Normalized kinetic energy as a hydrodynamical global quantity for inhomogeneous anisotropic turbulence

We introduce a hydrodynamical global quantity δ that characterizes turbulent fluctuations in inhomogeneous anisotropic flows. This time dependent quantity is constructed as the ratio of the instantaneous kinetic energy of the flow to the kinetic energy of the time-averaged flow. Such a normalization based on the dynamics of the flow makes this quantity comparable from one turbulent flow to any other. We show that δ(t) provides a useful quantitative characterization of any turbulent flow through generally only two parameters, its time average δ¯ and its variance δ2. These two quantities present topological and thermodynamical properties since they are connected, respectively, to the distance between the instantaneous and the time-averaged flow and to the number of degrees of freedom of the flow. Properties of δ¯ and δ2 are experimentally studied in the typical case of the von Kármán flow and used to characterize the scale by scale energy budget as a function of the forcing mode as well as the transition between two flow topologies.