# Numerical Simulation of Hydraulic Jumps. Part 2: Recent Results and Future Outlook

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods for Hydraulic Jump Research

#### 2.1. Overview

#### 2.2. One- and Two-Dimensional Modeling

#### 2.3. Three-Dimensional Modeling

#### 2.3.1. RANS Approach

- The Spalart–Allmaras model [30] solves a transport equation for an auxiliary turbulent viscosity ${\nu}_{t}\sim \tilde{\nu}$.
- The standard $k-\u03f5$ [31,32] and the RNG $k-\u03f5$ [33,34] solve transport equations for k and another for $\u03f5\to {\nu}_{t}\sim {k}^{2}/\u03f5$. The former is the most widely-used engineering turbulence model for industrial applications because of its robustness and reasonable accuracy for a wide range of flows. It is a two-equation transport model for k and $\u03f5$, their coefficients being empirically derived. The RNG version adds a term for the $\u03f5$ equation, which is known to be responsible for differences in its performance [8]. It is oftentimes mentioned that accuracy for rapidly-swirling flows is improved, despite fewer validations having been conducted [1].
- Standard $k-\omega $ (2008) and SST $k-\omega $ [35] solve transport equations for k and $\omega \to {\nu}_{t}\sim k/\omega $. The $k-\omega $ model is a two-equation transport model based on solving equations for k and $\omega $. The original version was known to be excessively sensitive to free stream quantities, which is an undesired feature [36], while the revised version of Wilcox [37] attempts to solve this issue. The SST $k-\omega $ is a variant of the former. The SST $k-\omega $ model uses a blending function to transition gradually from the standard $k-\omega $ model near the wall to a high Reynolds number version of the $k-\u03f5$ model in the outer portion of the boundary layer [35].

#### 2.3.2. LES and DES Approach

#### 2.3.3. DNS Approach

#### 2.4. Best Practices for CFD Modeling of Environmental Flows

## 3. Hydraulic Jump Numerical Modeling Studies

#### 3.1. General Comments

#### 3.2. Reynolds-Averaged Navier–Stokes Approach

^{®}codes was presented by Bayon et al. [5], investigating a steady hydraulic jump of a Froude number of 6.5. The RNG $k-\u03f5$ model and the VOF method were used. Performance assessment was conducted by comparing several hydraulic jump variables with the literature and their own experimental data [76,85,86,87,88,89,90]. Numerical uncertainty was assessed following Celik et al. [49], finding that FLOW-3D

^{®}presented a faster convergence with coarser meshes, whereas OpenFOAM had a more stable solution with refinement. All mean variables studied (not including aeration) obtained accuracies above 90%, except for the roller length, which remained at around 80%, and the negative velocities at the upper jump region. An autocorrelation analysis was conducted, finding good computation of jump toe frequencies, thus showing that RANS models can also reproduce the unsteadiness typical of hydraulic jumps.

#### 3.3. Detached Eddy Simulation and Large Eddy Simulation

#### 3.4. Direct Numerical Simulation

#### 3.5. Smoothed Particle Hydrodynamics

## 4. Conclusions

## 5. Future Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature and Abbreviations

## Nomenclature

${\mathrm{F}}_{1}$ | inlet Froude number |

f | fraction of fluid |

k | turbulence kinetic energy |

$\overline{p}$ | mean pressure |

${R}_{ij}$ | Reynolds stress tensor |

${R}_{ij}^{SGS}$ | sub-grid scale stresses |

T | time interval |

t | time |

${T}_{1}$ | characteristic time-scale of the velocity fluctuations |

${T}_{2}$ | characteristic time-scale of the unsteadiness |

T | time interval |

U | convection velocity |

u | instantaneous velocity |

$\overline{u}$ | mean velocity |

${u}^{\prime}$ | velocity fluctuation |

$\overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ | velocity fluctuation covariance |

${x}_{j}$ | coordinates ($x,y,z$) |

$\u03f5$ | turbulence dissipation rate |

$\mu $ | dynamic viscosity |

${\mu}_{t}$ | turbulent viscosity |

${\nu}_{t}$ | turbulent viscosity (${\mu}_{t}/\rho $) |

$\rho $ | fluid density |

$\omega $ | specific dissipation rate |

## Abbreviations

ASME | American Society of Mechanical Engineers |

CFD | Computational Fluid Dynamics |

DES | Detached Eddy Simulation |

DNS | Direct Numerical Simulation |

FAVOR | Fractional Area-Volume Obstacle Representation |

LES | Large Eddy Simulation |

N-S | Navier Stokes |

PDE | Partial Differential Equations |

RANS | Reynolds-Averaged Navier Stokes |

SPH | Smoothed Particles Hydrodynamics |

USBR | United States Bureau of Reclamation |

VOF | Volume of Fluid |

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**Figure 3.**Accuracy obtained in some of the past RANS studies for different turbulence models and flow variables: (

**a**) sequent depths relation; (

**b**) roller length; (

**c**) efficiency; and (

**d**) mean free surface. Inlet Froude numbers clustered by jump typology, as presented in Part 1 of this study.

Dimensional Scale | Unit | Equation |
---|---|---|

Turbulent kinetic energy | L${}^{2}$/T${}^{2}$ | $k=\overline{{u}_{i}^{\prime}{u}_{i}^{\prime}}/2$ |

Turbulence dissipation rate | L${}^{2}$/T${}^{3}$ | $\u03f5=\nu \overline{\frac{\partial {u}_{i}^{\prime}}{\partial {x}_{j}}\frac{\partial {u}_{i}^{\prime}}{\partial {x}_{j}}}$ |

Specific dissipation rate | 1/T | $\omega =\u03f5/k$ |

**Table 2.**Turbulence modeling, indicator for resolved/modeled turbulence (R: Resolved; M: Modeled), accuracy, and computational cost of RANS, LES, Detached Eddy Simulation (DES), and DNS.

RANS | DES | LES | DNS | |
---|---|---|---|---|

Modeling approach | Reynolds-averaged | RANS-LES coupling | Sub-grid scale | Not necessary |

Large scales/inertial Subrange/dissipation scale | M/M/M | R/M/M | R/R/M | R/R/R |

Expected accuracy | Mean variables | Allow inertial’ modeling | Allow inertial’ modeling | Up to the Kolmogorov scale |

Computational cost | Low | High | High | Extremely high |

Flow Field Description | Approach | References |
---|---|---|

Lagrangian | Smoothed Particle Hydrodynamics | Gingold and Monhagan [50] |

Monaghan [51] | ||

Violeau [52] | ||

Violeau and Rogers [53] | ||

Eulerian | Reynolds-Averaged Navier–Stokes | Reynolds [54] |

Rodi [28] | ||

Spalart [36] | ||

Wilcox [2] | ||

Detached Eddy Simulation Large Eddy Simulation | Spalart [55] | |

Labourasse et al. [56] | ||

Rodi et al. [39] | ||

Direct Numerical Simulation | Moin and Mahesh [42] | |

Prosperetti and Tryggvason [11] | ||

Alfonsi [41] |

Turbulence Model | References |
---|---|

Standard $k-\u03f5$ | Jones and Launder [31] |

Launder and Sharma [32] | |

RNG $k-\u03f5$ | Yakhot and Orszag [33] |

Yakhot et al. [34] | |

$k-\omega $ | Wilcox [2] |

Wilcox [37] | |

SST $k-\omega $ | Menter [35] |

**Table 5.**Numerical simulation studies of hydraulic jumps and inlet Froude number (${\mathrm{F}}_{1}$).

Flow Field Description | References | Year | Numerical Approach | Turbulence Model | Resolution Dependence Study | ${{F}}_{1}$ |
---|---|---|---|---|---|---|

Lagrangian Methods | Lopez et al. [57] | 2010 | SPH | $k-\u03f5$ | Meshless method | 3.25, 3.41, 3.58, 3.61, 3.64, 4.12, 4.45, 4.88, 5.13, 5.24, 6.62, 7.06, 7.16 |

De Padova et al. [58] | 2013 | XSPH | mixing length and $k-\u03f5$ | Meshless method | 3.90, 8.30 | |

De Padova et al. [59] | 2018 | XSPH | mixing length and $k-\u03f5$ | Meshless method | 2.58, 2.66, 2.70, 3.89, 4.99, 5.11 | |

Eulerian Methods | Chippada et al. [60] | 1994 | RANS | STD $k-\u03f5$ | − | 2.00, 4.00 |

Zhao et al. [61] | 2004 | RANS | STD $k-\u03f5$, $k-l$ | − | 1.46 | |

Gonzalez and Bombardelli [62] | 2005 | RANS | STD $k-\u03f5$ | √ | 2.00, 2.50, 3.32 | |

Carvalho et al. [14] | 2008 | RANS | RNG $k-\u03f5$ | − | 6.00 | |

Abbaspour et al. [63] | 2009 | RANS | STD $k-\u03f5$ RNG $k-\u03f5$ | − | 4.00, 4.70, 5.00, 5.70, 5.80, 6.10, 7.00, 7.20, 8.00 | |

Ma et al. [64] | 2011 | RANS | SST $k-\omega $ | − | 1.98 | |

Ebrahimi et al. [65] | 2013 | RANS | STD $k-\u03f5$ | − | 3.00, 3.30, 3.60, 3.60, 5.00, 5.70, 6.70, 8.00 | |

Bayon-Barrachina and [66] | 2015 | RANS RANS | STD $k-\u03f5$ RNG $k-\u03f5$ SST $k-\omega $ | √ | 6.10 | |

Witt et al. [67] | 2015 | RANS | realizable $k-\u03f5$ | √ | 2.43, 3.65, 4.82 | |

Bayon et al. [5] | 2016 | RANS | RNG $k-\u03f5$ | √ | 6.50 | |

Witt et al. [68] | 2018 | RANS | realizable $k-\u03f5$ | √ | 2.43, 3.65, 4.82 | |

Harada and Li [69] | 2018 | RANS | $k-\u03f5$, $k-\omega $ | √ | 5.80 | |

Valero et al. [6] | 2018 | RANS | RNG $k-\u03f5$, $k-\u03f5$ | √ | 3.12, 3.88, 4.20, 6.17, 6.37, 6.47, 8.27, 9.52 | |

Ma et al. [64] | 2011 | DES | √ | 1.98 | ||

Jesudhas et al. [70] | 2018 | DES | √ | 8.5 | ||

Gonzalez and Bombardelli [62] | 2005 | LES | √ | 2.00, 2.50, 3.32 | ||

Lubin et al. [71] | 2009 | LES | − | 5.09 | ||

Mortazavi et al. [19] | 2016 | DNS | Full solution of all the scales | √ | 2.00 |

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**MDPI and ACS Style**

Viti, N.; Valero, D.; Gualtieri, C.
Numerical Simulation of Hydraulic Jumps. Part 2: Recent Results and Future Outlook. *Water* **2019**, *11*, 28.
https://doi.org/10.3390/w11010028

**AMA Style**

Viti N, Valero D, Gualtieri C.
Numerical Simulation of Hydraulic Jumps. Part 2: Recent Results and Future Outlook. *Water*. 2019; 11(1):28.
https://doi.org/10.3390/w11010028

**Chicago/Turabian Style**

Viti, Nicolò, Daniel Valero, and Carlo Gualtieri.
2019. "Numerical Simulation of Hydraulic Jumps. Part 2: Recent Results and Future Outlook" *Water* 11, no. 1: 28.
https://doi.org/10.3390/w11010028