Calculation of interface curvatures with the level-set method for two-phase flow simulations and a second-order diffuse-domain method for elliptic problems in complex geometries
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This thesis considers in the first part the mathematical modelling of incompressible two-phase flow, in particular the calculation of interface curvatures and normal vectors with the level-set method. The main contribution is the development of two new numerical methods that enable a more robust calculation of the curvature and normal vectors in areas where the gradient of the level-set method is discontinuous. Incompressible two-phase flow is in this thesis modelled by the Navier- Stokes equations with a singular source term at the interface between the phases. The singular source term leads to a set of interface jump conditions. These jump conditions are used in the ghost-fluid method to solve two-phase flow in a sharp manner. The interface position is captured and evolved in time with the level-set method. The Navier- Stokes equations for two-phase flow are solved with projection methods and discretized by finite differences in space and Runge-Kutta methods in time. The advective terms in the governing equations are discretized by a weighted essentially non-oscillatory scheme. In the second part, the thesis considers the more general problem of solving partial-differential equations (PDEs) in complex geometries. An extension of a diffuse-domain method is presented, where the accuracy is improved by adding a correction term. The extension is derived for elliptic problems with Neumann and Robin boundary conditions. One of the advantages of the diffuse-domain methods is that they allow the use of standard tools and methods because they are based on solving PDEs reformulated in larger and regular domains.
Has partsLervåg, Karl Yngve. Calculation of interface curvature with the level-set method. MekIT’11: 171-187, 2011.
Lervåg, Karl Yngve; Ervik, Åsmund. Curvature calculations for the level-set method. Proceedings of ENUMATH 2011: 209-217, 2013. 10.1007/978-3-642-33134-3_23.
Lervag, Karl Yngve; Mueller, Bernhard; Munkejord, Svend Tollak. Calculation of the interface curvature and normal vector with the level-set method. Computers & Fluids. (ISSN 0045-7930). 84: 218-230, 2013. 10.1016/j.compfluid.2013.06.004.
Ervik, Å.; Lervåg, K. Y.; Munkejord, S. T.. A robust method for calculating interface curvature and normal vectors using an extracted local level set. .
Lervåg, K. Y.; Lowengrub, J.. TOWARDS A SECOND-ORDER DIFFUSE-DOMAIN METHOD FOR SOLVING PDES IN COMPLEX GEOMETRIES. .