Stability of generalized Kolmogorov flow in a channel

Abstract

The Kolmogorov flow is a paradigmatic model flow used to investigate the transition from laminar to turbulent regimes in confined and, especially, in unbounded domains. It represents a solution of the forced Navier–Stokes equation, where the forcing term is sinusoidal. The resulting velocity profile is also sinusoidal with the same wavenumber of the forcing term. In this study, we generalize the Kolmogorov flow making use of a generic forcing term defined by a Fourier series that bridges the classical Kolmogorov flow to an arbitrary even-degree power-law profile. Thereafter, we perform a linear stability analysis on the power-law profiles for exponents, α=2, 4, 6, 8, and 10, and the corresponding generalized Kolmogorov flows, varying the truncation index K of the Fourier series. Several neutral stability curves are computed numerically for wall-bounded flows and the relevant critical conditions are compared in terms of critical Reynolds number, critical wavelength, and eigenspectrum at criticality. The most dangerous perturbations are thoroughly characterized, and we identify three qualitatively different most dangerous modes, depending on α, K, the Reynolds number, and the perturbation wavelength.