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 Wic=12. 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 Wi 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 αt=3.69 for the temporal exponent and αs=3.18 for the spatial exponent, which are remarkably close to the values obtained in experiments.