The mathematical and numerical modelling of multi-phase flows is never ending challenge for the academia (astrophysics, geophysics, meteorology, etc.) and industry (combustion, nuclear, health etc.). This modelling often involves dispersed multiple phases (liquid, gas, solid, and their combinations) and/or components (individual particles, droplets, bubbles, etc.). The extreme complexity and variety of mixed phases interactions, such as pressure gradients, flow speeds, material properties often limits multi-phase oriented research to simplified or ideal cases while numerous importance physical phenomena have to be addressed, even if in approximate ways, such as bubbles and slugs in pipes, nuclear reactor safety, cavitation, dust combustion and explosions and many more. 

One of the most used approaches to describe such flows is multiphase or multi-fluid modelling based on a time, space or ensemble phase-conditioning average. A large variety of models exists based on the phases characteristics, geometries of dispersion, flow regimes, dissipation strategies, source terms, boundary conditions etc. However, all multifluid models are reduced to Euler-like fluid equations for mass, momentum and energy involving transport and pressure terms only when the dissipation effects are neglected. This is referred to as a ”backbone model”.

Apart of the model choice, the numerical discretization of multi-fluid equations is a constant challenge due to variety of reasons: (i) the pressure couplings often cause the non-conservative form of equations, (ii) strong contrast of fluid characteristics yields to the presence of stiff terms, (iii) possible ellipticity of the system for certain flows (with subsonic inter-fluid drift velocities, for instance), (iv) the loss of the thermodynamic consistency resulted by pressure calculations inconsistency. Thus, an appropriate numerical scheme is required in order to address these difficulties. To this extend, a novel and efficient multi-fluid numerical scheme for discretizing the ”backbone” equations over a moving grid (ALE or Arbitrary Lagrangian–Eulerian) has been developed through a “Geometry, Energy, and Entropy Compatible” mimicking procedure (GEEC) [1–4]. Starting from the discretized density fields, energy fields, and transport operators, the procedure yields the discrete evolution equations in a practically univocal way. With arbitrarily moving grids, number of fluids, contrasts of volume fractions and equations of state, the resulting scheme is fully conservative in masses, momentum, and energy, preserves isentropic behaviour to the scheme order, and ensures per-fluid thermodynamic consistency. Noticeably, optimal isentropic behaviour is obtained thanks to a non-standard downwind form of pressure gradient. This has been validated in an ”in-house” developped 1D/2D code on a classic set of academic validation problems.