# Advanced Lubrication Simulations of an Entire Test Rig: Optimization of the Nozzle Orientation to Maximize the Lubrication Capability

^{*}

## Abstract

**:**

## 1. Introduction

^{MC}Global Remeshing Approach with Mesh Clustering) combined with Rigid Mesh Motions (RMM) and an effective adoption of Arbitrary Mesh Interfaces (AMIs). The whole methodology was implemented in the open-source environment OpenFOAM

^{®}. Its GNU (General Public License) allowed the massive parallelization of the computation.

## 2. State of the Art

^{MC}—GRA with mesh clustering) using a recursive adoption of few meshes created a priori. This approach has led, for the reference configuration (a back-to-back test rig), to a speed up of the simulation by a factor of x328. This significant improvement in computational efficiency has paved the way for the possibility of simulating multistage gearboxes [34] and grease lubrication [35] in a reasonable amount of time. Moreover, the application of CFD not only to study gears but also to bearings seems to be a growing trend. Maccioni et al. published a review paper first [30], and successively the results of a numerical study of the lubricant aeration in a tapered roller bearing and their experimental validation [36]. Other records have been published by Bohnert et al. [37], Wen et al. [38], Feldermann et al. [39], Adeniyi et al. [40], Liebrecht et al. [41], Yan et al. [42] and by Profito et al. [43].

^{MC}approach. These simulations are aimed to understand the best nozzle inclination to ensure the optimal lubrication of the bearing.

## 3. Description of the System

## 4. Modelling of the System

#### 4.1. Governing Equations

^{®}v7 [46] was used to implement the CFD model of the system.

_{oil}+ α

_{air}= 1

_{oil}= 1, the control unit is full of oil; when α

_{oil}= 0, only air is present in the control unit; for any value between 0 and 1, in the control unit an oil–air interface is present. The solution of the continuity equation of the volume fraction allows to trace the oil–air interface.

**u**) = 0

_{c}) acting perpendicular to the interface is added in the conservation equation of α.

**u**) + ∇(

**u**

_{c}α(1 – α)) = 0

**u**

_{c}= min(c

_{α}|

**u**|, max (|

**u**|))⋅n), being c

_{α}the coefficient that controls the magnitude of the compression (usually set between 0 and 2).

_{eq}= α

_{oil}ϕ

_{oil}+ (1 – α

_{oil}) ϕ

_{air}

**u**) = 0

**u**))/∂t + ∇(ρuu) = −∇p + ∇ [μ(∇

**u**+ ∇

**u**

^{T})] + ρ

**g**+

**F**

#### 4.2. Implementation of the CFD Model

**is presented. A detailed description of the geometry partition, the meshing approach, and the mesh-handling technique are presented in the next chapters.**

^{®}#### 4.2.1. Geometry

#### 4.2.2. Domain Partitioning and Meshing Approach

#### 4.2.3. Mesh-Handling Technique

^{MC}) [33,35], which was implemented by the authors in the open-source environment OpenFOAM

^{®}to overcome the limitations of the previous approach. This algorithm foresees the solution of the Laplace smoothing equation and the pseudo solid equation (linearization of the motion equations for small deformations) to describe the motion of the boundaries.

**z**) = 0

^{(n+1)}= x

^{n}+ Δt⋅

**z**

**z**is the prescribed motion, γ is the diffusivity, x is the grid’s nodes position and t is the current timestep. The GRA

^{MC}is based on the preliminary computation of a set of mesh that covers one complete engagement cycle. Then, the wheels find themselves in the same position as the first mesh; thus, it possible to recursively use the computed mesh set for the entire simulation. In this way, the computational effort of the remeshing process is drastically reduced. Another advantage of this mesh-handling technique is related to the fact that the user has direct control of the elements size. Therefore, the mesh quality can be better controlled with respect to the LRA. Finally, the field variables are interpolated from mesh to mesh following an inverse distance weighting second order method. From the mesh deformation tests, it emerged that five meshes are necessary to complete one complete gear meshing. Afterwards, the first mesh could be reused, and the simulation could continue.

_{C}). pointMotion refers to the imposed motion of the points of the patches. In the GRA

^{MC}-domain, the points of the Gear- and the Shaft-boundaries are set into rotation (ω

_{G}/ω

_{S}). In the RMM-domain, the motion of the points of the Rollers-, InnerRing- and OuterRing-patches are read from the motion of the mesh (the relative pointMotion is zero pM

_{rel}= 0). The fourth column describes the artificial velocities. For the Gear- and the Shaft-boundaries, these velocities correspond to the ones used for moving the boundary points (U

_{rel}= 0). In the RMM-domain, the absolute velocity of the InnerRing was set to zero (independently from the motion of the grid—U

_{(abs)}= 0), while the one of the OuterRing was set considering the contribution given by the RMM, the one of the pointMotion (zero) and from U (U

_{rel}= {ω

_{OR}– ω

_{C}}∙R

_{OR}) resulting in the final value (Figure 6). Finally, the Rollers-velocity is constructed summing the pure translation (RMM ω

_{C}+ pM

_{rel}= 0) and the purely rotational contribution (U

_{rel}= −ω

_{RCIR}∙R

_{R}) around the roller axis.

#### 4.2.4. Operating Condition and Numerical Settings

^{®}CORE™ i7-6850K CPU, 6 Cores, 3.60 GHz machine requiring about one week to achieve the regime condition.

## 5. Discussion

## 6. Conclusions

^{MC}method for the mesh handling combined with a proper partitioning of the domain and the adoption of arbitrary mesh interfaces.

## Author Contributions

## Funding

## Conflicts of Interest

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Unit | Main Gear | Drive and Driven Gears | |
---|---|---|---|

Center distance (a) | mm | 91.5 | |

Normal module (m_{n}) | mm | 3 | |

Number of teeth (z) | - | 30 | |

Face width (b) | mm | 35 | |

Pressure angle (α_{n}) | ° | 20 | |

Helix angle (β) | ° | 0 | |

Profile shift coefficient (x) | - | 0.2648 | 0.2650 |

Unit | Main Bearing | |
---|---|---|

Bore diameter (d) | mm | 35 |

Outside diameter (D) | mm | 64 |

Width (z) | mm | 14 |

Basic static load rating, radial (C_{0r}) | kN | 65 |

Basic dynamic load rating, radial (C_{r}) | kN | 73 |

Limiting speed (n_{G}) | rpm | 12,300 |

Unit | Value | |
---|---|---|

PETRONAS ZC 601 FF 75W | - | - |

Kinematic viscosity (ν) at 40 °C | mm^{2}/s | 35 |

Kinematic viscosity (ν) at 100 °C | mm^{2}/s | 7 |

Density (ρ) at 15 °C | kg/m^{3} | 850 |

Number of Cells | Max. Non-Orthogonality [°] | Avg. Non-Orthogonality [°] | Max. Skewness |
---|---|---|---|

~2.5 M | 60.5 | 5.2 | 1.2 |

Patch | meshMotion | pointMotion | U | p | α |
---|---|---|---|---|---|

Gears (G) | 0 | pM = ω_{G}∙R_{G} | U_{rel} = 0 | ∇p = 0 | ∇α = 0 |

Shafts (S) | 0 | pM = ω_{S}∙R_{S} | U_{rel} = 0 | ∇p = 0 | ∇α = 0 |

Housing (H) | 0 | pM = 0 | U = 0 | ∇p = 0 | ∇α = 0 |

Rollers (R) | RMM ω_{C} | pM_{rel} = 0 | U_{rel} = −ω_{RCIR}∙R_{R} | ∇p = 0 | ∇α = 0 |

InnerRing (IR) | RMM ω_{C} | pM_{rel} = 0 | U = 0 | ∇p = 0 | ∇α = 0 |

OuterRing (OR) | RMM ω_{C} | pM_{rel} = 0 | U_{rel} = (ω_{OR−}ω_{C})∙R_{OR} | ∇p = 0 | ∇α = 0 |

OilJets (Inlet) (OJ) | 0 | pM = 0 | flow rate | p = const | α = 1 |

Outlet (O) | 0 | pM = 0 | ∇U = 0 | p = const | ∇α = 0 |

AMIs | - | - | - | - | - |

Convergence Criterion | 1 × 10^{−5} |

Maximum Courant number | 1 |

Pressure solver | PCG (preconditioned conjugate gradient) |

Velocity solver | PBiCG (stabilized preconditioned bi-conjugate gradient) |

Time step discretization. | First order implicit Euler scheme |

Velocity discretization | Second order linear-upwind scheme |

Convection for fraction | Second order vanLeer scheme |

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## Share and Cite

**MDPI and ACS Style**

Concli, F.; Mastrone, M.N.
Advanced Lubrication Simulations of an Entire Test Rig: Optimization of the Nozzle Orientation to Maximize the Lubrication Capability. *Lubricants* **2023**, *11*, 300.
https://doi.org/10.3390/lubricants11070300

**AMA Style**

Concli F, Mastrone MN.
Advanced Lubrication Simulations of an Entire Test Rig: Optimization of the Nozzle Orientation to Maximize the Lubrication Capability. *Lubricants*. 2023; 11(7):300.
https://doi.org/10.3390/lubricants11070300

**Chicago/Turabian Style**

Concli, Franco, and Marco N. Mastrone.
2023. "Advanced Lubrication Simulations of an Entire Test Rig: Optimization of the Nozzle Orientation to Maximize the Lubrication Capability" *Lubricants* 11, no. 7: 300.
https://doi.org/10.3390/lubricants11070300