  Numerical simulations of two-phase flow and fluid structure interaction
problems are of great interest in many environmental problems and engineering
applications. To capture the complex physical processes involved in these
problems, a high grid resolution is usually needed. However, one does not need
or maybe cannot afford a fine grid of uniformly high resolution across the
whole domain. The need to resolve local fine features can be addressed by the
adaptive mesh refinement (AMR) method, which increases the grid resolution in
regions of interest as needed during the simulation while leaving general
estimates in other regions.
  In this work, we propose a block-structured adaptive mesh refinement (BSAMR)
framework to simulate two-phase flows using the level set (LS) function with
both the subcycling and non-subcycling methods on a collocated grid. To the
best of our knowledge, this is the first framework that unifies the subcycling
and non-subcycling methods to simulate two-phase flows. The use of the
collocated grid is also the first among the two-phase BSAMR framework, which
significantly simplifies the implementation of multi-level differential
operators and interpolation schemes. We design the synchronization operations,
including the averaging, refluxing, and synchronization projection, which
ensures that the flow field is divergence-free on the multi-level grid. It is
shown that the present multi-level scheme can accurately resolve the interfaces
of the two-phase flows with gravitational and surface tension effects while
having good momentum and energy conservation.
