Abstract
This paper is devoted to the control problem of a nonlinear dynamical system obtained by a truncation of the two-dimensional (2D) Navier–Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along the x-direction. This special case of the 2D N-S equations is known as the 2D Kolmogorov flow. Firstly, the dynamics of the 2D Kolmogorov flow which is represented by a nonlinear dynamical system of seven ordinary differential equations (ODEs) of a laminar steady state flow regime and a periodic flow regime are analyzed; numerical simulations are given to illustrate the analysis. Secondly, an adaptive controller is designed for the system of seven ODEs representing the approximation of the dynamics of the 2D Kolmogorov flow to control its dynamics either to a steady-state regime or to a periodic regime; the value of the Reynolds number is determined using an update law. Then, a static sliding mode controller and a dynamic sliding mode controller are designed for the system of seven ODEs representing the approximation of the dynamics of the 2D Kolmogorov flow to control its dynamics either to a steady-state regime or to a periodic regime. Numerical simulations are presented to show the effectiveness of the proposed three control schemes. The simulation results clearly show that the proposed controllers work well.
Highlights
We study the dynamics as well as the adaptive and the sliding mode control problem of seven-mode truncation system of the 2D Navier–Stokes equations with periodic boundary conditions and a sinusoidal external force along the x-direction. is type of forcing is known as the Kolmogorov forcing and the resulting flow is known as the 2D Kolmogorov flow
Smaoui and Zribi [26,27,28] constructed reduced order ordinary differential equations (ODEs) systems that approximate the dynamics of the 2D Navier–Stokes equations using the truncated Fourier expansion method when the external force along the x-direction acts on the mode (0, k)
The dynamics of a steady state flow regime and a periodic regime flow observed in a dynamical system of a nonlinear dynamical system of seventh-order nonlinear differential equations truncated from the 2D Navier–Stokes equations with periodic boundary conditions and a sinusoidal external force known as 2D Kolmogorov flow is analyzed. en, an adaptive controller is designed to drag the dynamics of Kolmogorov flow either to a steady state flow regime or to a periodic flow regime
Summary
We study the dynamics as well as the adaptive and the sliding mode control problem of seven-mode truncation system of the 2D Navier–Stokes equations with periodic boundary conditions and a sinusoidal external force along the x-direction. is type of forcing is known as the Kolmogorov forcing and the resulting flow is known as the 2D Kolmogorov flow. We study the dynamics as well as the adaptive and the sliding mode control problem of seven-mode truncation system of the 2D Navier–Stokes equations with periodic boundary conditions and a sinusoidal external force along the x-direction. In 1997, using the Karhunen–Loeve decomposition and symmetry, Smaoui and Armbruster [11] used a computationally effective method to construct a reduced order system of nonlinear ODEs that approximates the dynamics of Kolmogorov flow when the external force acts on the mode (0, 2). Smaoui and Zribi [26,27,28] constructed reduced order ODE systems that approximate the dynamics of the 2D Navier–Stokes equations using the truncated Fourier expansion method when the external force along the x-direction acts on the mode (0, k). Where rx, ry, and ro are reflection symmetries across the xaxis, the y-axis, and the origin, respectively. erefore, it can be concluded that rx, ry, ro with the identity transformation i form an Abelian group: G rx, ry, ro, i
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