Full and reduced order model consistency of the nonlinearity discretization in incompressible flows

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