  This work focuses on steady and unsteady Navier-Stokes equations in a reduced
order modeling framework. The methodology proposed is based on a Proper
Orthogonal Decomposition within a levelset geometry description and the
problems of interest are discretized with an unfitted mesh Finite Element
Method. We construct and investigate a unified and geometry independent reduced
basis which overcomes many barriers and complications of the past, that may
occur whenever geometrical morphings are taking place. By employing a geometry
independent reduced basis, we are able to avoid remeshing and transformation to
reference configurations, and we are able to handle complex geometries. This
combination of a fixed background mesh in a fixed extended background geometry
with reduced order techniques appears beneficial and advantageous in many
industrial and engineering applications, which could not be resolved
efficiently in the past.
