Modeling of multiphase flows with interfaces

Flows including more than one-phase are ubiquitous in nature, e.g., rain drops in air, gas bubbles in water, dust carried by wind, surface of rivers and oceans. In particular, of interests are flows (often turbulent) with deformable interfaces between gas and/or liquid phases considered to be weakly miscible on the molecular level (water/air system is a good example). In Fig.1  turbulent surface of the stream is depicted showing complex evolution of the seemingly simple multiphase system.

Fig. 1. Turbulent surface of the Beerbach stream (private gallery).

The mathematical description of multi-phase flows becomes more complex than in one-phase cases as the deformable interface introduces additional time and length scales related to capillary forces originating on the molecular level. Moreover, the interface topology separating domains with different material properties is part of the flow problem solution. Hence, the successful  numerical  modeling of multi-phase/multi-scale flows requires the sound physical model of the interface itself.

In the literature, the sharp and diffusive interface models are recognized as two alternative ways of modeling of the interfacial region between the two phases. The sharp interface model is materialized in the Volume of Fluid  (VOF) and/or (standard) Level-Sets (SLS) methods, while the diffusive interface model is represented by the family of Phase Field methods (PHF). All these descriptions of the interface posses complementary mathematical/physical features and yet, they are (were ?)  considered as independent alternatives for the modeling of the interface evolution.

In our works IJHFF2014, JCP2015, IJMF2017 it is shown the sharp/diffusive interface models dichotomy is apparent. The sharp and diffusive interface models can be derived from the more general, statistical description introducing the sharp interface (defined on the molecular  level, see Science2004) oscillating around its expected position due to thermal motion of the gas/liquid phases molecules. We postulate, evolution and topology of the interface on the level of continuum mechanics (i.e. after removing thermal motion of particles by the ensemble averaging)  is described by the logistic distribution and its quantile function. The statistical model of the interface (SMI) is derived based on the ensemble averaging of the sharp interface advection equation and the conservative closure of the correlation between the instantaneous interface position and the normal vector describing the sharp interface spatial orientation.

The novel equations combine features of the sharp and diffusive interface models allowing to preserve mass and compute interface orientation and curvature in straightforward manner preserving the theoretical rates of convergence of the numerical schemes used for their discretization.

Fig. 2. Evolution of zero level-set of the signed-distance function in velocity field causing deformation of initially circular interface, for details see JCP2015.


Fig. 3. Second order convergence of the circular solid body interface curvature; E-Eulerian or L-Lagrangian schemes used to discretize the advection equation, for details see IJMF2017.

Further work on numerical solution of these equations is the subject of present studies in the project granted by National Science Center, Poland (Narodowe Centrum Nauki, Polska) in the project “Statistical modeling of turbulent two-fluid flows with interfaces”, ref. nr. 2016/21/B/ST8/01010, ID:334165.  .