Free Surface Flow Hands-On
by
Eric Valette ( mfn12 ) & Olivier Antibi ( mfn01 )
PART 1 : NUMERICAL COMPUTATION OF FREE SURFACE FLOW IN A CHANNEL WITH HUMP AT THE BOTTOM
I.
Presentation of the Work
II.
Boundary and initial conditions
A) Boundary
conditions
B) Constant
free surface initialisation
C)
Analytical solution
III. Influence
of bottom friction
IV. Transcritical
flow
V.
Problem of boundary conditions
PART II : Openning
a
mesh generator software MATISSE
INTRODUCTION
The purpose of this work is to compute the
surface flow of an horizontal canal using the TELEMAC2D Software. This
module solves the Shallow water equations for non-linear longwaves.
In the first part, we studied a simple case to analyse
software's reaction to different hydrodynamics regimes. A special attention
will be given to initial and boundary conditions.
A second part will be devoted to an overview of another Telemac module,
Matisse to study wave's propagation.
We considered a channel with a rectangular cross-section which is 20 m long and 10 m wide. At mid-lenght a hump is created at the bottom. THis hump has a height of 0.2 m and has a parabolic shape.
The lateral walls of the channel are supposed to shear-stress free whereas at the bottom we will consider both the case with or without shear-stress. The are two open boundaries at the inflow and the outflow where different boundary conditions have to be imposed depending on the flow regime.
II BOUNDARY AND INITIAL CONDITIONS
Shallow water problem admit two waves travelling in
the absolute reference frame with a celerity of U+C and U-C. Suppose now
that U>0 two different situations may be encountered:
* U<C called subcritical case (F<1)
* U>C called supercritical case (F>1)
Indeed, the froude number is a function of space so that a given flow may
be subcritical at some place and supercritical elsewhere. Such a flow is
sometimes called transcritical.
inflow boundaries:
* subcritical case:
Wave
1 : U+C > 0 inflow wave
Wave
2 : U-C< 0 outflow wave
We
must give ONE boundery condition : For example the flow Q
* supercritical case:
Wave
1 : U+C > 0 inflow wave
Wave
2 : U-C> 0 inflow wave
We
must give TWO boundery conditions : For example the flow Q, the height
H
outflow boundaries:
* subcritical case:
Wave
1 : U+C > 0 outflow wave
Wave
2 : U-C< 0 inflow wave
We
must give ONE boundery condition : For example the flow H
* supercritical case:
Wave
1 : U+C > 0 outflow wave
Wave
2 : U-C> 0 outflow wave
We
must give ZERO boundery condition.
II B CONSTANT FREE SURFACE INITIALISATION
The first possibility is to set the initial condition using the Key word : cote initiale = 'constante' in the parameter file ( in this case cas200 )
The constant initial solution is the easiest to start but she create oscilations and the transition time is very long.
An other solution to set the initial condition is to make a fortran program in the subroutine condini. this subroutine is adding to the main program of telemac2d telsa.f . In ourcase the key word is : cote initiale = 'particuliere' in the parameter file.
More other, this study case have an analytical solution that can be encoded easily
Ht= U**2/(2*g) + H0 +Zf)
Where
Ht is the total water-power
U is the horizontal speed
H is the hight of water
Zf is the bottom level
In the case of the rectangular conduit, we now that Q= U*H
So we have
Ht= Q**2/(2*g*H**2) + H + Zf
We prefer starting from the analytical solution because of the convergence time is shorter. moreother, we can compare the numerical solution given by telemac2d with the analytical solution.
III INFUENCE OF THE BOTTOM FRICTION
In Telemac, we have different possibilities for the friction law and the friction parameter. We studied 4 cases with the Chezy which use the Chezy number:
type of canal bottom |
chezy number |
concret |
80 |
maconned side |
51.5 |
ordinary ground |
30 |
grass, stone |
24 |
Moreother, we now that the more the Chezy number is small, the more the bottom friction is important.
Case I, Chezy number equal to 80
Case II, Chezy number equal to 51.5
Case III, Chezy number equal to 30
Case IV, Chezy number equal to 24
Telemac show that the energie-water decreas faster with an important bottom friction.
we can note too that the free surface isn't horizontal but have a incline more important with an important bottom friction.
As we want to produce a transcritical flow, the fluid
velocity at the inflow boundary should be increased. Keeping the bottom
friction at the same value ( Chezy number equal to 80), we have increased
gradually the inflow speed.
Note that you can start you computation from an old result file usin the
key words
SUITE DE CALCUL :oui
FICHIER DU CALCUL PRECEDENT :'******'
We have creat here a case with a transcritical flow. The boundary conditions are correct. One condition for the inlet boundary (subcritical flow) and one condition for the outlet boundary ( supercritical flow).
An other solution to create a supercritical region in the flow is to decrease the hight of the outlet condition :
V PROBLEM OF BOUNDARY CONDITION
The change of the boundary condition can create a supercritical flow at the outlet boundary :
We can see that Telemac must conect his solution to the boundary conditions. with a supercritical flow at the outlet boundary we mustn't put a boundary condition. if we let it, there a problem.
The same problem appear when we bring down the outlet boundary under the critical hight.
The bundary conditions are set of in the boundary file clim200 in our case.
the firs column set the high boundary,( 2 for a wall,
4 for a free hight, 5 imposed hight, 1 incident wave )
the second column set the horizontal velocity ( 0 null velocity , 1 incident
wave, 2 friction , 4 free, 5 fixed flow , 6 fixed celocity)
the third column set the vertical velocity
the last column give the boundary number of the node.
For example : 4 4 4 .... ............... .... .... ....12 give no condition on the boundary node 12
In addion to the first part in witch we learn to use the Telemac2D software, we learn how to use another module of telemac Artemis. This software caculates the wave propagation and turned to be very useful for our Industrial Hands-On. The different files required ( mesh file , boundary conditions file ) were generated thanks to Matisse also part of Telemac software modules. In this part, we will try to give a good overview of these two softwares with the help of precise examples and snapshots.
Matisse is a mesh generator software which use is easy to adapt with Telemac modules. First it needs a file which gives the depth of the ground both, under and below the sea. The cas we studied is directly link to our Industrial Hands-On and dealt with the St-Jean de Luz beach. This file might have been numerized automaticaly ( Sinusx format ".sx" ), but was manually obtained because of a failure. The format used was XYZ (".xyz") which is also recognised by Matisse :
Once digitalised, we obtain :
Then it was needed to define geometrical lines and borders. For that, we decided to use isobathymetric lines, defined hard points, and projected the line on the ground. This wasn't to forget to define the exterior lines, coast, liquid borders and islands as borders. :
To include reef bathymetric datas we chosed to change our technique modifying manually the XYZ file as if it was obtained by SINUSX. We could then, directly use lines without drawing isobathymetric ones :
To compute the mesh, we used a cell lenght linked to hydrodynamics theory. In fact, to well capture waves, you need almost ten cells for one wave length. The wave length is obtained with :
We used a cell length equals of the square root of the gravity multiply by the depth :
We can then compute the mesh viewed with RUBENS:
First mesh without the reef :
Detail of reef mesh :
This mesh will be used with Artemis software module in order to compute wave propagation.