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

In this paper, we derive a four-mode model for the Kolmogorov flow by employing Galerkin truncation and the 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 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 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 the numerical simulations very well.