  It is well known in the Reduced Basis approximation of saddle point problems
that the Galerkin projection on the reduced space does not guarantee the
inf-sup approximation stability even if a stable high fidelity method was used
to generate snapshots. For problems in computational fluid dynamics, the lack
of inf-sup stability is reflected by the inability to accurately approximate
the pressure field. In this context, inf-sup stability is usually recovered
through the enrichment of the velocity space with suitable supremizer
functions. The main goal of this work is to propose an alternative approach,
which relies on the residual based stabilization techniques customarily
employed in the Finite Element literature, such as Brezzi-Pitkaranta,
Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square. In the
spirit of \textit{offline-online} reduced basis computational splitting, two
such options are proposed, namely \textit{offline-only stabilization} and
\textit{offline-online stabilization}. These approaches are then compared to
(and combined with) the state of the art supremizer enrichment approach.
Numerical results are discussed, highlighting that the proposed methodology
allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer
enrichment) for which a modified inf-sup stability is still preserved at the
reduced order level.
