No need to stay positive: a practical approach to direct numerical simulations of elastic turbulence

Abstract

Abstract

Successfully performing direct numerical simulations of polymeric flows remains a major challenge in computational fluid mechanics. In addition to the velocity field, such simulations must resolve polymeric degrees of freedom, often expressed via the conformation tensor, $\mathbf{c}$, which captures the local stretch of polymer molecules. A key difficulty here lies in maintaining the physical requirement $\mathrm{Tr}\, \mathbf{c}>3$, which is not explicitly enforced by the governing equations. Consequently, simulations initiated from physical conditions may silently drift into unphysical states with $\mathrm{Tr}\, \mathbf{c} 3$ can still yield meaningful physical insight into the underlying dynamics. We simulate a model dilute polymer solution driven through a plane channel at low Reynolds number and observe the transition to elastic turbulence. Our simulations exhibit two threshold resolutions: below the first, they become numerically unstable and exhibit a finite-time blow-up; above the second, they maintain positive-definiteness. In between, simulations remain stable and chaotic despite local violations of $\mathrm{Tr}\, \mathbf{c}>3$. Surprisingly, these violations do not affect mid-plane statistics of velocity, its gradients, or polymer stretch, which match results from fully positive-definite simulations. This suggests that resolving flow structures or key flow statistics may not require the extreme resolutions needed to preserve positive-definiteness, potentially lowering computational barriers for studying elastic turbulence.