Abstract

Simulations of elastic turbulence, the chaotic flow of highly elastic and inertialess polymer solutions, are plagued by numerical difficulties: The chaotically advected polymer conformation tensor develops extremely large gradients and can loose its positive definiteness, which triggers numerical instabilities. While efforts to tackle these issues have produced a plethora of specialized techniques—tensor decompositions, artificial diffusion, and shock-capturing advection schemes—we still lack an unambiguous route to accurate and efficient simulations. In this work, we show that even when a simulation is numerically stable, maintaining positive-definiteness and displaying the expected chaotic fluctuations, it can still suffer from errors significant enough to distort the large-scale dynamics and flow-structures. We focus on two-dimensional simulations of the Oldroyd-B and FENE -P equations, driven by a large-scale cellular body-forcing. We first compare two positivity-preserving decompositions of the conformation tensor: symmetric square root (SSR) and Cholesky with a logarithmic transformation (Cholesky-log). While both simulations yield chaotic flows, only the latter preserves the pattern of the forcing, i.e., its fluctuating vortical cells remain ordered in a lattice. In contrast, the SSR simulation exhibits distorted vortical cells which shrink, expand and reorient constantly. To identify the accurate simulation, we appeal to a hitherto overlooked mathematical bound on the determinant of the conformation tensor, which unequivocally rejects the SSR simulation. Importantly, the accuracy of the Cholesky-log simulation is shown to arise from the logarithmic transformation. We then consider local artificial diffusion, a potential low-cost alternative to high-order advection schemes. Unfortunately, the diffusive smearing of polymer stress in regions of intense stretching is found to destabilize the adjacent vortices, thereby modifying the global dynamics. We end with an example, showing how the spurious large-scale motions, identified here, contaminate predictions of scalar mixing.