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

Question 1 - Gradually Varied Flow Profiles  

Water is flowing in a long prismatic channel of triangular cross section with side slopes of 19 degrees to the horizontal. The Manning n of plastic lined channel is 0.013.

The channel is conveying a steady flow rate of 6 + (2 x n₁) m³/s

The bed slope of the channel is 0.003 + (0.0001 x n₂)

Where, n₁ is the second last digit and n₂ is the last digit in your student number.

For example if your student number is 10005007648 then

Q = 6 + (2 x 4) = 14 m³/s

S₀ = 0.003 + (0.0001 x 8) = 0.0038

Upstream of this channel is a channel of the same cross section but of much lower slope. The sudden change in slope means that the depth at the upstream end of the channel of interest can be approximated as critical depth. Take alpha as being 1.1 (α = 1.1).

Your task:

a) Use the direct step method and the equation below to compute the water surface profile downstream of the change in slope. The water level will gradually end up at normal depth.

Δy/Δx = (S₀ - S^-_f)/(1 - F^-2_R)  where F_R = (√αV/√(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.

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 triangular 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

Background

You have been asked to investigate the flow behaviour of a large sporting field subjected to a short duration high intensity storm rainfall event.

As part of the process you will develop a computer simulation of the water depths and flow rates for a specified rainfall pattern. The kinematic wave approximation is a simple form of one dimensional flow model, which is deemed sufficient for this task.

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.

Attachment:- Assignment Files.rar