optimal design is elongated in the circumferential direction compared to the baseline geometry. Consequently, such configuration of the dimple can help eliminate the vortex formation inside it.

## Conclusions

A three-dimensional CFD model was constructed to analyze the effects of dimple geometry on the performance of a parallel pad thrust bearing. DOE and multi-objective optimization methods were applied to identify the relationships between dimple geometry parameters and bearing performance. The selected parameters described the size and distribution of the elliptical dimples. The load capacity and friction torque were considered as the main indicators of bearing performance.

The results showed that cavitation model had predicted higher load capacity of the thrust bearings with elliptical dimples, but the effect of cavitation model is strongly dependent on the operating condition of the bearing. Reverse flow pattern was observed at the bottom of the dimples. The regression models for load capacity and friction torque were both found to be statistically significant and accurate. It was shown that friction torque increases as the circumferential extension of the textured area increases. Furthermore, the load capacity has an inverse correlation with the length of circumferential axis of the elliptical dimples. The optimization process was conducted using an MOGA multi-objective algorithm where the load capacity and friction torque were selected as the objective functions. The results showed that the shape of the dimples has a crucial effect on the performance of the textured thrust bearings. Searching the design space for a proper combination among the design variables satisfying the constraints had the advantage of capturing the codependence among design variables and led to a surface patterning of the bearing, which showed a 42.7% improvement on the load capacity.

## Nomenclature

• a1 =

• a2 =

dimple minor axis on circumferential (axiscirc)

• c =

specific heat capacity

• d =

the distance from the dimple top surface to the cross-sectional plane

• dt =

dimple depth

• Nc =

dimple count on circumferential direction

• Nr =

• rpi =

• rpo =

• rti =

• rto =

• Sc =

dimple space on circumferential direction

• Sr =

• tf =

film thickness

• W =

• Wt =

textured width of a bearing pad direction

• β =

thermal expansivity

• ηi =

dynamic viscosity at 45 °C

• θ =

angular span of the bearing pad

• θt =

angular span of the textured zone

• ρ =

density (constant)

## References

Rahmani, R. , Shirvani, A. , and Shirvani, H. , 2007, “ Optimization of Partially Textured Parallel Thrust Bearings With Square-Shaped Micro-Dimples,” Tribol. Trans., 50(3), pp. 401–406.
Marian, V. G. , Kilian, M. , and Scholz, W. , 2007, “ Theoretical and Experimental Analysis of a Partially Textured Thrust Bearing With Square Dimples,” Proc. Inst. Mech. Eng. Part J, 221(771), pp. 771–778.
Untaroiu, A. , Liu, C. , Migliorini, P. J. , Wood, H. G. , and Untaroiu, C. D. , 2014, “ Hole-Pattern Seals Performance Evaluation Using Computational Fluid Dynamics and Design of Experiment Techniques,” ASME J. Eng. Gas Turbines Power, 136(10), p. 102501.
Migliorini, P. J. , Untaroiu, A. , and Wood, H. G. , 2014, “ A Numerical Study on the Influence of Hole Depth on the Static and Dynamic Performance of Hole-Pattern Seals,” ASME J. Tribol., 137(1), p. 011702.
Untaroiu, A. , Migliorini, P. , Wood, H. G. , Allaire, P. E. , and Kocur, J. A. , Jr., 2009, “ Hole-Pattern Seals: A Three Dimensional CFD Approach for Computing Rotordynamic Coefficient and Leakage Characteristics,” ASME Paper No. IMECE2009-11558.
Henry, Y. , Bouyer, J. , and Fillon, M. , 2015, “ An Experimental Analysis of the Hydrodynamic Contribution of Textured Thrust Bearings During Steady-State Operation: A Comparison With the Untextured Parallel Surface Configuration,” Proc. Inst. Mech. Eng. Part J, 229(4), pp. 362–375.
Gropper, D., Wang, L., Harvey, T. J., 2016, “ Hydrodynamic Lubrication of Textured Surfaces: A Review of Modeling Techniques and Key Findings,” Tribol. Int., 94, pp. 509–526.
Buscaglia, G. C. , Ciuperca, I. , and Jai, M. , 2005, “ The Effect of Periodic Textures on the Static Characteristics of Thrust Bearings,” ASME J. Tribol., 127(4), pp. 899–902.
Ozalp, A. A. , and Umur, H. , 2006, “ Optimum Surface Profile Design and Performance Evaluation of Inclined Slider Bearings,” Curr. Sci., 90(11), pp. 1480–1491.
Pascovici, M. D. , Cicone, T. , Fillon, M. , and Dobrica, M. B. , 2009, “ Analytical Investigation of a Partially Textured Parallel Slider,” Proc. Inst. Mech. Eng. Part J, 223(2), pp. 151–158.
Dobrica, M. B. , Fillon, M. , Pascovici, M. D. , and Cicone, T. , 2010, “ Optimizing Surface Texture for Hydrodynamic Lubricated Contacts Using a Mass-Conserving Numerical Approach,” Proc. Inst. Mech. Eng. Part J, 224(737), pp. 737–750.
Papadopoulos, C. I. , Kaiktsis, L. , and Fillon, M. , 2014, “ Computational Fluid Dynamics Thermohydrodynamic Analysis of Three-Dimensional Sector-Pad Thrust Bearings With Rectangular Dimples,” ASME J. Tribol., 136(4), p. 011702.
Gherca, A. , Fatu, A. , Hajjam, M. , and Maspeyrot, P. , 2016, “ Influence of Surface Texturing on the Hydrodynamic Performance of a Thrust Bearing Operating in Steady-State and Transient Lubrication Regime,” Tribol. Int., 102, pp. 305–318.
Marian, V. G. , Gabriel, D. , Knoll, G. , and Filippone, S. , 2011, “ Theoretical and Experimental Analysis of a Laser Textured Thrust Bearing,” Tribol. Lett., 44(3), pp. 335–343.
Papadopoulos, C. I. , Nikolakopoulos, P. G. , and Kaiktsis, L. , 2011, “ Evolutionary Optimization of Micro-Thrust Bearings With Periodic Partial Trapezoidal Surface Texturing,” ASME J. Eng. Gas Turbines Power, 133(1), p. 012301.
Papadopoulos, C. I. , Nikolakopoulos, P. G. , and Kaiktsis, L. , 2011, “ Geometry Optimization of Textured Three-Dimensional Micro-Thrust Bearings,” ASME J. Tribol., 133(10), p. 041702.
Fouflias, D. G. , Charitopoulos, A. G. , Papadopoulos, C. I. , Kaiktsis, L. , and Fillon, M. , 2015, “ Performance Comparison Between Textured, Pocket, and Tapered-Land Sector-Pad Thrust Bearings Using Computational Fluid Dynamics Thermohydrodynamic Analysis,” J. Eng. Tribol., 229(4), pp. 1–22.
Kennard, R. W. , and Stone, L. A. , 1969, “ Computer Aided Design of Experiments,” Technometrics, 11(1), pp. 137–148.
Morgan, N. R. , Untaroiu, A. , Migliorini, P. J. , and Wood, H. G. , 2014, “ Design of Experiments to Investigate Geometric Effects on Fluid Leakage Rate in a Balance Drum Seal,” ASME J. Eng. Gas Turbines Power, 137(3), p. 032501.
Untaroiu, C. D. , and Untaroiu, A. , 2010, “ Constrained Design Optimization of Rotor-Tilting Pad Bearing Systems,” ASME J. Eng. Gas Turbines Power, 132(12), p. 122502.
Fu, G., and Untaroiu, A., 2016, “ A Study of the Effect of Various Recess Shapes on Hybrid Journal Bearing Using CFD and Response Surface Method,” ASME Paper No. FEDSM2016-7907.
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## References

Rahmani, R. , Shirvani, A. , and Shirvani, H. , 2007, “ Optimization of Partially Textured Parallel Thrust Bearings With Square-Shaped Micro-Dimples,” Tribol. Trans., 50(3), pp. 401–406.
Marian, V. G. , Kilian, M. , and Scholz, W. , 2007, “ Theoretical and Experimental Analysis of a Partially Textured Thrust Bearing With Square Dimples,” Proc. Inst. Mech. Eng. Part J, 221(771), pp. 771–778.
Untaroiu, A. , Liu, C. , Migliorini, P. J. , Wood, H. G. , and Untaroiu, C. D. , 2014, “ Hole-Pattern Seals Performance Evaluation Using Computational Fluid Dynamics and Design of Experiment Techniques,” ASME J. Eng. Gas Turbines Power, 136(10), p. 102501.
Migliorini, P. J. , Untaroiu, A. , and Wood, H. G. , 2014, “ A Numerical Study on the Influence of Hole Depth on the Static and Dynamic Performance of Hole-Pattern Seals,” ASME J. Tribol., 137(1), p. 011702.
Untaroiu, A. , Migliorini, P. , Wood, H. G. , Allaire, P. E. , and Kocur, J. A. , Jr., 2009, “ Hole-Pattern Seals: A Three Dimensional CFD Approach for Computing Rotordynamic Coefficient and Leakage Characteristics,” ASME Paper No. IMECE2009-11558.
Henry, Y. , Bouyer, J. , and Fillon, M. , 2015, “ An Experimental Analysis of the Hydrodynamic Contribution of Textured Thrust Bearings During Steady-State Operation: A Comparison With the Untextured Parallel Surface Configuration,” Proc. Inst. Mech. Eng. Part J, 229(4), pp. 362–375.
Gropper, D., Wang, L., Harvey, T. J., 2016, “ Hydrodynamic Lubrication of Textured Surfaces: A Review of Modeling Techniques and Key Findings,” Tribol. Int., 94, pp. 509–526.
Buscaglia, G. C. , Ciuperca, I. , and Jai, M. , 2005, “ The Effect of Periodic Textures on the Static Characteristics of Thrust Bearings,” ASME J. Tribol., 127(4), pp. 899–902.
Ozalp, A. A. , and Umur, H. , 2006, “ Optimum Surface Profile Design and Performance Evaluation of Inclined Slider Bearings,” Curr. Sci., 90(11), pp. 1480–1491.
Pascovici, M. D. , Cicone, T. , Fillon, M. , and Dobrica, M. B. , 2009, “ Analytical Investigation of a Partially Textured Parallel Slider,” Proc. Inst. Mech. Eng. Part J, 223(2), pp. 151–158.
Dobrica, M. B. , Fillon, M. , Pascovici, M. D. , and Cicone, T. , 2010, “ Optimizing Surface Texture for Hydrodynamic Lubricated Contacts Using a Mass-Conserving Numerical Approach,” Proc. Inst. Mech. Eng. Part J, 224(737), pp. 737–750.
Papadopoulos, C. I. , Kaiktsis, L. , and Fillon, M. , 2014, “ Computational Fluid Dynamics Thermohydrodynamic Analysis of Three-Dimensional Sector-Pad Thrust Bearings With Rectangular Dimples,” ASME J. Tribol., 136(4), p. 011702.
Gherca, A. , Fatu, A. , Hajjam, M. , and Maspeyrot, P. , 2016, “ Influence of Surface Texturing on the Hydrodynamic Performance of a Thrust Bearing Operating in Steady-State and Transient Lubrication Regime,” Tribol. Int., 102, pp. 305–318.
Marian, V. G. , Gabriel, D. , Knoll, G. , and Filippone, S. , 2011, “ Theoretical and Experimental Analysis of a Laser Textured Thrust Bearing,” Tribol. Lett., 44(3), pp. 335–343.
Papadopoulos, C. I. , Nikolakopoulos, P. G. , and Kaiktsis, L. , 2011, “ Evolutionary Optimization of Micro-Thrust Bearings With Periodic Partial Trapezoidal Surface Texturing,” ASME J. Eng. Gas Turbines Power, 133(1), p. 012301.
Papadopoulos, C. I. , Nikolakopoulos, P. G. , and Kaiktsis, L. , 2011, “ Geometry Optimization of Textured Three-Dimensional Micro-Thrust Bearings,” ASME J. Tribol., 133(10), p. 041702.
Fouflias, D. G. , Charitopoulos, A. G. , Papadopoulos, C. I. , Kaiktsis, L. , and Fillon, M. , 2015, “ Performance Comparison Between Textured, Pocket, and Tapered-Land Sector-Pad Thrust Bearings Using Computational Fluid Dynamics Thermohydrodynamic Analysis,” J. Eng. Tribol., 229(4), pp. 1–22.
Kennard, R. W. , and Stone, L. A. , 1969, “ Computer Aided Design of Experiments,” Technometrics, 11(1), pp. 137–148.
Morgan, N. R. , Untaroiu, A. , Migliorini, P. J. , and Wood, H. G. , 2014, “ Design of Experiments to Investigate Geometric Effects on Fluid Leakage Rate in a Balance Drum Seal,” ASME J. Eng. Gas Turbines Power, 137(3), p. 032501.
Untaroiu, C. D. , and Untaroiu, A. , 2010, “ Constrained Design Optimization of Rotor-Tilting Pad Bearing Systems,” ASME J. Eng. Gas Turbines Power, 132(12), p. 122502.
Fu, G., and Untaroiu, A., 2016, “ A Study of the Effect of Various Recess Shapes on Hybrid Journal Bearing Using CFD and Response Surface Method,” ASME Paper No. FEDSM2016-7907.

## Figures

Fig. 1

The geometry of the thrust bearing with square dimples

Fig. 2

Thrust bearing model-configuration with elliptical dimples: (a) lateral view, (b) top view, (c) detailed elliptical dimple geometry, and (d) 3D model of the fluid domain

Fig. 3

Response surface plots of design variables for load capacity and friction torque: (a) load capacity versus major and minor axis lengths, (b) load capacity versus the distance between dimples in radial and circumferential directions, (c) load capacity versus texture extent in radial and circumferential directions, (d) friction torque versus major and minor axis lengths, (e) friction torque versus the distance between dimples in radial and circumferential directions, and (f) friction torque versus texture extent in radial and circumferential directions

Fig. 4

Sensitivity of design variables: (a) friction torque and (b) load capacity

Fig. 5

Goodness-of-fit of the response models for (a) friction torque and (b) load capacity

Fig. 6

Comparison between model with cavitation and without cavitation

Fig. 7

Pressure distribution of the baseline model: (a) without cavitation model and (b) with cavitation model

Fig. 8

Pareto front

Fig. 9

The pressure distribution of the optimal design

Fig. 10

Velocity vectors at different depths (d) of the elliptical dimple for baseline model and optimal model: (a) d = 0.001 mm, (b) d = 0.003 mm, (c) d = 0.006 mm, (d) d = 0.009 mm, (e) d = 0.001 mm, (f) d = 0.008 mm, (g) d = 0.016 mm, and (h) d = 0.027 mm

## Tables

Table 1 Results comparison between different CFD models and experimental data
Table 2 Geometry parameters of the base model with elliptical dimples
Table 3 Operating conditions and fluid properties
Table 4 Properties of the lubricant used in the CFD model
Table 5 Design parameters and boundaries
Table 6 Parameters used in the MOGA
Table 7 Statistical parameters of response surface model
Table 8 Geometry parameters of the optimal case compared with base model

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