IV.1 Exact solution

    The real solution is given by figure1 :



IV.2 Methodology

    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

                                    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.


IV.3 Test-case 1

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.


IV.4 Test-case 2

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


IV.5 Test-case 3

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


IV.6 Test-case 4

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 .