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
The Elements of Flood
Hydraulics
Flood routing is the process of
calculating backwater curves in
unsteady flow.
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


Graphical representation





(could be any program / language)
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
d
 d u
g
u

 g S o  S f 
x
 x t
A
u
d
u
A
b
0
x
x
t
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

Today’s 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
8
0
5
15
45
47.5
60
65
75
AV2296_11909
22.61
19.89
14.44
14.44
17
17
18.87
22.61
0.5
0.04
0.04
0.04
0.5
0.5
0.5
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
 (4×R = 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:
y
1
y
x

1
8
  sin  D 2
b  2 y 1  x2
1
2
D
b+2xy
sin  2D
Area, A
by
(b+xy)y
Wetted perimeter P
b  2y
Top width B
B
Hydraulic radius R
by b  2 y 
b  xy y
b  2y 1 x
depth Dm
D
b
b
mean Y
B
B
B
Hydraulic
Circle
Trapezoid
Rectangle
b  xy  y
b  2 xy
2
1  sin  
D
1 
 
4 
1    sin  
D

8  sin 1 / 2  
d
Steady Flow Equations

Conservation of energy
Energy / Bernoulli Equation
p1
V12
p2
V22
 1
 z1 
 2
 z2  H  constant
g
2g
g
2g
p
hydrostatic pressure distribution
 y  d cos
g
Bed slope small: tan θ ≈ sin θ ≈ θ in radians
H  
V2 
1 V2
   f
  z  d cos  
 S f
s s 
2g 
R 2g
Momentum Equation

When flow is not hydrostatic, steep,
discontinuous etc.

Hydraulic Jump
F  Q 2V2  1V1 
 =momentum correction factor
Velocity Distribution
Velocity Distribution on Bend
Hitoshi Sugiyama.
See animation. http://www.cc.utsunomiya-u.ac.jp/~sugiyama/avs4/avs4eng.html
Calculation of  and 
3
u
dA V13 A1  V33 A2  V33 A3

 3 
V A
V 3  A1  A2  A3 
V  Q/ A
2
u
 dA
V12 A1  V32 A2  V32 A3


2
V A
V 2  A1  A2  A3 
Q V1 A1  V2 A2  V3 A3
V  
A
A1  A2  A3
Function: Calculate the coefficients α and β for a given section and vel dist.
Reynolds Numebr

Using R as length scale
uR uD Re pipe
Re R channel 



4
4

Using DH as length scale
Re DH channel 

uDH uD

 Re pipe


For a wide river R = depth, DH = 4×depth.
Function: Calculate Re (ReR or ReDH) for a given fluid, section, depth and velocity.
Froude Number, Fr
V
Fr 
gd


 A
g 
B
Critical Depth Fr = 1
Fr < 1 sub-critical


Fr 
V
upstream levels affected by downstream controls
Fr > 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
o
So  sin  
gR
H z

 So
s s
So  S f
L
V
z

weight

s
bed
x
Function: Calculate bed shear stress, o for given section, depth and bed slope.
Chezy C

assuming rough turbulent flow

shear force is proportional to velocity squared
o  V 2
 o  kV 2

thus
V
g
k
RSo
V  C RSo
Functions:Calculate 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
S o = L / hf


and
4 fLV 2
hf 
2 gD
R  A/ P  D / 4
1 V2
So  f
R 2g
C
2g
f
2 gRS o
f 
V2
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:
hf 

lLV 2
2 gD
l
8 gRS o
V2
BE CAREFUL WITH FRICTION FORMULAE
Functions: Calculate f or λ 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
1
1.26

 4 log10


f
 14.8R Re R f






ks is effective sand grain size in mm
Implicit


Requires iterative solution
Use Altsul equation to start iteration
1/ 4

k
25 

f  0.4 0.365 s 
R Re R 

ks values

Some typical values of ks are
Material
ks (mm)
Plastic
0.01 – 0.02
Painted pipe
0.02
Riveted steel
1-10
Cast iron (new)
0.25
Cast iron (rusted)
1-1.5
Concrete
0.3-3
Concrete (rough)
3-10
Planed wood
0.6-2
Rubble
5-10
Straight earth channel
3
Function: Calculate f or λ from ReR depth, section and ks.
Manning’s n

Most commonly used expression for friction
R 2 / 3 S o1/ 2
V
n


n relates to C:
R1 / 6
C
n
In terms of discharge
1 A5 / 3 1 / 2
Q
S
2/3 o
nP
Function: Calculate Q from n, C from n, for given section.
Manning’s n values

Some typical values for n
Channel type
Surface material and form
Manning’s n range
River
earth, straight
0.02-0.025
earth, meandering
0.03-0.05
gravel (75-150mm), straight
0.03-0.04
gravel (75-150mm), winding
0.04-0.08
Grass / light growth
0.05
Trees
0.15
earth, straight
0.018-0.025
rock, straight
0.025-0.045
lined canal
concrete
0.012-0.017
lab. models
mortar
0.011-0.013
Perspex
0.009
Flood plain
unlined canal

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:
Q  AC RSo  K So
K  ACR1/ 2

Manning
A5 / 3
1
K  2 / 3  AR2 / 3
nP
n
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 α 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
H  
V2 
n 2V 2
   4 / 3
  z  d cos  
s s 
2g 
R
Vi2/2g
total head
2/2g
Vi+1
yi
di
free surface
y
i+1

zi
d i+1
s
zi+1
x
bed

H i1  H i
 0.5 S fi 1  S fi
si1  si

Backwater Calculation Procedure
1.
2.
3.
4.
5.
At point of known depth and Q, si. Calculate Ai,
Pi, Vi, Sf_i Hi,
Estimate di+1, calculate properties at i+1, H(1)i+1
Calculate H(*)i+1 using FD form of energy
equation
If H(1)i+1 not close to H(*)i+1 (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.