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Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow

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  • Nejib Smaoui

Abstract

The symmetries, dynamics, and control problem of the two-dimensional (2D) Kolmogorov flow are addressed. The 2D Kolmogorov flow is known as the 2D Navier-Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along the -direction. First, using the Fourier Galerkin method on the original 2D Navier-Stokes equations, we obtain a seventh-order system of nonlinear ordinary differential equations (ODEs) which approximates the behavior of the Kolmogorov flow. The dynamics and symmetries of the reduced seventh-order ODE system are analyzed through computer simulations for the Reynolds number range . Extensive numerical simulations show that the obtained system is able to display the different behaviors of the Kolmogorov flow. Then, we design Lyapunov based controllers to control the dynamics of the system of ODEs to different attractors (e.g., a fixed point, a periodic orbit, or a chaotic attractor). Finally, numerical simulations are undertaken to validate the theoretical developments.

Suggested Citation

  • Nejib Smaoui, 2018. "Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow," Complexity, Hindawi, vol. 2018, pages 1-15, May.
    • Handle: RePEc:hin:complx:4602485
    DOI: 10.1155/2018/4602485
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    References listed on IDEAS

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    1. Chen, Zhi-Min & Price, W.G., 2005. "Transition to chaos in a fluid motion system," Chaos, Solitons & Fractals, Elsevier, vol. 26(4), pages 1195-1202.
    2. Gambino, G. & Lombardo, M.C. & Sammartino, M., 2009. "Adaptive control of a seven mode truncation of the Kolmogorov flow with drag," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 47-59.
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