The real solution is given by figure1 :

__figure1__

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

Domain Extend :

x-coordinate (m) : min = 0
, max =10

y-coordinate (m) : min = 0
, max = 1.5

Obstacle:

define by the following equation y = 0.2
-0.05*(x-5)^2
for x in [3,7]

Grid size :

Number of nodes : 5184

Number of faces : 10223

Number of cells : 5040

detail of mesh

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 depend on the cases at T = 0 :

water
for x in [0,10] and y in [0,0.5]

air
for x in [0,10] and y in [0.5,1.5]

Boundaries conditions :

Superieurs and inferior sides are walls

right side is an outflow

left side is divided in
two parts :

for y in [0,0.5]
velocity inlet of water v1 = 0.3 m/s

for y in [0.5,1.5]
velocity inlet of air
v2 = 0.01 m/s

Initial repertition of pressure

the gauge pressure in the air is P=0

the initial pressure in the water is given by : P (y) = P + 10000*(h-y)
where h is the free surface of the water.

Time step : the hightest time step necessary to
have conergence of the method is Dt = 0.001s

It will be the time step choosen.

In this case the initial speed of the flow is taken contant
at V = 0.3 m/s

The V.O.F scheme choosen is the geo_reconstruc scheme

let's see the results

simulation at t=0,86s

This result is the last we can obtain with the condition
choosen because the method is divergent.

As in the first example the parameters and particulary
the scheme are not adapted to the problem.

For this test we have decided to input an initial condition
nearer of the final state

The V.O.F sceme is always the geo_reconstruct scheme

initial repartition of the phases

The results :

simulation at t=1s

simulation at t=1.86s

it seems that the results are better but the method
is still divergent.

The geo_reconstruct scheme is definitivly not adapted
to this sort of cases

In this case we have decided to show what gives the IMPLICIT
scheme with the initial conditions performed in the test-case 1

simulation at t=1s

If the solution is not better than in the test-case 1
the method is now convergent and we can think that we the initial condition
of the test-case 2 the result will be exellent

The initial conditions are the same that in the test-case 2but the V.O.F scheme is now the implicit scheme

The results follow :

simulation at t=1s

simulation at t=2s

simulation at t=3s

The profile of the solution is now realistic and the method
is convergent

We think that the problem observed at the end can be
avoided if the outlet height of the two phases are imposed .