  In this work we propose a novel Hybrid High-Order method for the
incompressible Navier--Stokes equations based on a formulation of the
convective term including Temam's device for stability. The proposed method has
several advantageous features: it supports arbitrary approximation orders on
general meshes including polyhedral elements and non-matching interfaces; it is
inf-sup stable; it is locally conservative; it supports both the weak and
strong enforcement of velocity boundary conditions; it is amenable to efficient
computer implementations where a large subset of the unknowns is eliminated by
solving local problems inside each element. Particular care is devoted to the
design of the convective trilinear form, which mimicks at the discrete level
the non-dissipation property of the continuous one. The possibility to add a
convective stabilisation term is also contemplated, and a formulation covering
various classical options is discussed. The proposed method is theoretically
analysed, and an energy error estimate in $h^{k+1}$ (with $h$ denoting the
meshsize) is proved under the usual data smallness assumption. A thorough
numerical validation on two and three-dimensional test cases is provided both
to confirm the theoretical convergence rates and to assess the method in more
physical configurations (including, in particular, the well-known two- and
three-dimensional lid-driven cavity problems).
