# A Novel Approach for Modeling Surface Effects in Hydrodynamic Lubrication

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

## 3. Numerical Method

#### 3.1. Geometry and Mesh

#### 3.2. Boundary Conditions

#### 3.3. Numerical Schemes

#### 3.4. Post Processing

## 4. Simulation Results and Discussion

#### 4.1. Influence of Orientation and Pressure/Shear Gradient

^{2}/s

^{2}) is used), which represent operating conditions such as found those in lower and a higher loaded areas in a journal bearing, were applied (${\frac{dp}{dx}}_{1}=5\times {10}^{4}\phantom{\rule{0.222222em}{0ex}}$mm/s${}^{2}$ and ${\frac{dp}{dx}}_{2}=5\times {10}^{8}\phantom{\rule{0.222222em}{0ex}}$mm/s${}^{2}$). Every calculated data point in the following figures represents one CFD-simulation of the lubrication gap. Due to the computational effort, an averaging over several simulations was not used. Moreover, by using large surfaces which cover several multiples of the correlation length, averaging was not seen as necessary to achieve a proof of principle.

#### 4.2. Influence of the Numerical Method on the Shear Flow Factor

## 5. Conclusions and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Variable | Name |

$\mathbf{u}$ | velocity vector |

∇ | Nabla operator |

t | time |

p | fluid pressure |

$\rho $ | fluid density |

$\nu $ | kinematic viscosity |

$\mathbf{n}$ | normal vector |

b | width of the control volume |

Q | volume flow rate |

${\chi}_{p}$ | pressure flow factor |

${\chi}_{s}$ | shear flow factor |

H | non-dimensional height factor ($=h/\sigma $) |

$\gamma $ | Peklenik (orientation) factor ($={\lambda}_{x}/{\lambda}_{y}$) |

$\lambda $ | correlation length |

${U}_{\Delta}$ | relative velocity |

$\sigma $ | standard deviation of the combined surface profile |

$x,y,z$ | coordinates of the three-dimensional space |

h | local gap height |

${h}^{*}$ | nominal gap height |

${h}_{1}$ | height changes of one surface measured from its mean level |

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**Figure 2.**Flow factors calculated by Patir & Cheng for two different $\gamma $ ($\gamma =9$ means roughness orientation is parallel to the flow direction, $\gamma =1/9$ means roughness orientation is cross the flow direction) [12,16]: (

**a**) Pressure flow factor (

**b**) Shear flow factor for a moving smooth surface and stationary rough surface.

**Figure 6.**Pressure flow factors calculated by the Navier-Stokes equations (N-S) for two different pressure gradients compared with the factors calculated by Patir & Cheng (P&C): (

**a**) Roughness orientation in flow direction ($\gamma =9$); (

**b**) Roughness orientation cross the flow direction ($\gamma =1/9$).

**Figure 7.**Shear flow factors for a smooth moving surface calculated by the Navier-Stokes equations (N-S) for two different velocities compared with the factors calculated by Patir & Cheng (P&C): (

**a**) Roughness orientation in flow direction ($\gamma =9$); (

**b**) Roughness orientation cross the flow direction ($\gamma =1/9$).

$\mathit{\gamma}$ | C | r | ${\mathit{A}}_{1}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ |
---|---|---|---|---|---|---|

1/9 | 1.48 | 0.42 | 2.046 | 1.12 | 0.78 | 0.03 |

9 | 0.87 | 1.5 | 1.011 | 0.54 | 1.07 | 0.08 |

**Table 2.**Boundary conditions for the calculation of the pressure flow factor [12].

Position | Boundary Condition |
---|---|

$x={x}_{1}$ | $p={p}_{1}$ |

$x={x}_{2}$ | $p={p}_{2}\ne {p}_{1}$ |

$y={y}_{1}$ | $\frac{dp}{dy}=0$ |

$y={y}_{2}$ | $\frac{dp}{dy}=0$ |

Surface 1 | ${U}_{1}=0$ |

Surface 2 | ${U}_{2}=0$ |

**Table 3.**Boundary conditions for the calculation of the shear flow factor [16].

Position | Boundary Condition |
---|---|

$x={x}_{1}$ | $p={p}_{1}={p}_{2}$ |

$x={x}_{2}$ | $p={p}_{2}={p}_{1}$ |

$y={y}_{1}$ | $\frac{dp}{dy}=0$ |

$y={y}_{2}$ | $\frac{dp}{dy}=0$ |

Surface 1 | ${U}_{1}=U\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{or}\phantom{\rule{0.222222em}{0ex}}0\right)$ |

Surface 2 | ${U}_{2}=0\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{or}\phantom{\rule{0.222222em}{0ex}}U\right)$ |

**Table 4.**Comparison between flow factors according to Patir and Cheng and flow factors calculated by “Mapping” and “Moving Mesh” for a smooth moving surface.

Patir & Cheng | Mapping | Moving Mesh | |
---|---|---|---|

Normalized gap height H (-) | 5.2 | 5.2 | 5.2 |

Orientation $\gamma $ (-) | 9 | 9 | 9 |

Surface velocity ${U}_{1}$ (mm/s) | $2\times {10}^{3}$ | $2\times {10}^{3}$ | $2\times {10}^{3}$ |

Shear flow factor ${\chi}_{s}$ (-) | 0.984 | 0.968 | 0.967 |

**Table 5.**Comparison between flow factors calculated by “Moving Mesh” and the method according to Patir and Cheng for a rough moving surface.

Patir & Cheng | Moving Mesh | |
---|---|---|

Normalized gap height H (-) | 5.2 | 5.2 |

Orientation $\gamma $ (-) | 9 | 9 |

Surface velocity ${U}_{2}$ (mm/s) | $2\times {10}^{3}$ | $2\times {10}^{3}$ |

Shear flow factor ${\chi}_{s}$ (-) | 1.097 | 1.086 |

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**MDPI and ACS Style**

Pusterhofer, M.; Bergmann, P.; Summer, F.; Grün, F.; Brand, C.
A Novel Approach for Modeling Surface Effects in Hydrodynamic Lubrication. *Lubricants* **2018**, *6*, 27.
https://doi.org/10.3390/lubricants6010027

**AMA Style**

Pusterhofer M, Bergmann P, Summer F, Grün F, Brand C.
A Novel Approach for Modeling Surface Effects in Hydrodynamic Lubrication. *Lubricants*. 2018; 6(1):27.
https://doi.org/10.3390/lubricants6010027

**Chicago/Turabian Style**

Pusterhofer, Michael, Philipp Bergmann, Florian Summer, Florian Grün, and Clemens Brand.
2018. "A Novel Approach for Modeling Surface Effects in Hydrodynamic Lubrication" *Lubricants* 6, no. 1: 27.
https://doi.org/10.3390/lubricants6010027