Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow
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 x-direction. First, using the Fourier Gale...
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Online Access: | http://dx.doi.org/10.1155/2018/4602485 |
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doaj-51bfe388c3014b0397671ace00dc7f552020-11-25T00:27:26ZengHindawi-WileyComplexity1076-27871099-05262018-01-01201810.1155/2018/46024854602485Symmetries, Dynamics, and Control for the 2D Kolmogorov FlowNejib Smaoui0Department of Mathematics, Kuwait University, P.O. Box 5969, 13060 Safat, KuwaitThe 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 x-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 0<Re<26.41. 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.http://dx.doi.org/10.1155/2018/4602485 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Nejib Smaoui |
spellingShingle |
Nejib Smaoui Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow Complexity |
author_facet |
Nejib Smaoui |
author_sort |
Nejib Smaoui |
title |
Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow |
title_short |
Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow |
title_full |
Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow |
title_fullStr |
Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow |
title_full_unstemmed |
Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow |
title_sort |
symmetries, dynamics, and control for the 2d kolmogorov flow |
publisher |
Hindawi-Wiley |
series |
Complexity |
issn |
1076-2787 1099-0526 |
publishDate |
2018-01-01 |
description |
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 x-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 0<Re<26.41. 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. |
url |
http://dx.doi.org/10.1155/2018/4602485 |
work_keys_str_mv |
AT nejibsmaoui symmetriesdynamicsandcontrolforthe2dkolmogorovflow |
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