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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 – PowerPoint PPT presentation

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


1
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

2
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

3
They cause disruption
4
They are dangerous
5
They Cause Financial and Personal loss
6
They cause structural damage
7
Human interference does not help
8
They are not new
9
Preventative Measures
  • build higher flood banks
  • divert the water with a relief channel
  • store the water
  • a combination of these

10
Design Considerations
  • Appearance
  • Effects on both upstream and downstream
  • The cost
  • The flood return period
  • Data availability

11
Consider that
  • Floods cannot be prevented
  • It is neither economic nor practical to design
    for exceptional floods

12
Flood routing is the process of calculating
backwater curves in unsteady flow.
The Elements of Flood Hydraulics
13
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

14
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

15
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

16
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

17
2-d Layout of Network
18
Section / Solution
19
Profile / Solution
20
3-d, gis?
21
Flood routing achieved using the St. Venant
Equations
22
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.

23
Dam Break real and dangerous
24
Dam break difficult to solve
  • Idealised case
  • Sharp gradients

25
Dam Break Animation
  • By the end of the course will be able to do
    something like this.

26
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.

27
How to represent channel network
  • Sections
  • Reach group of sections
  • Boundary conditions
  • Internal join reaches
  • External define inflow and outflow
  • Together define river system

28
Diagrammatic picture
29
Sections
  • Look downstream. Left bank, Right bank

30
Sections
  • Variable roughness, shape, across section

31
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

32
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)

33
Function for Section Properties
  • Any section defined by coordinates (in file)
  • Common sections

34
Steady Flow Equations
  • Conservation of energy

35
Energy / Bernoulli Equation
hydrostatic pressure distribution
Bed slope small tan ? sin ? ? in radians
36
Momentum Equation
  • When flow is not hydrostatic, steep,
    discontinuous etc.
  • Hydraulic Jump

b momentum correction factor
37
Velocity Distribution
38
Velocity Distribution on Bend
Hitoshi Sugiyama. See animation.
http//www.cc.utsunomiya-u.ac.jp/sugiyama/avs4/av
s4eng.html
39
Calculation of a and b
Function Calculate the coefficients a and ß for
a given section and vel dist.
40
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.
41
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.
42
Uniform Flow
  • Equilibrium Friction balances Gravity

Function Calculate bed shear stress, to for
given section, depth and bed slope.
43
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.
44
Friction Formulae
  • Darcy-Weisbach for pipe
  • Full pipe
  • So L / hf
  • and

45
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
46
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

47
ks values
  • Some typical values of ks are

Function Calculate f or ? from ReR depth,
section and ks.
48
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.
49
Mannings n values
  • Some typical values for n
  • Friction estimate great source of error

50
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.
51
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.
52
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
53
Exercises Calculations
  • Uniform flow exercise questions
  • ExerciseQuestions02.pdf on web page
  • Questions 1-7

54
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.

55
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).

56
Backwater finite difference
  • e.g. energy equation with Manning

57
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.
58
Backwater Exercise
  • Backwater integration exercise questions
  • ExerciseQuestions02.pdf
  • Question 8
  • Should be straight forward using developed
    functions.
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