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Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation

Published online by Cambridge University Press:  18 July 2013

Maciej J. Balajewicz ,
Earl H. Dowell  and
Bernd R. Noack
Maciej J. Balajewicz*
Affiliation:
Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA
Earl H. Dowell
Affiliation:
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA
Bernd R. Noack
Affiliation:
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 POITIERS CEDEX, France
*
†Email address for correspondence: [email protected]
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Abstract

We generalize the POD-based Galerkin method for post-transient flow data by incorporating Navier–Stokes equation constraints. In this method, the derived Galerkin expansion minimizes the residual like POD, but with the power balance equation for the resolved turbulent kinetic energy as an additional optimization constraint. Thus, the projection of the Navier–Stokes equation on to the expansion modes yields a Galerkin system that respects the power balance on the attractor. The resulting dynamical system requires no stabilizing eddy-viscosity term – contrary to other POD models of high-Reynolds-number flows. The proposed Galerkin method is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer. Generalizations for more Navier–Stokes constraints, e.g. Reynolds equations, can be achieved in straightforward variation of the presented results.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulence modelling
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 285 - 308
Copyright
©2013 Cambridge University Press 

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