#
Propeller Cavitation in Non-Uniform Flow and Correlation with the Near Pressure Field^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

#### 2.1. Facility and Propeller

#### 2.2. Wake Generator

#### 2.3. Flow Conditions

## 3. High-Speed Visualizations and Pattern Measurement

^{®}APX high-speed camera (San Diego, CA, USA) is used to record the cavitation pattern at a frame rate of 2000 images per second with a resolution of $1024\times 1024$ pixels. The rotation speed used in our experiments is 30 $\mathrm{Hz}$, corresponding to an angular resolution of 5.4°. The camera is oriented at an angle with respect to the test section window, as depicted in Figure 1. Since this arrangement would introduce strong optical aberrations, a glass tank in the form of a wedge and filled with water is placed against the window, such that the camera optical axis is normal to the wedge face. This solution minimizes the optical shortcomings of the non-normal viewing path. The illumination consists of a set of high-power flood lights to accommodate the short exposure time used to record the cavitation on the rotating blades, thus avoiding image blurring due to motion. A typical shutter time of 1/10,000th to 1/20,000th of a second is used. For the sake of completeness, we report in Appendix B the methodology developed by Pereira et al. [1] and used here to retrieve the extension of the attached cavity on the blade surface, hereafter referred to as ${S}_{c}$.

## 4. Pressure Measurements

#### 4.1. Instrumentation

^{®}model 8510C-15 piezo-resistive pressure transducers (Irvine, CA, USA), flush-mounted to the test section walls and in the propeller plane, as indicated in Figure 1. These pressure transducers are designed by ${P}_{1}$, ${P}_{2}$, ${P}_{3}$ and ${P}_{4}$ and are placed at 0°, 90°, 180° 270${}^{\xb0}$, respectively. Note that ${P}_{2}$ is at 12-o’clock (90°), above and closest to the propeller blade tips.

^{®}8103-type hydrophones (Nærum, Denmark) fixed to a streamlined strut are used for the pressure measurements in the fluid, and are referred to as ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ and ${H}_{4}$, see Figure 1. The sensors are located in the longitudinal, vertical plane of symmetry at a distance of about one diameter downstream of the propeller plane. They are placed radially at 80 $\mathrm{m}\mathrm{m}$ ≈ 0.7R (${H}_{1}$), 100 $\mathrm{m}\mathrm{m}$ $\approx R$ (${H}_{2}$), 120 $\mathrm{m}\mathrm{m}$ (${H}_{3}$) and 200 $\mathrm{m}\mathrm{m}$ (${H}_{4}$) from the propeller axis; see Figure 5. ${H}_{1}$ and $H3$ are respectively inside and outside the slipstream, while ${H}_{2}$ is roughly located at the tip vortex position, and ${H}_{4}$ is purposely distant from the flow perturbation created by the propeller.

#### 4.2. Harmonic Analysis of Pressure Data

## 5. Results and Discussion

#### 5.1. Cavitation Pattern

#### 5.2. Pressure Field

## 6. Correlations between Cavitation and Pressure Field

#### 6.1. Time Correlations

#### 6.2. Pressure from Vapor Volume

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Nomenclature

Symbol | Description | Units |

R | Propeller radius | $\mathrm{m}$ |

D | Propeller diameter, $D=2R$ | $\mathrm{m}$ |

${U}_{\infty}$ | Upstream axial inflow velocity | $\mathrm{m}\text{}{\mathrm{s}}^{-1}$ |

n | Propeller rotation frequency | $\mathrm{Hz}$ |

${p}_{0}$ | Pressure at propeller axis | $\mathrm{Pa}$ |

${p}_{v}$ | Vapor pressure | $\mathrm{Pa}$ |

$\mathsf{\rho}$ | Water density | $\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ |

${\mathsf{\sigma}}_{0}$ | Cavitation number referred to ${U}_{\infty}$, ${\mathsf{\sigma}}_{0}=({p}_{0}-{p}_{v})/\frac{1}{2}\mathsf{\rho}{U}_{\infty}^{2}$ | - |

$\mathsf{\theta}$ | Angular position | ${}^{\xb0}$ |

${S}_{0}$ | Blade area for $r/R\ge 0.3$ | ${\mathrm{m}}^{-2}$ |

${S}_{c}$ | Cavity extension | ${\mathrm{m}}^{-2}$ |

${l}_{c}$ | Cavity characteristic length, ${l}_{c}=\sqrt{{S}_{c}}$ | $\mathrm{m}$ |

${V}_{c}$ | Cavity volume | ${\mathrm{m}}^{-3}$ |

J | Advance coefficient, $J={U}_{\infty}/nD$ | - |

T | Thrust | $\mathrm{N}$ |

${K}_{T}$ | Thrust coefficient, ${K}_{T}=T/\mathsf{\rho}{n}^{2}{D}^{4}$ | - |

${A}_{n}$, ${\mathsf{\varphi}}_{n}$ | Amplitude and phase of the Fourier series | $\mathrm{Pa}$, ${}^{\xb0}$ |

${f}_{0}$ | Fundamental frequency | $\mathrm{Hz}$ |

BPF | Blade passage frequency | $\mathrm{Hz}$ |

## Appendix B. Cavitation Extension Measurement

**Figure B1.**Image cross-correlation procedure: (

**a**) template image; (

**b**) cavitation pattern image; (

**c**) correlation image; Mapping procedure: (

**d**) distorted image; (

**e**) plan-view image; (

**f**) processed image showing the cavitation extension ${S}_{c}$.

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**Figure 2.**Non-uniform wake generator: design (

**top**) and implementation in the cavitation tunnel (

**bottom**). The dynamometer is mounted upstream.

**Figure 3.**Non-uniform wake: mean velocity field normalized by the upstream flow velocity (

**top**); velocity fluctuations as a percentage of upstream flow velocity (

**bottom**). The propeller axis is at (0,0).

**Figure 4.**High-speed sequence of the cavitating propeller for ${\mathsf{\sigma}}_{0}=3.5$. Inter-frame time = 500 $\mathsf{\mu}\mathrm{s}$, angular step = 5.4°. Flow is from left to right, and the view is from the propeller back face, port side. The wake generator is on the left.

**Figure 6.**Cavity extension and corresponding root-mean-square fluctuations as a function of the propeller angle $\mathsf{\theta}$, for ${\mathsf{\sigma}}_{0}\in [2.5,7.5]$. The continuous lines are quadratic fit curves to the respective experimental data. The black circles indicate the location of the cavity maximum extension determined from the fit function.

**Figure 7.**Fluctuations of the cavity extension as a function of the propeller angle $\mathsf{\theta}$.

**Figure 8.**Energy integral of the fluctuating pressure versus the cavitation number at the wall (P-probes,

**top**) and in the fluid (H-probes,

**bottom**).

**Figure 9.**Angular history of the pressure signal from hydrophone ${H}_{2}$ and corresponding harmonic distribution, for different values of the cavitation number ${\mathsf{\sigma}}_{0}$. The polar graphs are non-dimensionalized. Grey area: ensemble mean; solid line: ensemble fluctuations.

**Figure 10.**Angular history of the pressure signal from transducer ${P}_{2}$ and corresponding harmonic distribution, for different values of the cavitation number ${\mathsf{\sigma}}_{0}$. The polar graphs are non-dimensionalized. Grey area: ensemble mean; solid line: ensemble fluctuations.

**Figure 11.**Phase-shift observed on the first harmonic component of the mean pressure, relative to the non-cavitating condition.

**Figure 12.**Angular history of pressure ${P}_{2}$ and cavity extension ${S}_{c}$: mean (

**top two rows**); fluctuations (

**bottom two rows**). Cavitation area is represented by the grayed area.

**Figure 13.**Comparison between the non-dimensional times ${\mathsf{\tau}}_{c}$ and ${\mathsf{\tau}}_{p}$, respectively calculated from the Rayleigh collapse time as per Equation (4), and from the time difference between cavity maximum extension and pressure peak. Labels report the value of the cavitation number ${\mathsf{\sigma}}_{0}$. The red line represents ${\mathsf{\tau}}_{c}={\mathsf{\tau}}_{p}$.

**Figure 14.**Comparison between reduced-order $\Delta p$ (―) from measurements and computed $\Delta p$ (°) from volume acceleration as per Equation (9).

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Alves Pereira, F.; Di Felice, F.; Salvatore, F. Propeller Cavitation in Non-Uniform Flow and Correlation with the Near Pressure Field. *J. Mar. Sci. Eng.* **2016**, *4*, 70.
https://doi.org/10.3390/jmse4040070

**AMA Style**

Alves Pereira F, Di Felice F, Salvatore F. Propeller Cavitation in Non-Uniform Flow and Correlation with the Near Pressure Field. *Journal of Marine Science and Engineering*. 2016; 4(4):70.
https://doi.org/10.3390/jmse4040070

**Chicago/Turabian Style**

Alves Pereira, Francisco, Fabio Di Felice, and Francesco Salvatore. 2016. "Propeller Cavitation in Non-Uniform Flow and Correlation with the Near Pressure Field" *Journal of Marine Science and Engineering* 4, no. 4: 70.
https://doi.org/10.3390/jmse4040070