Permeable plate modeling / transpiration cooling / porous wall
 

Dear All,

I need to simulate within ANSYS Fluent a two cases of flow along a permeable plate.
The sketch for a first case you can see below.

http://i.share.pho.to/27bb5687_o.jpeg

It is just flow along viscous wall with specified gas injection (mass flux). The main purpose of this calculation is determination of gas injection influence on the wall friction coefficient.
The obvious approach is to use porous-jump boundary condition, but unfortunately in this case there is no non-slip effect at the wall surface and, as consequence, the wall shear-stress equal zero. And of course this effect is unphysical.

The second case is similar to first one, but there are flows for both sides of permeable flat plate (see picture below).

http://i.share.pho.to/0381f692_o.bmp

Again, the main approach is to use porous solid and porous-jump boundary condition. But in this case there is no shear-stress at wall surfaces.

Could you please propose appropriate approach for correct modeling of specified above cases.

Thank you in andvance,

Any Ideas?

The first case is a physically unrealistic setup. But the obvious choice is to use a velocity or mass-flux boundary condition with specified direction. T̶h̶e̶r̶e̶ ̶i̶s̶ ̶n̶o̶ ̶w̶a̶l̶l̶-̶s̶h̶e̶a̶r̶ ̶s̶t̶r̶e̶s̶s̶ ̶i̶n̶ ̶t̶h̶i̶s̶ ̶c̶a̶s̶e̶ ̶b̶e̶c̶a̶u̶s̶e̶ ̶t̶h̶e̶ ̶s̶e̶t̶u̶p̶ ̶i̶t̶s̶e̶l̶f̶ ̶i̶s̶ ̶n̶o̶n̶-̶s̶e̶n̶s̶e̶. You have to understand that once the velocity of the injected gas is specified that the problem is no longer a flow over a plate problem but a mixing of two streams problem. Upstream of the injection is a traditional boundary layer problem, but with injection it becomes a mixing problem (a shear layer, which technically is also a boundary layer). As far as the fluid is concerned there is no plate there. The question remains how the heck did the injected fluid get there?

The second case is straightforward to implement as drawn by explicitly treating the porous plate. You just need a few parameters to describe permeability. If the plate is porous then you'll need porosity, and first and/or 2nd darcy coefficients.

Dear LuckyTran,

The first case posted above is classical boundary layer problem with mass transfer. It can be found in many handbooks (see H. Schlichting Boundary-Layer Theory, for example). For laminar boundary layer there are self-similar solutions. Which can be resolved in terms of ordinary differential equations.

Regarding wall shear stress...
http://ntrs.nasa.gov/search.jsp?R=19650014923

So, the questions are the same:

• how to implement injection at the wall: V=Vinj?
• how to implement wall shear stress at porous-jump?

Okay, there is still a notion of a wall shear stress as the mixing of streams does yield a velocity gradient.

The two cases are the same if the velocity injected is consistent with the velocity observed from the second case.

This is a boundary layer problem, yes. But this is not a wall-bounded flow problem (in the mean-flow sense). For velocity fluctuations the kinematic constraints to play a role, but that does not appear to be the concern here. You don't necessarily need to agree with my classification of this problem.

The porous jump condition is a simple pressure drop condition. You can model the porous plate using a pressure jump condition for a thin plate, but for a thick plate you should model the entire porous region explicitly and specify the coefficients to the Darcy-Forchheimer relation. I don't think you should be using the porous jump condition.

For the first case you should use a velocity inlet boundary condition. The no-slip condition is automatically satisfied because the velocity is explicitly declared. Fluent won't let you inject from a wall, which is a major inconvenience because you can't get the "wall shear stress." I don't disagree with this lack of capability, because the porous wall doesn't impose the same kinematic wall boundary condition that a wall should.

Dear LuckyTran,

I am not talking about the physics. Physics of this problem is well known and described in many handbooks, reports, dissertations and papers.
I am talking about an implementation of specific boundary conditions within ANSYS Fluent.
For the first case (which is obviously a particular case of the more common case two) it is just at the wall surface.

Velocity inlet is not appropriate BC for this case, since again there is no wall shear stress for this kind of BC.

For the second case it should be something like, so called, Beavers and Joseph boundary condition at interface between fluid and porous media. But as a first estimation it is acceptable to forget about slip velocity and use normal velocity component only.

Regarding porous jump BC. I am told about porous jump BC just like about interface between fluid and porous media.

So, all standard Fluent BCs are inconvenient for these cases. Moreover it is not clear how to implement required BC by using UDF. For the first case it is possible to set source of mass/energy/momentum but inside volume only (first near wall cell, for example), not at the wall itself. In case of source approach it is necessary to set source at wall surface.

For the second case I even have no idea how to implement it.
Any suggestions?

The second case is straightforward. You have a fluid zone for the upper fluid, fluid zone for lower fluid and porous zone for the plate. The first case, which is really trying to mimic the second case, is where you can introduce a lot of non-sense because you're trying to "model" certain behaviors of the flow coming from a porous media.

A velocity inlet does have a shear stress (you might not call it a wall shear stress, but shear stress nonetheless). Note that "wall shear stress" is a post-processing value such as (heat transfer coefficient) which you obtain by properly referencing properties of the flow (shear-stress of fluid right next to the wall). The inconvenience is that you can't just report "wall shear stress" from fluent.

The porous jump condition is used when you do not want to explicitly model porous region effects. The porous jump condition lumps the porous region into a infinitely thin membrane and assumes that the only influence of the porous region is a pressure drop across the membrane. The porous jump condition is not the porous-to-fluid interface. A correct porous-fluid interface should implement a condition such as that in Beavers & Joseph.

You cannot ignore slip/tangential velocity, unless the velocity truly is normal to the plate. If you are considering only the normal velocity component then it becomes a mixing of two streams problem and a velocity boundary condition is even more applicable.