(510i) Consistency and stiffness of multi-way couplings in multiphase dispersed flows in a Euler-Euler discretization

Eulerian models of multi-fluid flows consist in dividing the probability density function (PDF) of the flow in several groups (called fluids) which undergo separated but coupled transport and evolution. These models involve many features which challenge numerical simulation schemes. Notably, non-standard multi-modal PDF, compressibility with contrasted equations of state (EOS), and multi-way couplings at large volume fractions strain the schemes in terms of PDF resolution, entropy consistency, and coupling stiffness.

Stiff couplings can appear notably in mass and momentum exchanges and in pressure works. For instance, velocity relaxation (momentum exchange) can be stiff due to large drag forces. Stiffness in the pressure work can appear in energy equations when fluids follow different EOS with highly contrasted compressibility: global volume change of the mixture then impacts fluids very unevenly. An extreme example of high practical impact is the case of air bubbles in water with small volume fraction.

In addition to this stiffness issue it is also necessary to take into account the consistency issue raised by truncation residues on pressure work calculations. This issue is examined in another presentation by the authors in the same session.

In order to preserve robustness, numerical schemes cope with stiff terms by using implicit or explicit discretization: for instance relaxation terms such as drag require implicit schemes. This presumes that the sign of stiff terms is known beforehand. However, in the case of pressure work in the internal energy equation, the sign of the stiffness may vary depending on expansion and compression phases. In this case, an exponential scheme provides robustness with respect to stiffness and thermodynamic consistency (for instance forbidding negative densities): high contrasts of volume fractions, densities and EOS become accessible. This is a critical ingredient in situations of appearance or disappearance of highly contrasted fluids.

We illustrate this by simulating the partially elastic collision between two jets of particles with a four-binned PDF (see figure): two groups describing the incident particle jets and two groups capturing the post-collision particles. Here, the post-collision bins of the PDF strain the scheme with vanishing small void fractions before collision and high exchange rates during collision.