Analysis of the Kolmogorov model with an asymptotic-preserving method

•Kolmogorov linear stability problem solved by asymptotic matching.•Nonlinear steady state exhibits seemingly universal solutions.•Asymptotic-Preserving methods allow efficient integration to long times.•Asymptotic-Preserving schemes keep numerical accuracy at all times. We study the Kolmogorov model of a shear flow by means of a newly developed Asymptotic-Preserving method for the numerical resolution of the two-dimensional vorticity-Poisson (Navier-Stokes) system. The scheme is validated by comparing the results with those obtained with an explicit spectral code and with an analytic result about the linear instability regime. We show that the AP-properties of the method allow one to deal efficiently with the multi-scale nature of the problem by tuning the time step to the physical one and not by stability constraints. As a result, we investigate the long time scale evolution of the Kolmogorov flow, observing that it evolves into a final stable stationary state characterised by a seemingly universal relation between stream-function and vorticity.