# Bathymetry Development and Flow Analyses Using Two-Dimensional Numerical Modeling Approach for Lake Victoria

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- How are the systematic methods developed for lake bathymetry relevant for lake numerical and hydrodynamic modeling?
- How are lake bathymetry and flow models developed in CM?

- Identify specific challenges in building lake bathymetry and exemplify how to overcome them;
- illustrate the improvement of systematic methods for lake bathymetry using Matlab, and analyze 2D surface flow through the numerical model approach in CM;
- assess the reliability of the vertically integrated model by checking its prediction of mean water level;
- explore the relationship between lake outflow conditions and water levels to achieve a steady-state level and numerical accuracy of linear measurement of the lake’s water level;
- discuss the significance of systematic methods for lake bathymetry and the numerical hydrodynamic insights gained from model development.

## 2. Study Area and Methodological Approach

#### 2.1. Study Area: Lake Victoria

^{2}, of which 68,800 km

^{2}is the actual lake surface [20]. Its shoreline is shared by Kenya (6%), Uganda (45%), and Tanzania (49%) [21,22], and the tributary areas are composed of many sub-basins in riparian states of Kenya (21.5%), Uganda (15.9%), Tanzania (44.0%), Rwanda (11.4%), and Burundi (7.2%) [23]. Lake Victoria stretches 412 km from north to south, between latitudes 0°31′N and 3°12′S, and 355 km from west to east between longitudes 31°37′W and 34°53′E (Figure 1b). It has a volume of 2,760 km

^{3}[1] and is much shallower than other large African lakes (Malawi and Tanganyika) (Figure 1a).

#### 2.2. Methodological Approach

^{®}was chosen as a platform, since it allows dissimilar physics to be dealt with numerically within the same modeling framework. This is in constrast with the common approach, which treats problems from different disciplines with numerical models specifically developed for their respective discipline. Matlab

^{®}is a multi-paradigm numerical computing environment and proprietary programming language. Matlab libraries for Kriging [30] and triangulation are used to make a smooth lake topographical model.

- Documented equation-based modeling for all relevant physics. For example this supports validation against other simulation platforms, such that the effects of different numerical techniques can be distinguished;
- easy inclusion of equation-based dispersion, biological and chemical processes supported by the Multiphysics features;
- educational effect of a well-developed link between Multiphysics and Matlab with its eclectic numerical libraries;
- so-called COMSOL Server, where a COMSOL Multiphysics application can be run in major platforms such as Windows, Linux, etc. COMSOL Server is a multiuser service that can be run continuously on the host computer, and COMSOL can be run on multiple computers to support simultaneous users and more concurrent applications than could be supported by a single computer. The COMSOL Server service significantly facilitates cooperation between users distributed over different locations, as is the case in the present study where the participating researchers and collaborators are located in Sweden, Uganda, Tanzania and Kenya.

#### Numerical Simulation in Other Softwares

## 3. Materials and Methods

#### 3.1. Lake Bathymetry

#### 3.2. Lake Bathymetry Methods

#### 3.2.1. Digital Elevation Model (DEM) for Lake Bathymetry

^{2}GIS-raster maps from SRTM data sources to generate a high-resolution digital elevation model (DEM) of Lake Victoria using raster interferometry. Hamilton group has created a bathymetry model resolution for Lake Victoria. They used old data points (before 20th Century data manually digitized via a process of fitting admiralty maps to the lake shoreline) combined with GPS-enabled (2010) hydrographic survey systems. The raster model was created by the process of simple Kriging. This was unable to capture the deepest portions of Lake Victoria. Continued improvements will follow once more data is captured for Lake Victoria [14].

#### 3.2.2. Coordinates

#### 3.2.3. Depth Data from Several Sources

#### 3.2.4. Interpolation by Kriging and DACE

#### 3.2.5. Delaunay Triangulation and Interpolation

#### 3.2.6. Merging Iso-Curve Data with NaFIRRI Data

## 4. Governing Equations, Boundary Conditions and Mesh Geometry

#### 4.1. Saint–Venant Shallow Water Equations

**γ**(y), wind forcing in $\left({F}_{x},{F}_{y}\right)$, and the bottom friction is modelled by $\left(C\dots \right)$, with boundary conditions representing river inflow and outflow. Note that the horizontal components of the Coriolis acceleration are small.

#### 4.2. Boundary Conditions

#### 4.2.1. Model Boundary Conditions

#### 4.2.2. Shoreline Conditions

#### 4.2.3. Inflow and Outflow Conditions

#### As flux condition

#### As velocity condition

#### 4.3. COMSOL Model Setup

## 5. Results and Discussion

#### 5.1. Lake Victoria Topography in Matlab and COMSOL

#### 5.2. Lake Hydrodynamics

#### 5.3. Long-Time Simulations

#### 5.4. Outflow Boundary Allows a Steady-State Condition

#### 5.4.1. Weir Boundary Condition (WBC)

#### 5.4.2. Linear Outflow Conditions

- Net precipitation is a dominant source and is assumed to be homogeneous over the whole lake area;
- the travel time for a gravitational wave across the lake, assuming a mean depth of 60 m, is around $3.5$ h. Thus, water level differences would equalize over the lake in a much shorter time than a month.

#### 5.5. Water Level Validation

## 6. Conclusions

**Software and data sources:**The Comsol Multiphysics [48] software has many model libraries to focus on different areas, however, no special model libraries for surface waterbody analysis. The bathymetry, inflow/outflow, and precipitation/evaporation data for the model were obtained as described above: From our collaborators at Makerere University, Uganda, Water Resources Management Authority (WRMA), Kisumu, Kenya and Lake Victoria Basin Commission (LVBC), Kisumu, Kenya. Access to the data was granted by the sources referenced. Matlab was used for bathymetry data processing.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Shallowness of Lake Victoria compared with lake Malawi and Tanganyika, and (

**b**) Lake Victoria catchment area topography has taken from source [5] (Copyright @ 2004 Royal Meteorological Society).

**Figure 2.**Bathymetry and depth diagram where $z$ is the vertical coordinate, $\zeta $ is the wave height, $h$ is the water depth and $B(x,y)$ is the lake bathymetry.

**Figure 6.**Delaunay triangulation and interpolation to old iso-curve data and manual close-to-shore points.

**Figure 7.**Adjustment of shoreline points: (

**a**) Original data (red curve); and (

**b**) adjusted by bi-linear transformation (green curve).

**Figure 9.**Lake Victoria water depths based on iso-depth curves and NaFIRRI bathymetry data: (

**a**) Matlab model; (

**b**) Comsol model.

**Figure 10.**Flow patterns in 2000: (

**a**) Non-vanishing vorticity at edges of the Kagera inflow model; and (

**b**) wave height measured from the mean level.

**Figure 14.**Mean water-level simulations showing that exact conservation property of the COMSOL finite element model.

Parameters | Unit | Value | Expression |
---|---|---|---|

Differential equations | |||

x, y | m | - | East–viz. North coordinates |

U, V | m/s | - | x-viz. y-mean velocity of water |

$\Delta $ | m | - | Local element mesh size |

$\zeta $ | m | - | Free surface elevation |

${\mathit{F}}_{\mathit{x}},{\mathit{F}}_{\mathit{y}}$ | m/s^{2} | - | “Forces” (wind stress, etc.) |

g | m/s^{2} | 9.81 | Gravitational acceleration |

$\mathit{h}=\mathit{\zeta}+\mathit{B}\left(\mathit{x},\mathit{y}\right)$ | m | - | Water depth |

k | - | 0.0015 | Bottom friction coefficient |

C = k/h | m^{−}^{1} | - | Bottom force coefficient |

$\mathit{\mu}$ | - | 0.4 | Non-dimensional artificial viscosity/diffusion coefficient |

Q | ${\mathrm{m}}^{3}/\mathrm{s}$ | - | Source of water volume |

$\mathit{\gamma}\left(\mathit{Y}\right)$ | s^{−}^{1} | ${10}^{-11}\xb7Y$ | Coriolis parameter, depending on distance to the equator |

$\mathit{Y}$ | $\left(-300e3+y\right)$ | Distance to the equator | |

Boundaries | |||

$\mathit{U}{\mathit{n}}_{\mathit{x}}+\mathit{V}{\mathit{n}}_{\mathit{y}}$ | m/s | 0 | Wall normal velocity |

q | m^{3}/s | - | Normal in/outflow |

$\mathit{U}=\mathit{q}/\left(\mathit{h}\mathit{L}\right){\mathit{n}}_{\mathit{x}}$ | m/s | - | x-component wall normal velocity at flow q over length L of boundary with normal $\left({\mathrm{n}}_{\mathrm{x}},{\text{}\mathrm{n}}_{\mathrm{y}}\right)$ |

$\mathit{V}=\mathit{q}/\left(\mathit{h}\mathit{L}\right){\mathit{n}}_{\mathit{y}}$ | m/s | - | d:o, y-component |

$\mathit{B}\left(\mathit{x},\mathit{y}\right)$ | m | - | Bathymetry |

Initial data | |||

$\zeta \left(x,y,0\right)$ | m | ${\zeta}_{0}+d(x,y)$ | Initial constant water depth with a perturbation d |

$\mathit{U}=\mathit{V}$ | m/s | 0 | Initial value for U and V |

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**MDPI and ACS Style**

Paul, S.; Oppelstrup, J.; Thunvik, R.; Magero, J.M.; Ddumba Walakira, D.; Cvetkovic, V.
Bathymetry Development and Flow Analyses Using Two-Dimensional Numerical Modeling Approach for Lake Victoria. *Fluids* **2019**, *4*, 182.
https://doi.org/10.3390/fluids4040182

**AMA Style**

Paul S, Oppelstrup J, Thunvik R, Magero JM, Ddumba Walakira D, Cvetkovic V.
Bathymetry Development and Flow Analyses Using Two-Dimensional Numerical Modeling Approach for Lake Victoria. *Fluids*. 2019; 4(4):182.
https://doi.org/10.3390/fluids4040182

**Chicago/Turabian Style**

Paul, Seema, Jesper Oppelstrup, Roger Thunvik, John Mango Magero, David Ddumba Walakira, and Vladimir Cvetkovic.
2019. "Bathymetry Development and Flow Analyses Using Two-Dimensional Numerical Modeling Approach for Lake Victoria" *Fluids* 4, no. 4: 182.
https://doi.org/10.3390/fluids4040182