This paper analyzes inaccuracies of the numerical implementation of the con-servative level set model [68]. A circular bubble at rest is considered as a model problem. In this case the force due to surface tension should ideally be balanced by a jump in pressure. The study is performed by applying a complete Navier–
Stokes/two-phase flow solver to the problem and recording spurious velocities, see, e.g. [3], [25].
We observe that the velocity errors can be traced back to inaccuracies in the numerically calculated curvature that enter the continuous surface tension model. If the exact curvature is prescribed, there are no imbalances in the scheme, in case equal order elements are chosen for approximating pressure and the level set function. Choosing the element order of the level set function higher than the one for pressure introduces an additional imbalance. However, the use of higher order elements for the level set function has beneficial impact on the accuracy of curvature, so that the numerical model is still more accurate.
Moreover, we study the effect of the continuous representation of surface tension on the prediction of the jump in pressure between the inside and the outside of the bubble. This showed that the variation of the curvature along the finite width of the interface has the largest impact on pressure accuracy.
We perform the tests using two different flow solvers, in order to further eval-uate the accuracy of the force balance for finite element discretizations. The re-sults for a fully coupled implicit method and a decoupled (projection) solver are similar, which shows that choice of the solver does not affect the balance of forces.
6. Future Work
The method proposed in Paper III aims at providing phase-field-like features without having to use all its terms everywhere. This is done so that the phase field method is reduced to a level set formulation in regions far away from con-tact lines. It will be necessary to perform an in-depth convergence study of level set models and phase field models on several test cases in two-phase flow dy-namics. In particular, the resolution requirements of the two methods should be quantified in order to identify the precise benefits of the hybrid method. In our hybrid model, the equations have been coupled by a smoothly varying switch function. The influence of the switch function needs to be considered in an en-ergy estimate in order to ensure stability of the coupling of the equations, which has up to date only been verified numerically. Since the numerical implementa-tion of the equaimplementa-tions is based on a system of two equaimplementa-tions to avoid the direct appearance of fourth derivatives, the analysis needs to be performed in a way to correctly reflect the discretized state of the equations.
We also aim at using our models for larger problems, especially three-dimen-sional simulations. For this purpose, parallel implementations of the flow solver and the coupled level-set/Navier–Stokes system are necessary. Within the deal.II software that was used in Papers III and IV, it is possible to use parallel assembly and linear algebra routines. For a Stokes problem coupled to an advection equa-tion, a parallel version for up to about 50 processors has already been imple-mented by the author of this thesis.1The objective is to apply this implementa-tion framework to the coupled Navier–Stokes/level-set system and to eventually extend it to a massively parallel implementation for even larger systems.
The hybrid level-set/phase-field method is a potential tool for improving the simulation of multi-phase flow in porous media [17]. A first step in that direction is to apply the new model to more complicated geometries, like, e.g., channels with oscillating walls or flow around small obstacles. Small scale results are a helpful tool in the simulation of subsurface flow like the prediction of the flow in oil and gas reservoirs. The imposition of a pressure gradient on a small-scale material configuration induces a flow field, which can be used to determine val-ues for the permeability of that configuration. The permeability tensor is one of the main input parameter in coarse-scale simulations based on Darcy’s law.
In Paper IV, a study regarding error sources in the discrete approximation with the conservative level set implementation is presented. These results point out directions where the method needs to be improved. One such direction concerns
1See the deal.II tutorial program http://www.dealii.org/developer/doxygen/deal.II/step_32.html
the accuracy in the curvature calculations. A particularly easy approach is to use higher order finite elements for representing the level set function. When keep-ing the interface width constant, increaskeep-ing the order of approximation consid-erably increases resolution of the interface, and thus, accuracy of normals and curvature that rely on the interface representation. This improvement should be advantageous for the overall accuracy, even though a small imbalance in the force representation is introduced. Such a strategy increases the costs for the level set portion of the algorithm if the same mesh is used as for the Navier–
Stokes equations, though. An alternative would be to coarsen the level set mesh in regions away from the interface. This is reasonable since in these remote re-gions, only the momentum and continuity balances need to be represented ac-curately, while the level set equation does not contain any additional informa-tion. Similarly, the interface region can be resolved by more mesh points than there are used for the Navier–Stokes equations. However, this requires a special implementation for the evaluation of the surface tension force in the finite ele-ment algorithms because the variables are related to different meshes.
An alternative hybrid approach that might be taken into account is to not mix the level set formulation with the phase field formulation as has been done in Pa-per III, but to use the phase field model as a micro-scale input to a macro-scale formulation based on a level set description of two-phase flow. A heterogeneous multiscale approach has been proposed by Ren and E [75] for modeling the in-teraction of macro-scale fluid flow by micro-scale information from molecular dynamics. The response on molecular level provides information about the ex-pected motion of contact lines during one time step. This coupled model be-haves differently than large-scale level set model, which would assume zero ve-locity. It is conceivable that a similar interaction could be applied for a scale simulation done with the phase field model. The difficulty of such a micro-macro coupling is to find appropriate transfer operations from the micro scales to the macro scales. Suitable boundary conditions for the Cahn–Hilliard sub-problem given the solution of a macro-scale level set model are needed, as well as a shear velocity for the macro-scale Navier–Stokes equations, based on the results from a micro model.
Acknowledgments
I am very grateful to my adviser Gunilla Kreiss for introducing me to this inter-esting subject and for stimulating discussions on various aspects of multiphase flow. My thanks go to Volker Gravemeier who acquainted me with the subject of variational multiscale large eddy simulation for turbulent flow and stimulated ideas. I acknowledge discussions with Wolfgang Bangerth on implementation aspects of finite element programming. Finally, I would like to thank Katharina Kormann and Eddie Wadbro for countless discussions on computational math-ematics as well as proof-reading manuscripts of my articles and this thesis.
This work was supported by the Graduate School in Mathematics and Com-puting (FMB).
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