Energy, enstrophy and helicity transfers in polymeric turbulence

TL;DR

This work develops an exact scale-by-scale framework for polymeric turbulence by extending the Generalised Kolmogorov Equation to dilute polymer solutions and deriving budgets for $\langle \delta q^2 \rangle$, $\langle \delta \omega^2 \rangle$, and $\langle \delta h \rangle$ that include fluid–polymer interactions. Employing direct numerical simulations at $Re_\lambda \approx 460$ and $De \in [1/9,9]$, the study reveals a polymer-driven energy and helicity transfer channel that dominates at small scales for $De \ge 1$, coexisting with the classical inertial transfer at large scales. Polymerization suppresses extreme energy-transfer events, weakens vortex stretching, and shifts local flow topology toward two-dimensional straining, while increasing relative helicity and breaking mirror symmetry more at higher elasticity. The results illuminate the multiscale mechanisms underlying polymeric drag reduction and provide a robust framework for exploring transfers in more complex, inhomogeneous flows.

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 $Re_λ\approx 460$, and the Deborah number $De = τ_p/τ_f$ is varied between $1/9 \le De \le 9$; $τ_p$ is the polymeric relaxation time and $τ_f$ 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.

Figures (21)

• Figure 1: Instantaneous visualisations of energy, enstrophy and helicity fields for $De=0$ (top) and $De=1$ (bottom). Left: $q^2 = u_i u_i$. Centre: $\omega^2 = \omega_i \omega_i$. Right: $h = u_i \omega_i$. All quantities are normalised with the average value in the considered slice.
• Figure 2: Dependence of the velocity structure function $\left\langle {\delta q^2} \right \rangle$ (left), vorticity structure function $\left\langle {\delta \omega^2} \right \rangle$ (centre) and helicity structure function $\left\langle {\delta h} \right \rangle$ (right) on the Deborah number. For $\left\langle {\delta q^2} \right \rangle$ and $\left\langle {\delta h} \right \rangle$ the lines for the different $De$ are vertically shifted for increase the clarity.
• Figure 3: Instantaneous visualisations of the dissipation fields and the polymeric source/sink terms for $De=1$. Top: Dissipation fields $\varepsilon_f^{\delta q^2}$ (left), $\varepsilon_f^{\delta \omega^2}$ (centre) and $\varepsilon_f^{\delta h}$ (right). Bottom: Polymeric source/sink terms $\pi^{\delta q^2}$ (left), $\pi^{\delta \omega^2}$ (centre) and $\pi^{\delta h}$ (right). All the quantities are normalised with the modulus of their averaged value.
• Figure 4: Distribution of the dissipation of (left) energy, (centre) enstrophy, and (right) helicity for different Deborah numbers. The bottom panels plot the distribution of the three quantities normalised with their root-mean-square value.
• Figure 5: Distribution of the the $\pi$ terms for $\delta q^2$, $\delta \omega^2$ and $\delta h$ for different Deborah numbers. Left: $\pi^{\delta q^2}$, centre: $\pi^{\delta \omega^2}$ and right: $\pi^{\delta h}$. The bottom panels plot the distribution of the three quantities normalised with their root-mean-square value.