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Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows

Published online by Cambridge University Press:  04 February 2015

Bartosz Protas ,
Bernd R. Noack  and
Jan Östh
Bartosz Protas*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
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
Jan Östh
Affiliation:
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
*
†Email address for correspondence: bprotas@mcmaster.ca
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Abstract

We propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $\mathit{Re}=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $\mathit{Re}=300\,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Östh et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulent Flows Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 766 , 10 March 2015 , pp. 337 - 367
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© 2015 Cambridge University Press 

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