### Hydrodynamic instabilities

1 Objectives -------- 2 Mesh -------- 3 Simulation -------- 4 Results -------- 5 Conclusion -------- Exit

1. Objectives

The aim of this BE is to simulate with the software FLUENT a flow which develops an unsteady hydrodynamic instability. We choose to simulate surface tension instability of a liquid column. This instability can be seen in everyday life. When someone turns on the water from a tab, if the flow is not to big and laminar, before the water reaches the sink, the liquid column develops corrugation and ultimately breaks into discrete drops. Without any outside stress, the flow should be uniform and in a cylindrical shape. But the surface tension which actuate on the liquid column tends to accentuate the small differences of diameter of the flow. For instance where the diameter is smaller than the average diameter, the surface tension tends to reduce even more the diameter and where the diameter is bigger than the average diameter, the surface tension tends to increase the diameter. This can be seen in the diagram which follows.

The aim of this study is to simulate with a computer these drops which can be seen in everyday life.

2. Mesh

The first thing to do is to create a domain for the simulation. We first decided to use a 3 dimensional grid. We created a portion of a cylinder in order to reduce the number of nodes on the mesh and in order to reduce the time the computer would work. Here is the mesh we first used:

The main problem with this grid was its complexity: The number of nodes was far too high therefore the time of calculation between each time step was really too big (about 30 seconds) and furthermore, it was really hard to use it with fluent. It was really hard to patch the domain because the nodes used a Cartesian system and it would have been a lot easier with a cylindrical system. Therefore it was impossible for us to patch the right zone.
Another problem would have been the boundary conditions on the yellow sides of the cylinder.

We tried a lot of different grids in 3d but then we realised that it would be a lot easier to use a 2 dimensional grid. So instead of being a cylindrical problem, the problem became 2D and it changed in nature. It was the instabilities created in a thin falling film of water. According to the theorem of ???.. we know that a flow which develops an instability in 3d develops the same instability quicker in 2d. Therefore if an instablility was to be developed in the cylinder, it would be first developed in the film. In order to see this instability, we created another grid which was two dimension. It was a lot easier because it had a lot less of nodes, the Cartesian system could be used easily. This idea of using a 2D grid worked pretty well, so we kept it and we almost did not change the grid during the rest of the study except to make it bigger. Here is the grid we used during our study.

There are two velocity inlets as boundary conditions (it is the red line on the left). Indeed on one side of the velocity inlet, only water gets in the mesh and on the other side only air can get in the mesh. The yellow line on top is a symetry condition, it allows us to calculate on only one hafl of the domain. The blue lines are the pressure inlet and the velocity outlet.

3. Simulation

Many parameters have to be chosen for the simulation, so we decided to look in the tutorials of fluent in order to find a problem which looked like the problem we had. In the third tutorial, we found the problem of spinning water in a bowl. It is really similar to our problem because it is 3 dimensional, multiphase with air and water, the domain had to be patched and it depends on the gravity.

So here are the different models we used:

• 1st order implicit unsteady formulation for the solver in order to include the unsteady ness of the flow.
• a laminar viscous model in order not to have too many equations and therefore the time for the simulation wouldn't be too long
• the air and liquid water were chosen in the material panel
• the VOF scheme in the multiphase model and we also set the coefficient of surface tension to 0.0735 in order to include the surface tension which is the stress which destabilize the flow.
• the gravity was set to 9.81m/s^2
And here are the different boundary conditions we used:
• one velocity inlet with water coming out at the speed of 0.2m/s
• one velocity inlet with air coming out at the speed of 0.001m/s
• one axis which represents the axis of the column of water
• according to the tutorial, we used a pressure inlet on the side of the domain instead of a wall
• first we tried an outflow as a boundary condition for the outlet, but then we realised that in the fluent manual, they recommend to use a pressure outlet with the pressure inlet and not an outflow, so we finally used the pressure outlet.
We also changed the solution parameters according to the tutorial
• we set all under relaxation factors to 1
• we used the body force weighted scheme for the discretization of the pressure
• and we also selected PISO as the pressure-velocity coupling method
Up

4. Results

The first calculation we made was to develop the initial conditions. The calculation have been made out of a cylinder of falling water. This is the initial diagram of densityof the flow.

Instead of having a cylinder of liquid water surrounded with air, the gravity accelerate the fluid and the shape of the flow changes. Here is the shape of the fluid after a few time steps. The time step used was pretty big in order to converge quickly toward the solution with gravity. We can see here that we obtain the shape of the flow, and that no drops seem to be created.
As we can see here , the gravity accelerate the fluid, and because of the continuity equation, the section of the water flow, shrinks. We can see in the velocity diagram, that the speed of the water gets bigger as the flow gets lower.
It looks as if the flow widens near the bottom, but this is due to the air. Indeed this diagram, is just a velocity diagram, and it shows the velocity of any fluid particule using the same way. Therefore water and air are treated in the same way. The wide section of the flow near the bottom, is due to the water which, diffuse its speed to the air using the viscosity.

The above section are the only results which are correct. But we did obtain some other results which showed the appearance of a drop, but this results can not be right. They are not physical.

The growing drop

This different diagram, show the appearance of a drop in the flow. But we think that it is mainly due to the bottom boundary condition. We first used a pressure outlet as boundary condition. Here is the different diagram showing the growing drop.

We first thought that this drop could just be a drop which was beeing created. Then we realised that this drop was far to big to stay in this position. Therefore we exchanged the bottom boundary condition for an outflow, and thought that the drop would leave the domain and that we would start from this new flow. But instead of leaving the domain, drop went back up.

We realised that the drop was going up and that the drop was still growing, because it accumulated all the water which was coming from the velocity inlet. This was not physical at all and had to do with the initial counditions.

We tried a new experiment with a longer domain, because we thought that maybe, the first simulation which created the shape of falling water out of a tab, was right but that the water did not have the time to form drops and that the water went out of the domain before it created any instabilities. So we tried with a bigger domain.

The domain used is 10 cm long instead of 2, and all the other lengths are the same. in fact we just added some length to the domain. We worked with the first initialisation. We used the patched domain and not the gravity initialisation.

The time step used was pretty small, 1.10-3 second but we quickly realised that this time step was still to high. Indeed the residual went up really fast. We watched the result anyway and realised that it wasn't that bad. Here is a zoom of one part of the domain, we can see that the direction and magnitude of the velocity vectors were pretty good.

Here is a less zoomed section of the domain. We can see here that if the velocity vectors stay in this direction, it should create a few drops.

This diagram shows the whole domain, as we can see, the whole domain should create some drops.

These diagrams seem really nice, because they show the appearance of drops in the flow. It seems as if it was the surface tension and not the boundary coundition which created the drops. But they do have a problem: the magnitude of the velocity vectors. As we can see, the maximum magnitude is around 250 m/s which is far to quick for a flow coming out of a tab.

Because the residuals were going really fast up, we decided to reduce the time steps in order to keep a realistic speed for the flow. Here is the shape of the flow after a few time steps.

This shows that after a while, even the phase of the flow looked as if drops were beeing created. But it is the result of the high horizontal velocity of the flow, and it does not create a smooth surface.

5 Conclusion

Doing this study, we realised how hard it was to simulate instabilities with Fluent. Indeed using an unsteady flow complicates a lot the equations. For each time step, fluent has to calculate a converged solution before it goes to the next time step. Therefore Fluent spends a lot a time calculating. We had to do many simulations at the same time using many computers during the week ends in order to find something which look like the reality although we did not find what we are supposed to see in real life.

It also showed us that using the surface tension was really hard. First of all because we never studied it, and second of all because we do not know how the surface tension is coded within the program.

Here are some things that might have to be done in order to have a better simulation. First it would be interesting to do a simulation with water and almost water in order to see if the horizontal speed of the flow decreases. Then it could be interesting to leave the simulation on for a much longer time, using a really low time step. And finally it could also be interesting to use a second order for the unsteady option.

Although, we did not manage to see some drops with our simulation, this study allowed us to learn many things: How to use gambit, to use the VOF model in order to simulate the multiphase flow and to patch our domain.