We present in this double contribution two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.

%B Lecture Notes in Computational Science and Engineering %I Springer International Publishing %C Cham %P 245–264 %@ 978-3-030-30705-9 %G eng %R 10.1007/978-3-030-30705-9_22 %0 Journal Article %J Computers and Mathematics with Applications %D 2020 %T Stabilized reduced basis methods for parametrized steady Stokes and Navier–Stokes equations %A Shafqat Ali %A F. Ballarin %A Gianluigi Rozza %XIt 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 offline–online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and 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.

%B Computers and Mathematics with Applications %V 80 %P 2399-2416 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85083340115&doi=10.1016%2fj.camwa.2020.03.019&partnerID=40&md5=7ace96eee080701acb04d8155008dd7d %R 10.1016/j.camwa.2020.03.019