# A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{9}N/m

^{2}), while the gas density ratio changed from 1.0000 to 0.1754 [8,9,10]. Moreover, the stepwise nonlinearity of the density appearing at the oil film reformation boundary essentially increases the difficulty in the numerical solution.

## 2. AMR Algorithm

#### 2.1. Mesh and Data Storage

Algorithm 1 Mesh refining () |

for each cell if a cell is a leaf cell and the refinement flag is 1 add four lines at the end of the matrix table; fill mesh relation information (cell number, type, parent, level, xy coordinate, and so on); interpolate initial flow variable using scatteredInterpolant() function (MATLAB function); end end |

Algorithm 2 Mesh coarsening () |

for each cell if four leaf cells belong to a common parent cell and all coarse flags are 1 remove the lines of the four leaf cells; modify the parent cell into leaf cell; interpolate initial flow variable using the average mean of the four child cells; end end |

#### 2.2. Neighbor Finding

Algorithm 3 Neighbor finding () |

find neighbor of same size using ADJ function [34] according to a given direction; store query path simultaneously; find neighbor smaller size according to mirror query path using REFLECT function [34]; |

#### 2.3. Neighbor Fixing

Algorithm 4 Neighbor fixing () |

while 1 if 2:1 condition is satisfied (checked by Neighbor finding () algorithm) break; end for each direction (N S W E) if the number of neighboring cells exceed three set refinement flag to 1; end end for each diagonal (NW NE SW SE) if the absolute mesh level difference between central cell and diagonal cell exceeds 1 set refinement flag to 1; end end perform Mesh refining () algorithm; end |

#### 2.4. Handingnode Interpolating

_{2}, φ

_{3}, φ

_{4}, and x

_{i}is the x coordinate of φ

_{1}. The quadratic interpolation is implemented by MATLAB function [37]. If a boundary or fine grid is encountered, case (2) is considered a particular case. The values of φ

_{2}and φ

_{4}are, respectively, taken as the value of the boundary condition and the average value of the fine grid, as follows

_{b}is the value of the boundary condition. The present interpolation hybridizes linear and quadratic rules. Although all quadratic interpolations are feasible, more effort in programming is required. Other methods such as prolongation and restriction methods to handle interpolation can be seen in [38]. The Handingnode interpolating () algorithm is shown in Algorithm 5.

Algorithm 5 Handingnode interpolating () |

if the neighbor cell is a boundary (Neighbor finding () algorithm returns zero) return the value of boundary condition; else if the number of neighbor cells is two (case (1)) return the average value using Equation (1); end if the number of neighbor cells is one and the level of neighbor cells is small than the level of the central cell (case (2)) return the quadratic interpolation value using Equation (2); end if the neighbor cell is the particular case (2) return the quadratic interpolation value using Equation (3); end end |

#### 2.5. Periodic Matching

Algorithm 6 Periodic matching () |

detect the cells of west boundary and east boundary by Neighbor finding () algorithm; create periodic pair number table simultaneously; match the pair number according to the size of a cell; if 2:1 condition is not satisfying (checked by Neighbor finding () algorithm) preform Neighbor fixing () algorithm; end |

#### 2.6. Z-Order Filling Curve

Algorithm 7 Z-ordering () |

travel to the bottom NE cells as a start point; store the travel path layer by layer simultaneously; while the number of tagged cells is not equal to the total number of leaf cells travel the leaf cells on the same layer under the sequence of NW→NE→SW→SE; store the travel path layer by layer simultaneously; go back to ancestor cell once the SE leaf cell is reached; end |

## 3. Difference Schemes on Nonuniform Mesh

## 4. Error Estimation

_{1}and computed solution fc

_{1}; and a coarse grid with grid size h

_{2}and computed solution fc

_{2}, the extrapolated solution is evaluated as

_{2}/h

_{1}and p is the order of accuracy. In other words, the more accurate solution is obtained by fine grid solution f

_{c}

_{1}and coarse grid solution f

_{c}

_{2}with weight coefficients of r

^{p}/(r

^{p}− 1) and -1/(r

^{p}− 1). The discretization error for the fine grid solution f

_{c}

_{1}could be estimated as follows

## 5. Results and Discussion

_{c}= 28 kPa (a), and μ = 0.0127 Pa·s. The solution of the three-time refined mesh is compared with the solution of the corresponding uniform mesh and the experimental data. The initial mesh is taken as a very coarse mesh with 16 × 16 cells. This mesh is generated by recursively refining the whole mesh four times. In the first refinement, 30 cells in the liquid region and 40 cells in the cavitation region were tagged to be refined. The tagged cells were the cells of the coarsen mesh with 8 × 8 cells in the implementation of the Richardson extrapolation method. After the refinement operation, the number of the cells refined was 240. It was lower than the expected number of grids, 280, as calculated by 70 × 4. This is because some of the tagged cells with large errors were common in the cavitation and liquid regions. In the subsequent refinements, 90 and 300 cells in the liquid region and 40 and 180 cells in the cavitation region were tagged to be refined for the second and third refinements, respectively. It is noted that the number of cells tagged was empirically determined according to the initial numerical tests. Although the refinement strategy is empirical, it still shows the law that the ratio of the tagged number to the total number reduces with the increase in total cell numbers. As shown in Table 3, the refinement ratios are 91%, 65%, and 41% in descending order. The refinement strategy indicates that the discretization error of the coarse mesh should be reduced sufficiently. This is because the coarse mesh has the largest discretization error, where the discretization error is proportional to the power p of the grid spacing h.

_{0}= 750 μm, h

_{g}= 10 μm, L = 3 mm, h

_{0}= 4 μm, μ = 0.0035 Pa·s, U = 1.45 m/s, p

_{c}= 0.9 × 10

^{5}Pa (a), p

_{a}= 1.0 × 10

^{5}Pa (a). The refinement strategy for the bearing is listed in Table 3. The refinement ratios were 91%, 76%, and 48% for the three-time refinements. They are higher than those of case 1. A more computational cost was indicated to be paid. The solution of the three-times refined mesh is compared with the solution of the corresponding uniform mesh and the solution provided by Qiu and Khonsari [22], as shown in Figure 11 and Figure 12. It can be seen that the AMR solution is close to the solution of the uniform mesh (128 × 128 cells) with a small difference compared with the solution provided by Qiu and Khonsari [22]. The difference is mainly reflected by the value of maximum pressure. Ausas’ algorithm is sensitive to the number of grids. Insufficient number of grids often leads to high maximum pressure. The difference is acceptable under the current refinement strategy. The number of the final refined mesh is 9220, which is 56% of the uniform mesh. The AMR method provides almost the same accuracy solution while the computational cost is saved.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Different types of adaptive mesh refinement: (

**a**) block-based AMR method; (

**b**) cell-based AMR method.

**Figure 9.**Final refined mesh for journal bearing: (

**a**) pressure distribution; (

**b**) density ratio distribution.

**Figure 11.**Final refined mesh for thrust bearing: (

**a**) pressure distribution; (

**b**) density ratio distribution.

Table Index | Cell Number | Cell Type | NW Child Cell | NE Child Cell | SW Child Cell | SE Child Cell | Parent Cell Number | Level | Other Parameters |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0 | 2 | 3 | 4 | 5 | 0 | 1 | xy coordinate; refinement flag; coarse flag, and so on. |

2 | 2 | NW ^{1} | 6 | 7 | 8 | 9 | 1 | 2 | |

3 | 3 | NE | 0 | 0 | 0 | 0 | 1 | 2 | |

4 | 4 | SW | 0 | 0 | 0 | 0 | 1 | 2 | |

… | |||||||||

10 | 10 | NW | 14 | 15 | 16 | 17 | 9 | 4 | |

… | |||||||||

17 | 17 | SE | 0 | 0 | 0 | 0 | 10 | 5 |

^{1}NW, NE, SW, SE: northwest, northeast, southwest, southeast.

Cell Number of the West Boundary | Pair Number | Cell Number of the East Boundary | Pair Number |
---|---|---|---|

22 | 1 | 24 | 1 |

28 | 2 | 86 | 2 |

95 | 2 | 15 | 3 |

97 | 2 | ||

120 | 3 | ||

142 | 3 |

Bearings | Refinement Level | Number of Tagged Cells for Pressure | Number of Tagged Cells for Density Ratio | Number of Cells Actually Refined (A) | Total Number of Cells before Refinement (B) | Refinement Ratio (A/B) |
---|---|---|---|---|---|---|

Case 1: journal bearing | 1 | 30 | 40 | 240 | 265 | 91% |

2 | 90 | 40 | 672 | 1024 | 65% | |

3 | 300 | 180 | 1664 | 4096 | 41% | |

Case 2: thrust bearing | 1 | 30 | 40 | 252 | 265 | 95% |

2 | 150 | 60 | 776 | 1024 | 76% | |

3 | 350 | 180 | 1960 | 4096 | 48% |

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**MDPI and ACS Style**

Xu, W.; Li, K.; Geng, Z.; Zhang, M.; Yang, J.
A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes. *Appl. Sci.* **2021**, *11*, 9879.
https://doi.org/10.3390/app11219879

**AMA Style**

Xu W, Li K, Geng Z, Zhang M, Yang J.
A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes. *Applied Sciences*. 2021; 11(21):9879.
https://doi.org/10.3390/app11219879

**Chicago/Turabian Style**

Xu, Wanjun, Kang Li, Zhengyang Geng, Mingjie Zhang, and Jiangang Yang.
2021. "A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes" *Applied Sciences* 11, no. 21: 9879.
https://doi.org/10.3390/app11219879