# Validation of the LOGOS Software Package Methods for the Numerical Simulation of Cavitational Flows

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## Abstract

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## 1. Introduction

## 2. A Computational Method for Cavitation Problems

_{i}is the velocity vector component, i = {x, y, z}; x

_{i}is the Cartesian coordinate vector component, i = {x, y, z}; τ

_{ij}is the viscous stress tensor; ${\mathsf{\tau}}_{ij}^{t}$ is the Reynolds stress tensor; g

_{i}is the gravity acceleration vector component; p is the pressure; ρ is the resultant density in the given case, which is an averaged density for the two phases: ρ = ρ

_{l}α

_{l}+ ρ

_{v}α

_{v}; α is a volume fraction (the subscript v stands for the vapor phase; the subscript l stands for the liquid phase); and R

_{e}and R

_{c}are the mass sources describing the generation and breakdown of vapor inclusions.

_{ij}is Kronecker symbol.

_{t}and the Boussinesq hypothesis for the stress tensor calculation:

_{B}is the radius of a bubble, P

_{B}is the pressure on the bubble surface, and P is the pressure at a distance to the bubble surface.

_{e}and R

_{c}, the Schnerr–Sauer model uses the ratio between the volume fraction of vapor and the number of bubbles per unit volume:

_{B}of saturated vapors, the evaporation process goes and bubble radii R

_{B}increase; otherwise, the condensation process takes place with a decreasing radius of bubbles. According to the simplified Rayleigh–Plesset equation, the R

_{B}growth rate varies as

_{e}and R

_{c}:

_{v}and ρ

_{l}are the vapor and liquid phase densities, respectively, which are taken as constant.

_{B}= n$\frac{4}{3}$π${R}_{B}^{3}$, and the full gaseous phase mass variation in a unit volume corresponding to the interface mass transport describes the evaporation and condensation in the following way:

_{vap}and F

_{cond}. As a result, expressions for the evaporation and condensation sources take the form

_{vap}= 50, F

_{cond}= 0.01, bubble radius, and volume fraction of vaporization nuclei [11].

## 3. Simulation of a Flow around a Cylindrical Body with a Plane/Semispherical End

^{−3}Pa·s, ρ = 998.2 kg/m

^{3}for water and μ = 1.34 × 10

^{−5}Pa·s, ρ = 0.5542 kg/m

^{3}for vapor. The cavitation number served as an additional parameter for cavitating flows. In the problems of the flow around a body of a finite thickness, the cavitation number, σ, is defined as

_{sat}is the saturation pressure; ρ is the medium density; U

_{in}is the inflow rate, l is a typical size. For the mesh convergence assessment problems, the cavitation number was σ = 0.3 with P = 9864 Pa and P

_{sat}= 2736 Pa.

^{13}and radius of bubbles R = 10 × 10

^{−6}m. The cavitation numbers were 0.3 and 0.5 in flows around the cylinder with plane end and 0.4 and 0.5 in flows around the cylinder with semispherical end.

## 4. Simulation of a Rotating Propeller VP 1304

^{−3}m. Figure 6 shows the mesh model cross-section. The mesh includes 2.8 mln cells in total.

_{m}= J·n·D

_{m}.

^{3}for water; μ = 1.2676 × 10

^{−5}Pa·s and ρ = 0.5953 kg/m

^{3}for vapor.

_{0}is the propeller efficiency, ρ is a mean density of flow, n is the number of revolutions, and D

_{p}is the propeller diameter.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**The pressure coefficient distribution over the cylinder length: (

**a**) A cylinder with a plane end; (

**b**) A cylinder with a semispherical end.

**Figure 3.**The vapor volume fraction distribution: (

**a**) a cylinder with a plane end; (

**b**) a cylinder with a semispherical end.

**Figure 4.**The pressure coefficient distribution over the length of the cylinder with semispherical end.

**Figure 8.**Volume fractions of cavitation vapor above 50%: (

**a**) Test mode 1; (

**b**) Test mode 2; (

**c**) Test mode 3.

Meshes with Plane End: | Nx | Ny | Nz |
---|---|---|---|

mesh 1 | 300 | 200 | 2 |

mesh 2 | 450 | 350 | 4 |

mesh 3 | 600 | 450 | 6 |

Meshes with semispherical end: | |||

mesh 1 | 100 | 60 | 2 |

mesh 2 | 200 | 120 | 4 |

mesh 3 | 400 | 240 | 6 |

Test Mode | 1 | 2 | 3 |
---|---|---|---|

Pressure in tunnel | 43,071 | 31,353 | 42,603 |

Advance coefficient of propeller | 1.09 | 1.269 | 1.408 |

Cavitation number | 2.024 | 1.424 | 2.0 |

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**MDPI and ACS Style**

Kozelkov, A.; Kurkin, A.; Kurulin, V.; Plygunova, K.; Krutyakova, O.
Validation of the LOGOS Software Package Methods for the Numerical Simulation of Cavitational Flows. *Fluids* **2023**, *8*, 104.
https://doi.org/10.3390/fluids8030104

**AMA Style**

Kozelkov A, Kurkin A, Kurulin V, Plygunova K, Krutyakova O.
Validation of the LOGOS Software Package Methods for the Numerical Simulation of Cavitational Flows. *Fluids*. 2023; 8(3):104.
https://doi.org/10.3390/fluids8030104

**Chicago/Turabian Style**

Kozelkov, Andrey, Andrey Kurkin, Vadim Kurulin, Kseniya Plygunova, and Olga Krutyakova.
2023. "Validation of the LOGOS Software Package Methods for the Numerical Simulation of Cavitational Flows" *Fluids* 8, no. 3: 104.
https://doi.org/10.3390/fluids8030104