Single-Phase Level Set Method For Unsteady Viscous Free Surface Flows

Abstract

The level-set method has become a popular approach to tackle two-phase,
incompressible flow problems. In the standard level-set method the equations are solved in
both fluids with smoothed fluid properties across the interface. In contrast to the standard
level set method, the single-phase level set method is concerned with the solution of the flow
field in the denser phase only. Some of the advantages of such an approach are that the
interface remains sharp, the computation is performed within a fluid with uniform properties
and that only minor computations are needed in the air. The location of the interface is
determined using a signed distance function, exactly as done on the standard level-set
method, but appropriate interpolations and extrapolations are used at the fluid/fluid interface
to enforce the jump conditions. In our RANS solver with non-orthogonal grids, very large cell
aspect ratios appear on the near-wall regions of the flow, which causes the standard
reinitialization methods to fail. To overcome this problem, a reinitialization procedure has
been developed that works well with non-orthogonal grids with large aspect ratios. Since the
grid points in air don’t have a well defined velocity, the time derivatives cannot be treated in
the Eulerian fashion in points that change from air to water during a time step. This problem
is dealt with by using a convective extension to obtain the velocities at previous time-steps for
the grid points in air, which provides a good estimation of the total derivatives. In this paper
we discuss the details of such implementations. The method was applied to two unsteady
tests: sloshing in a two-dimensional tank and wave diffraction in a surface ship, and the
results compared against analytical solutions or experimental data. The method can in
principle be applied to any problem in which the standard level-set method works, as long as
the stress on the second phase can be specified and no bubbles appear in the flow during the
computation.