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Les équations de Navier-Stokes diphasiques jouent un rôle crucial dans plusieurs applications indus-trielles et environnementales. Cependant, leur résolution pose de nombreux problèmes et devient unenjeu important pour les mathématiciens ansi que les ingénieurs. Dans le contexte des écoulements di-phasiques incompressibles, les difficultés sont multiples : d’une part, le couplage vitesse-pression induitpar la contrainte d’incompressibilité, donne lieu à un problème de point selle pour lequel des solveursdédiés doivent être développés. Et, d’autre part, les forts rapports de densité et de viscosité en présencede grandes déformations de l’interface conduisent à des systèmes linéaires très mal conditionnés. Nousdécrivons ici un solveur tout-couplé parallèle 3D, didié à la simulation des écoulements diphasiquesincompressibles. Les équations linéarisées de quantité de mouvement et de continuité découlant de lasolution implicite des vitesses et de la pression dans les fluides sont résolues simultanément. Nous avonssimulé un cas de l’instabilité d’un jet visqueux pour vérifier les modèles, notamment en présence de fortsrapports de densité et de viscosité. Nous avons également réalisé un cas d’inversion de phase 3D, pourdémontrer la capacité de notre nouveau solveur tout-couplé à résoudre des problèmes diphasiques surun milliards de cellules.

The main objective of the present work is to further investigate the solution of the coupled system (without any time-splitting approach) by means of a preconditioned BiCGstab(2) solver. We obtain high performance with a new preconditioning strategy that combines a triangular block preconditionning for the velocity block with a pressure convection diffusion (PCD) preconditionner for the Schur complement. To provide the stability of the numerical scheme in the presence of high density and viscosity ratios, we use an algebric momentum preserving mass and momentum transport in the conservative form of discrete equations. The consistency between mass and momentum advection is achieved by resolving a new auxiliary continuity equation, using fifth-order WENO scheme and third Runga-Kutta SSP time integrator.

Two-phase flows with separated phases are prevalent in several industrial and environmental applications, where the conditions and regimes encountered spread wide and far. In the energy sector for example, the atomisation of liquid jets results from complex interactions between a coherent liquid jet, a gas-filled plenum and (potentially) a co-flowing gaseous jet. Breaking waves are another instance of two-phase flows, characterised by a high degree of turbulence, as are droplet impacts on liquid surfaces, nanofabrication by thermal plasma jets, to name a few.The high density and viscosity ratios encountered in these two-phase flows couple strongly the velocity components in the interface vicinity, and often result in strong shear at these interfaces. This yields in ill-conditioned linear systems (as a result of the semi-discretisation of the two-phase incompressible Navier-stokes equations) that pose significant problems together with the treatment of the coupled velocity-pressure saddle-point system. Solving these systems remains a challenging task for mathematicians as well as engineers. In addition, it has been demonstrated that solving theses equations in a non-conservative form with high density ratios leads to incorrect velocity field or volume fraction and often a failure of numerical scheme, due to numerical error accumulation near the highly deforming interface.The main objective of the present work is to further investigate the solution of the coupled system (without any time-splitting approach) by means of a preconditioned BiCGstab(2) solver. We obtain high performance with a new preconditioning strategy that combines a triangular block preconditionning for the velocity block with a pressure convection diffusion (PCD) preconditionner for the Schur complement. To provide the stability of the numerical scheme in the presence of high density and viscosity ratios, we use an algebric momentum preserving mass and momentum transport in the conservative form of discrete equations. The consistency between mass and momentum advection is achieved by resolving a new auxiliary continuity equation, using fifth-order WENO scheme and third Runga-Kutta SSP time integrator. Test cases such as the transport of a very-high density fluid sphere and the free fall of dense sphere are performed to validate the models, especially in the presence of strong density and viscosity ratios. Other cases, like the phase inversion, demonstrate the ability of the new fully-coupled solver to tackle problems of more than one billion cells, with excellent scalability.