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Proper EMA-balance (E: kinetic energy; M: momentum; A: angular momentum),pressure-robustness and $Re$-semi-robustness ($Re$: Reynolds number) are threeimportant properties of Navier-Stokes simulations with exactly divergence-freeelements. This EMA-balance makes a method conserve kinetic energy, linearmomentum and angular momentum under some suitable senses; pressure-robustnessmeans that the velocity errors are independent of the continuous pressure;$Re$-semi-robustness means that the constants appearing in the error bounds ofkinetic and dissipation energies do not explicitly depend on inverse powers ofthe viscosity. In this paper, based on the pressure-robust reconstructionmethods in [{A. Linke and C. Merdon, {\it Comput. Methods Appl. Mech. Engrg.}311 (2016), 304-326}], we propose a novel reconstruction method for a class ofnon-divergence-free simplicial elements which admits almost all the aboveproperties. The only exception is the energy balance, where kinetic energyshould be replaced by a properly redefined discrete energy. Some numericalcomparisons with exactly divergence-free methods, pressure-robustreconstructions and the EMAC scheme are provided to confirm our theoreticalresults.

An EMA-conserving, pressure-robust and Re-semi-robust reconstruction method for the unsteady incompressible Navier-Stokes equations