  We investigate both theoretically and numerically the consistency between the
nonlinear discretization in full order models (FOMs) and reduced order models
(ROMs) for incompressible flows. To this end, we consider two cases: (i)
FOM-ROM consistency, i.e., when we use the same nonlinearity discretization in
the FOM and ROM; and (ii) FOM-ROM inconsistency, i.e., when we use different
nonlinearity discretizations in the FOM and ROM. Analytically, we prove that
while the FOM-ROM consistency yields optimal error bounds, FOM-ROM
inconsistency yields additional terms dependent on the FOM divergence error,
which prevent the ROM from recovering the FOM as the number of modes increases.
Computationally, we consider channel flow around a cylinder and
Kelvin-Helmholtz instability, and show that FOM-ROM consistency yields
significantly more accurate results than FOM-ROM inconsistency.
