I'm simulating flow through a container having a porous medium and using Darcy's law, which states that:

V=(K/mu)*grad(P)

where V is the velocity of the permeating fluid, K is the permeability of the medium to flow, mu is the fluid viscosity and P is pressure.

Consider a box whose walls are impenetrable to flow, filled with a porous medium, with an inlet on one side and an outlet on the other side. The boundary conditions are pressure = Pin everywhere on the inlet and zero everywhere on the outlet. A natural boundary condition is applied to the walls of the box such that the outward normal of the derivative of pressure is zero. The equation is div((K/mu)*grad(P))=0 (essentially div(V)=0, assuming conservation of both mass at constant density). I'm using a very simple application, FlexPDE, as a general partial differential equation solver.

Since the fluid velocity is not a variable (only the pressure is, and the velocity is derived from the gradient of pressure), I cannot impose any boundary conditions on the velocity, only on pressure.

I find that the velocity profile of the permeating fluid appears to exhibit maxima at the edges of the surfaces defining both the inlet and the outlet. This seems to be an unphysical result. However, if the flow velocity is very low, this approximation might not be so bad.

Questions:

1) Isn't it unphysical to have a nonzero tangential fluid velocity at the wall?

2) Is this what you call a slip condition?

3) Is it reasonable to assume this slip condition in a situation involving permeation of fluid through a porous medium?

4) Is there a way to estimate the error involved by having this nonzero wall velocity?

5) Do experienced CFD experts not worry about such details?

6) Are there techniques using mixed boundary conditions that would somehow enable me to simulate a Darcy's Law flow situation and impose boundary conditions on velocity, and solve for velocity and pressure simultaneously? Any references?

Thanks in advance for any insights.

By the very nature of the PDE you're trying to solve, you cannot impose both no-flux and no-slip conditions at the walls. The only way is to impose no-flux, find how much slip there is at the wall, and then put vorticity sources with the same strength to cancel the problem with no-slip. Of course, you have to be careful with the implementation details.

Adrin Gharakhani

Thanks for your help. Your reply provides me some degree of encouragement that I am sort of on the right track. However, your proposal to apply vorticity sources seems like adding one nonphysical aspect to counteract another nonphysical aspect. What is the physical meaning of a vorticity source? I'm uncomfortable incorporating a ficticious source when I can't justify it on physical grounds. Is there an easy way to apprehend what physical reality such a vorticity source would correspond to? Or do I need to leave my gut behind at some point?

Thanks again.

There is nothing unphysical about putting vorticity source terms on the boundary to satisfy the no-slip boundary condition. In fact, by doing so, you would be replicating what nature (to the best of our understanding) actually does at the walls. Read Batchelor or other classic books on fluid mechanics which actually treat vorticity (some "fundamental" books on fluids shy away from this powerful concept)

Adrin Gharakhani

Hi!

I am not an expert for porous media flow, so what I say here does not claim to be correct, it is just what I think about it.

Using Darcy's law to compute the flow field in porous media is a simplified model. The correct equations would be the Navier-Stokes equations with full resolution of all grains in your porous media. This is uncomputable, of course. At least for a full scale problem. I am not sure if non-slip would be the correct boundary condition for this model. Porous media flow basically is flow through "microscopic channels" of random directions. Within such a channel non-slip would be correct as boundary condition. If you look at a wall you have microscopic channels again, only they are all aligned with the wall here. To me it feels that a no-flux condition across the wall is the right model.

Oliver

The "velocity" in Darcy's law is not a true local velocity, you might think of it more as a mass flux. The drag law in Darcy's law implies an averaging scale that is larger than many pore diameters. So there is no true single "wall" locally- there are "walls" everywhere and of course in the pore there are no-slip conditions. But all this detail is averaged away, so that you can get SOME kind of result. This has led to some controversy regarding corrections to Darcy's law, and there is a large literature on this subject.

Becasue Darcy's law only povides a notion of velocity in terms of some integral, or average scale, one must be acreful about the inherent scales in teh intended application. Permeability is a tensor, and the value can be scale dependent if the porous host is anisotropic.

Bejan has written good books on all this. Efforts to obtain Darcy's law by starting with the N-S equations are especially ad hoc, but you can do it if you want to waste a fine summer aftrnoon.

Yes, I think that is the way I understood this too. So, you would agree that there is no zero-velocity condition? This was the original point, because Paul Missel was concerned about his "unphysical" non-zero velocities on the boundary.