# A Numerical Study of Fluid Flow in a Vertical Slot Fishway with the Smoothed Particle Hydrodynamics Method

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## Abstract

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## 1. Introduction

## 2. SPH Formalism

#### 2.1. Governing Equations

#### 2.2. Viscosity Treatment and other Numerical Dissipation

#### 2.3. Time Stepping

## 3. Code Features and Boundary Conditions

## 4. Experimental and Numerical Set-Ups

## 5. Results and Discussion

#### 5.1. Discharge

#### 5.2. Water Elevation

#### 5.3. Velocity Profiles

#### 5.4. Effect of Particle Spacing and Artificial Viscosity Coefficient

#### 5.5. Use of the Laminar Viscosity and SPS Turbulence Model

#### 5.6. Relation between Particle Spacing and Artificial Viscosity Coefficient

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\alpha $ | coefficient of artificial viscosity (-) |

${\overline{{c}_{s}}}_{a,b}$ | average numerical speed of sound (m s${}^{-1}$) |

$dp$ | initial interparticle distance (m) |

$phi$ | Wilcoxon test constant (-) |

g | acceleration of gravity (m s${}^{-2}$) |

H | water elevation, depth (m) |

Q | discharge (m${}^{3}$ s${}^{-1}$) |

t | time (s) |

u | longitudinal velocity (m s${}^{-1}$) |

v | transverse velocity (m s${}^{-1}$) |

w | vertical velocity (m s${}^{-1}$) |

x | distance in longitudinal direction (m) |

y | distance in transverse direction (m) |

z | distance in vertical direction (m) |

## Abbreviations

2-D | Two-dimensional |

3-D | Three-dimensional |

ADV | Acoustic Doppler velocimeter |

CFD | Computational Fluid Dynamics |

CPU | Central processing unit |

CUDA | Compute unified device architecture |

DBC | Dynamic boundary conditions |

GPU | Graphics processing unit |

GPGPU | General-purpose graphics processing unit |

HPP | Hydropower plants |

LES | Large eddy simulation |

SPH | Smoothed particle hydrodynamics |

SPS | Sub-particle scales |

VSF | Vertical slot fishway |

WCSPH | Weakly compressible SPH |

## References

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**Figure 1.**Locations of flow velocity measurements by Bombač et al., (2015) [12].

**Figure 2.**Plan view of the VSF, showing the coordinate system with a base at the corner of the pool, locations of observed cross sections $x=0.5$ m, 1.0 m, 1.5 m, 2.0 m and 2.5 m, and profiles $y=0.55$ m, 1.10 m, 1.65 m. Dimensions are in meters.

**Figure 3.**Contour maps of flow variables at the free surface (top view). Left panels represent velocity magnitude, colored from 0 m/s (blue) to 1.5 m/s (red). Right panels show contours of vorticity in z, from −3 rad/s (blue) to 3 rad/s (red).

**Figure 4.**Contour maps of flow variables at $z=-0.5$ m from the free surface (top view). Left panels represent velocity magnitude, colored from 0 m/s (blue) to 1.5 m/s (red). Right panels show contours of vorticity in z, from −3 rad/s (blue) to 3 rad/s (red).

**Figure 9.**Velocity magnitude against vertical distance at different x locations. Red curves with circles are SPH velocities, blue curves with squares are experimental velocities.

**Figure 11.**Average velocity profile of u at different x positions and for several different values of $dp$ and $\alpha $ compared against field data.

**Figure 12.**Average velocity profiles with artificial and laminar + SPS treatment of viscosity, both compared against field data.

Location | $\mathit{x}=0.5$ m | $\mathit{x}=1.0$ m | $\mathit{x}=1.5$ m | $\mathit{x}=2.0$ m | $\mathit{x}=2.5$ m | |||||
---|---|---|---|---|---|---|---|---|---|---|

ADV | SPH | ADV | SPH | ADV | SPH | ADV | SPH | ADV | SPH | |

$y=1.70$ m | 1.43 | 1.23 | 1.23 | 0.99 | 1.18 | 1.02 | 1.14 | 1.03 | 1.11 | 1.15 |

$y=1.60$ m | 1.36 | 1.58 | 1.36 | 1.26 | 1.19 | 1.19 | 1.05 | 1.08 | 1.05 | 1.15 |

$y=2.10$ m | −0.08 | 0.00 | −0.06 | 0.02 | 0.31 | 0.15 | 0.64 | 0.34 | 0.88 | 0.51 |

$y=0.13$ m | −0.16 | −0.12 | −0.39 | −0.25 | −0.49 | −0.35 | −0.41 | −0.30 | −0.30 | −0.06 |

**Table 2.**Two-tailed Wilcoxon signed-ranks test for paired samples with $\varphi =0.05$. Values in cells represent the test statistic, critical statistic (in brackets), and whether there is a significant difference, i.e., yes/no.

$\mathit{x}=0.5$ m | $\mathit{x}=1.0$ m | $\mathit{x}=1.5$ m | $\mathit{x}=2.0$ m | $\mathit{x}=2.5$ m | |
---|---|---|---|---|---|

u | 40 (29); no | 52 (29); no | 52 (25); no | 66 (29); no | 31 (29); no |

v | 31 (25); no | 13 (25); yes | 14 (25); yes | 12 (29); yes | 46 (29); no |

**Table 3.**L${}_{2}$-norm for the average profile of u for various combinations of $dp$ and $\alpha $.

$\mathit{dp}$ [m] ($\mathit{\alpha}$) | 0.03 (0.07) | 0.04 (0.05) | 0.05 (0.03) | 0.06 (0.02) | 0.06 (0.01) |
---|---|---|---|---|---|

L${}_{2}$-norm | 12.1 | 11.7 | 11.9 | 14.1 | 15.6 |

**Table 4.**Relation between particle spacing, $dp$, number of fluid particles, and viscosity coefficient, $\alpha $.

$\mathbf{dp}$ [m] | # of Fluid Particles | $\mathit{\alpha}$ |
---|---|---|

0.06 | 86,482 | 0.01 |

0.05 | 158,188 | 0.03 |

0.04 | 306,128 | 0.05 |

0.03 | 761,824 | 0.07 |

© 2019 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Novak, G.; Tafuni, A.; Domínguez, J.M.; Četina, M.; Žagar, D. A Numerical Study of Fluid Flow in a Vertical Slot Fishway with the Smoothed Particle Hydrodynamics Method. *Water* **2019**, *11*, 1928.
https://doi.org/10.3390/w11091928

**AMA Style**

Novak G, Tafuni A, Domínguez JM, Četina M, Žagar D. A Numerical Study of Fluid Flow in a Vertical Slot Fishway with the Smoothed Particle Hydrodynamics Method. *Water*. 2019; 11(9):1928.
https://doi.org/10.3390/w11091928

**Chicago/Turabian Style**

Novak, Gorazd, Angelantonio Tafuni, José M. Domínguez, Matjaž Četina, and Dušan Žagar. 2019. "A Numerical Study of Fluid Flow in a Vertical Slot Fishway with the Smoothed Particle Hydrodynamics Method" *Water* 11, no. 9: 1928.
https://doi.org/10.3390/w11091928