  In this work, we investigate an original strategy in order to derive a
statistical modeling of the interface in gas-liquid two-phase flows through
geometrical variables. The con- tribution is two-fold. First it participates in
the theoretical design of a unified reduced- order model for the description of
two regimes: a disperse phase in a carrier fluid and two separated phases. The
first idea is to propose a statistical description of the in- terface relying
on geometrical properties such as the mean and Gauss curvatures and define a
Surface Density Function (SDF). The second main idea consists in using such a
formalism in the disperse case, where a clear link is proposed between local
statistics of the interface and the statistics on objects, such as the number
density function in Williams-Boltzmann equation for droplets. This makes
essential the use of topolog- ical invariants in geometry through the
Gauss-Bonnet formula and allows to include the works conducted on sprays of
spherical droplets. It yields a statistical treatment of populations of
non-spherical objects such as ligaments, as long as they are home- omorphic to
a sphere. Second, it provides an original angle and algorithm in order to build
statistics from DNS data of interfacial flows. From the theoretical approach,
we identify a kernel for the spatial averaging of geometrical quantities
preserving the topological invariants. Coupled to a new algorithm for the
evaluation of curvatures and surface that preserves these invariants, we
analyze two sets of DNS results conducted with the ARCHER code from CORIA with
and without topological changes and assess the approach.
