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Mass Conserving Simulations of Two Phase Flow
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2006 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

Consider a mixture of two immiscible, incompressible fluids e.g. oil and water. Since the fluids do not mix, an interface between the two fluids will form and move in time. The motion of the two fluids can be modelled by the incompressible Navier-Stokes equations for two phase flow with surface tension together with a representation of the moving interface. The parameters in the Navier-Stokes equations will depend on the position and other properties of the interface. The interface should move with the velocity of the flow at the interface. Since the fluids are incompressible, the density of each fluid is constant. Mass conservation then implies that the volume occupied by each of the two fluids should not change with time. The object of this thesis has been to develop a new numerical method to simulate incompressible two phase flow accurately that conserves mass and volume of each fluid correctly.

Numerical simulations of incompressible two phase flow with surface tension have been a challenge for many years. Several methods have been developed and used prior to the work presented in this thesis. The two most commonly used methods are volume of fluid methods and level set methods. There are advantages and disadvantages of both of the methods.

In volume of fluid methods the interface is represented by a discontinuity of a globally defined function. Because of the discontinuity it is hard both to move the interface as well as to calculate properties of the interface such as curvature. Specially designed methods have to be used, and all these methods are low order accurate. Volume of fluid methods do however conserve the volumes of the two fluids correctly.

In level set methods the interface is represented by the zero contour of the globally defined signed distance function. This function is smooth across the interface. Since the function is smooth, standard methods for partial differential equations can be used to advect the interface accurately. A reinitialization is however needed to make sure that the level set function remains a signed distance function. During this process the zero contour might move slightly. Because of this, the volume conservation of the method becomes poor.

In this thesis we present a new level set method. The method is designed such that the volume of each fluid is conserved, at least approximately. The interface is represented by the 0.5 contour of a regularized characteristic function. As for standard level set methods, the interface is moved first by an advective step, and then reinitialized. Unlike traditional level set methods, we can formulate the reinitialization as a conservation law. Conservative methods can then be used to move and to reinitialize the level set function numerically. Since the level set function is a regularized characteristic function, we can expect good conservation of the volume bounded by the interface.

The method is discretized using both finite differences and finite elements. Uniform and adaptive grids are used in both two and three space dimensions. Good convergence as well as volume conservation is observed. Theoretical studies are performed to investigate the conservation and the computational time needed for reinitialization.

Place, publisher, year, edition, pages
Stockholm: KTH , 2006. , v, 18 p.
Series
Trita-NA, ISSN 0348-2952 ; 0556
Keyword [en]
free boundary, two phase flow, level set method, conservative
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-3851ISBN: 91-7178-265-6 (print)OAI: oai:DiVA.org:kth-3851DiVA: diva2:9709
Presentation
2006-03-10, D31, Huvudbyggnaden, Lindstedtsvägen 3, Stockholm, 14:15
Opponent
Supervisors
Note
QC 20101122Available from: 2006-02-14 Created: 2006-02-14 Last updated: 2010-11-22Bibliographically approved
List of papers
1. A conservative level set method for two phase flow
Open this publication in new window or tab >>A conservative level set method for two phase flow
2005 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 210, no 1, 225-246 p.Article in journal (Refereed) Published
Abstract [en]

A conservative method of level set type for moving interfaces in divergence free velocity fields is presented. The interface is represented implicitly by the 0.5 level set of a function Phi being a smeared out Heaviside function, i.e., a function being zero on one side of the interface and one on the other. In a transition layer of finite, constant thickness Phi goes smoothly from zero to one. The interface is moved implicitly by the advection of Phi, which is split into two steps. First Phi is advected using a standard numerical method. Then an intermediate step is performed to make sure that the smooth profile of Phi and the thickness of the transition layer is preserved. Both these steps are performed using conservative second order approximations and thus conserving integral Phi. In this way good conservation of the area bounded by the 0.5 contour of Phi is obtained.

Numerical tests shows up to second order accuracy and very good conservation of the area bounded by the interface.

The method was also coupled to a Navier-Stokes solver for incompressible two phase flow with surface tension. Results with and without topological changes are presented.

Keyword
level set method, free boundary, two phase flow, incompressible flow, conservative method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-5362 (URN)10.1016/j.jcp.2005.04.007 (DOI)000231265500009 ()2-s2.0-23044452940 (Scopus ID)
Note
QC 20101122Available from: 2006-02-14 Created: 2006-02-14 Last updated: 2017-11-21Bibliographically approved
2. A conservative level set method for two phase flow II
Open this publication in new window or tab >>A conservative level set method for two phase flow II
2007 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 225, no 1, 785-807 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we continue to develop and study the conservative level set method for incompressible two phase flow with surface tension introduced in [J. Comput. Phys. 210 (2005) 225-246]. We formulate a modification of the reinitialization and present a theoretical study of what kind of conservation we can expect of the method. A finite element discretization is presented as well as an adaptive mesh control procedure. Numerical experiments relevant for problems in petroleum engineering and material science are presented. For these problems the surface tension is strong and conservation of mass is important. Problems in both two and three dimensions with uniform as well as non-uniform grids are studied. From these calculations convergence and conservation is studied. Good conservation and convergence are observed.

Keyword
level set method, two phase flow, conservative method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-5363 (URN)10.1016/j.jcp.2006.12.027 (DOI)000248854300040 ()2-s2.0-34447287141 (Scopus ID)
Note
QC 20101122. Uppdaterad från Submitted till Published (20101122).Available from: 2006-02-14 Created: 2006-02-14 Last updated: 2017-11-21Bibliographically approved

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