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PPT – Solution of the St Venant Equations / Shallow-Water equations of open channel flow PowerPoint presentation | free to download - id: 492b30-NTIzY

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Solution of the St Venant Equations /

Shallow-Water equations of open channel flow

- Dr Andrew Sleigh
- School of Civil Engineering
- University of Leeds, UK
- www.efm.leeds.ac.uk/CIVE/UChile

Background information

- Why should we model rivers?
- It is difficult and expensive to get data
- The flow changes from day to day
- Most of the time they are no problem

They cause disruption

They are dangerous

They Cause Financial and Personal loss

They cause structural damage

Human interference does not help

They are not new

Preventative Measures

- build higher flood banks
- divert the water with a relief channel
- store the water
- a combination of these

Design Considerations

- Appearance
- Effects on both upstream and downstream
- The cost
- The flood return period
- Data availability

Consider that

- Floods cannot be prevented
- It is neither economic nor practical to design

for exceptional floods

Flood routing is the process of calculating

backwater curves in unsteady flow.

The Elements of Flood Hydraulics

Why do we need to route floods?

To know

- Extent of flooding
- Effects hydraulic structures
- e.g. bridge piers, culverts, weirs
- Size of flood relief channels
- If flood relief measures will work
- Give flood warnings

For each return period

- Take the flood hydrograph
- Route this flow through the system
- Ask if your proposal will work
- Repeat for every proposal and return period

Objectives of this course

- Understand necessary computational components
- See different form of equations of unsteady flow
- Use appropriate solution techniques
- By the end will
- have programmed a model capable of simulating

passage of a flood wave on a simple river network - have programmed a model to simulate extreme open

channel flows and tested this with a dam break - But Today just steady flow like HEC-RAS steady

Functions / Programs

- We will develop programs
- Matlab functions equations
- (could be any program / language)
- Graphical representation
- 1-D and 2-D
- Input data
- Solution data
- Steady / Time dependent
- Put function together for complete model

2-d Layout of Network

Section / Solution

Profile / Solution

3-d, gis?

Flood routing achieved using the St. Venant

Equations

St Venant Assumptions of 1-D Flow

- Flow is one-dimensional i.e. the velocity is

uniform over the cross section and the water

level across the section is horizontal. - The streamline curvature is small and vertical

accelerations are negligible, hence pressure is

hydrostatic. - The effects of boundary friction and turbulence

can be accounted for through simple resistance

laws analogous to those for steady flow. - The average channel bed slope is small so that

the cosine of the angle it makes with the

horizontal is approximately 1.

Dam Break real and dangerous

Dam break difficult to solve

- Idealised case
- Sharp gradients

Dam Break Animation

- By the end of the course will be able to do

something like this.

Basics Consider Steady Flow

- Todays class will cover
- Components of a computational model
- How to represent a network
- Fundamental (steady) equations
- Section properties
- Friction formulas
- Conveyance
- Steady solutions
- uniform flow,
- backwater curve.

How to represent channel network

- Sections
- Reach group of sections
- Boundary conditions
- Internal join reaches
- External define inflow and outflow
- Together define river system

Diagrammatic picture

Sections

- Look downstream. Left bank, Right bank

Sections

- Variable roughness, shape, across section

Sections File Format

Local coordinates x along channel, y across, z

vertical

- SECTION AV2296_11909
- 8
- 0 22.61 0.5
- 5 19.89 0.04
- 15 14.44 0.04
- 45 14.44 0.04
- 47.5 17 0.5
- 60 17 0.5
- 65 18.87 0.5
- 75 22.61 0.5

Section Properties

- Depth (d or y) the vertical distance from the

lowest point of the channel section to the free

surface. - Stage (z) the vertical distance from the free

surface to a datum - Area (A) the cross-sectional area of flow,

normal to the direction of flow - Wetted perimeter (P) the length of the wetted

surface measured normal to the direction of flow. - Surface width (B) width of the channel section

at the free surface - Hydraulic radius (R) area to wetted perimeter

ratio (A/P) - Hydraulic mean depth (Dm) area to surface width

ratio (A/B) - Hydraulic diameter (DH) equivalent pipe

diameter - (4R 4A/P D for a circular pipe flowing full)
- Centre of gravity coordinates (centroid)

Function for Section Properties

- Any section defined by coordinates (in file)
- Common sections

Steady Flow Equations

- Conservation of energy

Energy / Bernoulli Equation

hydrostatic pressure distribution

Bed slope small tan ? sin ? ? in radians

Momentum Equation

- When flow is not hydrostatic, steep,

discontinuous etc. - Hydraulic Jump

b momentum correction factor

Velocity Distribution

Velocity Distribution on Bend

Hitoshi Sugiyama. See animation.

http//www.cc.utsunomiya-u.ac.jp/sugiyama/avs4/av

s4eng.html

Calculation of a and b

Function Calculate the coefficients a and ß for

a given section and vel dist.

Reynolds Numebr

- Using R as length scale
- Using DH as length scale
- For a wide river R depth, DH 4depth.

Function Calculate Re (ReR or ReDH) for a given

fluid, section, depth and velocity.

Froude Number, Fr

- Critical Depth Fr 1
- Fr lt 1 sub-critical
- upstream levels affected by downstream controls
- Fr gt 1 super-critical
- upstream levels not affected by downstream

controls

Function Calculate Fr, for a given section and

discharge. Also dcritical.

Uniform Flow

- Equilibrium Friction balances Gravity

Function Calculate bed shear stress, to for

given section, depth and bed slope.

Chezy C

- assuming rough turbulent flow
- shear force is proportional to velocity squared
- thus

FunctionsCalculate V or Q for a given section

and dn, C and bed slope. Also normal depth, dn

from Q, C, So, C from Q and So, dn, So from C, Q,

dn.

Friction Formulae

- Darcy-Weisbach for pipe
- Full pipe
- So L / hf
- and

Alternative form for f

- Some texts give the value f is 4 times larger

than quoted here - To clarify some text use l such that
- BE CAREFUL WITH FRICTION FORMULAE

Functions Calculate f or ? for a given section,

depth, slope and discharge. Calculate f from C

and vice versa

Colebrook-White equation for f

- Originally developed for pipes
- ks is effective sand grain size in mm
- Implicit
- Requires iterative solution
- Use Altsul equation to start iteration

ks values

- Some typical values of ks are

Function Calculate f or ? from ReR depth,

section and ks.

Mannings n

- Most commonly used expression for friction
- n relates to C
- In terms of discharge

Function Calculate Q from n, C from n, for given

section.

Mannings n values

- Some typical values for n
- Friction estimate great source of error

Computations in uniform flow

- Typical and common calculations
- Discharge from a depth normal flow
- Depth for a discharge normal depth
- Require iterative solution even for rectangular

channel

Function Calculate dn or flow for given section

and n, C or f , So, Q or dn.

Conveyance, K

- K measure of carrying capacity of a channel in

uniform flow - Chezy
- Manning

Function Calculate conveyance for a given

section and n, C or f.

Conveyance in Irregular Channels

- Split section into regions of uniform velocity
- Separate flood plain and main channel.
- Regions could be defined by roughness

Function Calculate conveyance for irregular

section must define a subdivision

method Calculate a for irregular channel with sub

division by specified roughness

Exercises Calculations

- Uniform flow exercise questions
- ExerciseQuestions02.pdf on web page
- Questions 1-7

Backwater Calculation

- Gradually varied flow surface profile
- Calculated from Energy / Bernoulli equation
- Basis of HEC-RAS Steady
- Backwater calculations are developed assuming
- Non-uniform flow
- Steady flow
- Flow is gradually varied
- That at any point flow resistance is the same as

for uniform flow i.e can use manning of Chezy

etc.

Backwater Calculation 2

- Start at known depth and Q, integrate up or down

stream - Control section Critical depth, change in slope,

structure, hydraulic jump - Super-critical at control section
- forward integration (downstream)
- Sub-critical at control section
- backwards integration (upstream).

Backwater finite difference

- e.g. energy equation with Manning

Backwater Calculation Procedure

- At point of known depth and Q, si. Calculate Ai,

Pi, Vi, Sf_i Hi, - Estimate di1, calculate properties at i1,

H(1)i1 - Calculate H()i1 using FD form of energy

equation - If H(1)i1 not close to H()i1 (e.g. 1mm) repeat

from step 2. - Else carry on integration further along channel

Functions Integrate backwater for a prismatic

channel.. Also a similar function for a channel

defined by a series of cross sections.

Backwater Exercise

- Backwater integration exercise questions
- ExerciseQuestions02.pdf
- Question 8
- Should be straight forward using developed

functions.