# Cavitation Model Calibration Using Machine Learning Assisted Workflow

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## Abstract

**:**

## 1. Introduction

## 2. CFD Setup

#### 2.1. Governing Equations

#### 2.2. Cavitation Model

#### 2.3. Geometry and Computational Domain

#### 2.4. Numerical Setup

**u**has been prescribed at the outlet. The Dirichlet boundary condition through explicitly prescribed fixed values or fixed values which account for mesh movement (movingWallVelocity) has been used for other patches. For pressure, prgh, the Dirichlet boundary condition was set at the outlet, thereby enforcing a fixed value for the total pressure. Remaining patches had a Neumann boundary condition specified with included gradient correction on walls due to body forces (fixedFluxPressure). A water fraction alpha was fixed at the inlet with the Neumann boundary condition used for other patches. Values for turbulent viscosity ${\nu}_{t}$, turbulent kinetic energy k and the specific rate of dissipation $\omega $ at the inlet were estimated based on domain size and inflow velocity [32]; at walls, appropriate wall functions were used due to enforced ${y}^{+}$ value. Boundary conditions for all test cases are briefly summarized in Table 2. Omitted boundaries included outer walls for which boundary conditions are equal to those used for stationary walls, apart from velocity, for which slip condition was set. Remaining patches were periodic faces and interfaces which used cyclicAMI and cyclicRepeatAMI boundary conditions respectively.

#### 2.5. Grid Convergence

## 3. Machine Learning Assisted Workflow

#### 3.1. Challenges in Cavitation Analysis

#### 3.2. ML Algorithm and Generalized Workflow

#### 3.3. Data Analysis and Random Forest Regression

## 4. Predictions and Validity

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Details of the numerical setup. (

**a**) Computational domain with relevant patches and (

**b**) cross-sectional detail of a coarse mesh.

**Figure 2.**Proposed workflow utilizing a random forest algorithm. Numerical simulations were used to create a dataset in which relevant performance parameters and cavitation observations were stored. An RF regression algorithm was employed to provide predictions for ${C}_{prod}$ and ${C}_{dest}$. Results were validated against experimental observations and further confirmed with DES analysis.

**Figure 4.**Cavity extents for vapor volume fraction of 0.5 when utilizing various values of parameters ${C}_{prod}$ and ${C}_{dest}$ from the literature. (

**a**) Bensow et al. [15], (

**b**) Morgut et al. [17], (

**c**) Kunz et al. [4], (

**d**) Kunz et al. [5], (

**e**) Vaz et al. [40], (

**f**) Zhou et al. [41], (

**g**) predicted values ${C}_{prod}=172$, ${C}_{dest}=5$ and (

**h**) experimental observations [30].

**Figure 5.**Cavitation patterns obtained using k-$\omega $ SST DES turbulence model. Results show good agreement with the experiment [30]. (

**a**) Results for $J=1.019$, $\sigma =2.024$ at 50% vapor volume fraction and (

**b**) 20% vapor volume fraction; (

**c**) $J=1.268$, $\sigma =1.424$ at 20% vapor volume fraction and (

**d**) 10% vapor volume fraction; (

**e**) $J=1.408$, $\sigma =1.999$ at 20% vapor volume fraction and (

**f**) 10% vapor volume fraction.

**Table 1.**VP1304 propeller geometry. [30].

Propeller diameter [m] | 0.250 |

Pitch ratio at $r/R=0.7$ | 1.635 |

Chord length at $r/R=0.7$ [m] | 0.104 |

Skew [${}^{\circ}$] | 18.837 |

Hub ratio | 0.300 |

Number of blades | 5 |

Rotation | right |

Field | Inlet | Outlet | Blades / Hub | Shaft |
---|---|---|---|---|

u | fixedValue | zeroGradient | movingWallVelocity | noSlip |

prgh | zeroGradient | prghPressure | fixedFluxPressure | fixedFluxPressure |

alpha | fixedValue | zeroGradient | zeroGradient | zeroGradient |

${\nu}_{t}$ | calculated | calculated | nutkWallFunction | nutkWallFunction |

k | fixedValue | zeroGradient | kqRWallFunction | kqRWallFunction |

$\omega $ | fixedValue | zeroGradient | omegaWallFunction | omegaWallFunction |

Mesh | Coarse | Medium | Fine |
---|---|---|---|

Grid Size | $1.1\xb7{10}^{6}$ | $1.93\xb7{10}^{6}$ | $3.59\xb7{10}^{6}$ |

Max. Cell Size (m) | $8\xb7{10}^{-3}$ | $6.5\xb7{10}^{-3}$ | $5\xb7{10}^{-3}$ |

Grid Refinement ratio | - | 1.231 | 1.300 |

Relative Error (%) | - | 0.983 | 0.758 |

GCI (%) | - | 2.175 | 1.244 |

Performance Metrics | MAE | RMSE | MAPE |
---|---|---|---|

Training Set (80%) | 0.1558 | 0.1982 | 0.2852 |

Test Set (20%) | 0.1583 | 0.2085 | 0.2966 |

Reference | ${\mathit{C}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$ | ${\mathit{C}}_{\mathit{d}\mathit{e}\mathit{s}\mathit{t}}$ |
---|---|---|

Bensow et al. [15] | 20,000 | 1000 |

Morgut et al. [17] | 455 | 4100 |

Kunz et al. [4] | 100 | 100 |

Kunz et al. [5] | 0.2 | 0.2 |

Vaz et al. [40] | 10,000 | 500 |

Zhou et al. [41] | 4328 | 3323 |

Predicted | 172 | 5 |

**Table 6.**Results for thrust (T) and torque (Q) obtained using RANS and DES models at different advance ratios J.

Case | ${\mathit{T}}_{\mathit{e}\mathit{x}\mathit{p}.}$ (N) | ${\mathit{T}}_{\mathit{R}\mathit{A}\mathit{N}\mathit{S}}$ (N) | ${\mathit{T}}_{\mathit{D}\mathit{E}\mathit{S}}$ (N) | ${\mathit{Q}}_{\mathit{e}\mathit{x}\mathit{p}.}$ (Nm) | ${\mathit{Q}}_{\mathit{R}\mathit{A}\mathit{N}\mathit{S}}$ (Nm) | ${\mathit{Q}}_{\mathit{D}\mathit{E}\mathit{S}}$ (Nm) |
---|---|---|---|---|---|---|

$J=1.019$ | 908.70 | 921.13 | 932.49 | 58.98 | 58.94 | 59.64 |

$J=1.268$ | 502.08 | 483.27 | 500.69 | 38.38 | 37.20 | 38.28 |

$J=1.408$ | 331.94 | 332.94 | 320.24 | 29.80 | 29.17 | 29.01 |

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**MDPI and ACS Style**

Sikirica, A.; Čarija, Z.; Lučin, I.; Grbčić, L.; Kranjčević, L. Cavitation Model Calibration Using Machine Learning Assisted Workflow. *Mathematics* **2020**, *8*, 2107.
https://doi.org/10.3390/math8122107

**AMA Style**

Sikirica A, Čarija Z, Lučin I, Grbčić L, Kranjčević L. Cavitation Model Calibration Using Machine Learning Assisted Workflow. *Mathematics*. 2020; 8(12):2107.
https://doi.org/10.3390/math8122107

**Chicago/Turabian Style**

Sikirica, Ante, Zoran Čarija, Ivana Lučin, Luka Grbčić, and Lado Kranjčević. 2020. "Cavitation Model Calibration Using Machine Learning Assisted Workflow" *Mathematics* 8, no. 12: 2107.
https://doi.org/10.3390/math8122107