  Geometrically parametrized Partial Differential Equations are nowadays widely
used in many different fields as, for example, shape optimization processes or
patient specific surgery studies. The focus of this work is on some advances
for this topic, capable of increasing the accuracy with respect to previous
approaches while relying on a high cost-benefit ratio performance. The main
scope of this paper is the introduction of a new technique mixing up a
classical Galerkin-projection approach together with a data-driven method to
obtain a versatile and accurate algorithm for the resolution of geometrically
parametrized incompressible turbulent Navier-Stokes problems. The effectiveness
of this procedure is demonstrated on two different test cases: a classical
academic back step problem and a shape deformation Ahmed body application. The
results show into details the properties of the architecture we developed while
exposing possible future perspectives for this work.
