Kolmogorov Flow: Seven Decades of History

Abstract

The Kolmogorov flow (k-flow) is generated by a stationary sinusoidal force that varies in space. This flow is rather academic since generating such a periodic forcing in an unbounded flow is difficult to appear in nature. Nevertheless, it allows for simple experimental measurements and for a fairly detailed analytical treatment. Although simple, the k-flow makes a good test case for investigating simultaneously inhomogeneous, sheared, and anisotropic features in a flow, and several studies concerning the stability, transition, and turbulence of the k-flow have been published. The present article reviews the most important published works incorporating the k-flow as a test-bed for studying fluid mechanics, testing numerical or experimental methods, or even studying the properties of the k-flow itself.

Kolmogorov flow: Linear stability and energy transfers in a minimal low-dimensional model

“…Mishra et al 30 studied the condensate regime using Ekman friction. There are more works on the Kolmogorov flow, including those by Chandler and Kerswell 31 , Lucas and Kerswell 32 & Fylladitakis 33 , and references therein.…”

On the Control of the 2D Navier–Stokes Equations with Kolmogorov Forcing

“…In this section, we analyze the dynamics of the seven-mode truncation ODE system given by (5) when the Reynolds number R e � 20 and R e � 24 corresponding to a steady-state laminar regime and a periodic regime, respectively.…”

“…At R e � 20, numerical simulations of system (5) shows that the system has seven fixed points. ese points are classified as four asymptotically stable points and three unstable points.…”

“…In this section, sliding mode controllers are proposed to control the dynamics of the seven-mode truncation ODE system presented in (5). e choice of this type of controllers is motivated by the fact that sliding mode controllers are known for their robustness and their insensitivity to modelling errors [43][44][45][46][47][48][49][50][51].…”

“…In the field of magnetohydrodynamics, Kolmogorov flow has been extensively used and was easily reproduced by suitably placed electrical and magnetic fields. Bondarenko et al [4] observed this flow inside a specific electrically conducting fluid driven by an electromagnetic field [5]. In the literature, linear and nonlinear stability analysis of this flow was investigated for different domain sizes and forcing wave numbers [6][7][8][9].…”