Statistical modeling of the gas–liquid interface using geometrical variables: Toward a unified description of the disperse and separated phase flows

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 contribution 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 interface relying on geometrical properties, such as the mean and Gauss curvatures, and to define an associated 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 of countable objects, such as a number density function. To this end we make essential the use of topological invariants in geometry through the Gauss-Bonnet formula. This strategy strictly includes the works conducted on sprays of spherical droplets, but it also yields a statistical treatment of populations of non-spherical objects, such as ligaments, as long as they are homeomorphic to a sphere. Second, we propose an original statistical post-processing of DNS data of interfacial flows. Starting from the proposed theoretical approach, we identify a kernel for the spatial averaging of geometrical quantities which preserves the topological invariants. Coupled to a new algorithm for the evaluation of the surface and its curvatures, that also preserves these invariants, we analyze two sets of DNS results obtained with the ARCHER code from CORIA, with and without topological changes, and assess the approach. Indeed, this procedure allows us to transform the interfacial information provided by a Level-Set function into a number distribution of a collection of objects in the proper geometrical phase space.