A Locally Adaptive Mesh Densification Scheme for Resolving Singularities in Multi-Scale Free Surface Flows

Simulating free surface flows near singularities, as in the thinning of a thread/neck in drop breakup or the growth of a bridge/neck in drop coalescence, presents unique challenges in computational fluid dynamics. In order to accurately capture the dynamics and deduce scaling laws of pinch-off (coalescence) in such problems, it is necessary to resolve many different length scales simultaneously at the incipience of the singularity. A successful strategy over the years has involved use of elliptic mesh generation coupled with Galerkin finite elements as the basis of a multi-dimensional, arbitrary Eulerian Lagrangian algorithm that can accurately track the deforming free surface as the neck radius varies by up to three orders in magnitude. Attaining smaller length scales while maintaining adequate mesh density near the singularity and also discretizing the domain far away had heretofore proved prohibitively expensive. Here we present a scheme to adaptively densify regions of the mesh near the singularity without perturbing the mesh far from it allowing simulations to span length scales that differ by six to seven orders of magnitude which had heretofore proved possible only with 1D algorithms and boundary integral simulations restricted to creeping or potential flows.