The distortion of the level set gradient under advection

Abstract

The practice of periodically reinitializing the level set function is well established in twophase 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 (|∇φ|) 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 |∇φ| along a fluid particle trajectory. This straining action not only affects the magnitude of |∇φ|, but the general character of φ, and consequently contributes to the growth in numerical error. These numerical consequences are examined by adopting the Gradient Augmented Level Set method. Specifically, it is shown that the local error for φ is directly connected to the size of |∇φ| and to the magnitude of the second and fourth order derivatives of φ. These analytical findings are subsequently supported by various examples.The role of reinitialization is discussed, where it is shown that in cases where the zero level set contour has a local radius of curvature that is below the local grid resolution, reinitialization exacerbates rather than diminishes the degree of error. For other cases, where the interface is well resolved, reinitialization helps stabilize the error as intended.

High Order Anchoring and Reinitialization of Level Set Function for Simulating Interface Motion

“…The velocity field is periodic in time with period T = 8. It induces tangential deformation in the level set function [20]. As mentioned in Sect.…”

“…Level set methods and the problem of reinitialization have been reviewed by Gibou et al [10]. An elaborate discussion on this topic has also been presented by Trujillo et al [20] by considering the classic Rider-Kothe single vortex test. The vortex test introduces severe interface deformation through advection and highlights the challenges in modeling dynamic interfaces.…”

“…It has been exercised as a benchmark for level set advection and reinitialization methods. Nourgaliev et al [25] present evaluations of a series of high order advection schemes based on the vortex test while Trujillo et al [20] analyze the necessity and consequences of reinitialization. The review of level set algorithms [10] also outlines the difficulties arising from reinitialization in terms of maintaining the interface position.…”

“…Our work also aims at exploring the influence of high order reinitialization on computation of geometrical properties of interface in addition to mass conservation. Trujillo et al [20] have shown an elaborate analysis on the distortion of level set function resulting from advection. The solution, i.e.…”

“…However, maintaining the signed distance property of level set function is challenging. Advection of level set function is known to distort the distance field [20] and Eq. 1 is not unconditionally satisfied after advection.…”