# Accuracy and Grid Convergence of the Numerical Solution of the Energy Equation in Fluid Film Lubrication: Application to the 1D Slider

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## Abstract

**:**

## 1. Introduction

## 2. The Numerical Solution of the Energy Equation Based on the Natural Discretization Method

## 3. The Numerical Solution of the Energy Equation Based on the Lobatto Points Collocation Method

## 4. Further Comparison of Numerical Results Obtained by NDM and LPCM

_{1}/h

_{2}= 2, imposed wall temperatures (Dirichlet boundary condition), and completely decoupled from the Reynolds equation. Other simple cases of the converging 1D slider were also investigated and bring interesting conclusions.

#### 4.1. Different Geometrical Configurations of the 1D Slider

_{1}/h

_{2}= 4 and h

_{1}/h

_{2}= 8, were investigated while keeping the same wall temperature boundary conditions and the decoupled Reynolds equation. The increased inlet/outlet film thickness ratio leads to a slower convergence of the NDM results with the y grid refinements. The best NDM solution obtained with Ny = 160 was considered as the reference results, and the number of computational cells in the x direction was kept the same (Nx = 80 cells).

_{1}/h

_{2}= 4 are depicted in Figure 9, Figure 10 and Figure 11. Again, the lower and upper wall results show different trends. The upper wall temperature gradient reaches grid convergence starting with the Legendre polynomials of 11 degrees, while the resolution of the lower wall gradient needs N = 16 Legendre polynomials. However, Figure 11b shows that even when using a high number of Legendre polynomials, the LPCM requires one order of magnitude less computational time than the NDM for the same accuracy.

_{1}/h

_{2}= 8 are depicted in Figure 12, Figure 13 and Figure 14. The same remarks as for h

_{1}/h

_{2}= 4 can be drawn, except for the fact that this time, a higher order of the Legendre polynomials was needed for the grid convergence solution, and this is related to the increased inlet/outlet film thickness ratio. Figure 14b shows that the computational time of the NDM with Ny = 160 increases for this calculation case. It can be seen from Figure 11b and Figure 14b that the computational effort of the NDM increases one order of magnitude with increasing the ratio h

_{1}/h

_{2}from 4 to 8. This was not the case for the LPCM, which required the same computational time for these two cases, one or two orders of magnitude lower than the NDM. Therefore, LPCM remains largely superior to NDM in terms of the computational time.

#### 4.2. Different Thermal Boundary Conditions Applied to 1D Slider

_{1}/h

_{2}= 4 and the inlet temperature ${\mathrm{T}}_{\mathrm{inlet}}=20\xb0\mathrm{C}$. For the LPCM, 10 and 14 Lobatto points were used and the results were compared to the NDM (Nx = 160 cells and Ny = 160 cells, all equidistant). Figure 15 and Figure 16 show the consistent resolution of the upper wall temperature gradient and of the lower wall temperature.

## 5. Results for the Energy Equation Coupled with the Reynolds Equation

## 6. Example of a Two-Lobe Journal Bearing

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Temperature Fields

## Appendix B. Reference Values

$\overline{\mathit{X}}[-]$ | 0.00625 | 0.11900 | 0.24400 | 0.36900 | 0.49400 | 0.61900 | 0.74400 | 0.86900 | 0.99400 | |
---|---|---|---|---|---|---|---|---|---|---|

Case 1 | $\mathit{d}\overline{\mathit{T}}/\mathit{d}\overline{\mathit{y}}\_\mathit{L}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathit{W}$ | 1.55718 | 5.21441 | 6.77197 | 7.44975 | 7.53430 | 7.14248 | 6.36616 | 5.34304 | 4.35192 |

$\mathit{d}\overline{\mathit{T}}/\mathit{d}\overline{\mathit{y}}\_\mathit{U}\mathit{p}\mathit{p}\mathit{e}\mathit{r}\mathit{W}$ | −0.13495 | −0.96439 | −1.92252 | −3.09524 | −4.61156 | −6.64528 | −9.46179 | −13.49330 | −19.48457 | |

Case 2 | $\mathit{d}\overline{\mathit{T}}/\mathit{d}\overline{\mathit{y}}\_\mathit{L}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathit{W}$ | 1.43830 | 5.35156 | 7.30481 | 8.42416 | 8.87026 | 8.57660 | 7.37678 | 5.21935 | 3.57358 |

$\mathit{d}\overline{\mathit{T}}/\mathit{d}\overline{\mathit{y}}\_\mathit{U}\mathit{p}\mathit{p}\mathit{e}\mathit{r}\mathit{W}$ | −2.46419 | −2.34816 | −2.24448 | −2.22769 | −2.42486 | −3.13238 | −5.03152 | −10.04817 | −24.05486 | |

Case 3 | $\mathit{d}\overline{\mathit{T}}/\mathit{d}\overline{\mathit{y}}\_\mathit{L}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathit{W}$ | 0.830646 | 3.894458 | 5.609075 | 6.852488 | 7.746559 | 8.195229 | 7.845554 | 5.840858 | 3.052226 |

$\mathit{d}\overline{\mathit{T}}/\mathit{d}\overline{\mathit{y}}\_\mathit{U}\mathit{p}\mathit{p}\mathit{e}\mathit{r}\mathit{W}$ | −5.17655 | −4.86225 | −4.47603 | −4.02043 | −3.49895 | −2.97106 | −2.75667 | −4.76198 | −25.4916 | |

Case 4 | $\mathit{T}\_\mathit{L}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathit{W}$[°C] | 20.276 | 22.293 | 24.839 | 27.649 | 30.669 | 33.738 | 36.470 | 38.132 | 38.967 |

−0.41141 | −0.33314 | −0.28279 | −0.32754 | −0.59665 | −1.37867 | −3.38145 | −8.58824 | −23.9895 | ||

Case 5 | 0.938225 | 5.39037 | 7.32993 | 8.39752 | 8.86711 | 8.55326 | 7.32938 | 5.15078 | 3.62624 | |

−0.42745 | −0.34021 | −0.28590 | −0.32154 | −0.59344 | −1.36122 | −3.32850 | −8.46418 | −23.39503 |

**Table A2.**Coefficients of Legendre polynomials ${\widehat{T}}_{j}\left(\overline{X}\right)$ describing temperature variations for Case 1, N = 12.

$\overline{\mathit{X}}[-]$ | 0.00625 | 0.11900 | 0.24400 | 0.36900 | 0.49400 | 0.61900 | 0.74400 | 0.86900 | 0.99400 | |
---|---|---|---|---|---|---|---|---|---|---|

jth Degree | ||||||||||

1 | 20.39 | 23.98 | 27.32 | 30.51 | 33.61 | 36.64 | 39.70 | 42.97 | 46.43 | |

2 | −0.09 | −1.09 | −1.70 | −1.73 | −1.04 | 0.54 | 3.24 | 7.36 | 12.66 | |

3 | −0.16 | −2.41 | −5.04 | −7.85 | −10.74 | −13.60 | −16.28 | −18.67 | −20.42 | |

4 | −0.07 | −0.07 | 0.28 | 0.43 | 0.09 | −1.00 | −3.10 | −6.52 | −11.08 | |

5 | −0.10 | −0.99 | −1.76 | −2.35 | −2.77 | −3.14 | −3.62 | −4.55 | −6.18 | |

6 | 0.03 | 0.78 | 1.18 | 1.20 | 0.93 | 0.47 | −0.14 | −0.90 | −1.72 | |

7 | −0.04 | −0.37 | −0.43 | −0.30 | −0.11 | 0.06 | 0.17 | 0.20 | 0.12 | |

8 | 0.06 | 0.32 | 0.24 | 0.12 | 0.03 | 0.00 | 0.00 | 0.05 | 0.12 | |

9 | −0.04 | −0.19 | −0.10 | −0.02 | 0.02 | 0.03 | 0.03 | 0.04 | 0.05 | |

10 | 0.04 | 0.07 | 0.01 | −0.01 | −0.01 | −0.01 | 0.00 | 0.00 | 0.01 | |

11 | −0.04 | −0.02 | 0.01 | 0.01 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | |

12 | 0.03 | 0.00 | −0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

13 | −0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

## References

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**Figure 3.**Dimensionless temperature gradient at the lower wall for the natural discretization method (NDM) solution and imposed wall temperatures (h

_{1}/h

_{2}= 2, Nx = 80 cells).

**Figure 4.**Dimensionless temperature gradient at the upper wall for the NDM solution and imposed wall temperatures (h

_{1}/h

_{2}= 2, Nx = 80 cells).

**Figure 5.**(

**a**) Relative difference and (

**b**) computational time of the NDM solution for y direction grid refinements (h

_{1}/h

_{2}= 2, Nx = 80 cells).

**Figure 6.**Dimensionless temperature gradient at the lower wall for the Lobatto Points Collocation Method (LPCM) solution with imposed wall temperatures (h

_{1}/h

_{2}= 2, Nx = 80 cells).

**Figure 7.**Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed wall temperatures (h

_{1}/h

_{2}= 2, Nx = 80 cells).

**Figure 8.**(

**a**) Relative difference and (

**b**) computational time of the LPCM solution for Y direction grid refinements (h

_{1}/h

_{2}= 2, Nx = 80 cells)

**Figure 9.**Dimensionless temperature gradient at the lower wall for the LPCM solution with imposed wall temperatures (h

_{1}/h

_{2}= 4, Nx = 80 cells).

**Figure 10.**Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed wall temperatures (h

_{1}/h

_{2}= 4, Nx = 80 cells).

**Figure 11.**(

**a**) Relative difference and (

**b**) computational time of the LPCM solution for Y direction grid refinements (h

_{1}/h

_{2}= 4, Nx = 80 cells).

**Figure 12.**Dimensionless temperature gradient at the lower wall for the LPCM solution with imposed wall temperatures (h

_{1}/h

_{2}= 8, Nx = 80 cells).

**Figure 13.**Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed wall temperatures (h

_{1}/h

_{2}= 8, Nx = 80 cells).

**Figure 14.**(

**a**) Relative difference and (

**b**) computational time of the LPCM solution for Y direction grid refinements (h

_{1}/h

_{2}= 8, Nx = 80 cells).

**Figure 19.**(

**a**) Pressure variation in the 1D slider (h

_{1}/h

_{2}= 2) coupled thermo-hydrodynamic (THD) solution, and (

**b**) outlet temperature variation across the film thickness in the 1D slider (h

_{1}/h

_{2}= 2) coupled THD solution.

**Figure 20.**(

**a**) Relative difference ${\epsilon}_{K}$ of the NDM for different grid refinements, and (

**b**) computational time of the NDM for different numbers of cells, over the film thickness for the 1D THD slider.

**Figure 21.**(

**a**) Relative difference ${\epsilon}_{N}$ between the LPCM and the reference NDM Ny = 120, and (

**b**) computational time of the LPCM compared to the reference NDM Ny = 120 results, for the 1D THD slider.

**Figure 22.**(

**a**) Comparison of measured pressures and current numerical results in the mid plane of the loaded lobe (3500 rpm, 6 kN load), and (

**b**) comparisons of measured temperatures and current numerical results in the mid plane of the loaded lobe.

Physical Characteristics | Values | Physical Characteristics | Values |
---|---|---|---|

Density ${\rho}_{0}$ | $800\left[\mathrm{kg}/{\mathrm{m}}^{3}\right]$ | Inlet thickness ${h}_{1}$ | $1.8288{e}^{-4}\left[\mathrm{m}\right]$ |

Specific Heat Capacity ${C}_{p}$ | $2000\left[\mathrm{J}/\mathrm{kgK}\right]$ | Outlet thickness ${h}_{2}$ | $0.9144{e}^{-4}\left[\mathrm{m}\right]$ |

Thermal Conductivity $\lambda $ | $0.14\left[\mathrm{W}/\mathrm{mK}\right]$ | Slider length $L$ | $0.18288\left[\mathrm{m}\right]$ |

Dynamic Viscosity ${\eta}_{0}$ | $0.081\left[\mathrm{Pa}\xb7\mathrm{s}\right]$ | Ambient temperature $Ta$ | $20\left[\xb0\mathrm{C}\right]$ |

Lower Wall Operating Speed $U$ | $31.946\left[\mathrm{m}/\mathrm{s}\right]$ | Ambient pressure $Pa$ | $1\left[\mathrm{bar}\right]$ |

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**MDPI and ACS Style**

Zhang, S.; Hassini, M.-A.; Arghir, M.
Accuracy and Grid Convergence of the Numerical Solution of the Energy Equation in Fluid Film Lubrication: Application to the 1D Slider. *Lubricants* **2018**, *6*, 95.
https://doi.org/10.3390/lubricants6040095

**AMA Style**

Zhang S, Hassini M-A, Arghir M.
Accuracy and Grid Convergence of the Numerical Solution of the Energy Equation in Fluid Film Lubrication: Application to the 1D Slider. *Lubricants*. 2018; 6(4):95.
https://doi.org/10.3390/lubricants6040095

**Chicago/Turabian Style**

Zhang, Silun, Mohamed-Amine Hassini, and Mihai Arghir.
2018. "Accuracy and Grid Convergence of the Numerical Solution of the Energy Equation in Fluid Film Lubrication: Application to the 1D Slider" *Lubricants* 6, no. 4: 95.
https://doi.org/10.3390/lubricants6040095