# Numerical Insight into the Kelvin-Helmholtz Instability Appearance in Cavitating Flow

^{*}

## Abstract

**:**

## 1. Introduction

- Re-entrant jet: the flow that passes over the attached cavity, deviates towards the surface because of the discrepancies in the pressure in and out of the attached cavity. Just downstream from the attached cavity closure line, a stagnation point forms and divides the flow into a downstream and an upstream portion. The latter enters the attached cavity and causes its separation—creating a detached cavitation cloud that is carried downstream, where it reaches and collapses into a higher-pressure zone. The cycle continues periodically.
- Shock wave: a shock wave that travels through the flow region causes the collapse of the cavitation cloud. It suppresses the attached cavity as it moves upstream. A large vapor cloud is shed when the distortion in the void fraction enters the area of cavity separation at the top of the wedge. The cavity expands again later on, and the cycle continues.

## 2. Experimental Observations

## 3. Numerical Procedure

#### 3.1. Mesh

#### 3.2. Reynolds-Averaged Navier-Stokes (RANS) Equations

**U**velocity vector, defined as $\mathit{U}={\left[U,V,W\right]}^{\mathrm{T}}$:

**U**, while S

_{Mx}and S

_{My}represent momentum source terms in x and y directions, respectively

#### 3.3. Turbulence Modeling

#### 3.4. Two-Phase Flow Modeling

_{e}and R

_{c}represent mass transfer source terms for evaporation and condensation, which account for the mass transfer between the phases of liquid and vapor in cavitation and are thus related to the growth and collapse of the vapor bubbles. According to the cavitation model used, their formulation varies.

#### 3.5. Cavitation Model

_{e}and R

_{c}(see Equation (7) are defined as:

_{evap}and condensation F

_{cond}are 1 and 0.2, respectively. To link the fraction of the vapor volume to the number of bubbles per liquid volume n

_{b}, the Schnerr-Sauer cavitation model uses:

_{b}and bubble number density n. The compressibility of water and water vapor was not considered in the modeling.

#### 3.6. Boundary Conditions

#### 3.7. Physics and Solver Settings

^{−5}of the iterative numerical solution of the individual equations at each time step of the simulation. The iteration error was estimated to be less than 0.02%. By assessing its effect against the average pressure difference and the cavity length, the size of the time step was obtained. There was little difference in these parameters if the time step was shorter than 5 μs, but a shorter one—1 μs was ultimately chosen for observation of the Kelvin-Helmholtz instability. We conducted 50 ms of computational modeling for each case, where the last 30 ms was applicable for further analysis.

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Development of the Kelvin-Helmholtz instability in the microchannel ($\stackrel{\xb7}{\mathrm{m}}$ = 9.15 g/s, Δp = 4.00 bar, σ = 1.24).

**Figure 3.**The geometry of the microchannel computational domain and the mesh detail in the wedge apex area.

**Figure 4.**Separation of cavitation clouds and Kelvin-Helmholtz instability formation (experiment: $\stackrel{\xb7}{\mathrm{m}}$ = 9.15 g/s, Δp = 4.00 bar, σ = 1.24; simulation: $\stackrel{\xb7}{\mathrm{m}}$ = 9.03 g/s, Δp = 3.71 bar, σ = 1.26). The time difference between the images is Δt = 100 µs.

**Figure 5.**Development of Kelvin-Helmholtz instability with velocity profiles at individual cross-sections. The time difference between the images is Δt = 100 µs.

**Figure 6.**A close-up view of the initial development of Kelvin-Helmholtz instability. The time difference between the images is Δt = 20 µs. The black line shows a 10% vapor fraction region limit.

Mesh Size | Δp (bar) | l (mm) |
---|---|---|

~80,000 | 3.56 | 25.0 |

~160,000 | 3.71 | 25.6 |

~320,000 | 3.73 | 25.7 |

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**MDPI and ACS Style**

Pipp, P.; Hočevar, M.; Dular, M.
Numerical Insight into the Kelvin-Helmholtz Instability Appearance in Cavitating Flow. *Appl. Sci.* **2021**, *11*, 2644.
https://doi.org/10.3390/app11062644

**AMA Style**

Pipp P, Hočevar M, Dular M.
Numerical Insight into the Kelvin-Helmholtz Instability Appearance in Cavitating Flow. *Applied Sciences*. 2021; 11(6):2644.
https://doi.org/10.3390/app11062644

**Chicago/Turabian Style**

Pipp, Peter, Marko Hočevar, and Matevž Dular.
2021. "Numerical Insight into the Kelvin-Helmholtz Instability Appearance in Cavitating Flow" *Applied Sciences* 11, no. 6: 2644.
https://doi.org/10.3390/app11062644