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Coursework 2: Godunov-Type Scheme for Free Surface Flow

1 Important Notes

This final coursework is worth 40% of the total module mark

The deadline for submission is on Friday 12 April 2024 at 14:00

2 Tasks

Develop a Godunov-type model for solving the following 1D shallow water equations:

Test different aspects of your numerical model by applying it to simulate all test cases

listed at the end of this document.

You should produce figures, tables, etc. to visualize and interpret your results.

Attach your code setup for test case 2: Tidal Wave over a Varying Bed

3 Submission/Report

You should write a short academic essay no more than 7 pages to clearly introduce your

model and present the results. You may use the following template:

A Godunov-Type Model for Free Surface Flow

Shannon Leaky

School of Civil Engineering & Geosciences

Newcastle University

Introduction

Herein, you should give a background for computational hydraulics, introduce different

numerical methods (finite difference, finite element, finite volume, etc.) and explain why you choose

a finite volume Godunov-type scheme (e.g. Toro 2001) to construct your model. A brief literature

review may be necessary.

Godunov-Type Shallow Flow Model

In this section, you should introduce the governing equations, i.e. the 1D shallow water

equations, and your numerical scheme.

Results and Discussion

You should present your results for all test cases using figures, tables, etc. Detailed discussion

should be provided to interpret the results. The analytical solutions which are provided should be

used to validate your model.

Conclusions

Draw brief conclusions here.

References

Toro EF (2001) Shock-capturing methods for free-surface shallow flows, John Wiley & Sons, Chichester.

Appendix

Attach your code set up for test case 2. The appendix will not be counted into the page limit

4 Test Cases

Test 1: Still water test

The bed elevation of the frictionless 1D channel is described by

where L = 14,000 m is the length of the channel.

Uniform computational grid: 50 cells;

Initial conditions:

Boundary conditions: transmissive / reflective;

Output results (water surface and velocity profiles) at t = 5000 s.

Test 2: Tidal Wave over a Varying Bed

In the same channel as Test 1, the analytical solutions of a tidal flow are given by

q = 0 throughout the channel

Boundary conditions: transmissive/open at both ends

Output results (water surface and velocity profiles) at t = 5 s.

Test 4: Tidal Wave over Steps

A tidal wave flow occurs in a 1500m long frictionless channel with two vertical steps with the bed

profile defined by

An asymptotic analytical solution of the flow is provided by

Uniform computational grid: 200 cells;

Initial conditions:

Boundary conditions: upstream

th ),0(

; and downstream reflective;

Output results at t = 10,800s and t = 32,400s.

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