Divergence free velocity fields

Divergence free velocity fields

sebgei 03 May 2011 20:08

CSMP seems to have problems creating divergence free velocity fields for quadrilateral FE meshes. Has anyone realised this? We solve, after all

(1)

\begin{align} \nabla\cdot\mathbf{v}=0 \end{align}

or

(2)

\begin{align} \nabla\cdot\left(\frac{k}{\mu}\nabla p\right)=0 \end{align}

and the resulting divergence, as computed with the FV scheme (Stephan's generic and my old one) must be numerically zero (this is the case for triangular meshes, see below). Now, I only get this for a homogeneous permeability field and quadrilateral finite elements, as shown in this image:

If I include a low-permeability region in the centre such that flow is around it, then the divergence becomes immediately non-zero.

Has somebody noticed this and/or a good explanation? We only realised this problem when Christine's Newton iterator got stuck trying to resolve these local source/sink terms that arise from the divergence.

Last edited on 03 May 2011 22:05 by sebgei Show more

Re: Divergence free velocity fields

shaho 03 May 2011 20:38

I haven't worked with generic node center finite volume transport for a long time (more than 30 years, means practically not at all).
One thing that I want to address here is the divergence of velocity is equal to the total volume sink/source per volume per time, and it is equal to zero when you don't have sink or source in the domain, which I think it hasn't been considered in CSMP.
I have my own patch class which is my given name for the control volume in my new scheme and I have computed the flux balance over a patch and the divergence is in the range of numerical precision (epsilon) for triangular elements when you don't have a crazy permeability variation (like 1e-14 for matrix and 1e-8 for fracture). But for quadrilateral elements I have problem, and it comes from the pressure solution and it seems that the solver has not converged properly. I think if we tighten the convergence criteria for the solver the flux mismatch will vanish for the quadrilateral as well.
I never agree with adding or removing any mass to the CVs, and in my scheme I never do that and it is fine and I don't have any problem and it is conservative. I think there is something fishy with generic node center finite volume transport class. I propose to check it for a single phase case to find out if the problem is because of the mobilities that has been used in the transport algorithm or it comes from somewhere else.

Re: Divergence free velocity fields

shaho 03 May 2011 20:48

Another thing that is strange for me in your pictures is, why the divergence is always positive and you don't have any negative value. It means that you are destroying mass in every control volume. This can't be correct!!!

Re: Divergence free velocity fields

sebgei 03 May 2011 20:55

As for the positive divergence: I am plotting in log-scale, hence only the values > 0 are shown. There are values < 0 though. But as I said, for triangular FV I get divergence free fields.

Here is the volume flux for one of Mandefro's Bristol meshes (with 3 orders permeability difference between fractures and matrix)

and the essentially zero divergence field

So, all seems well for triangular meshes and the new FV scheme but not necessarily quadrilateral meshes and the old scheme.

Last edited on 03 May 2011 22:10 by sebgei Show more