Assignment

Gradually Varied Flow Profiles and Numerical Solution of the Kinematic Equations:

Objectives

1. Evaluate and apply the equations available for the description of open channel flow
2. Solve the equations governing unsteady open channel flow
3. Apply the equations of unsteady flow to practical flow problems

Special Instructions

a. Computer programs or spreadsheets must be the work of the individual student.

b. Assignments submitted without adequate proof of program validation will not be eligible for greater than a C grading.

c. A proportion of the marks is allocated to the communication aspects of the assignment. Marks will be deducted for untidy and poorly presented work, poor English expression, and failure to cite sources of information.

Question 1 - Gradually Varied Flow Profiles

Your task:

a) Use the direct step method, and the equation below to compute the water surface profile upstream of the outfall.

Δy/Δx = S₀ - S‾_f/ 1 - F_R²

where F_R = √α V/√(gy‾)

b) Plot the water depth against distance

c) Plot the longitudinal bed, normal depth, critical depth, water surface and energy line over the length of this profile. (All on the same set of axis e.g. Fig 5.22 Chadwick)

d) Include sample hand calculation in the report

Hints:

- The size of the step is up to you.

- Use of computers for this task (Matlab, Excel etc is encouraged)

- When computing the water surface profile you should stop just short of normal depth

- The Froude number and critical depth for a trapezoidal channel are different to that of a rectangular channel (e.g. need average depth (y¯) instead of max depth (y) in F_R)

Question 2 - Kinematic Wave Model

Your Task

1) Complete the tutorial problems 5.1, 5.2, 5.3 and 5.4 in Module 5. You will find full solutions of first three questions in the study book that will help you to solve 5.4.

2) OPTIONAL: If you are not confident about your answer to 5.4 you may submit your working (formulas/equations) to the examiner using the link provided on studydesk before proceeding with the numerical scheme. Your examiner will be able to guide you through.

3) Build your model (using any programming language or spreadsheet) for solving the kinematic wave equations for computing depth and flow rate resulting from the storm events. You must configure your model according to the specifications above.

4) Validate your mathematical model by modelling the runoff under steady rainfall (constant rainfall depth) and compare results with the theoretical results for steady rainfall. The analytical procedure for theoretical results has been discussed at the end of this problem (The section Model Validation)
You are required to check all three conditions

5) Modify the model to accommodate the design storm hyetograph. Then use this program to calculate water depth and flow rate at uniform distance interval dx along the unit width channel for the given storm event.

6) Write up all equations, model development, validation, results and discussion in a report format

Presentation of Results
The final report should include as a minimum:
- Introduction and background, a description of the problem
- Formulation of the finite difference solution,
- Basic description of model;
- Validation of the model for a constant slope by:
   - plot of steady depth profile for a constant lateral inflow of sufficiently long duration, and
   - comparison of your program output with the results from the analytic solutions of the kinematic equations given below
- Evidence that the Courant condition for stability has been satisfied for both the steady inflow simulation and the simulation of the example storm.
- Plots (for the given variable rainfall) of:

- the runoff hydrograph (q vs t) at the lower end of the channel (taken to at least 30 min after the cessation of rainfall), and
- the water surface profile (y vs x) when the discharge from the end of the channel is a maximum. Also note the time at which this maximum discharge occurs.
- Appropriate discussion. Some points that you should cover include:
i. What are your assumptions and how they might impact on the ability of your results to replicate flow in the real world?
ii. Your results for the runoff hydrograph and depth profiles.
iii. General conclusions of the practical use of your model.
- Acknowledgement of any sources of information in a reference list

Attachment:- Hydrulics.pdf