Supercritical transition to turbulence in an inertially driven von Kármán closed flow

Abstract

We study the transition from laminar flow to fully developed turbulence for an inertially driven von Kármán flow between two counter-rotating large impellers fitted with curved blades over a wide range of Reynolds number (102–106). The transition is driven by the destabilization of the azimuthal shear layer, i.e. Kelvin–Helmholtz instability, which exhibits travelling/drifting waves, modulated travelling waves and chaos before the emergence of a turbulent spectrum. A local quantity – the energy of the velocity fluctuations at a given point – and a global quantity – the applied torque – are used to monitor the dynamics. The local quantity defines a critical Reynolds numberRecfor the onset of time-dependence in the flow, and an upper threshold/crossoverRetfor the saturation of the energy cascade. The dimensionless drag coefficient, i.e. the turbulent dissipation, reaches a plateau above this finiteRet, as expected for ‘Kolmogorov’-like turbulence forRe→∞. Our observations suggest that the transition to turbulence in this closed flow is globally supercritical: the energy of the velocity fluctuations can be considered as an order parameter characterizing the dynamics from the first laminar time-dependence to the fully developed turbulence. Spectral analysis in the temporal domain, moreover, reveals that almost all of the fluctuation energy is stored in time scales one or two orders of magnitude slower than the time scale based on impeller frequency.