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

At this 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.