Computation of the flow with Fluent I . BOUNDARIES CONDITIONS

We decided to impose the following boundaries conditions :

• Velocity inlet for the inlet flow
• Axis for the two bottom boundary around the bubble
• Outflow for the top boundary
• Outflow condition for the outlet condition

With the boundaries conditions on the bubble, it was quite more difficult . Indeed, we had to impose two conditions :

• the normal velocity equal to zero
• no shear stress
The industrial code Fluent gives two possibilities for these conditions   : symmetry  or  wall  with specified shear stress. II . SIMULATIONS WITH FLUENT

We've had several problems to simulate the flow around the air bubble.  We think the problem comes from the boundary condition on the bubble and the mesh. We have tested the symmetry condition and the wall with specified shear stress condition. Both of them gave us wrong results.
But, it is not so surprising. Indeed, the first one condition of symmetry doesn't insure that the shear stress is null. Only the derivative in r direction is null. A term with tangential composant of velocity stays . For the second one (boundary condition of wall with specified shear stress), there are two problems : It is possible FLUENT considers the shear stress in the (x,y) directions and not in the (r,teta) directions (local reference or not); moreover, the wall boundary condition  impose that normal and tangentiel velocity are null  and yet we only want the cancellation of the normal velocity.
We think this is the reason why the two cases we have tested gave us wrong results.

Indeed,  for Re << 1, we must have  : and for Re >> 1 : Morever, from a qualitatif point of view, our results seems to be in "good" agreements with the theoritical prediction. Indeed, as it is shown on the following graph, we don't observe any circulation after the bubble. (but as boudaries conditions are not acceptable this result has not real signification) Graph of velocity with the bubble (Re = 1000 with v = 0.1 m/s )

Hence, we have decided to study the problem of a flow around a solid sphere. The new boundary condition on the sphere is a wall. As previously, our results seems to be accurate from a qualitative point of view due to the phenomena of circulation in downstream of the air bubble. From a quantitative point of view, the results are far from the theorical results (a removing appears for Re=1000 while we knows it is for Re=20); So, we deduce there is an other problem, it's not probably only the bad boundaries conditions which gave bad results for the bubble. Graph of velocity with the sphere (Re = 1000 with v = 0.1 m/s )