Bifurcations of a plane parallel flow with Kolmogorov forcing

We study the primary bifurcations of a two-dimensional Kolmogorov flow in a channel subject to boundary conditions chosen to mimic a parallel flow, i.e., periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. The control parameter is the Reynolds number based on the friction coefficient, denoted as Rh. We find that as we increase Rh the laminar steady flow can display different types of bifurcation depending on the forcing wave number of the base flow. This is in contrast with the case of doubly periodic boundary conditions for which the primary bifurcation is stationary. In the present case, both stationary and Hopf bifurcations are observed. In addition, we discover a type of bifurcation with both the oscillation frequency and the amplitude of the growing mode being zero at the threshold, that we call a stationary drift bifurcation. A reduced four-mode model captures the scalings that are obtained from the numerical simulations. As we increase Rh further we observe a secondary instability which excites the largest mode in the domain. The saturated amplitude of the largest mode is found to scale as a 3 /2 power law of the distance to the threshold which is also explained using a low-dimensional model.