Upper bounds on functions of the dissipation rate in turbulent shear flow

The search for a statistical stability criterion characterizing steady-state turbulent shear flows ‘has led us to the study of a function proportional to the ratio of the fluctuation dissipation rate and the dissipation rate of the mean. The first Euler–Lagrange equations for an upper bound of this function have optimal solutions with the observed scaling laws and an asymptotic velocity defect for turbulent channel flow. As in the case of maximum transport, the optimal solution for the model equations is a discrete spectrum of streamwise vortices. It is shown how these solutions can be brought closer to the realized flow with additional constraints on the smallest vortex.