Jump conditions for filtered quantities at an under-resolved discontinuous interface. Part 1: Theoretical development

In this paper, we study turbulent two-phase flow. We consider the level of description where only the large scales of turbulence and the large deformations of bubbles are explicitly described: all scale of turbulence are not represented and we are close to the Large Eddy Simulation concept, an geometry of each bubble is explicitly described but the small deformations of the bubbles are not represented. The bubble interface is still supposed to be infinitely thin (i.e. interfaces are supposed to be under-resolved and discontinuous). s level of description, there is no reason that the well known jump conditions are still valid. Using a two-step methodology, we determine the jump conditions for filtered quantities (i.e. local mean velocity and pressure) at the under-resolved discontinuous interface (i.e. small deformations of the interface are not represented). In particular, we express the velocity of the under-resolved discontinuous interface as a function of the filtered velocity, a scale similarity hypothesis and the time evolution of the interface mean curvature.