Finite Element Method for Two Phase Flow

[Sumedh Yadav]

Hello,
In my project I am modeling two-phase flow using phase-field model (an interface capturing method). Now I am done with first draft of my code and I am facing the well-known issue of 'spurious currents in two-phase flow using FEM' (paper 1 and paper 2). The take home message of paper 2 is to use a finite element space known as extended finite element space (XFEM) for pressure dofs. I wonder is this fe is available in deal.ii? If not then I would request this feature since it shall come in handy for two-phase flow researchers.

Secondly, since there is no tutorial demonstrating full two-phase formulation (I meant complete modeling i.e. Navier-Stokes + Phase-Field/Level-Set) I need suggestions (in any form, say any group using deal.ii or even other FEM package for two-phase flows currently or in past or any available code et cetera) on how to address this issue.

[Daniel Arndt]

Sumedh,

In my project I am modeling two-phase flow using phase-field model (an interface capturing method). Now I am done with first draft of my code and I am facing the well-known issue of 'spurious currents in two-phase flow using FEM' (paper 1 and paper 2). The take home message of paper 2 is to use a finite element space known as extended finite element space (XFEM) for pressure dofs. I wonder is this fe is available in deal.ii?

the unfinished example step-47 (completed in [1]) considers an XFEM approach within deal.II. Using the recently introduced FE_Enriched class [2] it should be possible to simplify this a bit.

If not then I would request this feature since it shall come in handy for two-phase flow researchers.

There seems to be some interest by others in using FE_Enriched for XFEM (https://groups.google.com/forum/#!topic/dealii/GbARfRzwNb4), but deal.II ist still a community project.

 

Secondly, since there is no tutorial demonstrating full two-phase formulation (I meant complete modeling i.e. Navier-Stokes + Phase-Field/Level-Set) I need suggestions (in any form, say any group using deal.ii or even other FEM package for two-phase flows currently or in past or any available code et cetera) on how to address this issue.

There is an application in the code gallery [3] and two example programs [4,5] on two-phase flow problems and multiple others, e.g. step-22, step-35, step-57, that deal with flow problems. These should serve as a goog starting point.
Do you have any specific problems?

Best,
Daniel

[1] https://journals.ub.uni-heidelberg.de/index.php/ans/article/view/22317
[2] https://www.dealii.org/8.5.0/doxygen/deal.II/classFE__Enriched.html
[3] https://dealii.org/developer/doxygen/deal.II/code_gallery_two_phase_flow.html
[4] https://www.dealii.org/8.5.0/doxygen/deal.II/step_21.html
[4] https://www.dealii.org/8.5.0/doxygen/deal.II/step_43.html

[Sumedh Yadav]

Daniel,
I had went through step-21 and it appears step-43 extends step-21 with the intentions of improving preconditioning et cetera. In essence both the tutorials solve the same set of equations, namely Darcy's Law, conservation of mass and saturation equation. But for this set of approximate model equations the issue of 'spurious currents' doesn't arise (as far as I know). The set of equations I am solving are essentially same in nature to the ones solved in the 'Two phase flow interaction' code gallery program. These are very strong equations to model the two phase phenomenon and require sophisticated numerical treatment.
The exact problem I am currently facing in my code is 'spurious currents in the velocity field'. The test case is as follows. The problem is 2D and I am using homogeneous Dirichlet BC throughout the boundary. There is a droplet/bubble at the center of domain. So ideally my velocity vector should be homogeneous throughout the domain for all the time steps. Please have a look at the images of first few time steps - here. You can see in the images that with increment in time steps a non-physical velocity appears and the solution numerically diverges. This non-physical phenomenon is discussed in the papers I mentioned in the earlier post (just the abstracts are sufficient to know the reasons for 'spurious currents').

You mentioned the 'Two phase flow interaction' code gallery program and it shall be a good starting point for me. I shall then have to learn MPI as it is heavily used in this code. Also the material on XFEM page shall be helpful. Thank you!

 

[Sumedh Yadav]

I would like to add an idea I intend to apply but right now am clueless of how to apply.
In the earlier post I mentioned issue of 'spurious currents'. In the past researchers have tried quite a few approaches to handle this. One of them involves reconstruction of computed pressure field. In essence if you look carefully at the images I shared in the last post you can see the failure of taylor-hood elements to handle the discontinuity in pressure which in turn makes the velocity field oscillatory/spurious. Reconstruction of pressure is done only in the interface region by interpolating the local normal direction values of pressure inside and outside this interface region. I am able to construct the normal field necessary for this local reconstruction. But I need to traverse to neighboring cells in both directions of this normal vector in the interface cells. I am clueless to how this can be done in deal.ii. The normal field image -

[luca.heltai]

Or, you could use as a stable pair of finite elements the FE_Q(k)-FE_DGP(k-1) pair:

D. Boffi and L. Gastaldi. On the quadrilateral Q2-P1 element for the Stokes problem. Int. J. Numer. Meth. Fluids, 39 (2002), 1001-1011.

This pair of finite elements is known to produce consistently better results for discontinuous pressures (after all, p is only supposed to be in L2).

L.

> On 7 Apr 2017, at 23:51, Sumedh Yadav <sumedh...@gmail.com> wrote:
>
>
> I would like to add an idea I intend to apply but right now am clueless of how to apply.
> In the earlier post I mentioned issue of 'spurious currents'. In the past researchers have tried quite a few approaches to handle this. One of them involves reconstruction of computed pressure field. In essence if you look carefully at the images I shared in the last post you can see the failure of taylor-hood elements to handle the discontinuity in pressure which in turn makes the velocity field oscillatory/spurious. Reconstruction of pressure is done only in the interface region by interpolating the local normal direction values of pressure inside and outside this interface region. I am able to construct the normal field necessary for this local reconstruction. But I need to traverse to neighboring cells in both directions of this normal vector in the interface cells. I am clueless to how this can be done in deal.ii. The normal field image -
>
>
>

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