Hybrid High-order formulations with turbulence modelling capabilities for incompressible flow problems

• A Hybrid High-Order method for incompressible Navier-Stokes is presented. • The method is Reynolds semi-robust and pressure-robust by design. • Conservation of volume is enforced cell-wise up to machine precision. • ESDIRK time stepping enables local high-order accurate time integration. • Validated on 2D/3D cases including the Taylor-Green vortex at Re = 1600. We propose a Hybrid High-Order (HHO) formulation of the incompressible Navier–Stokes equations, that is well suited to be employed for the simulation of turbulent flows. The spatial discretization relies on hybrid velocity and pressure spaces and the temporal discretization is based on Explicit Singly Diagonal Implicit Runge-Kutta (ESDIRK) methods. The formulation possesses some attractive features that can be fruitfully exploited when high-fidelity computations are required, namely: pressure-robustness, conservation of volume enforced cell-by-cell up to machine precision, robustness in the inviscid limit, implicit high-order accurate time stepping with local time step adaptation, reduced memory footprint thanks to static condensation of both velocity and pressure, possibility to exploit inherited p -multilevel solution strategies to improve performance of iterative solvers. After demonstrating the relevant properties of the scheme in practice, performing challenging 2D and 3D test cases, we consider the simulation of the Taylor–Green Vortex flow problem at Reynolds 1 600.