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Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial

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Abstract

The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier–Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.

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Authors and Affiliations

  1. Control and Dynamical Systems, California Institute of Technology, Mail Stop 107-81, 1200 E. California Blvd, Pasadena, CA, 91125, U.S.A.

    TROY R. SMITH

  2. Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA, 93106, U.S.A.

    JEFF MOEHLIS

  3. Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, U.S.A.

    PHILIP HOLMES

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  1. TROY R. SMITH
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SMITH, T.R., MOEHLIS, J. & HOLMES, P. Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial. Nonlinear Dyn 41, 275–307 (2005). https://doi.org/10.1007/s11071-005-2823-y

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