Combining Discrete Equations Method and Upwind Downwind-Controlled Splitting for non-reacting and reacting two-fluid computations: Two dimensional case

• High-order (Reactive) Discrete Equations Methods on unstructured meshes.
• Reactive Riemann solver for both deflagration and detonation.
• Original formulation of a multi-D anti-diffusive approach for unstructured grids.
• Multi-D extension of a new coupling between (R)DEM and an anti-diffusive approach.
• Robust and accurate computation of impermeable/permeable fronts.

This work deals with impermeable and permeable interfaces and the design of numerical strategies allowing multi-dimensional propagation of these interfaces on general unstructured grids. The numerical context is the (Reactive) Discrete Equations Method (DEM/RDEM) for the Baer–Nunziato type non-equilibrium multiphase model allowing a diffused interface, and meanwhile preserving the global conservation, which is of fundamental importance for studying long term combustion phenomena in large-scale geometries. Another advantage of RDEM for combustion lies in its ability to compute both deflagration and detonation, provided an appropriate reactive Riemann solver is inserted within the method. The present paper is a sequel to the recent publication (Tang et al., 2014) [35] where an anti-diffusive approach and an original Upwind Downwind-Controlled Splitting method (UDCS) were combined with the 1D formulation of the DEM and RDEM. The method successfully developed in 1D for computing inert interfaces (e.g. impermeable water gas shock tube problem) and flame interfaces (e.g. Chapman–Jouguet deflagration and strong detonation wave) with excellent robustness and accuracy properties is extended here to two dimensional problems. The proposed low- and anti-diffusive versions of the multi-D UDCS strategy form an original contribution to the modeling of multifluid flows on unstructured grids. This multi-D extension relies on a general derivation of the Downwind Factors involved in the formulation of UDCS. In particular, the proposed UDCS anti-diffusive algorithm represents a new alternative to the “Extended–Vofire” solver (Faucher and Kokh, 2013) [15] for unstructured meshes. Numerical experiments performed for non-reacting gas–gas and liquid–gas shock bubble interactions as well as for a model combustion problem demonstrate the combination of DEM/RDEM with UDCS yields excellent robustness/accuracy properties. Some remaining issues linked to the modeling of flame propagation in multi-dimensional cases are eventually discussed.