Characterizing elastic turbulence in the three-dimensional von Kármán swirling flow using the Oldroyd-B model

Abstract

We present a comprehensive three-dimensional numerical investigation of the von Kármán swirling flow between two parallel plates using the Oldroyd-B model and characterize the onset and development of elastic turbulence. We quantify the flow state with the secondary-flow strength, a measure of the average strength of the velocity fluctuations, and then define an order parameter as the time average of the secondary-flow strength. The order parameter displays a subcritical transition from the laminar to a bistable flow that switches between weakly chaotic flow and elastic turbulence. The transition to the bistable flow occurs at the critical Weissenberg number [Formula: see text]. In the elastic turbulent state, we observe a strong increase in velocity fluctuations and flow resistance which we define as the total work performed on the fluid. Upon starting simulations in the turbulent state and subsequently lowering [Formula: see text] below its critical value, we observe hysteretic behavior in the order parameter and the flow resistance, which is a common feature of a subcritical transition. Hysteresis has also been found in experiments. Additionally, we find power-law scaling in the spatial and temporal power spectra of the velocity fluctuations, a characteristic for elastic turbulence. The maximum values of the power-law exponents in our simulations are [Formula: see text] for the temporal exponent and [Formula: see text] for the spatial exponent, which are remarkably close to the values obtained in experiments.