# A Computation Fluid Dynamics Methodology for the Analysis of the Slipper–Swash Plate Dynamic Interaction in Axial Piston Pumps

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*Fluids*

**2023**,

*8*(9), 246; https://doi.org/10.3390/fluids8090246 (registering DOI)

## Abstract

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## 1. Introduction

- Single-land slipper: it represents the original slipper design. It shows a single pocket at the center of the slipper sliding surface, directly connected to the main pressure source through the slipper hole. This design ensures the minimum possible leakage flow through slipper–swash plate clearance;
- Grooved slipper: one or more circular grooves are obtained on the running surface of the slipper. This device brings much more stability to the slipper as it periodically switches from a high-tilt position to a nearly flat one over each revolution of the pump. In this case, the leakage flow is always increased with respect to the single-land configuration, while the lift force may increase or decrease depending on the groove position central radius according to [18];
- Vented grooved slipper: the more external groove is connected to the drain pressure through one or more radial connections with the intention of reducing the slipper spin. However, in this configuration, the external groove does not produce hydrostatic lift, thus leading to a significant reduction in the lift force with respect to both previous designs.

## 2. Materials and Methods

_{0}was considered between the slipper and the swash plate. In Figure 3, a section view of the entire computational domain highlights the fluid regions associated with the main elements of the slipper.

_{0}.

_{0}was forced at the interfaces with the gap, and two boundary layers were generated near the solid surfaces. The extruded mesh approach was used for the axial discretization of regular-shape volumes, such as the leakage and the grooves, where five and fifteen cell layers were respectively defined. Ultimately, the three-dimensional grid resulted in almost 3.4 million cells, corresponding to a 14-day computational time on a modern cluster architecture with 160 processors for the simulation of an entire revolution of the shaft. In Figure 4a, a central section of the slipper highlights the fluid domain discretization, while two separate views of the mesh within the slipper pocket and inside grooves are shown in Figure 4b,c, respectively. In Figure 4d, a very expanded view of the mesh within the gap reveals the 0.2h

_{0}layers’ height.

_{pi,i}produced by the pressurization of the piston effective area is further transmitted to the slipper through the piston–slipper ball joint as in (1):

_{pi,i-sl,i}represents the net piston force pressing the slipper towards the swash plate, while β symbolizes the swash-plate inclination angle. Moreover, the constraint force produced by the slippers retaining ring F

_{rp-sl,i}and the centrifugal force generated by slipper rotation F

_{sl,icen}were added to the model in the form of external forces acting on the rigid body. On the other hand, the weight F

_{sl,ig}, the force of inertia F

_{sl,iI}, the viscous friction force F

_{sl,ivisc}, and the contact force F

_{sw-sl,i}produced by the slipper–swash plate collision were all included in the results of the simulation. In particular, the weight contribution was computed from the gravity acceleration included in the continuum properties along with the peculiar mass of the slipper. The lumped parameters model of the full pump was exploited to derive the evolution of the external forces as a function of the slipper angular position φ. The resulting dataset was imported in the CFD simulation through bi-dimensional file tables, while the corresponding plots were normalized with respect to the maximum force acting on the piston, F

_{pi,max}, which are reported in Figure 7.

_{0}for this specific application. Furthermore, it is worth stressing that the spinning motion of the slipper, i.e., the revolution of the slipper about its own axis, was not addressed in this study as it would have produced high mesh distortions that are still not supported by the morphing methodology.

_{0}represents the fluid density at the atmospheric pressure, while c is the speed of sound of the oil.

## 3. Results

_{0}, the slipper progressively moved towards the swash plate after reaching a dynamic equilibrium after a 25-degree initial transient. In Figure 9b, the same plot outlines the evolution of the slipper–swash plate distance over the second revolution of the driving shaft.

_{0}was obtained at the suction-to-supply transition interval, during which the highest-pressure peak occurred. Moreover, in Figure 10, a zoomed view of the line integral convolution at the slipper outer radius gives a qualitative description of the gap maximum compression.

_{lim}represents the maximum tilt angle under nominal conditions, i.e., the angle Ψ, which produces slipper–swash plate contact for a central distance h

_{0}between parts. A sinusoidal trend was observed in the rotations about both the X-axis and the Y-axis, while a mainly linear characteristic interspersed with abrupt variations at the transition intervals was extracted for the slipper translation along the Z-axis. In absolute terms, the variable distance between the slipper and the swash plate is primarily influenced by the slipper axial displacement with a secondary effect produced by the tilting motions.

_{pi,max}was estimated over the suction period, while 0.62F

_{pi,max}was measured at the supply phase due to the greater contribution of the hydrostatic lift. Moreover, a maximum force of 0.76F

_{pi,max}was reached at the transition from the low-pressure to the high-pressure side of the pump, whereas an absolute minimum of 0.2F

_{pi,max}was registered at the suction opening. On the other hand, the normalized repulsive force produced by the slipper–swash plate collision, which revealed specular behavior to that of the slipper translation, thus showing a linear increase in the contact force for the entire suction stroke followed by a descending ramp over the delivery phase. Once again, the occurrence of force peaks in the regions of outer and inner dead points of the piston were noted, where 0.4F

_{pi,max}and 0.06F

_{pi,max}forces were detected, respectively.

_{G}was defined as in (3) to estimate the slipper hydraulic balancing:

_{Gab}represents the lift force acting on the slipper, whereas F

_{NS}is the sum of the external forces pushing the slipper towards the swash plate. Therefore, three main conditions fully describe the instantaneous behavior of the slipper-swash plate interface.

- B
_{G}< 1: the thrust force is not compensated by the lift component, thus suggesting slipper–swash plate contact. In this configuration, the contact force is enabled. - B
_{G}= 1: a perfect hydraulic balancing of the slipper is reached, thus preventing metal-to-metal contact between parts. Consequently, contact force does not apply in this case; - B
_{G}> 1: the lift contribution exceeds the thrust force, thus losing the fundamental coupling between the slipper and the swash plate.

_{G}< 1 and a certain slipper–swash plate contact is accepted.

_{0}effective range at which the contact force activates. Indeed, a more accurate prediction of the hydraulic force, as well as of the clearance height, would have required a reduction in the user-defined effective range. However, starting from h

_{0}, this operation would have led to higher cells’ distortion with remarkable effects on the mesh stability. Therefore, the proposed solution is the best tradeoff between the results’ accuracy and the modeling limitations of the software.

_{max}, and it highlights the combined effect of both the working pressure and the clearance height. In effect, larger flow losses were measured at pump delivery due to the higher pressure drop across the gap, and a linear increase in the curve in this region was related to the progressive expansion of the flow area. A peak of 0.0079% was finally achieved before the complete closing of the outlet port, when the maximum flow area was reached over the entire supply interval. Instead, the minimum leakage was registered at the supply-to-suction transition, where the simultaneous presence of the lowest working pressure and the largest gap height produced an instantaneous flow rate equal to 0.0044% of Q

_{max}. Hence, from the point of view of the volumetric losses at the slipper–swash plate interface for the proposed slipper design, this result emphasized a greater contribution of the inlet pressure with respect to the flow area.

## 4. Discussion and Conclusions

_{G}was observed with a decreasing trend along the suction interval and a corresponding increment during supply. At each stage, this result was further suggested by constant hydraulic lift opposing the variable piston force pushing the slipper against the swash plate. The largest discrepancy between the external thrust force and the hydraulic lift on the slipper was measured at the high-pressure transition, in which the maximum piston force was transmitted to the slipper through the ball joint. At this point, a global minimum value of B

_{G}, i.e., 0.62, was measured, indicating the highest slipper instability. Moreover, the primary correlation between the leakage flow and the working pressure was pointed out, with a secondary effect produced by the flow area or, in other terms, the clearance height. A non-uniform pressure distribution was observed through the gap due to the tilted position of the slipper during actual pump-operating conditions, but no significant effects were registered on the hydraulic lift because of the short distance between the grooves, which resulted in a concentrated pressure drop. Therefore, the serious possibility of neglecting secondary hydrodynamic effects induced by slipper tilting in the design stages of vented grooved slippers was outlined in this paper. Furthermore, the laminar behavior of the flow inside the gap was confirmed by a null intermittency along an entire revolution of the pump shaft. In conclusion, despite having known limitations related to the impossibility of simulating an effective zero gap between colliding surfaces, the proposed numerical approach represents an extremely advanced and powerful tool for the slippers’ design in axial piston machines. More works are expected in the future to further investigate the effects of different pump-operating conditions, e.g., high working pressure and variable displacement operations, and different slipper designs.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ivantysyn, J.; Ivantysynova, M. Hydrostatic Pumps and Motors: Principles, Design, Performance, Modelling Analysis, Control and Testing; Akademia Books International: New Delhi, India, 2001; ISBN 8185522162. [Google Scholar]
- Jiang, J.H.; Wang, Z.B.; Wang, K.L. Power loss of slipper/swashplate based on elastohydrodynamic lubrication model in axial piston pump. IOP Conf. Ser. Earth Environ. Sci.
**2018**, 188, 012025. [Google Scholar] [CrossRef] - Haidak, G.; Wang, D.; Shiju, E.; Liu, J. Study of the influence of slipper parameters on the power efficiency of axial piston pumps. Adv. Mech. Eng.
**2018**, 10, 1687814018801460. [Google Scholar] [CrossRef] - Koc, E.; Hooke, J. Considerations in the design of partially hydrostatic slipper bearings. Tribol. Int.
**1997**, 30, 815–823. [Google Scholar] [CrossRef] - Zhang, J.; Chao, Q.; Xu, B.; Pan, M.; Wang, Q.; Chen, Y. Novel three-piston pump design for a slipper test rig. Appl. Math. Model.
**2017**, 52, 65–81. [Google Scholar] [CrossRef] - Rizzo, G.; Massarotti, G.P.; Bonanno, A.; Paoluzzi, R.; Raimondo, M.; Blosi, M.; Veronesi, F.; Caldarelli, A.; Guarini, G. Axial piston pumps slippers with nanocoated surfaces to reduce friction. Int. J. Fluid Power
**2015**, 16, 1–10. [Google Scholar] [CrossRef] - Chao, Q.; Zhang, J.; Xu, B.; Wang, Q. Discussion on the Reynolds equation for the slipper bearing modelling in axial piston pumps. Tribol. Int.
**2018**, 118, 140–147. [Google Scholar] [CrossRef] - Wieczorek, U.; Ivantysynova, M. Computer aided optimization of bearing and sealing gaps in hydrostatic machines—The simulation tool CASPAR. Int. J. Fluid Power
**2002**, 3, 7–20. [Google Scholar] [CrossRef] - Mukherjee, S.; Sarode, S.; Mujumdar, C.; Shang, L.; Vacca, A. Effect of dynamic coupling on the performance of piston pump lubricating interfaces. MM Sci. J.
**2022**, 2022, 5783–5790. [Google Scholar] [CrossRef] - Shen, H.; Zhou, Z.; Guan, D.; Liu, Z.; Jing, L.; Zhang, C. Dynamic contact analysis of the piston and slipper pair in axial piston pumps. Coatings
**2020**, 10, 1217. [Google Scholar] - Karpenko, M.; Stosiak, M.; Šukevičius, Š.; Skačkauskas, P.; Urbanowicz, K.; Deptuła, A. hydrodynamic processes in angular fitting connections of a transport machine’s hydraulic drive. Machines
**2023**, 11, 355. [Google Scholar] - Yao, Y.; Qamhiyah, A.Z.; Fang, X.D. Finite element analysis of the crimping process of the piston-slipper component in hydraulic pumps. J. Mech. Des.
**2000**, 122, 337–342. [Google Scholar] [CrossRef] - Milani, M.; Montorsi, L.; Muzzioli, G.; Lucchi, A. A CFD approach for the simulation of an entire swash-plate axial piston pump under dynamic operating conditions. In Proceedings of the ASME 2020 International Mechanical Engineering Congress and Exposition, Virtual, 16–19 November 2020. [Google Scholar]
- Corvaglia, A.; Rundo, M. Comparison of 0D and 3D hydraulic models for axial piston pumps. Energy Procedia
**2018**, 148, 114–121. [Google Scholar] [CrossRef] - Frosina, E.; Marinaro, G.; Senatore, A. Experimental and numerical analysis of an axial piston pump: A comparison between lumped parameter and 3D CFD approaches. In Proceedings of the ASME-JSME-KSME 2019 8th Joint Fluids Engineering Conference, San Francisco, CA, USA, 28 July–1 August 2019. [Google Scholar]
- Milani, M.; Montorsi, L.; Venturelli, M. A combined numerical approach for the thermal analysis of a piston water pump. Int. J. Thermofluids
**2020**, 7–8, 100050. [Google Scholar] [CrossRef] - Canbulut, F.; Koc, E.; Sinanoglu, C. Design of artificial neural networks for slipper analysis of axial piston pumps. Ind. Lubr. Tribol.
**2009**, 61, 67–77. [Google Scholar] [CrossRef] - Bergada, J.M.; Watton, J.; Haynes, J.M.; Davies, D.L. The hydrostatic/hydrodynamic behaviour of an axial piston pump slipper with multiple lands. Meccanica
**2010**, 45, 585–602. [Google Scholar] [CrossRef] - Bergada, J.M.; Watton, J. Axial piston pump slipper balance with multiple lands. In Proceedings of the ASME 2002 International Mechanical Engineering Congress and Exposition, New Orleans, LA, USA, 17–22 November 2002. [Google Scholar]
- Bergada, J.M.; Watton, J. Force and flow through hydrostatic slippers with groove. In Proceedings of the International Symposium on Fluid Control, Measurement and Visualization, Chengdu, China, 22–25 August 2005. [Google Scholar]
- Siemens PLM Software. Star-CCM+ 2021.1.1 User Guide; Siemens PLM Software: Plano, TX, USA, 2021. [Google Scholar]
- Hernandez-Perez, V.; Abdulkadir, M.; Azzopardi, B.J. Grid generation issues in the CFD modelling of two-phase flow in a pipe. J. Comput. Multiph. Flows
**2011**, 3, 13–26. [Google Scholar] [CrossRef] - ISO3448:1992; Industrial Liquid Lubricants—ISO Viscosity Classification. Available online: https://www.iso.org/standard/8774.html (accessed on 20 July 2023).
- Muzzioli, G.; Montorsi, L.; Polito, A.; Lucchi, A.; Sassi, A.; Milani, M. About the influence of eco-friendly fluids on the performance of an external gear pump. Energies
**2021**, 14, 799. [Google Scholar] [CrossRef] - Menter, F.R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J.
**1994**, 32, 1598–1605. [Google Scholar] [CrossRef] - Menter, F.R. Zonal two equation k-ω turbulence models for aerodynamic flows. In Proceedings of the 23rd Fluid Dynamics, Plasmadynamics, and Lasers Conference, Orlando, FL, USA, 6–9 July 1993. [Google Scholar]
- Menter, F.R.; Langtry, R.B.; Likki, S.R.; Suzen, Y.B.; Huang, P.G.; Völker, S. A correlation-based transition model using local variables—Part I: Model formulation. J. Turbomach.
**2006**, 128, 413–422. [Google Scholar] [CrossRef]

**Figure 2.**Main slipper designs: (

**a**) single-land slipper; (

**b**) grooved slipper; (

**c**) vented grooved slipper.

**Figure 4.**Fluid domain discretization: (

**a**) mesh section through the entire 3D geometry; (

**b**) discretization of the slipper’s central pocket; (

**c**) extruded mesh within the grooves; (

**d**) extruded mesh within the clearance.

**Figure 7.**User-defined external forces: (

**a**) piston force; (

**b**) retaining ring force; (

**c**) centrifugal force.

**Figure 10.**Clearance compression: (

**a**) initial condition; (

**b**) maximum compression at the high-pressure transition.

Fluid Regions | Cell Shape | Base Size |
---|---|---|

Central pocket | Polyhedral | 16.7h_{0} |

Grooves | Polyhedral | 6.7h_{0} |

Slipper | Polyhedral | 16.7h_{0} |

Environment | Polyhedral | 33.3h_{0} |

Parameter | Symbol | Value |
---|---|---|

Rotational speed | n_{pump} | 2500 rpm |

Swash-plate inclination angle | β | 18 degrees |

Supply pressure | p_{out} | 30 bar |

Suction pressure | p_{in} | 20 bar |

Drain pressure | p_{drain} | 0.5 bar |

Fluid Regions | Cell Shape | Base Size |
---|---|---|

Density at atmospheric pressure | ρ_{0} | 834 kg/m^{3} |

Dynamic viscosity | µ | 8.14 mPa*s |

Speed of sound | c | 1300 m/s |

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**MDPI and ACS Style**

Muzzioli, G.; Paltrinieri, F.; Montorsi, L.; Milani, M.
A Computation Fluid Dynamics Methodology for the Analysis of the Slipper–Swash Plate Dynamic Interaction in Axial Piston Pumps. *Fluids* **2023**, *8*, 246.
https://doi.org/10.3390/fluids8090246

**AMA Style**

Muzzioli G, Paltrinieri F, Montorsi L, Milani M.
A Computation Fluid Dynamics Methodology for the Analysis of the Slipper–Swash Plate Dynamic Interaction in Axial Piston Pumps. *Fluids*. 2023; 8(9):246.
https://doi.org/10.3390/fluids8090246

**Chicago/Turabian Style**

Muzzioli, Gabriele, Fabrizio Paltrinieri, Luca Montorsi, and Massimo Milani.
2023. "A Computation Fluid Dynamics Methodology for the Analysis of the Slipper–Swash Plate Dynamic Interaction in Axial Piston Pumps" *Fluids* 8, no. 9: 246.
https://doi.org/10.3390/fluids8090246