A diffuse interface approach for disperse two-phase flows involving dual-scale kinematics of droplet deformation based on geometrical variables

¹ ONERA, DMPE, 8 Chemin de la Hunière, 91120 Palaiseau, France
² CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau Cedex, France
³ CNRS UMR 6614 - CORIA, 76801, Saint Etienne du Rouvray cedex, France
⁴ DEN/DANS/DM2S/STMF/LMEC, CEA Saclay, 91191 Gif-sur-Yvette, France

Abstract

The purpose of this contribution is to derive a reduced-order two-phase flow model in- cluding interface subscale modeling through geometrical variables based on Stationary Action Principle (SAP) and Second Principle of Thermodynamics in the spirit of [6, 14]. The derivation is conducted in the disperse phase regime for the sake of clarity but the resulting paradigm can be used in a more general framework. One key issue is the definition of the proper potential and kinetic energies in the Lagrangian of the system based on geometrical variables (Interface area density, mean and Gauss curvatures...), which will drive the subscale kinematics and dissipation, and their coupling with large scales of the flow. While [14] relied on bubble pulsation, that is normal deformation of the interface with shape preservation related to pressure changes, we aim here at tackling inclusion deformation at constant volume, thus describing self-sustained oscillations. In order to identify the proper energies, we use Direct Numerical Simulations (DNS) of oscillating droplets using ARCHER code and recently devel- oped library, Mercur(v)e, for mean geometrical variable evaluation and analysis preserving topological invariants. This study is combined with historical analytical studies conducted in the small perturba- tion regime and shows that the proper potential energy is related to the surface difference compared to the spherical minimal surface. A geometrical quasi-invariant is also identified and a natural definition of subscale momentum is proposed. The set of Partial Differential Equations (PDEs) including the conservation equations as well as dissipation source terms are eventually derived leading to an original two-scale diffuse interface model involving geometrical variables.