A reduced order variational multiscale approach for turbulent flows

  • Giovanni Stabile
  • Francesco BallarinEmail author
  • Giacomo Zuccarino
  • Gianluigi Rozza
Part of the following topical collections:
  1. Model reduction of parametrized Systems


The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case, the VMS stabilization method is used at both the full order and the reduced order levels. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.


Navier-Stokes equations Variational multiscale Reduced order methods High Reynolds number flows 

Mathematics Subject Classification (2010)

65N12 65N30 76D05 76F65 


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We acknowledge Prof. Guglielmo Scovazzi from Duke University for the fruitful discussions concerning the implementation of the VMS method. The computations in this work have been performed with RBniCS [1] library, developed at SISSA mathLab, which is an implementation in FEniCS [40] of several reduced order modeling techniques; we acknowledge developers and contributors to both libraries.

Funding information

This study received support from the European Union Funding for Research and Innovation - Horizon 2020 Program - in the framework of European Research Council Executive Agency: H2020 ERC Consolidator Grant 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics.” We also acknowledge the INDAM-GNCS projects “Metodi numerici avanzati combinati con tecniche di riduzione computazionale per PDEs parametrizzate e applicazioni” and “Numerical methods for model order reduction of PDEs.”


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.mathLab, Mathematics Area, SISSATriesteItaly

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