Solution of the St Venant Equations / Shallow-Water equations of open channel flow

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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

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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

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They cause disruption
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They are dangerous
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They Cause Financial and Personal loss
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They cause structural damage
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Human interference does not help
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They are not new
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Preventative Measures

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

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Design Considerations

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

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Consider that

• Floods cannot be prevented
• It is neither economic nor practical to design
  for exceptional floods

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Flood routing is the process of calculating
backwater curves in unsteady flow.
The Elements of Flood Hydraulics
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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

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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

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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

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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

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2-d Layout of Network
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Section / Solution
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Profile / Solution
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3-d, gis?
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Flood routing achieved using the St. Venant
Equations
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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.

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Dam Break real and dangerous
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Dam break difficult to solve

• Idealised case
• Sharp gradients

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Dam Break Animation

• By the end of the course will be able to do
  something like this.

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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.

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How to represent channel network

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

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Diagrammatic picture
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Sections

• Look downstream. Left bank, Right bank

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Sections

• Variable roughness, shape, across section

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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

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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)

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Function for Section Properties

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

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Steady Flow Equations

• Conservation of energy

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Energy / Bernoulli Equation
hydrostatic pressure distribution
Bed slope small tan ? sin ? ? in radians
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Momentum Equation

• When flow is not hydrostatic, steep,
  discontinuous etc.
• Hydraulic Jump

b momentum correction factor
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Velocity Distribution
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Velocity Distribution on Bend
Hitoshi Sugiyama. See animation.
http//www.cc.utsunomiya-u.ac.jp/sugiyama/avs4/av
s4eng.html
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Calculation of a and b
Function Calculate the coefficients a and ß for
a given section and vel dist.
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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.
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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.
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Uniform Flow

• Equilibrium Friction balances Gravity

Function Calculate bed shear stress, to for
given section, depth and bed slope.
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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.
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Friction Formulae

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

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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
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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

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ks values

• Some typical values of ks are

Function Calculate f or ? from ReR depth,
section and ks.
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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.
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Mannings n values

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

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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.
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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.
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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
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Exercises Calculations

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

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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.

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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).

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Backwater finite difference

• e.g. energy equation with Manning

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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.
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Backwater Exercise

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