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Colour of turbulence

Published online by Cambridge University Press:  05 January 2017

Armin Zare ,
Mihailo R. Jovanović Open the ORCID record for Mihailo R. Jovanović [Opens in a new window]  and
Tryphon T. Georgiou
Armin Zare
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Mihailo R. Jovanović*
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Tryphon T. Georgiou
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: mihailo@umn.edu
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Abstract

In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier–Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for coloured-in-time forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that coloured-in-time excitation of the Navier–Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization and control design.

JFM classification

Flow Control: Control theory Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent Flows
Type
Papers
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Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 636 - 680
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© 2017 Cambridge University Press 

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