Efficiency of turbulence

TL;DR

This paper investigates the efficiency of turbulence, showing it is bounded and can saturate following power laws, with implications for understanding energy transfer and dissipation in turbulent flows.

Contribution

It introduces the concept of efficiency saturation in turbulence and analyzes its behavior across different flow types using numerical and experimental data.

Findings

Efficiency is bounded from above and can saturate following a power law.

Saturation of efficiency in von Kármán flow is insensitive to forcing details.

In Rayleigh-Bénard convection, efficiency saturates in the inviscid limit.

Abstract

We consider the efficiency of turbulence, a dimensionless parameter that characterises the fraction of the input energy stored into 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 K{\'a}rm{\'a}n flow the efficiency saturation is insensitive to the details of the forcing impellers. In the case of Rayleigh-B{\'e}nard convection, we show that within the Grossman 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 the 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.