Conservation of energy in the direct numerical simulation of interface-resolved multiphase flows

In the context of deformable bubbles, surface tension produces a dynamic exchange between kinetic and surface elastic energy. This exchange of energy is relevant to bubble dynamics, like bubble induced turbulence or drag reduction. Unfortunately, the underlying physical mechanism is not exactly explained by the state-of-the-art numerical methods. In particular, the numerical violation of energy conservation results in an uncontrolled evolution of the system and yields well-known numerical pathologies. To remedy these problems, we tackle two of the most problematic terms in the numerical formulation: convection and surface tension. We identify the key mathematical identities that imply both physical conservation and numerical stability, present a semi-discretization of the problem that is fully energy preserving, and assess their stability in terms of the discrete energy contributions. Numerical experiments showcase the stability of the method and its energy evolution for stagnant and oscillating inviscid bubbles. Results show robust as well as bounded dynamics of the system, representing the expected physical mechanism.