Large-scale instability of a generalized turbulent Kolmogorov flow

We present an analytical study of the large scale instability of a generalized turbulent Kolmogorov flow, i.e. a periodic shear flow where the molecular viscosity has been substituted by an eddy viscosity parameterized with the Clark-Smagorinsky model and where the external forcing is adapted to maintain the flow against this dissipation. We employ multiscaling technique assuming a scale separation between the basic scale of such a generalized turbulent Kolmogorov flow and the largest scales of the flow. The main result is that an amplitude equation for the large-scale secondary flow is obtained which exhibits, like for the standard Kolmogorov flow, an instability of the negative viscosity type. We find that the presence of mirror symmetry in the basic flow is a necessary condition and that further propagative and nonlinear contribution are produced otherwise. The result is encouraging for the generic existence of large-scale instabilities of the negative viscosity type in fully turbulent flows.