#### III. HORIZONTAL CHANNEL WITH A GROUND SILL

III.1 Exact solution

The real solution is given by the following figure : figure 1

III.2 Methodology

The meshing of the domain was realized with GAMBIT mesher as pictures below.

Domain Extend :
x-coordinate (m) :     min = 0            , max = 20
y-coordinate (m) :     min = -0.2      , max = 2

Obstacle:
define by the following equation      y = -0.05*(x-10)^2           for x in [8,12]        (E1)

Grid size :
Number of nodes : 2202
Number of faces : 4522
Number of cells : 5040  detail of the meshing

Such a problem is resolved with FLUENT 5.0 by using V.O.F methods :
The first phase is water_liquid.
The second phase is air.

Repartition of the two phases at T = 0 :
water  for   x in [0,20]   and   y in [-0.2,1.8]
air        for   x in [0,20]   and   y in [1.8,2] Boundary conditions :
Superieurs and inferior sides are walls
right side is an outflow
left side is divided in two parts :
for y in [-0.2,1.8]  velocity inlet of water   v1 = 4.429 m/s
for y in [1.8,2]        velocity inlet of air          v2 = 0.01 m/s

Initial repertition of pressure:
the gauge pressure in the air is P = 0
the pressure in the water is given by the equation     P(y) = 10000*(1.8-y) Time Step :
In all the cases,  the highest time step necessary to have convergence of the method is Dt = 0.001
It will be the time step choosen.

III.3 Test-case 1

In this case the initial speed of the flow is taken contant at V = 4.429 m/s
The V.O.F scheme choosen is the geo_reconstruc scheme
As we will realize it is two big mistakes.

Let's see the results simulation at t=1s simulation at t=2s simulation at t=3s

As we can see the solution is far from the real solution and furthermore it is not steady.

- the first mistake made is the choise of the V.O.F sheme :
with V.O.F method if the result is steady you have to choice the IMPLICIT scheme and  not the default_scheme which is
GEO_RECONSTRUCT.
with the IMPLICIT scheme you have two possibilities:
make the simulation in steady if the result does not depend on the initial flow
make the simulation in unsteady if the result depends on the initial  flow
- the second mistake is the choise of the initial water velocity :
in fact the velocity is constant but the flow is not constant because of the obstacle.

III.4 Test-case 2

In this case the V.O.F scheme is the IMPLICIT scheme.
the initial condition is not on the velocity of the wather but on the flow which is taken constant :
As to leave of the equation (E1) is not difficult to find the relation that has to respect the initial velocity to keep constant the flow :

V(x) = Q/(1.8+0.059x-10)^2    for x in [8,12]            Q = 8.859 is the flow Let's see the results simulation at t=0.3 simulation at t= 0.1

The result is better than in the first case but it is not the real solution.
I think do perform better there is two parameters upon which we will have to play:
impose the outlet height of the two phase.
impout a more realistic initial condition of the water.