A fully decoupled, iteration-free, unconditionally stable fractional-step scheme for dispersed multi-phase flows

Volume-averaged flow equations model fluid systems with two or more interpenetrating phases, as used in various engineering and science applications. Each fluid obeys its own set of Navier- Stokes equations, and the interphase coupling occurs via mass conservation, drag forces, and a common pressure shared by all phases. Therefore, designing decoupling schemes to avoid costly monolithic solvers is a complex, yet very relevant task. In particular, it requires treating the pressure explicitly in a stable way. To accomplish that, this article presents an incremental pressure-correction method built upon the fact that the mean (volume-averaged) flow field is incompressible, even though each individual phase may have a non-solenoidal velocity. To completely and stably decouple the phase equations, the drag is made implicit-explicit (IMEX). Furthermore, by treating all nonlinear terms in a similar IMEX fashion, the new method completely eliminates the need for Newton or Picard iterations. At each time step, only linear advection-diffusion-reaction and Poisson subproblems need to be solved as building blocks for the multi-phase system. Unconditional temporal stability is rigorously proved for the method, i.e., no CFL conditions arise. The stability and first-order temporal accuracy of the scheme are confirmed via two-phase numerical examples using finite elements for the spatial discretisation.