Numerical Methods for Interfacial Flows

Underlying causes for level set deformation

The practice of periodically reinitializing the level set function is well established in two- phase flow applications as a way of controlling the growth of anomalies and/or numerical errors. In the present work, the underlying roots of this anomalous growth are studied, where it is established that the augmentation of the magnitude of the level set gradient is directly connected to the nature of the flow field; hence, it is not necessarily the result of some type of numerical error. More specifically, for a general flow field advecting the level set function, it is shown that the eigenpairs of the strain rate tensor are responsible for the rate of change of the level set gradient along a fluid particle trajectory. This straining action not only affects the magnitude of this gradient, but the general character of level set field, and consequently contributes to the growth in numerical error. The role of reinitialization exacerbates the problem in cases where the zero level set contour has a local radius of curvature that is below the local grid resolution.  For other cases, where the interface is well resolved, reinitialization helps stabilize the error as intended.

The association between increasing numerical errors with the well known single vortex case, is generally attributed to the diminishing width of the interfacial region, often below the grid spacing. While this is definitely a symptom of a numerical difficulty, the underlying contributor, which is not mentioned in the literature, is the significant growth in norm of the level set gradient. This quantity has direct negative effect on the error magnitude during level set advection.