# Evaluation of the Turbulence Model Influence on the Numerical Simulation of Cavitating Flow with Emphasis on Temperature Effect

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formation

#### 2.1. Governing Equations

_{m}is the dynamic viscosity; μ

_{t}is the turbulent dynamic viscosity; ${k}_{eff}$ is the effective thermal conductivity; T is temperature; ${C}_{p}$ is specific heat; and L is the latent heat.

#### 2.2. Turbulence Model

#### 2.2.1. $k-\omega $ Turbulence Model

_{kb}and P

_{εb}are the influence source terms of buoyancy, and P

_{k}is the turbulent kinetic energy generation term.

#### 2.2.2. RNG $k-\omega $ Turbulence Model

_{ε2RNG}= 1.68, β

_{RNG}= 0.012, C

_{μRNG}= 0.085.

#### 2.2.3. SST $k-\epsilon $ Turbulence Model

_{1}and F

_{2}are the blending functions; and S is the shear tensor. The constant coefficients are taken separately: a

_{1}= 0.31, σ

_{k}= 2, β’ = 0.09, σ

_{ω}= 2, α = 5/9, β = 0.075, and σ

_{ω}

_{2}= 1/0.856.

_{1}and F

_{2}are as follows:

#### 2.2.4. Modification of Turbulence Model

_{3}is the empirical coefficients, which is assigned to be 1.0.

#### 2.3. Cavitation Model for Thermosensitive Fluids

^{+}and sink term m

^{−}, respectively. To capture the cavitation characteristics, liquid volume fraction and vapor volume fraction needs to be obtained. A representative method is to employ a transport equation to determine the liquid volume fraction and vapor volume fraction, respectively. For the Merkle cavitation model [29], the transfer rate between the liquid phase and vapor phase is supposed to be proportional to the local pressure difference. The governing equation for the Merkle cavitation model is shown below:

_{v}is the saturated vapor pressure. The default values for ${C}_{vap}$ and ${C}_{cond}$ are listed as follows:

## 3. Numerical Setup and Validation

#### 3.1. NACA0015 Hydrofoil Geometry Model

#### 3.2. Mesh Implementation

^{+}value is usually employed to judge the mesh quality [31,32]. There is no specific range for the distribution of the y

^{+}value for different turbulence models. The y

^{+}value is usually maintained under 60 to keep the accuracy of the simulation results in the research community. In order to improve the quality of the mesh grid, a refined C-shaped structure grid is used around the leading edge and trailing edge of the hydrofoil. The distribution of the y

^{+}value on the upper and lower surfaces of the hydrofoil is shown in Figure 4.

#### 3.3. Boundary Conditions

#### 3.4. Mesh Independence Study

## 4. Results and Discussion

#### 4.1. Influence of Different Turbulence Models on NACA0015 at 25 °C

#### 4.2. Influence of Different Turbulence Models on NACA0015 at 50 °C

#### 4.3. Influence of Different Turbulence Models on NACA0015 at 70 °C

#### 4.4. Influence of Modified RNG k-ε Model on NACA0015 Hydrofoil at Different Temperatures

## 5. Conclusions

- (1)
- At 25 °C, the correction effect is significant for the modified k-ε model, and the vortex is eliminated in the closed area of the cavity tail. The simulation results obtained from the modified RNG k-ε model and the SST k-ω model showed reasonably good agreement with the experimental results.
- (2)
- At 50 °C, the modified RNG k-ε model and the modified SST k-ω model have a small difference between numerical results and experimental results for the RMS error and the maximum deviation.
- (3)
- At 70 °C, the modified RNG k-ε model is smaller than the result of the modified SST k-ω model in terms of the RMS error and the maximum deviation. The turbulent kinetic energy of the modified SST k-ω model near the wall is significantly larger than that obtained by the modified RNG k-ε model, and the cavitation is more serious, which is quite different from the experimental results.
- (4)
- The feasibility of the modified RNG k-ε turbulence model is demonstrated using this model to simulate cavitating flow around the NACA0015 hydrofoil at different temperatures of water.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Mesh implementation around the hydrofoil. (

**a**) Mesh around leading edge, (

**b**) Mesh around trailing edge.

Fluid | Temperature T_{∞} (K) | Inlet Speed u_{in} (m/s) | Outlet Pressure p_{out} (Pa) |
---|---|---|---|

Water | 298 (25 °C) | 8 | 51,025 |

Water | 323 (50 °C) | 8 | 59,768 |

Water | 343 (70 °C) | 8 | 78,110.4 |

Mesh | Mesh Nodes | Min Angle | Max Aspect Ratio | Min Determinant | Min Quality |
---|---|---|---|---|---|

1 | 1,138,840 | 46.08 | 145 | 0.794 | 0.72 |

2 | 2,847,100 | 46.08 | 56.1 | 0.794 | 0.72 |

3 | 4,555,360 | 46.08 | 34.8 | 0.794 | 0.72 |

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**MDPI and ACS Style**

Deng, Y.; Feng, J.; Wan, F.; Shen, X.; Xu, B.
Evaluation of the Turbulence Model Influence on the Numerical Simulation of Cavitating Flow with Emphasis on Temperature Effect. *Processes* **2020**, *8*, 997.
https://doi.org/10.3390/pr8080997

**AMA Style**

Deng Y, Feng J, Wan F, Shen X, Xu B.
Evaluation of the Turbulence Model Influence on the Numerical Simulation of Cavitating Flow with Emphasis on Temperature Effect. *Processes*. 2020; 8(8):997.
https://doi.org/10.3390/pr8080997

**Chicago/Turabian Style**

Deng, Yilin, Jian Feng, Fulai Wan, Xi Shen, and Bin Xu.
2020. "Evaluation of the Turbulence Model Influence on the Numerical Simulation of Cavitating Flow with Emphasis on Temperature Effect" *Processes* 8, no. 8: 997.
https://doi.org/10.3390/pr8080997