Change search
ReferencesLink to record
Permanent link

Direct link
Numerical Methods for Fluid Interface Problems
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.ORCID iD: 0000-0002-4911-467X
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns numerical techniques for two phase flowsimulations; the two phases are immiscible and incompressible fluids. Strategies for accurate simulations are suggested. In particular, accurate approximations of the weakly discontinuousvelocity field, the discontinuous pressure, and the surface tension force and a new model for simulations of contact line dynamics are proposed.

In two phase flow problems discontinuities arise in the pressure and the gradient of the velocity field due to surface tension forces and differences in the fluids' viscosity. In this thesis, a new finite element method which allows for discontinuities along an interface that can be arbitrarily located with respect to the mesh is presented. Using standard linear finite elements, the method is for an elliptic PDE proven to have optimal convergence order and a system matrix with condition number bounded independently of the position of the interface.The new finite element method is extended to the incompressible Stokes equations for two fluid systemsand enables accurate approximations of the weakly discontinuous velocity field and the discontinuous pressure.

An alternative way to handle discontinuities is regularization. In this thesis, consistent regularizations of Dirac delta functions with support on interfaces are proposed. These regularized delta functions make it easy to approximate surface tension forces in level set methods.

A new model for simulating contact line dynamics is also proposed. Capillary dominated flows are considered and it is assumed that contact line movement is driven by the deviation of the contact angle from its static value. This idea is used together with the conservative level set method. The need for fluid slip at the boundary is eliminated by providing a diffusive mechanism for contact line movement. Numerical experiments in two space dimensions show that the method is able to qualitatively correctly capture contact line dynamics.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology , 2011.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:07
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-33111ISBN: 978-91-7415-969-1OAI: oai:DiVA.org:kth-33111DiVA: diva2:413402
Public defence
2011-05-20, Sal D3, Lindstedtsvägen 5, KTH, Stockholm, 14:21 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note
QC 20110503Available from: 2011-05-03 Created: 2011-04-28 Last updated: 2012-05-24Bibliographically approved
List of papers
1. A Conservative Level Set Method for Contact Line Dynamics
Open this publication in new window or tab >>A Conservative Level Set Method for Contact Line Dynamics
2009 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 228, no 17, 6361-6375 p.Article in journal (Refereed) Published
Abstract [en]

A new model for simulating contact line dynamics is proposed. We apply the idea of driving contact-line movement by enforcing the equilibrium contact angle at the boundary, to the conservative level set method for incompressible two-phase flow [E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (2005) 225-246]. A modified reinitialization procedure provides a diffusive mechanism for contact-line movement, and results in a smooth transition of the interface near the contact line without explicit reconstruction of the interface. We are able to capture contact-line movement without loosing the conservation. Numerical simulations of capillary dominated flows in two space dimensions demonstrate that the model is able to capture contact line dynamics qualitatively correct.

Keyword
level set method, contact line, conservative, two--phase flow, wetting
National Category
Computer and Information Science
Identifiers
urn:nbn:se:kth:diva-10505 (URN)10.1016/j.jcp.2009.05.043 (DOI)000268898200016 ()2-s2.0-67650147952 (ScopusID)
Note
QC 20100818Available from: 2009-05-19 Created: 2009-05-19 Last updated: 2011-05-03Bibliographically approved
2. Delta Function Approximations in Level Set Methods by Distance Function Extension
Open this publication in new window or tab >>Delta Function Approximations in Level Set Methods by Distance Function Extension
2010 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, no 6, 2199-2219 p.Article in journal (Refereed) Published
Abstract [en]

 In [A.-K. Tornberg, B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys. 200 (2004) 462-488], it was shown for simple examples that the then most common way to regularize delta functions in connection to level set methods produces inconsistent approximations with errors that are not reduced with grid refinement. Since then, several clever approximations have been derived to overcome this problem. However, the great appeal of the old method was its simplicity. In this paper it is shown that the old method - a one-dimensional delta function approximation extended to higher dimensions by a distance function - can be made accurate with a different class of one-dimensional delta function approximations. The prize to pay is a wider support of the resulting delta function approximations.

Keyword
Level set method, Delta function, Consistent approximations, Discretization, Distance function
National Category
Computer and Information Science
Identifiers
urn:nbn:se:kth:diva-10506 (URN)10.1016/j.jcp.2009.11.030 (DOI)000275092000015 ()2-s2.0-73649126152 (ScopusID)
Funder
Knut and Alice Wallenberg Foundation
Note
QC 20100907. Uppdaterat från Manuskript till Artikel (20100907)Available from: 2009-05-19 Created: 2009-05-19 Last updated: 2011-05-03Bibliographically approved
3. Spurious currents in finite element based level set methods for two-phase flow
Open this publication in new window or tab >>Spurious currents in finite element based level set methods for two-phase flow
2011 (English)In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 69, no 9, 1433-1456 p.Article in journal (Refereed) Published
Abstract [en]

A study of spurious currents in continuous finite element based simulations of the incompressible Navier-Stokes equations for two-phase flows is presented on the basis of computations on a circular drop in equilibrium. The conservative and the standard level set methods are used. It is shown that a sharp surface tension force, expressed as a line integral along the interface, can give rise to large spurious currents and oscillations in the pressure that do not decrease with mesh refinement. If instead a regularized surface tension representation is used, exact force balance at the interface is possible, both for a fully coupled discretization approach and for a fractional step projection method. However, the numerical curvature calculation introduces errors that cause spurious currents. Different ways to extend the curvature from the interface to the whole domain are discussed and investigated. The impact of using different finite element spaces and stabilization methods is also considered.

Keyword
Curvature calculation, Finite element methods, Level set method, Spurious currents, Surface tension, Two phase flow
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-33108 (URN)10.1002/fld.2643 (DOI)000305450000001 ()2-s2.0-84862622190 (ScopusID)
Note

QC 20120716

Available from: 2011-04-28 Created: 2011-04-28 Last updated: 2012-07-16Bibliographically approved
4. A uniformly well-conditioned, unfitted Nitsche method for interface problems: PartI
Open this publication in new window or tab >>A uniformly well-conditioned, unfitted Nitsche method for interface problems: PartI
(English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170Article in journal (Other academic) Submitted
Abstract [en]

A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented.The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and a variant of Nitsche's method enforces the jump conditions at the interface.In this method, the solutions on each side of the interface are extended to the entire domain, which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. Numerical experiments are presented showing optimal convergence order in the energy and $L^2$ norms, and also for pointwise errors. The presented results also show that the condition number of the system matrix is independent of the position of the interface relative to the grid.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-33103 (URN)
Note
QS 20120328Available from: 2011-04-28 Created: 2011-04-28 Last updated: 2012-03-28Bibliographically approved
5. A uniformly well-conditioned, unfitted Nitsche method for interface problems
Open this publication in new window or tab >>A uniformly well-conditioned, unfitted Nitsche method for interface problems
2013 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, 791-820 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-33105 (URN)10.1007/s10543-012-0417-x (DOI)000323729800012 ()2-s2.0-84883463552 (ScopusID)
Note

QC 20160816

Available from: 2011-04-28 Created: 2011-04-28 Last updated: 2016-08-16Bibliographically approved
6. An unfitted Nitsche method for the incompressible two fluid Stokes system
Open this publication in new window or tab >>An unfitted Nitsche method for the incompressible two fluid Stokes system
(English)Manuscript (preprint) (Other academic)
Abstract [en]

An easy-to-use finite element method for two fluid Stokes flow, with accurate treatment of jumps in pressure and in velocity gradients at the fluid-fluid interface, is presented. The method allows for an interface not aligned with the grid, and is based on continuous linear finite elements. The jumps at the interface are enforced by a variant of Nitsche's method. Numerical experiments demonstrate optimal convergence order.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-33110 (URN)
Note
QS 2011Available from: 2011-04-28 Created: 2011-04-28 Last updated: 2011-05-03Bibliographically approved

Open Access in DiVA

fulltext(1281 kB)814 downloads
File information
File name FULLTEXT01.pdfFile size 1281 kBChecksum SHA-512
144800bd831c19fc7dde3a9b5d9648b4793ea5b6cb3318cb726c0677ff117b5ace29863b2d67f0ecb15c7565d0f7317f48cc35d04e7937f6ba5a1cde131f0826
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Zahedi, Sara
By organisation
Numerical Analysis, NA
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 814 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 875 hits
ReferencesLink to record
Permanent link

Direct link