# A New Mass-Conservative, Two-Dimensional, Semi-Implicit Numerical Scheme for the Solution of the Navier-Stokes Equations in Gravel Bed Rivers with Erodible Fine Sediments

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Model

#### 2.1.1. Governing Equations

#### 2.1.2. Staggered Mesh

#### 2.1.3. Numerical Method

#### 2.1.4. Crank-Nicholson Time Discretization

#### 2.2. Validation Tests

#### Numerical Experiments

## 3. Results and Discussion

#### 3.1. Validation Tests

#### 3.2. Numerical Experiments

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the simulated gravel bed topographies: (

**a**) in-line arrangement, view from above; (

**b**) closest-packing arrangement, view from above; (

**c**) real gravel bed, 3D view; (

**d**) in-line arrangement, longitudinal section; (

**e**) closest-packing arrangement, longitudinal section; (

**f**) real gravel bed, longitudinal section. In panels (

**d**–

**f**), the horizontal dashed lines indicate the fine sediment filling rate, while the vertical lines indicate the gravel crest (dotted) and inter-grain cavity (continuous) sections.

**Figure 3.**(

**a**) Field map of the horizontal velocity component u, showing the formation of the Blasius boundary layer; (

**b**) comparison between the numerical solution and the analytical Blasius solution at three different locations of the domain; and (

**c**) magnitude of the horizontal velocity component u in the $(x,\xi )$ plane.

**Figure 4.**(

**a**) Streamlines in the cavity at the final time $t=100$ s; and (

**b**) comparison with the reference solution [44] for $Re=400$.

**Figure 5.**(

**a**) Streamlines in the cavity at the final time $t=100$ s; and (

**b**) comparison with the reference solution [44] for $Re=1\phantom{\rule{0.166667em}{0ex}}000$.

**Figure 6.**(

**a**–

**d**) Evolution of the streamlines in the cavity with the erodible bed at times $t=20,\phantom{\rule{0.166667em}{0ex}}50,\phantom{\rule{0.166667em}{0ex}}100,\phantom{\rule{0.166667em}{0ex}}300$ s and coloured velocity magnitude $|\overrightarrow{v}|$, where $\overrightarrow{v}$ denotes the velocity vector; and (

**e**–

**h**) volume concentration of suspended sediment.

**Figure 7.**Time evolution of (

**a**) sediment and (

**b**) water mass, rescaled with respect to the initial total mass.

**Figure 8.**Horizontal velocity component u and streamlines (

**left panels**), and magnitude of the vorticity $\left|\overrightarrow{\omega}\right|$ (

**right panels**) for the spheres in-line arrangement.

**Figure 9.**Horizontal velocity component u and streamlines (

**left panels**), and magnitude of the vorticity $\left|\overrightarrow{\omega}\right|$ (

**right panels**) for the spheres closest-packing arrangement.

**Figure 10.**Horizontal velocity component u and streamlines (

**left panels**), and magnitude of the vorticity $\left|\overrightarrow{\omega}\right|$ (

**right panels**) for the real gravel topography configuration.

**Figure 11.**Time evolution of the horizontal (u) and vertical (w) velocity component intensities, averaged over the region below the gravel crest level, and relative to the real gravel bed configuration with fine sediment filling rate $Z=0.5$.

**Figure 12.**Vertical profiles of the horizontal velocity component u at chosen sections representative of the gravel crest and of the inter-grain cavity (see Figure 2), for the in-line arrangement (

**left panels**), closest-packing arrangement (

**central panels**), and real gravel bed configuration (

**right panels**). Thick lines indicate profiles averaged from 10 to 15 s of simulation, while thin lines indicate instantaneous profiles every $0.1$ s.

**Figure 13.**Vertical profiles of the vertical velocity component w at chosen sections representative of the gravel crest and of the inter-grain cavity (see Figure 2), for the in-line arrangement (

**left panels**), closest-packing arrangement (

**central panels**), and real gravel bed configuration (

**right panels**). Thick lines indicate profiles averaged from 10 to 15 s of simulation, while thin lines indicate instantaneous profiles every $0.1$ s.

© 2020 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/).

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**MDPI and ACS Style**

Tavelli, M.; Piccolroaz, S.; Stradiotti, G.; Pisaturo, G.R.; Righetti, M. A New Mass-Conservative, Two-Dimensional, Semi-Implicit Numerical Scheme for the Solution of the Navier-Stokes Equations in Gravel Bed Rivers with Erodible Fine Sediments. *Water* **2020**, *12*, 690.
https://doi.org/10.3390/w12030690

**AMA Style**

Tavelli M, Piccolroaz S, Stradiotti G, Pisaturo GR, Righetti M. A New Mass-Conservative, Two-Dimensional, Semi-Implicit Numerical Scheme for the Solution of the Navier-Stokes Equations in Gravel Bed Rivers with Erodible Fine Sediments. *Water*. 2020; 12(3):690.
https://doi.org/10.3390/w12030690

**Chicago/Turabian Style**

Tavelli, Maurizio, Sebastiano Piccolroaz, Giulia Stradiotti, Giuseppe Roberto Pisaturo, and Maurizio Righetti. 2020. "A New Mass-Conservative, Two-Dimensional, Semi-Implicit Numerical Scheme for the Solution of the Navier-Stokes Equations in Gravel Bed Rivers with Erodible Fine Sediments" *Water* 12, no. 3: 690.
https://doi.org/10.3390/w12030690