Modeling of dam failure and Numerical Solutions of twodimensional Saint-Venant Equations.
Abdallah TAMRABET & Abdelouahab KADEM
Mathematics Department,University of Batna,Algeria.
E-mail : aktamrabet@yahoo.fr
1. Introduction:
Floods are natural phenomena exceptional, characterized by a sudden rise of
water levels and overflow of rivers. The origin of this rise is due either to a rain
event across a pond or a sudden rupture of a barrage.This study focuses on
this type of failure that can lead to waves whose behavior in a first is identical
to that of dynamic waves by the predominant effect of inertia, then as a wave
of continuity as the flooding subsides by moving further downstream.
The determination of the propagation of rupture should be able to predict their
evolution in real time from data stored upstream in order to take appropriate
protective measures against this phenomenon.
The most accurate approach to this phenomenon and requires the least
information relating to flooding, based on numerical modeling of the SaintVenant equations. These equations describe the unsteady flows in open
channels established, consisting of a description of the depth averaged flow.
Due to their mathematical complexity, the analytical integration of these
equations in the case of an unsteady flow is virtually impossible except in
some idealized situations. The presence of hydraulic jumps and obstacles in
transient flow makes the problem even more difficult.
The numerical solution of system equations (PDEs) of Saint-Venant for
natural channels can be addressed using either the method of characteristics
or other methods such as finite differences, finite elements or the finite volume
method is generally preferred for its ability to reproduce numerical solutions.
In this study we opted for the explicit finite difference scheme with no fixed
time that is a direct method for solving the Saint-Venant equations. It is clear
that the description of a load of shallow flow does not really fit in the case
where the curvature of current lines is significant, which can happen in waves
induced by an instantaneous dam break. For example, the velocity field in the
area near the dam is complex with significant vertical components. But such
features are fortunately limited in time and space and their influence on the
behavior of distant fields of the flow is relatively low. Properties such as
conservation, hyperbolicity, the assumption of jumps are accepted as obvious
properties.
2. Computational models of wave propagation at break :
Setting equation of the phenomenon of rupture and flood wave propagation:
Flooding of dam failure are free surface flows, non-permanent, non-uniform
horizontal principal components. The Saint-Venant equations describe their
transition and transformation through the various sections of the canal. The
formulation of Saint-Venant is written as a system of two equations, one
representing the mass conservation of fluid, the other conservation of its
momentum. These conservations imply a vertical distribution of hydrostatic
pressure along the zero vertical velocity and vertical acceleration low. Their
validity is limited to relatively slow variations in space and time. The calculated
local velocities are horizontal and represent an average estimate on the
depth. Neglecting the velocity transverse to longitudinal velocities and the
differences in water levels cross, these equations integrated over the
transverse imension can be reduced to a one-dimensional form. The
calculation of the shape-dimensional means in practice that further simplifies
data calculation. The spread of the flow is along the canal. The laws apply
pressure drops to a kinetic energy term based on the average speed, so
unique in each section.
3. Equation of conservation of mass free surface:
For an incompressible fluid and a supposed one-dimensional flow (x,t), the
equation of mass conservation reads:
Q S

 qx  (3-1)
x t
The function represents the lateral flow per unit length of the curvilinear
abscissa of the channel.
Considering the particular derivative of the volume occupied within a control
surface, the equations are written as follows:
 
x2
Dv
  V  n dS   q x dx
(3.2)
S
x1
Dt
  y
On the free surface V  n 
since yx, t  the draft means. The flow is still
t
one-dimensional element of the surface ds of the free surface is written
dS  Bdx as Bx, t  the mirror is the width of the cross section line to the
abscissa x and wet at the moment t. The flow balance is written:
x2
x2
y
Q x 2 , t   Q x1 , t    B dx   qx dx
(3.3)
x1
x1
t
Where:
Qx 2 , t   Q x1 , t   
x2
x1
Q
dx
x
(3.4)
The medium is assumed continuous, the theorem of the integral zero leads to
Q
y
B
 qx 
(3.5)
the equation:
x
t
Note that Bdy represents the increase ds in the wetted area on the curvilinear
abscissa x. The previous equation becomes:
Q S

 qx 
x t
(3.6)
4. Different forms of equation of mass conservation:
4.1 Taking account of the speed:
The equation of mass conservation is written, given the relationship (3.1):
U
S
U S
S

 q x 
x
x t
(3.7)
4.2 Taking account of y and B:
Is the differential dS  Bdy and dividing both members of the above equation
by B, necessarily different from zero, since there flow into the channel, we
obtain the following expression:
y S U y q
U



(3.8)
x B x t B
4.3 Taking into account the speed of the wave:
The report
c2
S
S
is by definition the average draft ym. By substituting.
by
B
B
g
We obtain the equation of conservation of mass for a prismatic channel:
y c 2 U y q
U



x g x t B
(3.9)
The general equation of mass conservation written below:
S
U S
U
S

 q x 
(3.7)
x
x t
5.1 Introduction of speed instead of speed for the writing of Saint Venant
equation:
Since the flow is unidimensional and considering the equation Q  SU . The
general equation of conservation of mass, taking into account any input side
S Q

q
(3.11)
is written:
x x
The resulting term
1 U
in the stationary flow is written in view of the
g t
equation of mass conservation:
1 U
1 Q Qq
Q Q

 2 2
g t
gS t gS
gS x
The term
(3.12)
H
H
y
Q Q Q 2 S
to be   1 written:
 J f 


x
x gS 2 x gS 3 x
x
(3.13)
Saint Venant's equation takes the following form, after regrouping of terms:
y  Q 2 L 
1 Q Qq
Q Q Q 2  S 
1  h S 
qV
1 


J

J



2
 3    G  
f
3 
2
2
x 
gS t gS
S  x  y ,t gS
gS 
gS x gS  x  y ,t
Since the flow is steady
1 Q
 0 , the draft and flow do function as the
gS t
abscissa of the flow.
The channel is prismatic 
Q2
gS 3
1  h S 
 S 
  G   0 , the terms
  0
S  x  y ,t
 x  y ,t
and
and conditions.
The flow is gradually growing in the channel
The previous equation becomes:
dQ
 q , then we can write.
dx
dy  Q 2 L 
Qq qV
1 
Jf  J  2 
3 
dx 
gS
gS 
gS
(3.15)
6. Numerical method of solving the system of Saint-Venant:
6.1 Discretization of the domain:
In the finite difference method, the channel is divided into a number of
sections having a length equal to x . The end of each section is called
compute node. If the channel is divided into N sections and the first node
(upstream) is numéroté1, so the last node (downstream) will be N  1 .
Nodes upstream and downstream boundary are called boundary nodes and
the remaining nodes are called interior nodes. The calculations are performed
in discrete time. The difference between two consecutive time instants is
called time steps. Thus, x  t the plan divided into a network of lines that
intersect each other in the computational nodes, is called roasting.
Solving the problem of dam failure is possible by using the equations of onedimensional Saint-Venant following:
(3.14)
 y
y
U
 t  U x  y x  0

 U  U U  g y  g J f  J
 t
x
x
6.1
6.2
Where y is the height of water U , represents the average velocity of flow.
Both quantities depend on each time t and space x.
By replacing the partial derivatives of the governing equations by finite
differences we can solve the resulting algebraic equations at each point in the
network or at each node, assuming the initial values at the time of the speed
and depth of flow at all points of this network.
In determining the
t 0  t
following values corresponding to this interval of time.
Several explicit schemes have been used for flows in stationary free surface
with diffusive Lax scheme for which we opted for the fact that it is conditionally
stable numerically based on the value of the dimensionless parameter
u = Cr
≤ 1.0 . In this scheme, the governing partial differential equations
t
x
are replaced by quotients of finite differences as
follows:
y yij 1  yi*
6.3

t
t
U U i j 1  U i*

t
t
6.4
U U i j 1  U i j1

x
2x
6.5
Where:

y i*  0.5 y ij1  y ij1


6.6

6.7 
U i*  0.5 U i j1  U i j 1
6.2 Discretization in space:
Discretizing the equations by a centered scheme:
yi
y  yi 1
U  U i 1
 U i 1
 yi i 1
t
2x
2x
U i
U  U i 1
y  yi 1
 U i i 1
 g i 1
 g J f  J i 
t
2x
2x
6.8
6.9
For a variable a i  axi  , where.a xi  i  x
This equation is applicable only for nodes that are not at the edges, but as in
any boundary value problem we know the edges so the Riemann problem can
be solved.
In addition: J i 
Ui Ui
Rh i
4
6.10
n2
3
Where: n - Manning coefficient
6.3 Discretization in time:
Using an explicit scheme that is that we know U , y and J e now, we can
calculate these values at once j  1 .

y j  yij1
U j  U i j1 
yij 1  yi*  t  U i j i 1
 yij i 1

2x
2x 

6.11


U j  U i j1
y j  yij1
U i j 1  U i*  t  U i j i 1
 g i 1
 g J f  J i j 1 
2x
2x



Ji
j 1

U i j 1 U i j 1
(Rh  )
4
3 j 1
i
By asking:  1 

6.12
6.13
n2
n2
Rh  
4
3
j 1
gt and multiplying the equation 4.104 by  1 , we
i




obtain the following equation: U i j 1   U i j 1    0
2
6.14


U j  U i j 1 gt y ij1  y ij1
t
U i j i 1

 gtJ f 
where:   U i* 
2x
2x
2x
2x


This is a quadratic equation U i j 1 and the result:
U i j 1 
1
    2  4
2 


1
2



6.15
6.4 Conditions to the upstream limit:
At the upstream boundary is assumed that,
4.103
then the equation
y y ij1  y ij

x
x
whose depth is modified and contains only terms with indices i and i +1
becomes:
y ij 1  y ij 
 
t
U i j y ij  y ij1  y ij U i j  U i j 1
x


The speed can then be obtained with

6.16
, Where
U i j 1 
Qi
j 1
S i j 1
: - wetted area
S i j 1
corresponding to the depth
hi j 1
6.5 Conditions at the downstream limit:
For the downstream end of the channel, the situation is the opposite of that
upstream.
Assuming that
: the equation
4.103 whose depth is modified
y y i  y i 1

x
x
and contains only terms with indices and when written to her as follows:
y ij 1  y ij 
 
t
U i j y ij1  y ij  y ij U i j 1  U i j
x



(6,17)
6. 6 Stability of the scheme:
It is usually necessary in the explicit finite difference schemes of the report x
and t satisfies a stability condition.Un A scheme is stable if an error made in
the solution does not increase when the calculations are progressing in time.
In the case of unstable scheme the error is amplified quickly and hides the
true solution in some time intervals. For the Lax scheme is stable, with no
time and space must meet the following condition, called stability condition of
current: t 
x
U c
6.18
7. Numerical results:
The main characteristics of the dam and canal were studied:
Maximum height of dam: 55.0 m;
The crest length 609.7 m.
The crest width 10 m
The maximum width of the natural ground level 367.62 m.
The channel length equal to. 14737.29m
The channel slope equal to. 0.001
The Manning coefficient equal to. 0.05
The maximum flow rate at time of rupture depends on the geometry of the
dam was obtained by Ritter's formula is Qmax = 0.81 609.7.553/2m3/s
Figure 1.Variation height at the foot of the dam.
Figure 2.Variation the speed of the wave breaking at the foot of
Figure 3.Variation speed at the foot of the dam.
Figure 4. Variation of the height at x = 7368.645 m from the dam
Figure 5. Change the speed at x = 7368.645 m dam.
Figure 6. Change the speed of the wave breaking at x = 7368.645 m from the dam.
Figure 7. Variation of the height at t = 100 s.
Fig8.Variation speed at t = 100 s.
Fig9.Variation height at t = 5000 s.
Fig10.Variation speed at t = 5000 s.
Fig11. Change the speed of the wave breaking at t = 5000 s dam
Fig.12 Influence of the coefficient of Manning on the depth of water x = 7368,645 m
Fig.13 Influence of the Manning coefficient on velocity x = 7368,645 m
Fig14.Influence Manning coefficient on the velocity x = 7368,645 m
Fig15. Influence of filling height of the water level x = 7368.645 m.
Fig16. Influence of filling height on the speed x = 7368.645 m
Fig17.Influence of filling height on the speed x = 7368,645 m
Fig18. Influence of slope on the water level x = 7368,645 m
Fig19. Influence of slope on speed x = 7368,645 m
Fig20.Influence slope on expeditiously x = 7368,645 m
Fig21. Reducing the height of water.
Fig22. Attenuation of flow velocity
Fig23. Reducing the speed of the wave breaking.
Interpretation of results :
In the present study on immediate rupture, and as shown in the above figures
including those heights, speeds or flow rates ( fig5 fig13) and the wave
velocity and considering the variation slope, roughness of the walls and fill
heights inducing dynamic loads variables at time t = 1390.1 s and at the foot
of the dam studied, the downstream area is suddenly submerged. The water
depth and flow reach respective values of 39.2136 and 190808.2 m m3 / s,
speed is of the order of 7980 m / s, the speed of the wave to when it reaches
a value of 19.6134 m / s, reflecting the extremely violent and aggressive (F =
≈ 2.0) flow whose impact on the shape of the channel will be irreversibly very
devastating. A
X = 7368.645 m and at time equal to 1590.1 s, we observe a sharp increase
in flows and speeds at the expense of height has decreased approximately
13.63031 m. This decrease is explained by the effect of wall friction has
caused a loss of relatively high pressure. Finally at t = 3.47 hours and at a
distance x = m 14737.29.0 all hydraulic parameters of flow cancel out the fact
of loss of hydraulic power flow.
Water levels calculated at all points downstream of the dam are particularly
dependent resistance factor used in this case the Manning coefficient n to
characterize the active canal.Plus bed resistance to flow and losses linear
load and the most important local velocity of the flood decreases thus causing
a higher level of flooding and further erosion.
It should be noted that the simulation results obtained are dependent
boundary condition used to limit the domain which
is obtained by
Qmax
breaking the formula of Ritter.
Breaking waves subside gradually moving from the downstream toe of the
dam, that is to say at a distance x = 2653m, reaching at time t = 2000s the
maximum value of 35m to go completely extinct in end of term at x = 12379m.
Other features of the hydrodynamic flow also undergo the same process,
namely a decrease in the output of the dam and a total extinction at the end.
Conclusion:
Breaking studies are fundamental to the analysis of dam safety. They provide
a fairly accurate portrait of flows that must be propagated to downstream
areas receiving flood waves break and how long should this wave to reach the
areas where flooding could cause very serious consequences.
Often breaking studies provide the opportunity to collect data that can cover
the reservoir and its downstream.
These data and topography are often used by numerical models. Being a
reliable source that transmits the various computer codes for reliable data and
consistent with the degree of precision required by the calculations, we
ourselves used these data to perform calculations.
The study results are very useful in breaking the security structures. They
should therefore be optimally exploited for maximum secure such structures
against flooding can cause their destruction.
The security of coastal populations depends on the quality of information they
receive. It is appropriate to interpret the numerical results or graphs such as
those presented in this study and pass avoiding any ambiguity, but taking into
account the limitations inherent in scientific studies.
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