# CFD Modelling of Particle-Driven Gravity Currents in Reservoirs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Governing Equations and Model Setup

#### 2.1.1. Hydrodynamic Model

#### 2.1.2. Sediment Transport Model

^{−1}equals the particle diameter in m to the power of 1.3 [48]. In addition Stokes fall velocity [49] has been added to the code in a three-dimensional form. The equation is piecewise continuous and has discontinuities at ${d}_{s}$ = 100 µm and ${d}_{s}$ = 1000 µm.

#### 2.1.3. Buoyant Forces

#### 2.1.4. Discretization

#### 2.2. Code Validation and Mesh Sensitivity Analysis

^{−3}[19]. Sediment density, volumetric sediment concentration and channel width as characteristic length scale correspond to a Grashof number of about 1.9 × 10

^{8}.

- Case A: one grain fraction with ${d}_{s}$ = 25 µm and ${v}_{s}$ = 0.8 mm s
^{−1}. - Case D: two grain fractions to equal amounts with ${d}_{s1}$ = 25 µm and ${d}_{s2}$ = 69 µm and ${v}_{s1}$ = 0.8 mm s
^{−1}and ${v}_{s2}$ = 5.8 mm s^{−1}, respectively. - Case G: one grain fraction with ${d}_{s}$ = 69 µm and ${v}_{s}$ = 5.8 mm s
^{−1}. - Case R: five grain fractions to equal amounts with ${d}_{s1}$ = 17 µm, ${d}_{s2}$ = 37 µm, ${d}_{s3}$ = 63 µm, ${d}_{s4}$ = 88 µm and ${d}_{s5}$ = 105 µm and ${v}_{s1}$ = 0.3 mm s
^{−1}, ${v}_{s2}$ = 1.7 mm s^{−1}, ${v}_{s3}$ = 4.8 mm s^{−1}, ${v}_{s3}$ = 9.4 mm s^{−1}and ${v}_{s5}$ = 11.3 mm s^{−1}, respectively.

^{−5}m s

^{−1}(0.001 L s

^{−1}) was applied at the inlet for stability reasons.

#### 2.3. Test Case for the Study of Turbidity Current Venting Efficiency

^{3}s

^{−1}of a sediment suspension with a volumetric concentration of 2.3% (≙27 g L

^{−1}) was constantly fed into a laboratory channel. The channel representing a reservoir is initially filled with clear water so that the water depth amounts to 0.8 m. The dimensions of the channel are 6.7 m in length and 0.27 m in width. A bottom outlet 0.12 m high and 0.09 m wide is placed at the end of the channel through which water is vented at a specific flow rate [24,25,26].

^{−3}[24,25,26]. Concentration, density and channel width correspond to a Grashof number of 4.1 × 10

^{8}. This is of the same order of magnitude as the Grashof number in the lock exchange experiments [19] (see above). For the numerical model eight fractions of sediment size listed in Table 2 were used.

^{3}s

^{−1}. The outlet situation including the bottom outlet for venting have been modelled in the following way: A quadratic channel with 4.4 cm edge length was attached to the end wall of the flume. This represents the venting pipe with a diameter of 5.0 cm which has the same cross sectional area. The edges between the end wall of the flume and the venting channel were rounded so that a 9 cm wide section is left open in the end wall.

## 3. Results

#### 3.1. Flow Front Advance in Lock Exchange Experiments

#### 3.2. Computational Effort of the Numerical Models of the Lock Exchange Experiments

#### 3.3. Venting

^{−1}at time $t$ = 150 s. It can be seen that the velocity of the turbidity current decays along its length. At the beginning of the channel the velocity is almost 60 times the fall velocity. In the current head the velocity is constant with a magnitude of approximately 30 times the fall velocity. In the front of the current head the velocity drops sharply. At all points along the channel the normalized velocity of the current was slightly higher in the numerical model than in the physical model [24].

#### 3.4. Sediment Deposition Along the Channel

## 4. Discussion

#### 4.1. Code Verification and Validation

#### 4.2. Mesh Sensitivity Analysis and Quantitative Error Analysis

`3vl20`(42,160 cells) and

`3vl30`(63,240 cells). Between those two meshes a refinement is only made in the vertical direction. This suggests that sufficient fine vertical spatial discretization is particularly crucial for modelling turbidity currents. Further discussion of the mesh convergence on the basis of the grid convergence index [58,59] has been dismissed. The complexity of the mesh and different refinement ratios in horizontal and vertical direction make this approach impractical.

`3vl20`. This can be explained with the overestimation of the flow front advance in this model (see above and Figure 4c, Figure 10c). Despite this, the $p2$-norm is approaching a constant value on the fine meshes which shows that mesh convergence has been achieved (see above).

#### 4.3. Venting

^{−1}to 3.7 mm s

^{−1}(see also Table 2). The normalized velocity in the numerical model would agree with the measured velocity in the physical model, when a fall velocity of 2.3 mm was used for normalization.

#### 4.4. Applicability of the Code to Real Reservoirs

`5vl12`, 7280 cells) and between two and three days (depending on the convergence) using the finest mesh (

`2vl40`, 172,200 cells, see Table 3). This is particularly relevant considering the problem of long computation times with 3D models of turbidity currents in reservoirs mentioned by Lee et al. [11]. Although the computation time is still too long for real-time studies, the model may allow a detailed analysis of the turbidity current in a reservoir in reasonable time. In order to finally confirm the applicability of the RSim-3D code to model turbidity in real reservoirs a test using a real reservoir geometry remains to be done as future work.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | computational fluid dynamics |

GVE | global venting efficiency |

VENT | venting |

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**Figure 1.**Plan view on the initial situation of the lock exchange experiments carried out by Gladstone et al. [19] on a hexahedral mesh with 3 cm cell diameter; red: $c$ = 3490 ppm, blue: $c$ = 0 ppm.

**Figure 2.**Scheme visualising the definition of the flow front position in the numerical model results.

**Figure 3.**Formation of the turbidity current after release of the sediment suspension into clear water in the lock-exchange experiment (case A, longitudinal section [19]); flow direction from left to right, concentration contours in ppm.

**Figure 4.**Flow front advance; the time steps correspond to a Courant number of approximately 0.5: (

**a**) Case A $d$ = 25 µm. (

**b**) Case D ${d}_{1}$ = 25 µm, ${d}_{2}$ = 69 µm. (

**c**) Case G $d$ = 69 µm. (

**d**) Case R ${d}_{1}$ = 17 µm, ${d}_{2}$ = 37 µm, ${d}_{3}$ = 63 µm, ${d}_{4}$ = 88 µm, ${d}_{5}$ = 105 µm. In cases including more than one grain size the total sediment concentration is uniformly distributed across all fractions.

**Figure 5.**Sediment concentration contours in ppm at the start of venting ($t$ = 150 s, maximum concentration: 23,000 ppm).

**Figure 6.**Results of the venting experiments [24].

**Figure 7.**Longitudinal section through the sediment concentration field in the channel with suspended sediment concentration contours in ppm at different venting degrees ${\varphi}_{\mathrm{VENT}}$. The plots display a situation 350 s after the start of venting when steady state conditions have been reached. The velocity field is displayed by velocity vectors (unscaled).

**Figure 9.**Sediment deposition along the channel at a venting degree of ${\varphi}_{\mathrm{VENT}}$ = 80%.

**Figure 10.**Mesh convergence of the numerical models of lock exchange experiments. The cases differ in their composition of particle sizes in the suspension: (

**a**) Case A $d$ = 25 µm. (

**b**) Case D ${d}_{1}$ = 25 µm, ${d}_{2}$ = 69 µm. (

**c**) Case G $d$ = 69 µm. (

**d**) Case R ${d}_{1}$ = 17 µm, ${d}_{2}$ = 37 µm, ${d}_{3}$ = 63 µm, ${d}_{4}$ = 88 µm, ${d}_{5}$ = 105 µm. In cases including more than one grain size the total sediment concentration is uniformly distributed across all fractions.

**Table 1.**Meshes, time step and approximate maximum Courant number for the mesh sensitivity analysis.

Δt | Meshes | 5vl10 | 3vl20 | 3vl30 | 2vl40 |
---|---|---|---|---|---|

horizontal spacing in cm | 5 | 3 | 3 | 2 | |

s | number of vertical layers | 10 | 20 | 30 | 40 |

total number of cells | 7280 | 42,160 | 63,240 | 172,200 | |

1.00 | max. Courant number | 2.2 | 4.7 | 7.2 | – |

0.50 | 1.2 | 2.5 | 3.9 | – | |

0.20 | 0.5 | 1.1 | 1.7 | 2.2 | |

0.10 | 0.3 | 0.5 | 0.8 | 1.1 | |

0.05 | 0.1 | 0.3 | 0.4 | 0.6 |

**Table 2.**Characteristic grain size distribution used for the numerical model of the venting experiments [24].

Grain Size in µm | 67 | 80 | 90 | 110 | 140 | 150 | 160 | 214 |
---|---|---|---|---|---|---|---|---|

Stokes fall velocity Equation (2) in mm s^{−1} | 0.4 | 0.6 | 0.7 | 0.9 | 1.5 | 1.7 | 2.0 | 3.5 |

volumetric fraction at inflow in ppm | 2328 | 2328 | 2328 | 4655 | 4655 | 2328 | 2328 | 2328 |

share of total sediment in % | 10 | 10 | 10 | 20 | 20 | 10 | 10 | 10 |

**Table 3.**Computation time for Case D of the lock exchange experiment on different meshes and with different time steps.

Δt | Meshes | 5vl10 | 3vl20 | 3vl30 | 2vl40 |
---|---|---|---|---|---|

horizontal spacing in cm | 5 | 3 | 3 | 2 | |

s | number of vertical layers | 10 | 20 | 30 | 40 |

total number of cells | 7280 | 42,160 | 63,240 | 172,200 | |

1.00 | Computation time in h | 0.5 | 12.8 | 17.0 | – |

0.50 | 0.6 | 10.8 | 17.7 | – | |

0.20 | 0.7 | 11.5 | 17.0 | 57.0 | |

0.10 | 0.7 | 12.1 | 17.5 | 60.7 | |

0.05 | 0.8 | 13.4 | 18.0 | 67.0 |

**Table 4.**Euclidean norms of the difference between the flow front in the physical [19] and numerical model with different spatial and temporal discretization. The cases differ in their composition of particle sizes in the suspension: Case A $d$ = 25 µm; Case D ${d}_{1}$ = 25 µm, ${d}_{2}$ = 69 µm; Case G $d$ = 69 µm and Case R ${d}_{1}$ = 17 µm, ${d}_{2}$ = 37 µm, ${d}_{3}$ = 63 µm, ${d}_{4}$ = 88 µm, ${d}_{5}$ = 105 µm. In cases including more than one grain size the total sediment concentration is uniformly distributed across all fractions.

Δts | 5vl10 | 3vl20 | 3vl30 | 2vl40 | 5vl10 | 3vl20 | 3vl30 | 2vl40 |
---|---|---|---|---|---|---|---|---|

Case | A | Case | D | |||||

1.00 | 0.515 | 0.315 | 0.224 | – | 0.405 | 0.245 | 0.128 | – |

0.50 | 0.396 | 0.216 | 0.110 | – | 0.309 | 0.145 | 0.049 | – |

0.20 | 0.318 | 0.127 | 0.035 | 0.017 | 0.237 | 0.072 | 0.017 | 0.021 |

0.10 | 0.282 | 0.091 | 0.018 | 0.011 | 0.213 | 0.051 | 0.013 | 0.024 |

0.05 | 0.273 | 0.077 | 0.013 | 0.011 | 0.230 | 0.049 | 0.011 | 0.022 |

Case | G | Case | R | |||||

1.00 | 0.026 | 0.018 | 0.006 | – | 0.348 | 0.186 | 0.080 | – |

0.50 | 0.014 | 0.006 | 0.002 | – | 0.276 | 0.107 | 0.028 | – |

0.20 | 0.008 | 0.003 | 0.004 | 0.007 | 0.221 | 0.056 | 0.010 | 0.020 |

0.10 | 0.006 | 0.002 | 0.006 | 0.010 | 0.210 | 0.045 | 0.008 | 0.021 |

0.05 | 0.005 | 0.002 | 0.007 | 0.010 | 0.229 | 0.047 | 0.007 | 0.016 |

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**MDPI and ACS Style**

Wildt, D.; Hauer, C.; Habersack, H.; Tritthart, M.
CFD Modelling of Particle-Driven Gravity Currents in Reservoirs. *Water* **2020**, *12*, 1403.
https://doi.org/10.3390/w12051403

**AMA Style**

Wildt D, Hauer C, Habersack H, Tritthart M.
CFD Modelling of Particle-Driven Gravity Currents in Reservoirs. *Water*. 2020; 12(5):1403.
https://doi.org/10.3390/w12051403

**Chicago/Turabian Style**

Wildt, Daniel, Christoph Hauer, Helmut Habersack, and Michael Tritthart.
2020. "CFD Modelling of Particle-Driven Gravity Currents in Reservoirs" *Water* 12, no. 5: 1403.
https://doi.org/10.3390/w12051403