# Accurate Numerical Modeling for 1D Open-Channel Flow with Varying Topography

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

**U**represents the conserved variables in the form of a vector; $A\left(x,\eta \right)$ is a function of the space and water surface elevation, which are predetermined and arbitrary, representing the cross-sectional area of the channel; Q is the flow discharge;

**F**denotes the flux in the x direction; g is the gravitational acceleration; ${I}_{1}$ and ${I}_{2}$ are integral terms that are related to the hydrostatic pressure forces caused by the varying channel breadth; $\eta $ is the water surface elevation; ${\eta}_{b}$ denotes the bottom elevation; $B\left(x,\tau \right)$ is the channel width;

**S**represents the source terms; ${S}_{0}$ represents the inclination or gradient of the channel bed; and S

_{f}denotes the inclination or gradient of the energy line within the channel.

## 3. Riemann Solver

## 4. Numerical Methodology

**I**. Obtain the Riemann solutions ${\mathbf{W}}_{i-\frac{1}{2}}(x/t)$ and ${\mathbf{W}}_{i+\frac{1}{2}}(x/t)$ of the Riemann problems RP (${\mathbf{W}}_{i-1}^{n},{\mathbf{W}}_{i}^{n}$) and RP (${\mathbf{W}}_{i}^{n},{\mathbf{W}}_{i+1}^{n}$) at a general point (x, t).

**II**. Perform a random sampling of solutions in the cell i to obtain a state at time level ${t}_{n+1}$. The selected state is determined based on a random or quasi-random number ${R}^{n}$, which falls within the range of [0, 1].

**I**. Solve the local Riemann problems $\left({\mathbf{W}}_{i-1}^{n},{\mathbf{W}}_{i}^{n}\right)$ and $\left({\mathbf{W}}_{i}^{n},{\mathbf{W}}_{i+1}^{n}\right)$, which are defined by the constant states ${\mathbf{W}}_{i-1}^{n}$, ${\mathbf{W}}_{i}^{n}$ and ${\mathbf{W}}_{i+1}^{n}$. These Riemann problems are solved to determine the intermediate states for each cell interface.

## 5. Discretization of Source Terms and Stability Constraints

- Pure advection:$${\mathbf{W}}_{i}^{n+1}={\mathbf{W}}_{i}^{n}-\frac{\mathsf{\Delta}t}{\mathsf{\Delta}x}\left({\mathbf{G}}_{i+1/2}^{n+1/2}-{\mathbf{G}}_{i-1/2}^{n+1/2}\right)$$
- Update with the source term by $\mathsf{\Delta}t/2$:$${\overline{\mathbf{W}}}_{i}^{n+1}={\mathbf{W}}_{i}^{n+1}+\frac{\mathsf{\Delta}t}{2}\left(\right)close\; separators="|">\mathbf{J}\left({\mathbf{W}}_{i}^{n+1}\right)$$
- Re-update with the source term by $\mathsf{\Delta}t$:$${\stackrel{\u033f}{\mathbf{W}}}_{i}^{n+1}={\mathbf{W}}_{i}^{n+1}+\mathsf{\Delta}t\left(\right)close\; separators="|">\mathbf{J}\left({\overline{\mathbf{W}}}_{i}^{n+1}\right)$$$${\left(\frac{\mathit{\partial}B\left(x\right)}{\mathit{\partial}x}\right)}_{i}=\frac{{B}_{i+1/2}-{B}_{i-1/2}}{{x}_{i+1/2}-{x}_{i-1/2}}=\frac{\mathsf{\Delta}{B}_{i}}{\mathsf{\Delta}{x}_{i}}\phantom{\rule{0ex}{0ex}}{\left(\frac{\mathit{\partial}{\eta}_{b}\left(x\right)}{\mathit{\partial}x}\right)}_{i}=\frac{{{\eta}_{b}}_{i+1/2}-{{\eta}_{b}}_{i-1/2}}{{x}_{i+1/2}-{x}_{i-1/2}}=\frac{\mathsf{\Delta}{{\eta}_{b}}_{i}}{\mathsf{\Delta}{x}_{i}}$$$${\left({S}_{f}\right)}_{i}=\frac{{{n}_{m}}^{2}{u}_{i}\left|{u}_{i}\right|}{{R}_{i}^{4\u22153}}$$

## 6. Numerical Validations and Discussion

^{2}, and the Courant number value was 0.8. The model was tested against a range of scenarios to evaluate its ability to accurately simulate open-channel flows and to ensure its suitability for future applications.

#### 6.1. Dam Break Tests

#### 6.2. Lobovsky et al.’s Dam Break ExperimentA

^{1/2}.

#### 6.3. Laboratory Experiment of a Dam Break Flow over a Triangular Obstacle

#### 6.4. Dam Break Flow onto a Channel with a Local Constriction

- Gauge 1: 1.00 m upstream from the dam;
- Gauge 2: 6.10 m downstream from the dam;
- Gauge 3: 8.60 m downstream from the dam;
- Gauge 4: 10.50 m downstream from the dam.

#### 6.5. Channel Contraction Test Problem

^{3}/s and a downstream water depth of 1.5 m. The discretization of the channel is accomplished using a uniform grid spacing of $\mathsf{\Delta}x$ = 2.5 m. The simulation was initialized using these specific boundary and initial conditions, and the simulation ran until a steady state was reached. The exact bed elevation, water surface level, and velocity profiles obtained from the numerical simulation are plotted in Figure 12, and they show a good agreement with the expected analytical profiles. The evaluation of numerical and analytical profiles is an essential step in validating the precision of the numerical approach used in the simulation. The predicted free surface level matches the analytical profiles, indicating that the current model has achieved a satisfactory level of accuracy for simulating steady flows.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Water depth profile simulations compared with the exact solution at time = 36 s ($\mathsf{\Delta}x=10\mathrm{m}$).

**Figure 6.**The simulated water levels at the designated points in relation to time are compared with the experimental data ($\mathsf{\Delta}x=0.061\mathrm{m}$).

**Figure 8.**Dam break flow over an elevated section: chronological records of water depth at various measuring points: G4, G10, G13, and G20 ($\mathsf{\Delta}x=0.76=\mathrm{m}{n}_{m}=0.0125$).

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**MDPI and ACS Style**

Xue, Z.; Zhou, L.; Liu, D.
Accurate Numerical Modeling for 1D Open-Channel Flow with Varying Topography. *Water* **2023**, *15*, 2893.
https://doi.org/10.3390/w15162893

**AMA Style**

Xue Z, Zhou L, Liu D.
Accurate Numerical Modeling for 1D Open-Channel Flow with Varying Topography. *Water*. 2023; 15(16):2893.
https://doi.org/10.3390/w15162893

**Chicago/Turabian Style**

Xue, Zijian, Ling Zhou, and Deyou Liu.
2023. "Accurate Numerical Modeling for 1D Open-Channel Flow with Varying Topography" *Water* 15, no. 16: 2893.
https://doi.org/10.3390/w15162893