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For the discretization of the convective term in the Navier-Stokes equations(NSEs), the commonly used convective formulation (CONV) does not preserve theenergy if the divergence constraint is only weakly enforced. In this paper, weapply the skew-symmetrization technique in [B. Cockburn, G. Kanschat and D.Sch\"{o}tzau, Math. Comp., 74 (2005), pp. 1067-1095] to conforming finiteelement methods, which restores energy conservation for CONV. The crucial ideais to replace the discrete advective velocity with its a$H(\operatorname{div})$-conforming divergence-free approximation in CONV. Weprove that the modified convective formulation also conserves linear momentum,helicity, 2D enstrophy and total vorticity under some appropriate senses. Its aPicard-type linearization form also conserves them. Under the assumption$\boldsymbol{u}\in L^{2}(0,T;\boldsymbol{W}^{1,\infty}(\Omega)),$ it can beshown that the Gronwall constant does not explicitly depend on the Reynoldsnumber in the error estimates. The long time numerical simulations show thatthe linearized and modified convective formulation has a similar performancewith the EMAC formulation and outperforms the usual skew-symmetric formulation(SKEW).

Longer time simulation of the unsteady Navier-Stokes equations based on a modified convective formulation