A large collection of fluid problems involve moving interfaces.
Applications include air-water dynamics, breaking surface waves,
solidification melt dynamics, and combustion and reacting flows.
In many such applications, the interplay between the interface
dynamics and the surrounding fluid motion is subtle, with factors
such as density ratios and temperature jumps across the interface,
surface tension effects, topological connectivity and boundary
conditions playing significant roles in the dynamics.
Over the past fifteen years, a class of numerical
techniques known as level set methods have been built to tackle some of the
most complex problems in fluid interface motion.
Level set methods, introduced by Osher and Sethian, are computational
techniques for tracking moving
interfaces; they rely on an implicit representation of the interface
whose equation of motion is numerically approximated using schemes
built from those for hyperbolic conservation laws. The resulting
techniques are able to handle problems
in which the speed of the
evolving interface may sensitively depend on local properties such as
curvature and normal direction, as well as complex physics off the front
and internal jump and boundary conditions determined by the interface
location. Level set methods are particularly designed for problems in
multiple space dimensions in which the topology of the evolving
interface changes during the course of events, and problems in which
sharp corners and cusps are present.
In this review, we discuss the numerical development of
these techniques and their application to a collection of problems in
fluid mechanics, including incompressible and compressible flow, and
applications to bubble dynamics, ship hydrodynamics, and inkjet
printhead design. We note that there already exists a collection of review
articles and books on these techniques, and refer the
interested reader to works by Osher and Fedkiw (2001) and
Sethian (1996b 1996c, 1999a, 1999b, 2001).