Kolmogorov flow: Linear Stability and Energy Transfers in a minimal low-dimensional model

TL;DR

This paper introduces a minimal four-mode model for Kolmogorov flow that accurately captures bifurcation behavior and energy transfer dynamics, validated by direct numerical simulations.

Contribution

A novel low-dimensional four-mode model for Kolmogorov flow that reproduces key bifurcation features and energy transfer mechanisms.

Findings

Model accurately predicts critical Reynolds number.

Reproduces flow structures observed in previous studies.

Energy transfers from intermediate to large scales.

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.