# CFD Analysis of a Hydrostatic Pressure Machine with an Open Source Solver

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Description of Flow Solver caffa3d

#### 2.1.1. Turbulence Modeling

#### 2.1.2. Free Surface Flows

^{2}is the gravitational acceleration, $\overrightarrow{{F}_{\sigma}}$ is the force at the interface due to surface tension $\sigma $ (in the case of water–air interface $\sigma =73\times {10}^{-3}$ N/m), calculated according to the expression proposed by [33]:

#### 2.1.3. Turbomachinery Modeling

#### 2.1.4. Discretization of Equations

#### 2.2. Case Study

#### 2.3. Domain, Mesh, Boundary Conditions and Other Parameters

^{3}and ${\mu}_{{H}_{2}O}=1\times {10}^{-3}$ Pa·s). Air density corresponds to atmospheric pressure and a temperature of 20 °C (${\rho}_{air}=1.2$ kg/m

^{3}), while ${\mu}_{air}=1.7\times {10}^{-3}$ Pa·s, which is 100 times higher than that corresponding to the mentioned conditions, in order to reduce spurious velocities in air [17]. For the force due to surface tension, the value of $73\times {10}^{-3}$ N/m corresponding to the tension between water and air was used. The properties of water, together with the simulated inlet velocities (0.056 m/s to 0.169 m/s) and the length scale of the HPM ($D=1.2$ m), give a Reynolds number ($\frac{vD}{\nu}$) between $6.7\times {10}^{4}$ and $2.0\times {10}^{5}$.

#### 2.4. Power and Efficiency Calculation

#### 2.5. Convergence Study

#### 2.6. Grid Dependency Study

## 3. Results and Discussion

#### 3.1. Turbulence Charactaristics of the Flow

#### 3.2. Instantaneous Power

#### 3.3. Mean Power and Efficiency

#### 3.4. Channel Flow Analysis

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagram of the sliding interface, showing the relative displacement angle $\alpha $ between a rotating block and other static ones (particular case).

**Figure 2.**Diagram of the HPM with straight blades tested in the HYLOW project. Taken from [38].

**Figure 3.**Operational curves of the HPM of medium-scale and radial straight blades tested in the HYLOW project (taken from [38]). (

**a**) shows Power—Discharge and Efficiency—Discharge curves while (

**b**) shows the relationship between discharge and rotating speed.

**Figure 8.**Residual drop along iterative steps, for the u velocity component momentum equation, the mass balance equation and the VOF scalar transport equation.

**Figure 9.**Part of the computational mesh around the HPM, for the coarse (

**left**) and fine (

**right**) grids.

**Figure 10.**Power evolution for operating point at $Q=97.8$ L/s, for three grids with different densities (medium, coarse and fine).

**Figure 11.**Time series of the three velocity components (longitudinal-u, vertical-v and transversal-w). Broken lines represent the averages of each component.

**Figure 13.**Instantaneous power over 4 turns of the HPM for three operating points: (

**a**) $Q=58.9$ L/s, (

**b**) $Q=97.8$ L/s and (

**c**) $Q=120$ L/s.

**Figure 14.**Power—Discharge curve of the HPM. Comparisson of numerical with experimental results presented in Schneider et al. (2011) [38].

**Figure 15.**Efficiency—Discharge curve of the HPM. Comparisson of numerical with experimental results presented in Schneider et al. (2011) [38].

**Figure 16.**Relationship between discharge and rotating speed of the HPM. Comparisson of numerical with experimental results presented in Schneider et al. (2011) [38].

**Figure 20.**Lateral image of the instantaneous flow of water through the HPM for low discharge.

**Left**image shows an instantant of simulations with caffa3d for $Q=58.9$ L/s, while

**right**image shows an instant of experimental tests for $Q=60$ L/s [38].

**Figure 23.**Lateral image of the instantaneous flow of water through the HPM for low discharge.

**Left**image shows an instantant of simulations with caffa3d for $Q=120$ L/s, while

**right**image shows an instant of experimental tests for $Q=180$ L/s with the level downstream of the wheel 200 mm below the edge of the hub [38].

**Table 1.**Operating points of the HPM with straight blades of 1200 mm in diameter, based on experimental tests.

Q (L/s) | $58.9$ | 77 | $97.8$ | 120 | 137 |

N (rpm) | $2.5$ | $3.5$ | $5.0$ | $6.2$ | $7.2$ |

${P}_{m}$ (W) | $123.4$ | 155 | $186.7$ | 149 | 80 |

$\eta $ (%) | 53 | 51 | 49 | 29 | 14 |

**Table 2.**Operating points of the HPM of straight blades of $1200\mathrm{mm}$ in diameter, from the simulations in caffa3d.

Q (L/s) | 25 | 40 | $58.9$ | 77 | $97.8$ | 110 | 120 |

N (rpm) | 0 | $1.0$ | $2.069$ | $3.333$ | $4.615$ | $5.2174$ | $6.25$ |

${P}_{m}$ (W) | 0 | $59.5$ | $117.5$ | 159 | 177 | $84.8$ | $34.9$ |

$\eta $ (%) | 0 | $38.0$ | $50.9$ | $52.7$ | $46.2$ | $19.7$ | $7.4$ |

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**MDPI and ACS Style**

Pienika, R.; Cataldo, J.; Ramos, H.M.
CFD Analysis of a Hydrostatic Pressure Machine with an Open Source Solver. *Fluids* **2023**, *8*, 9.
https://doi.org/10.3390/fluids8010009

**AMA Style**

Pienika R, Cataldo J, Ramos HM.
CFD Analysis of a Hydrostatic Pressure Machine with an Open Source Solver. *Fluids*. 2023; 8(1):9.
https://doi.org/10.3390/fluids8010009

**Chicago/Turabian Style**

Pienika, Rodolfo, José Cataldo, and Helena M. Ramos.
2023. "CFD Analysis of a Hydrostatic Pressure Machine with an Open Source Solver" *Fluids* 8, no. 1: 9.
https://doi.org/10.3390/fluids8010009