The real solution is given by the following
figure :

__figure 1__

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.

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.

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.