The effect of periodicity in the elastic turbulence regime

Abstract

In this work, we show that the double-periodic boundary conditions typically applied in numerical simulations of elastic turbulence can lead to significantly incorrect results if not treated properly. This is demonstrated by simulating elastic turbulence using the popular four-roll mill benchmark at different levels of periodicity, namely, 16, 36 and 64 rolls using the popular Oldroyd-B model with added artificial diffusivity. We find that the initial onset of elastic turbulence causes a breakdown in symmetry independent of periodicity, which is characterised by a leading vortex and is known to be attributed to artificial diffusivity. Beyond this initial transition, the standard four-roll mill case transitions into a periodic state, a well-known characteristic from the literature. On the other hand, the cases with higher levels of periodicity quickly overcome the effects of a leading vortex and experience purely chaotic flow fluctuations, characterised by a broadband spectrum and steep power law behaviour. Certain qualities of the flow at higher levels of periodicity are reminiscent of the true solutions of elastic turbulence obtained numerically without any artificial diffusivity (Gupta & Vincenzi, J. Fluid Mech., vol. 870, 2019). These results suggest that the well-known periodic states observed for the four-roll mill are due to insufficient periodicity as the problem transitions into the elastic turbulence regime, leading to a dominant vortex cycling around all four quadrants of the unit cell throughout time unable to recover the initial symmetry. This work demonstrates the importance and caution required when applying periodic boundary conditions in numerical experiments of the elastic turbulence regime and further emphasises the impact and care required for artificial diffusivity.