Energy, enstrophy and helicity transfers in polymeric turbulence

Abstract

We characterise the scale-by-scale transfers of energy, enstrophy and helicity in homogeneous and isotropic polymeric turbulence using direct numerical simulations. The microscale Reynolds number is set to $Re_\lambda \approx 460$, and the Deborah number $De = \tau_p/\tau_f$ is varied between $1/9 \le De \le 9$; $\tau_p$ is the polymeric relaxation time and $\tau_f$ is the turnover time of the largest scales of the flow. The study relies on the exact scale-by-scale budget equations (derived from the the governing model equations) for energy, enstrophy and helicity, which account for the back-reaction of the polymers on the flow. Polymers act as a sink/source in the flow, and provide alternative routes for the scale-by-scale transfers of the three quantities, whose relevance changes with $De$. We find that polymers deplete the nonlinear energy cascade mainly at smaller scales, by weakening both the extreme forward as well as reverse local events. The new polymer-driven energy flux dominates at small scales for $De \ge 1$, and on average transfers energy from larger to smaller scales with localised backscatter events. Polymers weaken the stretching of vorticity with the enstrophy being mainly generated by the fluid-polymer interaction, especially when $De \ge 1$. Accordingly, an inspection of the small-scale flow topology shows that polymers favour events with two-dimensional state of straining, and promote/inhibit extreme extension/rotation events: in polymeric turbulence shear and planar extensional flows are more probable. The helicity injected at the largest scales shows a similar transfer process to as energy, being mainly driven by the nonlinear cascade at large scales and by the polymer-driven flux at small scales. Polymers are found to favour events that break the small-scale mirror symmetry, with the relative helicity monotonically increasing with $De$ at all scales.