Efficiency of turbulence

We consider the efficiency of turbulence, a dimensionless parameter that characterises the fraction of the input energy stored in a turbulent flow field. We first show that the inverse of the efficiency provides an upper bound for the dimensionless energy injection in a turbulent flow. We analyse the efficiency of turbulence for different flows using numerical and experimental data. Our analysis suggests that efficiency is bounded from above, and, in some cases, saturates following a power law reminiscent of phase transitions and bifurcations. We show that for the von Kármán flow the efficiency saturation is insensitive to the details of the forcing impellers. In the case of Rayleigh–Bénard convection, we show that within the Grossmann and Lohse model, the efficiency saturates in the inviscid limit, while the dimensionless kinetic energy injection/dissipation goes to zero. In the case of pipe flow, we show that saturation of the efficiency cannot be excluded, but would be incompatible with the Prandtl law of the drag friction coefficient. Furthermore, if the power-law behaviour holds for the efficiency saturation, it can explain the kinetic energy and the energy dissipation defect laws proposed for shear flows. Efficiency saturation is an interesting empirical property of turbulence that may help in evaluating the ‘closeness’ of experimental and numerical data to the true turbulent regime, wherein the kinetic energy saturates to its inviscid limit.