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We propose two Hybrid High-Order (HHO) methods for the incompressibleNavier-Stokes equations and investigate their robustness with respect to theReynolds number. While both methods rely on a HHO formulation of the viscousterm, the pressure-velocity coupling is fundamentally different, up to thepoint that the two approaches can be considered antithetical. The first methodis kinetic energy preserving, meaning that the skew-symmetric discretization ofthe convective term is guaranteed not to alter the kinetic energy balance. Theapproximated velocity fields exactly satisfy the divergence free constraint andcontinuity of the normal component of the velocity is weakly enforced on themesh skeleton, leading to H-div conformity. The second scheme relies on Godunovfluxes for pressure-velocity coupling: a Harten, Lax and van Leer (HLL)approximated Riemann Solver designed for cell centered formulations is adaptedto hybrid face centered formulations. The resulting numerical scheme is robustup to the inviscid limit, meaning that it can be applied for seekingapproximate solutions of the incompressible Euler equations. The schemes arenumerically validated performing steady and unsteady two dimensional test casesand evaluating the convergence rates on h-refined mesh sequences. In additionto standard benchmark flow problems, specifically conceived test cases areconducted for studying the error behaviour when approaching the inviscid limit.

HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations