Hello there, I would like to get some info about numerical simulation for incompressible flow in real porous medium with finite volume method. Can anyone help me to find the appropiate boundary conditions outside + inside (the obstacle in porous medium)? Thanks in advance.

Dear Jack,

I suppose your question is quite general. Could you give some details so that I may be of help?

Regards,

Mahesh

Dear Mahesh,

Thanks for the reply.

I want to do the simulation of incompressible flows in domain \G using finite volume method. The domain \G is divided in the n-rectangular (cubical) cells, which consist of fluid cells (the pure fluid part) and obstacle cells (the pure solid part) and perhaps mushy cells (mixed of fluid and solid). The Navier-Stokes equations are solved only in the fluid cells and perhaps in the mushy cells.

My first question is what kind of grid should I better use? (the staggered or non-staggered one).

Then, how should I implement the outer boundary conditions, I mean that just given the velocity or velocity + pressure (let's say) on the west boundary and what kind of boundary condition for the east boundary. For the other boundaries (North and South), can I just take the no-slip conditions (for wall bondary) or I have to take into account the pressure too? I think it depends on the grid (staggered or non-staggered). How can I realize in the discrete equations? Do I need the ghost (boundary) cells?

What should I give for the inner boundary conditions (for the obstacle cells)? Just the velocity equals 0 or I should give the pressure boundary condition, too? How about the single obstacle cell which is surrounded by the fluid cells?

How about the mushy cells?

I hope that I describe it quite clearly.

Thanks again.

Best regards, Jack

Dear Jack,

It seems to me that you are using some kind of a commercial CFD package for your problem. The staggered or non-staggered mesh have nothing to do with boundary conditions as far as I know. You will use the same boundary conditions as you would normally use for a solid wall. As far as the mushy zone or porous zone is concerned, you will have to incorporate additional source terms in your momentum equations(ie: the Navier Stokes equation). The type of source terms employed will depend on the nature of your flow. If your flow is very slow (typically a Reynolds number of less than 1, where Re is calculated on the basis of the pore diameter or the size of the particles constituting the porous medium) then you can use only the Darcy source term. Otherwise you will have to incorporate the Forchheimer term as well. For the interface between the clear fluid and the porous medium, if you are using the Brinkman term in the equations for flow in the porous medium (the Brinkman term is the diffusion like term for flow in a porous medium) you will not have to specify any explicit interface condition. Otherwise you will have to use something like the Beavers and Joseph condition.

I would advice you to look at some Journal articles on flow in a porous medium. I am listing a couple here:

Beckeremann et al., "Natural convection in vertical enclosures containing simultaneously fluid and porous layers", Journal of Fluid Mechanics, 1988, 186, 257-284.

Neale, G. and Nader, W., "Practical significance of Brinkman's extension of Darcy's law", Canadian Journal of Chemical Engineering, 1974, 52, 475-478.

I hope this helps you.

Cheers

Mahesh

Dear Mahesh,

Thanks for the hint of journal articles. I'll look it.

No, I don't use any commercial CFD-packages, I have to write my own code. Because of that and since I don't have so much experience with CFD, I want to make sure first before I beginning to write the code. I think I have to solve only the Navier-Stokes equation at the microscale or is it wrong? At the macrosacle (after homogenization or something like that), then, I can apply the Darcy-term or Brinkman-term in my equation.

Thanks again for the articles.

regards, Jack

Dear Jack,

When you say microscale I suppose you mean "Before the averaging operator" is applied. When you are using the equations for flow in a porous medium you are essentially using a volume averaged form of an equation which is made to resemble the Navier-Stokes equation. Strictly speaking there is still no general (universal) goverening equation for flow in a porous medium like the Navier Stokes equation.

The Darcy and Forchheimer terms acutally are a result of the volume averaging procedure. There are several papers by Professor Stephen Whitaker which explain at length these issues. If you are interested I can give you the references to these papers.

Dear Mahesh,

Yes, you're right. We can do either homogenization or averaging to get the D'Arcy and Forchheimer terms. I have some papers and a book by Professor Whitaker, but I would be very happy to have some more references. The book called "The Method of Volume Averaging", in which he discussed the averaging processes widely and deeply.

I would be most grateful to have some references or information about simulation the flow in porous medium (before averaging processes, just use the Navier-Stokes Equation (something like flow simulation in complex geometries)) using finite volume method or other methods.

Thanks a lot and best wishes, Jack

Hi, I will start working on CFD on porous media also, I would thank the references of Prof. Whitaker and any other references or source. Thanks