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Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics pp 117–145Cite as

Unstable Periodically Forced Navier–Stokes Solutions–Towards Nonlinear First-Principle Reduced-Order Modeling of Actuator Performance

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Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 50))

Abstract

We advance the computation of physical modal expansions for unsteady incompressible flows. Point of departure is a linearization of the Navier–Stokes equations around its fixed point in a frequency domain formulation. While the most amplified stability eigenmode is readily identified by a power method, the technical challenge is the computation of more damped higher-order eigenmodes. This challenge is addressed by a novel method to compute unstable periodically forced solutions of the linearized Navier–Stokes solution. This method utilizes two key enablers. First, the linear dynamics is transformed by a complex shift of the eigenvalues amplifying the flow response at the given frequency of interest. Second, the growth rate is obtained from an iteration procedure. The method is demonstrated for several wake flows around a circular cylinder, a fluidic pinball, i.e. the wake behind a cluster of cylinders, a wall-mounted cylinder, a sphere and a delta wing. The example of flow control with periodic wake actuation and forced physical modes paves the way for applications of physical modal expansions. These results encourage Galerkin models of three-dimensional flows utilizing Navier–Stokes based modes.

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Acknowledgements

The authors acknowledge support by the Polish National Science Center (NCN) under the Grant No.: DEC-2011/01/B/ST8/07264 and by the Polish National Center for Research and Development under the Grant No. PBS3/B9/34/2015 and travel support of the Bernd Noack Cybernetics Foundation.

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Authors and Affiliations

  1. Chair of Virtual Engineering, Poznań University of Technology, Jana Pawła II 24, 60-965, Poznań, Poland

    Marek Morzyński & Wojciech Szeliga

  2. Poznań Supercomputing and Networking Center, Jana Pawła II 10, 61-139, Poznań, Poland

    Wojciech Szeliga

  3. LIMSI-CNRS, UPR 3251, Campus Univeristaire d’Orsay, Bât 508, Rue John von Neumann, 91405, Orsay, France

    Bernd R. Noack

  4. Graduate School Shenzhen, Institute for Turbulence-Noise-Vibration Interactions and Control, Harbin Institute of Technology, HIT Campus, University Town, Xili, Shenzhen, 58800, China

    Bernd R. Noack

  5. Hermann-Föttinger Institute, Technische Universtät Berlin, Straße des 17. Juni 135, 10623, Berlin, Germany

    Bernd R. Noack

Authors
  1. Marek Morzyński
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  2. Wojciech Szeliga
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  3. Bernd R. Noack
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Correspondence to Marek Morzyński .

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Editors and Affiliations

  1. School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel

    Alexander Gelfgat

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Morzyński, M., Szeliga, W., Noack, B.R. (2019). Unstable Periodically Forced Navier–Stokes Solutions–Towards Nonlinear First-Principle Reduced-Order Modeling of Actuator Performance. In: Gelfgat, A. (eds) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-91494-7_4

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  • DOI: https://doi.org/10.1007/978-3-319-91494-7_4

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  • Online ISBN: 978-3-319-91494-7

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