Multiphase flows with sharp interfaces

Multiphase flows with sharp interfaces

Two-phase flows usually have solutions with low regularity, i.e. they feature kinks and jumps due to jumps in density and viscosity, and because of surface tension. The challenge is to handle these jumps and kinks in the numerical method, without unphysical smearing. For an ellipsoidal droplet, velocity and pressure may look like this:

xdg_air_water_drop

In BoSSS, an eXtended discontinuous Galerkin (XDG, aka. unfitted DG, aka. XFEM-DG, aka. cut-cell DG) is used to describe velocity and pressure with sub-cell accuracy. In cells where both phases are present, individual DG basis polynomials for both phases are introduced. By doing so, it is possible to retain the spectral convergence properties of the DG method for solutions with low regularity (see examples below).

xdg_explained

Oscillating droplet

by: Florian Kummer

Simulation of a droplet with initially ellipsoidal shape.

Kummer, Florian ; Utz, Thomas (2015):
High order discretizations for two-phase flow: spatial discretization, curvature evaluation and solver strategies.
In: European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs: Theory and Applications (HONOM-2015), Trento, Italy, 2015, [Konferenzveröffentlichung]

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Long term behavior and stability

by: Florian Kummer

One issue with respect to surface tension is long-term stability. This is because the pressure jump is proportional to the curvature, which is a property that is nonlinear and depends on second derivatives -- an unfortunate combination. By using sophisticated filtering techniques, BoSSS acquires long-term stability

xdg_deformed_droplet
Velocity magnitudes for an initially strongly distorted droplet

Kummer, Florian ; Utz, Thomas (2015):
High order discretizations for two-phase flow: spatial discretization, curvature evaluation and solver strategies.
In: European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs: Theory and Applications (HONOM-2015), Trento, Italy, 2015, [Konferenzveröffentlichung]

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Convergence results

by: Dr.-Ing. Florian Kummer

By using the XDG approach, BoSSS is able to provide spectral convergence for low-regularity solutions. I.e. the error in the velocity and pressure behave approximately as hp+1 and hp, where p is the order of the DG polynomials. This is in fact one of the most important results obtained by the BoSSS code so far.

xdg_convergence_droplet
Results of an h-convergence study for an initially ellipsoidal droplet.
xdg_convergence_taylor_couette
Results of an h-convergence study for a two-phase Taylor-Couette flow.

Kummer, Florian ; Utz, Thomas (2015):
High order discretizations for two-phase flow: spatial discretization, curvature evaluation and solver strategies.
In: European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs: Theory and Applications (HONOM-2015), Trento, Italy, 2015, [Konferenzveröffentlichung]

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Rising bubble

by: Martin Smuda, M.Sc.

Smuda, Martin ; Kummer, Florian ; Utz, Thomas (2017):
An extended Discontinuous Galerkin method for transient two-phase problems: Application in gravity/surface tension driven flows.
In: XIX International Conference on Finite Elements in Flow Problems (FEF 2017), Rome, Italy, April 5 - 7, 2017, [Online-Edition: http://congress.cimne.com/FEF2017/admin/files/fileabstract/a...],
[Konferenzveröffentlichung]

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