# A Review on the Erosion Mechanism in Cavitating Jets and Their Industrial Applications

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cavitation Mechanism and Flow Dynamics in Cavitating Jets

#### 2.1. Cavitation Number

_{e}= the mean jet exit velocity.

_{e}in such two-phase flow and the numerical prediction of the jet exit velocity should be considered with precaution, an estimation of V

_{e}can be obtained from standard fluid mechanics principles. A theoretical prediction of V

_{e}using a standard pipe flow model [19] can be used, assuming a steady state flow regime. Using the energy balance equation and the conservation of mass, the following equation can be derived [20] for a cavitating jet with a convergent section and a throat:

_{e}= jet exit velocity;

_{1}and D

_{2}= diameter just upstream of the nozzle and at the nozzle throat exit, respectively;

_{B}= the static pressure measured just upstream of the nozzle;

_{L}= 0.05 [21], and the friction factors ${f}_{1}$ and ${f}_{2}$ can be calculated assuming a smooth inner pipe surface and using the Haaland equation [19].

_{e}) can be estimated using Equations (2) and (3) once the static pressure is measured just upstream of a cavitating jet. Then, the jet Reynolds number and cavitation number can be calculated. It should be noted, however, that the above model assumes that cavitation is not present inside the nozzle. Tarasenko [22] showed that if cavitation occurs inside the nozzle, there would be a velocity increase in the nozzle (Vena contracta effect).

#### 2.2. Self-Resonating Cavitating Jets

_{d}is the critical Strouhal number;

_{N}is the model parameter.

_{d}is about 0.3 [30].

#### 2.3. Rules for the Design of Self-Excited Structured Jets

_{N}is the mode number):

#### 2.4. Cavitating Jet Flow Dynamics

^{2}.

## 3. Erosion Mechanism in Cavitating Jets

^{3}s

^{−1}corresponds to the transition between low and high strain rate sensitive plastic flow. It was shown that for $\dot{\epsilon}$ < 10

^{3}s

^{−1}, where the plasticity is controlled by thermally activated processes and the dislocation dynamics, the effect of strain rate is weak. In contrast, for $\dot{\epsilon}$ > 10

^{3}s

^{−1}, where the plasticity is controlled by viscous drag on dislocation motion (velocity dependent), the effect of strain rate is more pronounced [49,50].

#### 3.1. Effect of Nozzle Surface Roughness

#### 3.2. Effect of Standoff Distance

**Figure 4.**Mass loss vs. Standoff distance [51].

#### 3.3. Effect of Angle of Attack

_{1}= 14.6 ± 0.1 MPa, P

_{2}= 0.321 ± 0.001 Mpa, V = 113.6 ± 0.5 m/s, σ = 0.00342 ± 0.001, T = 25 °C. The experimental results are presented in Figure 10. It can be seen that the angle of attack significantly affects the erosion rate and the mass loss. Due to the inclination, the distance between the nozzle and the target varied along the sample. In the upper part of the sample, the longer distance allowed more bubbles to be produced before they collapsed along the surface. Therefore, most of the bubbles will collapse on the upper region of the target (Figure 11), leading to more pronounced damage in that region. The optimal angle that results in a higher erosion rate could be related to the influence on the two most important cavitation erosion parameters: the shape of the bubbles and the pressure distribution around them.

#### 3.4. Effect of Exit Velocity

#### 3.5. Effect of Fluid Temperature

## 4. Industrial Applications of Cavitating Jets

#### 4.1. Jet Hydrodynamics for Cleaning Applications

#### 4.2. CAVIJET Performance for Well Drilling

^{3}.

#### 4.3. Rock-Cutting Ability of Self-Resonating Jets

#### 4.4. Well Drilling Test Using Cavitating Jets

#### 4.5. Bitumen Separation Using Cavitating Jets

#### 4.6. Borehole Cavitating Jet Hydraulic for Oil-Shale Mining

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## List of Symbols

α | nozzle exit angle |

β | nozzle inlet angle |

d | nozzle outlet diameter |

D | nozzle diameter |

D_{1} | diameter just upstream of the nozzle |

D_{2} | diameter at the nozzle throat exit |

${f}_{1}$, ${f}_{2}$ | friction factors (upstream and at the nozzle throat, respectively) |

H | liquid height above the jet exit |

$\Gamma $ | circulation around the vortex |

h | height difference between inlet and exit of the nozzle |

K_{L} | loss coefficient for the nozzle convergent section |

K_{N} | mode parameter |

λ | acoustic wavelength |

L | the organ-pipe length |

L_{1} | length of nozzle’s convergent part |

L_{2} | nozzle’s throat length |

L_{3} | nozzle’s divergent length |

M | Mach number |

${P}_{1}$ | upstream pressure |

${P}_{2}$ | downstream pressure |

${P}_{a}$ | ambient pressure |

P_{B} | static pressure measured just upstream of the nozzle |

${P}_{v}$ | vapor pressure of liquid |

ρ | liquid density |

r_{c} | vortex core radius |

σ | cavitation number |

S_{d} | critical Strouhal number |

T | working fluid temperature |

θ | angle of attack |

V_{e} | the mean jet exit velocity |

V | jet velocity |

V_{j} | exit-jet velocity |

x | distance from nozzle exit to the target |

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**Figure 1.**Flow comparison of impinging submerged cavitating jets [23].

**Figure 2.**Organ-pipe CAVIJET nozzle [23].

**Figure 3.**Vapor volume fraction in submerged cavitation jet; numerical prediction and flow visualization [33]. (

**a**) growing, (

**b**) shed-ding, (

**c**) collapsing.

**Figure 5.**Jet Nozzle geometry [52].

**Figure 6.**Erosion rate vs. standoff distance [52].

**Figure 7.**Erosion pattern for different jet diameters [52].

**Figure 8.**Erosion pits; NSD: (

**a**) 8.3, (

**b**) 33.3, (

**c**) 66.7, (

**d**) 91.7, (

**e**) 100, (

**f**) 133.3, (

**g**) 166.7, (

**h**) 200 [56].

**Figure 9.**Erosion pits observed by laser microscope; NSD: (

**a**) 33.3, (

**b**) 66.7, (

**c**) 100, (

**d**) 200 [56].

**Figure 10.**Erosion rate vs. angle of attack [52].

**Figure 11.**Erosion pattern vs. angle of attack [52].

**Figure 12.**Mass loss vs. exposure time at different temperatures [61].

**Figure 13.**Mass loss rate vs. relative temperature [62].

**Figure 14.**Hydrodynamic nozzle [33].

**Figure 15.**Effect of divergent angle on erosion performance [33]: (

**a**) α = 40°, (

**b**) α = 60°, (

**c**) α = 80°.

**Figure 16.**(

**a**) Cavitation nozzles geometry (all sizes are in mm): nozzles A and B, respectively. (

**b**) Cavitation erosion pattern [68].

**Figure 17.**Typical cavitating water-jet nozzle configurations [73].

**Figure 19.**Schematic diagram of the self-resonating cavitating jet nozzle structure and operating principle [80].

**Table 1.**Experimental parameters for erosion tests [52].

Nozzle Diameter (d) [mm] | Upstream Pressure (P1) [MPa] ±0.1 | Downstream Pressure (P2) [MPa] ±0.01 | Exit-Jet Velocity (Vj) [m/s] ±0.5 | Cavitation Number (σ) ±0.001 | Working Fluid Temperature (T) [°C] ±1 |
---|---|---|---|---|---|

0.40 | 12.36 | 0.309 | 101.1 | 0.040 | 19 |

0.45 | 12.1 | 0.31 | 101.4 | 0.040 | 19 |

0.55 | 14.54 | 0.31 | 101.3 | 0.040 | 19 |

Study | Cavitation Number | Nozzle Type | Standoff Distance | Attack Angle | Nozzle’s Surface Roughness | Temperature | Fluid |
---|---|---|---|---|---|---|---|

Li et al. [44] | 0.01–0.0017 | organ-pipe self -resonating jet | 25-30 nozzle’s diameter (diameter 2 mm) | jet is normal to target | 0.8, 1.6, 3.2, 6.3 (optimal), 12.5 (optimal) and 24 μm | 20 °C | tap water |

Yamauchi et al. [51] | 0.003 | conical, cylindrical and venturi (horn) types | ≈20 nozzle’s diameters (optimal) (diameter: 1 mm) | jet is normal to target | N/A (standard stainless steel SUS 304) | 20 °C | tap water |

Hulti et al. [52,53] | 0.04–0.014 | cylindrical nozzle | ≈47 nozzle’s diameters (optimal) (diameters: 0.4, 0.45, 0.55 mm) | 90°, 100°, 105° (optimal), 110°, 115°, 120° | N/A (stainless steel) | 19 °C | tap water |

Peng et al. [56] | 0.038 | cylindrical nozzle | 8.3–200 nozzle’s diameters (diameter: 0.3mm) | jet is normal to target | N/A (stainless steel) | 20–23 °C | tap water |

Hattori et al. [61,62] | 0.03–0.015 | cylindrical nozzle | optimum stand-off distances at σ = 0.03, 0.025, 0.02 and 0.015 were 11, 15, 21, and 25 mm, respectively (nozzle’s diameters: N/A) | jet is normal to target | N/A (stainless steel) | 25–120 °C (optimal temperature at average of freezing and boiling temperatures) | deionized water (optimal), Ethanol |

Yang et al. [33] | 0.01 | venturi (horn) type | optimum stand-off distance: 66–72 mm | jet is normal to target | N/A (stainless steel) | 25 °C | tap water |

Bukharin et al. [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66] | 0.19–0.46 | conical, cylindrical, organ-pipe self -resonating jet (optimal) | Optimal distance in tests with plates ≈ 7 nozzle’s diameters (nozzles diameters: 10–15 mm | jet is normal to target | N/A (stainless steel, bronze) | 4–70 °C | water and mixture with oil sand (slurry) |

**Table 3.**Nozzle parameters [33].

L_{2} (mm) | d (mm) | L_{3} (mm) | β (^{o}) |
---|---|---|---|

2 | 0.5 | 2 | 40 |

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El Hassan, M.; Bukharin, N.; Al-Kouz, W.; Zhang, J.-W.; Li, W.-F.
A Review on the Erosion Mechanism in Cavitating Jets and Their Industrial Applications. *Appl. Sci.* **2021**, *11*, 3166.
https://doi.org/10.3390/app11073166

**AMA Style**

El Hassan M, Bukharin N, Al-Kouz W, Zhang J-W, Li W-F.
A Review on the Erosion Mechanism in Cavitating Jets and Their Industrial Applications. *Applied Sciences*. 2021; 11(7):3166.
https://doi.org/10.3390/app11073166

**Chicago/Turabian Style**

El Hassan, Mouhammad, Nikolay Bukharin, Wael Al-Kouz, Jing-Wei Zhang, and Wei-Feng Li.
2021. "A Review on the Erosion Mechanism in Cavitating Jets and Their Industrial Applications" *Applied Sciences* 11, no. 7: 3166.
https://doi.org/10.3390/app11073166