Séance Séminaire

NEMESIS Seminar

HHO methods for variable density incompressible flows

We discuss a Hybrid High-Order formulation of the variable density incompressible Navier-Stokes equations designed to simulate mixtures of immiscible incompressible fluids [1]. The formulation provides exact conservation of volume at the discrete level and produces a pointwise divergence-free H-div conforming velocity field that serves as the advection velocity in the mass conservation equation. H-div conformity is achieved through pressure hybridization starting from broken polynomial spaces with no continuity requirements at interelement boundaries. In particular, the pressure space face unknowns impose continuity of the normal velocity components at cell interfaces while the pressure space cell unknowns enforce the divergence-free constraint within each mesh cell. Note that the role of the pressure at the continuous level is closely replicated at the discrete level. The pressure-velocity coupling was analysed in the context of the incompressible Stokes equation demonstrating stability, convergence and pressure-robustness of the formulation [2]. The turbulence modelling capabilities have been investigated in context of the constant density incompressible Navier-Stokes equations focusing on robustness in the inviscid limit and tackling the Taylor-Green vortex problem [3]. The variable density formulation is provably stable and bounds preserving at the lowest polynomial degree (k=0), accordingly, the density variable can be employed as a phase field variable for tracking the diffuse interface. The numerical test cases demonstrate that optimal convergence rates are achieved and the Rayleigh-Taylor instability has been successfully simulated considering different Atwood and Reynolds numbers. [1] Botti, L., Massa, F.C., A HHO formulation for variable density incompressible flows where the density is purely advected, submitted to CMAME, https://arxiv.org/abs/2510.15733v3}. [2] Botti, L., Botti, M., Di Pietro, D.A., Massa, F.C., Stability, convergence, and pressure-robustness of numerical schemes for incompressible flows with hybrid velocity and pressure, Mathematics of Computation, Vol. 95 (2025). [3] Botti, L., Di Pietro, D.A., Massa, F.C., Hybrid high-order formulations with turbulence modelling capabilities for incompressible flow problems, Computers \& Fluids, Vol. 305, (2026).