  Proper EMA-balance (E: kinetic energy; M: momentum; A: angular momentum),
pressure-robustness and $Re$-semi-robustness ($Re$: Reynolds number) are three
important properties of Navier-Stokes simulations with exactly divergence-free
elements. This EMA-balance makes a method conserve kinetic energy, linear
momentum and angular momentum under some suitable senses; pressure-robustness
means that the velocity errors are independent of the continuous pressure;
$Re$-semi-robustness means that the constants appearing in the error bounds of
kinetic and dissipation energies do not explicitly depend on inverse powers of
the viscosity. In this paper, based on the pressure-robust reconstruction
methods 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 of
non-divergence-free simplicial elements which admits almost all the above
properties. The only exception is the energy balance, where kinetic energy
should be replaced by a properly redefined discrete energy. Some numerical
comparisons with exactly divergence-free methods, pressure-robust
reconstructions and the EMAC scheme are provided to confirm our theoretical
results.
