Symmetries, dynamics, and control for the 2D Kolmogorov flow
type: Item
instance of: scholarly article
title: Symmetries, dynamics, and control for the 2D Kolmogorov flow
zbMATH Open document ID: 1390.93403
DOI: 10.1155/2018/4602485
author: Nejib Smaoui
published in: Complexity
publication date: 2018-07-06T00:00:00Z
review text: Summary: 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 \(0 < R_e < 26.41\). 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.
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 93B17
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 93D05
zbMATH DE Number: 6899114
zbMATH Keywords: 2D Kolmogorov flow
zbMATH Keywords: symmetries
zbMATH Keywords: dynamics
zbMATH Keywords: control
zbMATH Keywords: 2D Navier-Stokes (N-S) equations
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: sinusoidal external force
zbMATH Keywords: Fourier Galerkin method
describes a project that uses: DSTool
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/4602485
OpenAlex ID: W2801247914
cites work: On approximate inertial manifolds to the Navier-Stokes equations
cites work: Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds
cites work: On the construction of families of approximate inertial manifolds
cites work: Construction of inertial manifolds by elliptic regularization
cites work: The algebraic approximation of attractors: The finite dimensional case
cites work: Induced trajectories and approximate inertial manifolds
cites work: Model reduction on inertial manifolds for N-S equations approached by multilevel finite element method
cites work: Inertial manifolds for nonlinear evolutionary equations
cites work: Q4695261
cites work: Bifurcation computations on an approximate inertial manifold for the 2D Navier-Stokes equations
cites work: A five-dimensional truncation of the plane incompressible Navier-Stokes equations
cites work: A five-mode truncation of the Navier-Stokes equations on a three- dimensional torus
cites work: Common periodic behavior in larger and larger truncations of the Navier-Stokes equations.
cites work: Bifurcations of a steady-state solution to the two-dimensional Navier-Stokes equations
cites work: Transition to chaos in a fluid motion system
cites work: Chaotic behavior of a Galerkin model of a two-dimensional flow
cites work: Instability for the Navier-Stokes equations on the 2-dimensional torus and a lower bound for the Hausdorff dimension of their global attractors
cites work: Instabilities in two-dimensional spatially periodic flows. Part I: Kolmogorov flow
cites work: Q3351045
cites work: Turbulence and the dynamics of coherent structures. I. Coherent structures
cites work: Symmetries and dynamics for 2-D Navier-Stokes flow
cites work: Phase-space analysis of bursting behavior in Kolmogorov flow
cites work: Timely Communication:Symmetry and the Karhunen--Loève Analysis
cites work: A model for the unstable manifold of the bursting behavior in the 2D Navier-Stokes flow
cites work: Linear versus Nonlinear Dimensionality Reduction of High-Dimensional Dynamical Systems
cites work: On some control problems in fluid mechanics
cites work: Finite-dimensional approximation and control of non-linear parabolic PDE systems
cites work: Nonlinear control of incompressible fluid flow: Application to Burgers' equation and 2D channel flow
cites work: Adaptive control of a seven mode truncation of the Kolmogorov flow with drag
cites work: Dynamics and control of the 2-d Navier-Stokes equations
cites work: Dynamics and control of the seven-mode truncation system of the 2-d Navier-Stokes equations
cites work: On the control of the chaotic attractors of the 2-d Navier-Stokes equations
cites work: Q5576913
Recommended article: Symmetries and dynamics for 2-D Navier-Stokes flow
Recommended article: Dynamics and control of the 2-d Navier-Stokes equations
Recommended article: Some controllability results for the 2D Kolmogorov equation
Recommended article: Instabilities, bifurcations, and nonlinear dynamics in two-dimensional generalizations of Kolmogorov flow
Recommended article: Symmetries and conservation laws of two-dimensional hydrodynamics
Recommended article: Asymptotic analysis of the attractors in two-dimensional Kolmogorov flow
Recommended article: On the controllability of 2-D incompressible perfect fluids
Recommended article: Symmetries in vakonomic dynamics: Applications to optimal control
Recommended article: Q5487518
description: scientific article; zbMATH DE number 6899114