# SPH Modelling of Hydraulic Jump Oscillations at an Abrupt Drop

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Set Up

_{1}is the inflow water depth; y

_{t}is the water depth downstream of the jump; F

_{1}= V

_{1}/(gy

_{1})

^{0.5}is the inflow Froude number and Re is the Reynolds number defined as Re = V

_{1}y

_{1}/ν = V

_{t}y

_{t}/ν where V

_{1}and V

_{t}are the flow velocities at water depths y

_{1}and y

_{t}, respectively, and ν the kinematic water viscosity at the run temperature. Figure 3 shows the locations where y

_{1}and y

_{t}were measured for each flow pattern i.e., the A-jump and B-jump.

## 3. SPH Numerical Method

^{−1}guarantees a numerical Mach number everywhere lower than 0.07.

_{i}. The summations are extended to all the particles j at a distance from i smaller than 2h, i.e., lying within the circle where the adopted C2 Wendland kernel function W

_{ij}[59] is defined.

**v**= (u, v) is the velocity vector, p is pressure, ρ is density,

**g**is the gravity acceleration vector, $\mathcal{T}$ is the turbulent shear stress tensor, c is the speed of sound in the weakly compressible fluid, μ

_{T}is the dynamic eddy viscosity, $\mathcal{S}$ is rate-of-strain tensor and the subscript 0 denotes a reference state for pressure computation. All the variables are assumed to be Reynolds-averaged.

_{ij}, renormalized through a procedure which enforces consistency on the first derivatives to the 1st order [60], leading to a 2nd order accurate discretization scheme in space. The kernel renormalization is applied everywhere, apart from the pressure gradient term, where the form originally proposed by [58] is retained to guarantee momentum conservation.

_{v}is a velocity smoothing coefficient and

**v**

_{i}

^{n+1}is the value obtained by solution of the second equation in Equation (1). The use of XSPH leads to a satisfactory regularization of particle distribution within the computational domain: this regular particle pattern, together with the kernel renormalization procedure described above, leads to computed pressure fields whose energy content at the higher frequencies (i.e., the frequencies mostly connected with numerical noise) is sufficiently low, as shown in the following Section 4 (see, for instance, Figure 17). The alternative of making use of particle-shifting algorithms, such as those developed for Incompressible SPH ([61] and [62]) and recently extended to WCSPH ([63,64]) was therefore not followed, also considering the problems which might arise at the jump toe, where physical voids arising from wave breaking and air entrainment may be obliterated by an unsuitable particle shifting.

_{T}:

- (1)
- A mixing-length model ([29,46]), in which ${\mu}_{T}={c}_{\mu}\rho {l}^{2}\Vert \mathcal{S}\Vert $, where c
_{μ}= 0.09, the mixing-length for each particle is evaluated as:$${l}_{i}=\mathrm{min}\left[1,{\left|{\displaystyle {\displaystyle \sum}_{j}}\frac{{m}_{j}}{{\rho}_{j}}\nabla {W}_{ij}\right|}^{-3}\right]\mathrm{min}\left(\kappa y,{l}_{max}\right)$$

_{max}is a cutoff maximum value and the damping function in the first factor of the RHS avoids a non-physical growth of l near the free-surface when the particle distribution is irregular and the SPH evaluation of the gradient of a constant function departs sharply from zero: this function plays therefore a different role than the one of a wake function, such as the one included in the mixing-length model by [69] to simulate turbulent, open-channel uniform flows; the use of a wake function was not considered here because relevant turbulence effects occur mostly close to the hydraulic jump, where the flow conditions are in any case far from uniform;

- (2)
- A SPH version of the standard k-ε turbulence model by [70], in which ${\mu}_{T}={c}_{\mu}\frac{{k}^{2}}{\epsilon}$ and the two equations for the turbulent kinetic energy k and for the turbulent dissipation rate ε are:$$\frac{D{k}_{i}}{Dt}={P}_{{k}_{i}}+\frac{1}{{\sigma}_{k}}{\displaystyle {{\displaystyle \sum}}_{j}}{m}_{j}\frac{{\nu}_{{T}_{i}}+{\nu}_{{T}_{j}}}{{\rho}_{i}+{\rho}_{j}}\frac{{k}_{i}+{k}_{j}}{{r}_{ij}^{2}+0.01{h}^{2}}{\mathit{r}}_{ij}\xb7\nabla {\widehat{W}}_{ij}-{\epsilon}_{i}\phantom{\rule{0ex}{0ex}}\frac{D{\epsilon}_{i}}{Dt}={C}_{{\epsilon}_{1}}\frac{{\epsilon}_{i}}{{k}_{i}}{P}_{{k}_{i}}+\frac{1}{{\sigma}_{\epsilon}}{{\displaystyle \sum}}_{j}{m}_{j}\frac{{\nu}_{{T}_{i}}+{\nu}_{{T}_{j}}}{{\rho}_{i}+{\rho}_{j}}\frac{{\epsilon}_{i}+{\epsilon}_{j}}{{r}_{ij}^{2}+0.01{h}^{2}}{\mathit{r}}_{ij}\xb7\nabla {\widehat{W}}_{ij}+{C}_{{\epsilon}_{2}}\frac{{\epsilon}_{i}}{{k}_{i}}{{\displaystyle \sum}}_{j}\frac{{m}_{j}}{{\rho}_{j}}{\epsilon}_{j}{\widehat{W}}_{ij}$$
_{T}is the eddy viscosity. As in [71], the values originally proposed by [70] for the model constants (σ_{k}= 1, σ_{ε}= 1.3, C_{ε}_{1}= 1.44, C_{e2}= 1.92) were maintained here.

_{1}and head y

_{1}: a new row of particles is created upstream of the layer at each Σ/V

_{1}time interval, Σ being the initial particle spacing. The k and ε values at the inflow are computed by assuming a constant 10% turbulence intensity and a mixing length equal to 0.5 y

_{1}.

_{t}and head y

_{t}are imposed to each particle crossing the outflow boundary, and these values are kept frozen in the 2h-wide outflow buffer layer, so that their motion is maintained at constant speed: when the particles exit the buffer layer, they are removed from the computation; k and ε values are also frozen in the buffer layer.

## 4. Numerical Tests and Results

_{v}= 0.01. The ratio of the smoothing length to the initial particle spacing Σ was maintained to a constant value of η/Σ = 1.5 [73] for all the simulations. A convergence analysis was carried out by choosing different initial particle spacing Σ ranging from 0.015 to 0.005 m. The related number of SPH particles N

_{P}in the computational domain ranged from about 1000 to 9000, respectively. It can be seen that the simulation at the lowest resolution is not able to predict the oscillating characteristics and cyclic mechanisms in hydraulic jumps (Figure 5).

_{max}= 0.5 h

_{2}and the two-equation model (10). Table 2 summarizes the principal characteristics of the simulations in the sensitivity analysis.

#### Analysis of Stable vs. Oscillating Flow Behaviour

_{t}/y

_{1}and the upstream Froude number F

_{1}as a function of the relative step height s/y

_{1}. The results are presented in the form of diagram (Figure 15) as function of the relative step height s/y

_{1}equal to 0.06 to 0.1. In the diagram (Figure 15), the basic flow pattern is indicated in the legend (A-jump, B-jump) including the oscillatory flow patterns (B-wave and A-wave).

_{1}and x

_{2}are the two variables values, and the bar denotes an average of the two variables values.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Flow conditions (from Ohtsu and Yasuda, 1991): (

**a**) A-jump; (

**b**) wave jump; (

**c**) wave train; (

**d**) B-jump (maximum plunging condition); and (

**e**) minimum B-jump (limited jump).

**Figure 2.**Channel at the Hydraulics laboratory of the Department of Civil, Environmental, Land, Building Engineering and Chemistry of the Polytechnic University of Bari: (

**a**) sketch; and (

**b**) picture of the channel.

**Figure 4.**Schematic figure of the geometrical setup. Solid black lines indicates solid walls, dashed blue lines the initial free surface and dashed red lines show the position of the periodic open boundaries.

**Figure 5.**Instantaneous SPH vorticity field in the SPH simulation of Test T1a with an initial particle spacing equal to 0.015: (

**a**) t = 6 s; (

**b**) t = 9 s; (

**c**) t = 12 s; and (

**d**) t = 15 s. Vorticity values in the colour scale are expressed in s

^{−1}.

**Figure 6.**Instantaneous SPH vorticity field (t = 8 s) in the SPH simulation of Test T1 with different particle resolutions: (

**a**) Σ = 0.015 m; (

**b**) Σ = 0.010 m; and (

**c**) Σ = 0.005 m. Vorticity values in the colour scale are expressed in s

^{−1}.

**Figure 7.**Instantaneous SPH vorticity field in the SPH simulation of Test T1a: (

**a**) t = 8 s; (

**b**) t = 17 s; (

**c**) t = 22 s; and (

**d**) t = 27 s. Vorticity values in the colour scale are expressed in s

^{−1}.

**Figure 8.**Instantaneous SPH vorticity field in the SPH simulation of Test T1b: (

**a**) t = 15 s; (

**b**) t = 21 s; (

**c**) t = 26 s; and (

**d**) t = 30 s. Colour scale is the same as in Figure 7.

**Figure 9.**Amplitude spectrum of pressure fluctuations under the hydraulic jump (configuration B32 of Table 1) for the SPH simulations of test T1 and two different turbulence models: mixing-length (T1a) and k-ε (T1b).

**Figure 10.**Instantaneous SPH vorticity field in the SPH simulation of Test T4: (

**a**) t = 15 s; (

**b**) t = 18 s; (

**c**) t = 22 s; and (

**d**) t = 24 s. Colour scale is the same as in Figure 7.

**Figure 11.**Instantaneous SPH vorticity field in the SPH simulation of Test T6: (

**a**) t = 30 s; (

**b**) t = 40 s; (

**c**) t = 50 s; (

**d**) t = 60 s; (

**e**) t = 70 s; and (

**f**) t = 80 s. Colour scale is the same as in Figure 7.

**Figure 12.**Instantaneous SPH vorticity field in the SPH simulation of Test T2: (

**a**) t = 5 s; (

**b**) t = 10 s; (

**c**) t = 12 s; (

**d**) t = 16 s; Colour scale is the same as in Figure 7.

**Figure 13.**Test T1: Amplitude spectrum of the time series of the surface elevations: (

**a**) upstream; and (

**b**) downstream of the jump.

**Figure 14.**Test T2: Amplitude spectrum of the time series of the surface elevations: (

**a**) upstream; and (

**b**) downstream of the jump.

**Figure 15.**Regime chart for flow configurations with 0.6 < s/y

_{1}< 1.1. The dashed line shows the approximate boundaries between A and B jump types.

**Figure 16.**Time series of the pressure under hydraulic jump (configuration: T1) at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 17.**Amplitude spectrum of pressure fluctuations under hydraulic jump (configuration: T1) at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 18.**Time series of the pressure under hydraulic jump (configuration: T2) at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 19.**Amplitude spectrum of pressure fluctuations under hydraulic jump (configuration: T2) at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 20.**Time series of the horizontal velocity component under hydraulic jump (configuration: T1) measured at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 21.**Time series of the vertical velocity component under hydraulic jump (configuration: T1) measured at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 22.**Time series of the horizontal velocity component under hydraulic jump (configuration: T2) measured at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 23.**Time series of the vertical velocity component under hydraulic jump (configuration: T2) measured at a distance of: (

**a**) 7 cm; (

**b**) 10 cm; (

**c**) 20 cm; and (

**d**) 100 cm, from the time-averaged position of the hydraulic jump toe.

**Figure 24.**Plot of

**r**versus

**d**for: (

**a**) (p–u); (

**b**) (p–v); and (

**c**) (u–v) pairs of data (configuration: T1).

**Figure 25.**Plot of

**r**versus

**d**for: (

**a**) (p–u); (

**b**) (p–v); and (

**c**) (u–v) pairs of data (configuration: T2).

TEST | Run no. (Mossa 2002) | y_{1} (cm) | y_{t} (cm) | V_{1} (m/s) | V_{t} (m/s) | F_{1} | y_{1}/y_{t} | Re | s (cm) | s/y_{1} | Jump Type |
---|---|---|---|---|---|---|---|---|---|---|---|

T1 | B32 | 3.5 | 16.63 | 1.93 | 0.41 | 3.3 | 4.75 | 6.10 × 10^{4} | 3.2 | 0.9 | B-wave |

T2 | B37 | 3.7 | 17.65 | 1.81 | 0.38 | 3 | 4.76 | 5.80 × 10^{4} | 3.2 | 0.9 | A-wave |

T3 | B38 | 3.48 | 1818 | 1.87 | 0.36 | 3.2 | 5.22 | 5.90 × 10^{4} | 3.2 | 0.9 | A-wave |

T4 | B39 | 3.14 | 17.97 | 2.09 | 0.36 | 3.8 | 5.72 | 5.70 × 10^{4} | 3.2 | 0.9 | A-jump |

T5 | B30 | 3.19 | 18.2 | 2.04 | 0.36 | 3.6 | 5.71 | 5.90 × 10^{4} | 3.2 | 0.9 | A-jump |

T6 | B28 | 3.78 | 16.1 | 1.79 | 0.42 | 2.8 | 4.26 | 6.10 × 10^{4} | 3.2 | 0.9 | B-jump (Max.plung.condit.) |

T7 | B30 | 3.39 | 16.78 | 2.02 | 0.41 | 3.5 | 4.95 | 6.00 × 10^{4} | 3.2 | 0.9 | B-jump (Max.plung.condit.) |

TEST | Turbulence Model | η/Σ | N_{P} |
---|---|---|---|

T1a | mixing-length model | 1.5 | 3000 |

T1b | k-ε turbulence model | 1.5 | 3000 |

© 2017 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**

De Padova, D.; Mossa, M.; Sibilla, S.
SPH Modelling of Hydraulic Jump Oscillations at an Abrupt Drop. *Water* **2017**, *9*, 790.
https://doi.org/10.3390/w9100790

**AMA Style**

De Padova D, Mossa M, Sibilla S.
SPH Modelling of Hydraulic Jump Oscillations at an Abrupt Drop. *Water*. 2017; 9(10):790.
https://doi.org/10.3390/w9100790

**Chicago/Turabian Style**

De Padova, Diana, Michele Mossa, and Stefano Sibilla.
2017. "SPH Modelling of Hydraulic Jump Oscillations at an Abrupt Drop" *Water* 9, no. 10: 790.
https://doi.org/10.3390/w9100790