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

Abstract

In this paper, we derive a four-mode model for the Kolmogorov flow by employing Galerkin truncation and Craya-Herring basis for the decomposition of velocity field. After this, we perform a bifurcation analysis of the model. Though our low-dimensional model has fewer modes than the past models, it captures the essential features of the primary bifurcation of the Kolmogorov flow. For example, it reproduces the critical Reynolds number for the supercritical pitchfork bifurcation and the flow structures of the past works. We also demonstrate energy transfers from intermediate scales to large scales. We perform direct numerical simulations of the Kolmogorov flow and show that our model predictions match with the numerical simulations very well.

Transitions of turbulent superstructures in generalized Kolmogorov flow

“…Interestingly, this wavenumber is also expected based on a linear stability analysis of laminar 2D Kolmogorov flow [23,34]. By increasing µ, this mode can be stabilized, triggering different flow patterns.…”

Inertial instability of the time-periodic Kolmogorov flow in a rotating fluid with the full account of the Coriolis force

“…Chatterjee and Verma (2020) derive a minimal low-dimensional four-mode model for Kolmogorov flow using Galerkin truncation and Craya-Herring basis for velocity field decomposition. The latter is an alternative to the stream function formalism used in this work.2 Gerkema et al (2008) give a thorough review of the effect of abandoning the traditional approximation in various geophysical and astrophysical problems.…”