  The need for accelerating the repeated solving of certain parametrized
systems motivates the development of more efficient reduced order methods. The
classical reduced basis method is popular due to an offline-online
decomposition and a mathematically rigorous {\em a posterior} error estimator
which guides a greedy algorithm offline. For nonlinear and nonaffine problems,
hyper reduction techniques have been introduced to make this decomposition
efficient. However, they may be tricky to implement and often degrade the
online computation efficiency.
  To avoid this degradation, reduced residual reduced over-collocation (R2-ROC)
was invented integrating empirical interpolation techniques on the solution
snapshots and well-chosen residuals, the collocation philosophy, and the
simplicity of evaluating the hyper-reduced well-chosen residuals. In this
paper, we introduce an adaptive enrichment strategy for R2-ROC rendering it
capable of handling parametric fluid flow problems. Built on top of an
underlying Marker and Cell (MAC) scheme, a novel hyper-reduced MAC scheme is
therefore presented and tested on Stokes and Navier-Stokes equations
demonstrating its high efficiency, stability and accuracy.
