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Computational Hydrodynamic Stability and Flow Control Based on Spectral Analysis of Linear Operators

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Abstract

This paper considers the analysis and control of fluid flows using tools from dynamical systems and control theory. The employed tools are derived from the spectral analysis of various linear operators associated with the Navier–Stokes equations. Spectral decomposition of the linearized Navier-Stokes operator, the Koopman operator, the spatial correlation operator and the Hankel operator provide a means to gain physical insight into the dynamics of complex flows and enables the construction of low-dimensional models suitable for control design. Since the discretization of the Navier-Stokes equations often leads to very large-scale dynamical systems, matrix-free and in some cases iterative techniques have to be employed to solve the eigenvalue problem. The common theme of the numerical algorithms is the use of direct numerical simulations. The theory and the algorithms are exemplified on flow over a flat plate and a jet in crossflow, as prototypes for the laminar-turbulent transition and three-dimensional vortex shedding.

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Notes

  1. Friction drag constitutes more than half of the total aircraft drag, with 18 %,4 %,3 % and 3 % for wing, horizontal tail plane, fin and nacelles, respectively. If the flow were laminar on 40 % of the surfaces, the total drag would be reduced by 16 % [92].

  2. Similar to bounded states in quantum mechanics, resulting in a discrete energy spectrum.

  3. A note on the basic notation used through out the paper is appropriate at this point. Square brackets [ and ] are used to construct matrices and vectors, i.e. [1 2]T is a column vector ∈ℝ2×1 which is abbreviated as ℝ2. Curved brackets ( and ) are used surrounding lists of entries, delineated by commas as an alternative method to construct (column) vectors, (1,2)=[1 2]T.

  4. The analysis requires some measure theory, but here we make no attempt to be mathematically precise and refer to [56] for rigorous treatment on the subject. Most importantly, we need to introduce an invariant measure, essentially meaning that we can find a measure μ such that the value of a integral,

    $$ \int_\mathbb{U}a(\mathbf{u}) d\mu= \int_\mathbb{U}a \bigl(\mathbf {g}(\mathbf {u}) \bigr)d\mu. $$

    is invariant. Henceforth we drop μ and use the notation dμ=d u. Such a measure can always be found [56] if g satisfies certain properties (that g is a measure-preserving operator). Observables are thus elements in the space

    $$ L^2(\mathbb{U}) = \biggl\{a:\mathbb{U}\rightarrow\mathbb{R} \, |\, \int_\mathbb{U}|a|^2 d\mathbf {u}< \infty \biggr\}. $$

  5. Here, we consider only the point spectrum of U, see [65] for the continuous spectrum.

  6. In the literature, \(\hat{n}\) is refereed to as the McMillan degree.

  7. The eigenvectors are the same if the sampling period Δt is chosen properly, i.e. so that it reflects the characteristic time scale of the physical structures in the flow. More specifically, to avoid aliasing Δt must be small enough such that two sampling points in one period of the highest frequency mode are obtained (the Nyquist criterion).

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Acknowledgements

I wish to thank Dan Henningson, Philipp Schlatter and Luca Brandt at KTH for the countless discussions and great ideas. Thanks to Peter Schmid and Clancy Rowley for my stays in LadHyx/Ecole Polytechnique and Princeton, where part of this work was performed. Financial support by the Swedish Research Council (VR-DNR 2010-3910) and computer time from the Swedish National Infrastructure for Computing (SNIC) is gratefully acknowledged.

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  1. DICAT, Universita‘ di Genova, Via Montallegro 1, 16145, Genova, Italy

    Shervin Bagheri

  2. Linné Flow Centre, Royal Institute of Technology (KTH), 10044, Stockholm, Sweden

    Shervin Bagheri

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Bagheri, S. Computational Hydrodynamic Stability and Flow Control Based on Spectral Analysis of Linear Operators. Arch Computat Methods Eng 19, 341–379 (2012). https://doi.org/10.1007/s11831-012-9074-0

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