__Simulations with Fluent 5__

In this section we are going to present the results obtained with Fluent 5 concerning the air filter meshed with Icem. In fact, we are going to study the flow inside two different filters. More particularly, we will analyse the pressure and the velocity vectors for different values of the viscous resistance in the porous media and for different velocity inlets.

It is a 2D problem which can be sum up by the picture below:

As we can see on the picture above, this filter has a porous media (in green) between two walls (in white). The inlet corresponds to the blue line and the oultlet to the red one. The two yellow lines represent symmetries.

The dimensions of the domain are:

- lenght (X): 5m

- width(Y): 1m

The dimensions of the porous media are:

- lenght (X): 3m

- width (Y): 0.1m

This does not represent the reality but by choosing a right
value for the velocity inlet we would be able to have a similary
flow .

The kinematic viscosity is about 1e-5 for the air and the
caracteristic lenght of the flow is about 1. So, we chose an axial velocity
inlet of 1e-3 to have a final Reynolds number of 200 corresponding
to a laminar flow.

Concerning the porous media, we had to determine a vector
and the viscous resitance in each directions. The porous media
was isotropic so we chose the vector (1,1) and we select the same viscous
resistance for both directions. In general, the permeability of a porous
media is between 1e-9 m2 (hight permeability) and 1e-12 m2 (low permeability).
The viscous resistance is the inverse of the permeability. Finally, we
decided to take a viscous resistance equals to 1e+10.

It is a 2D axisymmetric problem. The filter is in reality a cylinder in wich there is a smaller porous cylinder .

Contrary to the picture representing the first filter, in this picture green colour shows the fluid domain. In black we can observe the three porous zone. In fact there is only two porous media: the two small zones which were previously considered as wall are gathered into the same porous zone. The axis of symmetry is in yellow, the inlet in green and the outlet in red. Now, the upper frontier has been changed into a wall.

The dimensions of the domain are:

- lenght (X): 1m

- width(Y): 0.1m

In this case which represents more accurately the reality of the use of an air filter we chose a velocity inlet equals to 0.01 (10 times more important than in the previous filter) but the final Reynolds number is the same. Re=200.

In this case, we select for the biggest porous zone a viscous resistance equals to 1e+10 and for the other zone a viscous resistance equals to 1e+13. We tried different values for this last value and the corresponding results are in the results.

__Influence of the velocity
inlet__

In this section we are going to compare fluent results with two different velocity inlet. We used the first air filter and the two velocities are: 0.001 m/s and 0.005 m/s it is to say a velocity 5 times more important than the first.

We analysed first the pressure in the flow.The picture belows shows the pressure in the flow with a velocity inlet equals to 0.001m/s:

We can see the underpressure inside the porous media which splits the flow in two parts. It is the loss of pressure which will trigger the motion of the air throught the porous media.

Now, we are going to analyse the velocity magnitude in the domain. For a velocity equals to 0.001m/s the results are:

Before and after the walls, the velocities have a low intensity. According to the continuity equation the mass which entrers the domain must be equals to the mass which goes out. This is the reason why there is a high velocity above the first wall and below the second one. We can also observe that the velocity inside the porous media has a low intensity.

We are going to see now the velocity magnitude in the domain for a velocity inlet 5 times more important. V=0.005m/s

The most important difference comes from the dissymetry of the velocity magnitude above or below the walls. In fact, when the velocity inlet is important at the end of the domain there is a reversed flow. Indeed, some air enters the domain through the pressure outlet limit condition and creates vortex inside the fluid domain. As we sais before the mass must be kept in the domain. Finally, the reversed flow triggers a very high velocity near the axis in order to keep the mass.

We can see more precisely what happens at the end of the domain by looking at the velocity vectors in this zone:

It shows the velocity profile at the end of the mesh.

__Influence of the viscous resistance__

In this section we used the second air filter. The aim is to change the permeability of the new porous media which replace the former walls in order to see its impact on the flow. In these cases we selected a velocity inlet equlas to 0.01. Do not forget that the dimensions of the domain are smaller!

First of all, have a look to the the pressure inside the domain:

This picture is similar to the one obtained previously with the first air filter. The difference is that in this case the underpressure is in the two porous media, there is no more any wall zone.

The velocity vectors picture was not quite clear. This is the reason why we are going to analyse the axial velocity profile at precise coordinate X. In particular, we are going to place this X coordinate inside the two porous media. Finally, for a permeability of the new porous zone equals to 1e-10 - the viscous resistance= 1e+10 - the results are:

This graph shows the axial vitesse in function of Y coordinate for different X coordinate. When X=0.18 we are for Y<0.06 in the porous media this is why the velocity is lower than for Y>0.06. The same remark can be applied for X=0.82, in this case we can also notice the high velocity below the porous media due to the continuity equation.

When we define a lower permeability for the new porous zone, it is to say for example a viscous resistance equals to 1e+12 these profiles became:

We can see thanks to this picture that the axial velocities inside the porous media equal to zero whereas there were not so weak in the previous test. In fact, the lower permeability involves no motion through the new porous media.

The last profile represents the radial velocity for Y=0.05 ( the line Y=0.05 goes through the three porous zones) :

We can notice the negative axial velocity inside the long porous media corresponding to the motion of the fluid across this zone. Before this zone the high axial velocity represents the "decollement" of the flow due to the porous zone which is not quite permeable. There is a negative axial velocity at the begining of the domain. We did not find any physical explanation to this phenomenun. At the end of the mesh the axial velocity tends to reach 0. If the distance between the second porous media and the end of the mesh has been more important we could have seen the profile reaching 0.