Global vs local energy dissipation: The energy cycle of the turbulent von Kármán flow

Abstract

In this paper, we investigate the relations between global and local energy transfers in a turbulent von Kármán flow. The goal is to understand how and where energy is dissipated in such a flow and to reconstruct the energy cycle in an experimental device where local as well as global quantities can be measured. In order to do so, we use particle image velocimetry (PIV) measurements and we model the Reynolds stress tensor to take subgrid scales into account. This procedure involves a free parameter that is calibrated using angular momentum balance. We then estimate the local and global mean injected and dissipated powers for several types of impellers, for various Reynolds numbers, and for various flow topologies. These PIV estimates are then compared with direct injected power estimates provided by torque measurements at the impellers. The agreement between PIV estimates and direct measurements depends on the flow topology. In symmetric situations, we are able to capture up to 90% of the actual global energy dissipation rate. However, our results become increasingly inaccurate as the shear layer responsible for most of the dissipation approaches one of the impellers and cannot be resolved by our PIV setup. Finally, we show that a very good agreement between PIV estimates and direct measurements is obtained using a new method based on the work of Duchon and Robert [“Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations,” Nonlinearity 13, 249–225 (2000)] which generalizes the Kármán-Howarth equation to nonisotropic, nonhomogeneous flows. This method provides parameter-free estimates of the energy dissipation rate as long as the smallest resolved scale lies in the inertial range. These results are used to evidence a well-defined stationary energy cycle within the flow in which most of the energy is injected at the top and bottom impellers and dissipated within the shear layer. The influence of the mean flow geometry and the Reynolds number on this energy cycle is studied for a wide range of parameters.