2.2.1. Homogeneous isotropic turbulence

For homogeneous statistically steady flows, the three structure functions and the terms of the corresponding budget equations lose their dependence on $\boldsymbol{X}$ and $t$ , and are functions of only the separation vector $\boldsymbol{r} \equiv (r_x,r_y,r_z)$ . In HIT, the mean-flow is absent, and $\boldsymbol{U} = \boldsymbol{0}$ and $\boldsymbol{\varOmega } = \boldsymbol{0}$ , implying $\mathcal{P}^{\delta g} = 0$ . Exploiting isotropy, we integrate (2.8) in the $\boldsymbol{r}$ space over spherical shells of radius $r = |\boldsymbol{r}|$ , with surface $\mathcal{S}(r)$ and volume $\mathcal{V}(r)$ , and obtain the generalised Kármán–Howarth equation for PHIT (see Chiarini, Singh & Rosti Reference Chiarini, Singh and Rosti2025, for further details on the energy equation), which in a compact form reads

(2.31) \begin{align} \left\langle \varPhi _{\!f}^{\delta g} \right\rangle (r) + \left\langle \varPhi _p^{\delta g} \right\rangle (r) - \left\langle D_{\!f}^{\delta g} \right\rangle (r) - \left\langle V_{st}^{\delta g} \right\rangle (r) - \left\langle F^{\delta g} \right\rangle (r) = - \frac {4}{3} \left\langle \varepsilon _{\!f}^{\delta g} \right\rangle r - \frac {4}{3} \left\langle \pi ^{\delta g} \right\rangle r. \end{align}

On the left-hand side, $\varPhi _{\!f}^{\delta g}$ and $\varPhi _p^{\delta g}$ represent the contributions of fluid–fluid and fluid–polymer interactions, respectively, to the scale-space transfer of $\delta g$ , and are given by

(2.32) \begin{align} \left\langle \varPhi _{\!f}^{\delta q^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} \left \langle { \delta u_{\!j} \delta q^2 }\right \rangle n_{\!j} \,\text{d} \varSigma , \end{align}

(2.33) \begin{align} \left\langle \varPhi _{\!f}^{\delta \omega ^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} \left \langle { \delta u_{\!j} \delta \omega ^2 }\right \rangle n_{\!j} \,\text{d} \varSigma , \end{align}

(2.34) \begin{align} \left\langle \varPhi _{\!f}^{\delta h} \right\rangle = & \underbrace { \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} \left \langle { \delta u_{\!j} \delta h}\right \rangle n_{\!j} \,\text{d} \varSigma }_{ \left \langle { \varPhi _{f,a}^{\delta h} }\right \rangle } + \underbrace { \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} -\left \langle { \delta \omega _{\!j} \delta q^2 }\right \rangle n_{\!j} \,\text{d} \varSigma }_{ \left \langle { \varPhi _{f,b}^{\delta h} }\right \rangle } \end{align}

and

(2.35) \begin{align} \left\langle \varPhi _p^{\delta q^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} - 4 \left \langle { \delta u_i T_{\textit{ij}}^* }\right \rangle /\rho \ n_{\!j} \,\text{d} \varSigma , \end{align}

(2.36) \begin{align} \left\langle \varPhi _p^{\delta \omega ^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} - 4 \left \langle { \delta \omega _i T_{\textit{ij}}^{\omega ,*} }\right \rangle /\rho \ n_{\!j} \,\text{d} \varSigma , \end{align}

(2.37) \begin{align} \left\langle \varPhi _p^{\delta h} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} - 2 ( \left \langle { \delta u_i T_{\textit{ij}}^{\omega ,*} }\right \rangle + \left \langle { \delta \omega _i T_{\textit{ij}}^* }\right \rangle)/\rho \ n_{\!j} \,\text{d} \varSigma , \end{align}

where $\boldsymbol{n}$ is the outward-pointing unit vector normal to the spherical surface.

Instead, $V_{st}$ denotes the vortex stretching term, which contributes to enstrophy generation across all scales ( $V_{st}^{\delta q^2} = V_{st}^{\delta h} = 0$ ), and is expressed as

(2.38) \begin{align} \left \langle { V_{st}^{\delta \omega ^2} }\right \rangle = \frac {1}{\mathcal{S}(r)} \int _{\mathcal{V}(r)} 2 \left ( \left \langle { \delta \omega _i \omega _{\!j}^* \delta \left ( \frac {\partial u_i}{\partial x_{\!j}} \right) }\right \rangle + \left \langle { \delta \omega _i \delta \omega _{\!j} \left ( \frac {\partial u_i}{\partial x_{\!j}} \right)^* }\right \rangle \right) \,\text{d} \varOmega . \end{align}

Viscous diffusion at scale $r$ is expressed as

(2.39) \begin{align} \left\langle D_{\!f}^{\delta q^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} 2 \nu \frac {\partial \left \langle {\delta q^2}\right \rangle }{\partial r_{\!j}} n_{\!j} \,\text{d} \varSigma , \end{align}

(2.40) \begin{align} \left\langle D_{\!f}^{\delta \omega ^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} 2 \nu \frac {\partial \left \langle {\delta \omega ^2}\right \rangle }{\partial r_{\!j}} n_{\!j}\, \text{d} \varSigma , \end{align}

(2.41) \begin{align} \left\langle D_{\!f}^{\delta h } \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \oint _{\mathcal{S}(r)} 2 \nu \frac {\partial \left \langle {\delta h}\right \rangle }{\partial r_{\!j}} n_{\!j}\, \text{d} \varSigma , \end{align}

while the forcing term that accounts for the injection of $\langle {\delta q^2}\rangle$ , $\langle {\delta \omega ^2}\rangle$ and $\langle {\delta h}\rangle$ at scale $r$ reads

(2.42) \begin{align} \left\langle F^{\delta q^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \int _{\mathcal{V}(r)} 2 \left \langle { \delta f_i \delta u_i }\right \rangle\, \text{d}\varOmega , \end{align}

(2.43) \begin{align} \left\langle F^{\delta \omega ^2} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \int _{\mathcal{V}(r)} 2 \left \langle { \delta f_i^{\omega } \delta \omega _i }\right \rangle\, \text{d}\varOmega , \end{align}

(2.44) \begin{align} \left\langle F^{\delta h} \right\rangle = & \ \frac {1}{\mathcal{S}(r)} \int _{\mathcal{V}(r)} \left( \left \langle { \delta f_i \delta \omega _i}\right \rangle + \left \langle { \delta f_i^{\omega } \delta u_i }\right \rangle \right)\, \text{d}\varOmega . \end{align}

These equations highlight the fundamental differences between Newtonian and polymeric turbulence. In Newtonian fluids, dissipation occurs solely through fluid viscosity, represented by $\varepsilon _{\!f}$ , and the fluxes arise exclusively from the nonlinear term $\varPhi _{\!f}$ . In the intermediate range of scales where the viscous diffusion $D_{\!f}$ and forcing $F$ terms are typically negligible, a positive production term associated with vortex stretching appears only in the enstrophy equation. In contrast, polymeric flows feature an additional flux component, $\varPhi _p$ , which (also) accounts for the transfer across scales mediated by the polymers. This term appears in (2.8) under a divergence in scale space ( $\partial /\partial r_{\!j}$ ), analogous to $\varPhi _{\!f}$ , and its net contribution averages to zero over the entire triperiodic domain. Physically, this reflects how stretched polymers facilitate transfers across scales, providing an alternative cascade pathway that ultimately leads to dissipation. Furthermore, the presence of polymers introduces an additional source/sink term, $\pi$ , on the right-hand side of the equations. For the energy equation, $\langle \pi ^{\delta q^2} \rangle$ matches the polymeric dissipation rate $\langle \epsilon _p \rangle \equiv \langle { \mu _p (R_{ii} - 3)/(2 \tau _p^2)} \rangle$ and, on average, acts as a net sink for the fluid phase (De Angelis et al. Reference De Angelis, Casciola, Benzi and Piva2005). For the enstrophy and the helicity, the average $\langle \pi ^{\delta \omega ^2} \rangle$ and $\langle \pi ^{\delta h} \rangle$ can be both positive and negative, depending on the small-scale flow topology.

In the case of homogeneous and isotropic flows, the quantities $\mathcal{E}$ and $\varPi$ reduce to the single-point terms $\varepsilon _{\!f}$ and $\pi$ , which are independent of the scale $r$ . The corresponding equations can therefore be interpreted as balance relations for the average total dissipation rates. At each scale $r$ , the sum of the scale-dependent terms on the left-hand side must balance the scale-independent average total dissipation rate, which acts as an invariant of the system. This interpretation is fully consistent with the classical Kármán–Howarth equation for the second-order velocity structure function in HIT. However, as discussed previously, a key distinction in the case of PHIT is that the invariant quantity is no longer the fluid dissipation alone, but the combined contribution $\langle \varepsilon _{\!f} \rangle + \langle \pi \rangle$ (Chiarini et al. Reference Chiarini, Singh and Rosti2025). In this context, it is worth noting that the equation for $\langle \delta q^2 \rangle$ closely resembles the classical $-4/5$ law for Newtonian homogeneous and isotropic turbulence. Specifically, in the absence of polymers, in the high-Reynolds-number limit ( $Re \to \infty$ ) and for scales in the inertial range ( $r/L \to 0$ ), the energy equation reduces to $1/S(r) \oint _{S(r)} \langle \delta u_{\!j} \delta q^2 \rangle n_{\!j} \,\text{d} \varSigma = - (4/3) \langle \varepsilon _{\!f}^{\delta q^2} \rangle r$ , which, under assumptions of incompressibility and isotropy (Hill Reference Hill1997), simplifies further to the well-known result $S_3(r) \equiv \langle ( ( \boldsymbol{u}(\boldsymbol{x}+\boldsymbol{r},t) - \boldsymbol{u}(\boldsymbol{x},t)) \boldsymbol{\cdot }\boldsymbol{r}/r)^3 \rangle = -(4/5) \langle \varepsilon _{\!f}^{\delta q^2} \rangle r$ .

To better characterise the influence of the polymers on the transfers of energy and helicity, we decompose $\varPhi _p$ into an inertial transfer term $\varPhi _{p,i}$ and a dissipative term $D_p$ , following the approach introduced by Rosti et al. (Reference Rosti, Perlekar and Mitra2023) and informed by earlier work by Perlekar, Mitra & Pandit (Reference Perlekar, Mitra and Pandit2006). Perlekar et al. (Reference Perlekar, Mitra and Pandit2006) proposed modelling this behaviour via a scale-dependent effective viscosity. Alternatively, Rosti et al. (Reference Rosti, Perlekar and Mitra2023) separated the dissipative part by assuming that it shares the same asymptotic scale dependence as the fluid viscous term $D_{\!f}$ and requiring that, at vanishing scales, it matches the polymeric dissipation $\langle \pi ^{\delta q^2} \rangle \equiv \langle \varepsilon _p \rangle$ . Here, we adopt this second approach for the budgets of energy and helicity. We split the polymeric flux $\varPhi _p$ into two contributions, such that

(2.45) \begin{align} \varPhi _{p} = \varPhi _{p,i} + D_{p}, \quad \text{with} \ \left \langle {D_{p}}\right \rangle = \frac {\left \langle {\pi }\right \rangle }{\left \langle {\varepsilon _{\!f}}\right \rangle } \left \langle {D_{\!f}}\right \rangle \ \text{and} \ \left \langle {\varPhi _{p,i}}\right \rangle = \left \langle {\varPhi _{p}}\right \rangle - \left \langle {D_{p}}\right \rangle . \end{align}

To simplify the notation, hereafter, we use $\varPhi _{p}$ instead of $\varPhi _{p,i}$ to indicate the inertial contribution to the polymeric flux.

We do not apply the same decomposition to the budget for $\langle \delta \omega ^2 \rangle$ , as at large ${\textit{De}}$ , the fluid–polymer interaction yields, on average, a positive enstrophy source term; that is, $\langle \pi ^{\delta \omega ^2} \rangle \lt 0$ (see § 4.2). Unlike in the cases of energy and helicity, $\langle \pi ^{\delta \omega ^2} \rangle$ cannot be thus interpreted, on average, as a purely dissipative contribution.

4.1.1. Scale-by-scale budget

Figure 6 represents the terms in the budget equation for $\langle {\delta q^2}\rangle$ for both Newtonian HIT and PHIT.

Figure 6.

All contributions in budget equation for $\langle {\delta q^2}\rangle$ . The panels are for (a) ${\textit{De}}=0$ , (b) ${\textit{De}}=1/3$ , (c) ${\textit{De}}=1$ and (d) ${\textit{De}}=9$ . The yellow and blue shaded regions identify the inertial and elastic ranges of scales. Here, $\varPhi _t = \varPhi _{\!f} + \varPhi _p$ and $D_t = D_{\!f} + D_p$ .

We begin by examining the ${\textit{De}}=0$ case, which represents the classical Richardson–Kolmogorov turbulence scenario for Newtonian fluids. Energy is injected at the largest scales, $r\gt r_{\textit{IE}} \approx 1.56$ , through external forcing, with $\langle F^{\delta q^2} \rangle \approx (4/3) \langle \varepsilon _{\!f}^{\delta q^2} \rangle r$ . From there, energy is transferred from larger to smaller scales within the inertial range of scales $0.028 \approx r_{\textit{DI}} \leqslant r \leqslant r_{\textit{IE}}$ via the classical inertia-driven cascade. Here, $r_{\textit{IE}}$ marks the scale above which forcing dominates, separating the inertial and energetic ranges, while $r_{\textit{DI}}$ denotes the scale below which viscous dissipation dominates, separating the dissipative and inertial ranges. Consistent with the Kármán–Howarth equation, the energy transfer rate remains constant within the inertial range and equal in magnitude to the dissipation rate as evidenced by the plateau in figure 6(a); here, $\langle \varPhi _{\!f}^{\delta q^2} \rangle \approx -(4/3) \langle \varepsilon _{\!f} \rangle r$ . At scales smaller than $r_{\textit{DI}}$ , viscous effects $\langle D_{\!f}^{\delta q^2} \rangle$ dominate and $\langle \delta q^2 \rangle$ is dissipated into heat through viscous friction.

Polymeric additives introduce an additional dissipative process $D_p^{\delta q^2}$ and an alternative route for energy transfer among scales $\varPhi _p^{\delta q^2}$ . For small ${\textit{De}} \ll 1$ , the global picture resembles what is observed in Newtonian HIT. In this case, indeed, the polymers are only marginally stretched as their relaxation time is small compared with the characteristic time scales of the flow and they quickly relax back to their equilibrium lengths; here, $\varPhi _{p}^{\delta q^2} \approx D_{p}^{\delta q^2} \approx \pi ^{\delta q^2} \approx 0$ .

As ${\textit{De}}$ increases towards $\mathcal{O}(1)$ , the relevance of the non-Newtonian contribution increases and the polymeric additives modify the energy cascade in a non-trivial way. Energy enters the system at the large scales ( $r\gt r_{\textit{IE}}$ ) and is dissipated away at the same rate, via both the fluid dissipation and the polymeric sink term, i.e. $ \langle F^{\delta q^2} \rangle = (4/3) (\langle \varepsilon _{\!f}^{\delta q^2} \rangle + \langle \pi ^{\delta q^2} \rangle) r$ . In the intermediate range of scales ( $r_{\textit{DI}} \lt r \lt r_{\textit{IE}}$ ), energy is then transferred from larger to smaller scales by two alternative routes, associated respectively with (i) the classic inertia-driven term $\varPhi _{f}^{\delta q^2}$ and (ii) the fluid–polymer interaction term $\varPhi _{p}^{\delta q^2}$ . At these scales, the viscous effects are negligible, and the total energy transfer rate is an invariant and matches the total dissipation rate (Chiarini et al. Reference Chiarini, Singh and Rosti2025); here, $\langle \varPhi _{\!f}^{\delta q^2} \rangle + \langle \varPhi _{\!f}^{\delta q^2} \rangle \approx - (4/3) ( \langle \varepsilon _{\!f}^{\delta q^2} \rangle + \langle \pi ^{\delta q^2} \rangle) r$ . Eventually, at the smallest scales ( $r \lt r_{\textit{DI}}$ ), energy is dissipated away by both the fluid and the polymers as $\text{d} \langle D_{\!f}^{\delta q^2} \rangle /\text{d}r \rightarrow (4/3) \langle \varepsilon _{\!f}^{\delta q^2} \rangle$ and $\text{d} \langle D_p^{\delta q^2} \rangle /\text{d}r \rightarrow (4/3) \langle \pi ^{\delta q^2} \rangle$ for $r \rightarrow 0$ .

The range of scales over which the energy cascade is significantely altered by the presence of polymers varies with ${\textit{De}}$ , as indicated by the blue shaded region in figure 6. To quantify this, we define for each ${\textit{De}}$ , a characteristic scale $r_p^*$ below which elastic effects are expected to dominate. This scale is estimated by comparing the local eddy turnover time at scale $r$ , i.e. $\tau _{\!f}(r) \sim r/\left \langle {\delta V^2}\right \rangle ^{1/2}$ , where $\delta V(r) = ( u_i(\boldsymbol{x}+\boldsymbol{r}) - u_i(\boldsymbol{x})) \boldsymbol{\cdot }r_i/r)$ , with the polymeric relaxation time $\tau _p$ . Thus, $r_p^*$ is estimated as

(4.1) \begin{equation} \frac {r_p^*}{\sqrt {\langle {\delta V^{2}} \rangle ^{*}}} = \tau _p, \quad \text{where} \, \langle {\delta V^2} \rangle ^* = \langle {\delta V^2}\rangle (r_p^*). \end{equation}

For $r\gt r_p^*$ , the time scale of the fluid fluctuations is larger than $\tau _p$ ( $\tau _p/\tau _{\!f}(r)\lt 1$ ): the polymers are only marginally stretched and thus their influence on the flow is weak. In contrast, when $r\lt r_p^*$ , the time scale of the fluid is smaller than the polymeric relaxation time ( $\tau _p/\tau _{\!f}(r)\gt 1$ ): the polymers interact more effectively with the fluid and thus modify the organisation of the corresponding fluctuations. For the present cases, we measure $r_p^* \approx 0.04, 0.32, 1.34$ and $2.81$ for ${\textit{De}}=1/9,1/3,1$ and $3$ : the range of scales where elastic effects significantly modify the energy cascade increases with ${\textit{De}}$ , in agreement with the widening of the elastic range of scales shown in figure 6. For ${\textit{De}}=9$ , $\tau _p\gt r/\left \langle {\delta V^2}\right \rangle ^{1/2}$ for all $r$ , suggesting that the polymeric additives effectively modify the energy cascade in the whole range of scales. The scaling behaviour observed in figure 2 aligns with $r_p^*$ lying well within the inertial range $r_{\textit{DI}} \lt r \leqslant r_{\textit{IE}}$ when ${\textit{De}}$ and $Re$ are sufficiently large. For small ${\textit{De}}$ and/or low $Re$ , the $\langle \delta q^2 \rangle \sim r^{1.3}$ scaling disappears as $r_p^*$ falls within the dissipative range (De Angelis et al. Reference De Angelis, Casciola, Benzi and Piva2005; Perlekar et al. Reference Perlekar, Mitra and Pandit2010). We reiterate that when ${\textit{De}}$ is very large and $\tau _p \gg r/\left \langle {\delta V^2}\right \rangle ^{1/2}$ , the polymeric chains decouple from the carrier phase and their influence on the carrier flow is rather low.

This suggests that the relevant control parameter is not the global Deborah number, but rather the scale-dependent or local Deborah number, defined as ${\textit{De}}(r) = \tau _p / \tau _{\!f}(r)$ , where $\tau _{\!f}(r)$ is indeed the characteristic time scale of eddies of size $r$ . This perspective is consistent with the observation that different bulk flow quantities exhibit a change in behaviour at different values of (the global) ${\textit{De}}$ ; see table 1. Specifically, the inversion in the behaviour of a given quantity due to the change of regime is expected to occur when ${\textit{De}}(r)$ becomes sufficiently large at the scale $r$ where that quantity is most active or sensitive.

A last comment regards the influence of ${\textit{De}}$ on the dissipative range of scales. Our data show that $r_{\textit{DI}}$ (i.e. scale below which the total dissipative term $D_t^{\delta q^2} = D_{\!f}^{\delta q^2} + D_p^{\delta q^2}$ dominates) decreases with ${\textit{De}}$ , thus indicating that the dissipative range of scales progressively shrinks as the polymeric relaxation time increases.

4.1.2. Nonlinear and non-Newtonian fluxes

The average picture of the cascade and interscale exchanges in the $r$ -space is not representative of the actual physical processes. In the previous section, we have shown the influence of the polymers on the energy cascade in an average sense only, even though it is known that the interscale energy transfer is highly intermittent (see for example Piomelli et al. Reference Piomelli, Cabot, Moin and Lee1991; Domaradzki, Liu & Brachet Reference Domaradzki, Liu and Brachet1993; Cerutti & Meneveau Reference Cerutti and Meneveau1998). Moreover, even though energy cascade in HIT is, on an average, from larger to smaller scales, there exist localised regions in the flow where energy actually cascades from smaller to larger scales, opposite to the average sense. We now investigate how the presence of polymeric additives modifies this picture and whether this localised inverse energy transfer is also detected in the transfer mechanism driven by the fluid–polymer interaction.

Figures 7 and 8 show the probability distribution functions of the nonlinear energy fluxes $\varPhi _{\!f}^{\delta q^2}$ and the polymeric energy fluxes $\varPhi _p^{\delta q^2}$ for various values of ${\textit{De}}$ . We analyse three different separations: $r=0.1$ , where $\varPhi _p^{\delta q^2}$ dominates for ${\textit{De}} \gt 1$ , and $r=0.3$ and $r=0.9$ , where $\varPhi _{\!f}^{\delta q^2}$ dominates for all ${\textit{De}}$ . In our convention, a positive flux $\varPhi \gt 0$ corresponds to backscatter events, i.e. local transfers of energy from smaller to larger scales, whereas a negative flux $\varPhi \lt 0$ indicates forward cascade events, transferring energy from larger to smaller scales. The distributions reveal the highly intermittent and strongly non-Gaussian nature of both $\varPhi _{\!f}^{\delta q^2}$ and $\varPhi _p^{\delta q^2}$ , being characterised by heavy tails (Ishihara, Gotoh & Kaneda Reference Ishihara, Gotoh and Kaneda2009; Yasuda & Vassilicos Reference Yasuda and Vassilicos2018). Both extreme forward transfers and backscatter events occur with significantly higher probability than predicted by a normal distribution. The distributions are skewed towards negative values, confirming that on average, both fluxes transfer energy forward across scales. Quantitatively, for ${\textit{De}}=1$ at $r=0.9$ ( $r=0.1$ ), the probability that $\varPhi _{\!f}^{\delta q^2}$ exceeds its root-mean-square value is $0.0672$ ( $0.0274$ ), while the probability that it falls below minus its root-mean-square value is $0.1640$ ( $0.0767$ ). Similarly, for $\varPhi _p^{\delta q^2}$ , the probabilities of exceeding and being less than minus its root-mean-suqare value are $0.0323$ ( $0.0105$ ) and $0.2402$ ( $0.1545$ ), respectively.

Figure 7.

Dependence of the distribution of the nonlinear flux $\varPhi _{f}$ on the Deborah number, for (a,d) $r=0.1$ , (b,e) $r=0.3$ and (c,f) $r=0.9$ . (a–c) distribution of $\varPhi _{\!f}^{\delta q^2}$ . (d–f) Distribution of $\varPhi _{\!f}^{\delta q^2}/\varPhi _{f,rms}^{\delta q^2}$ . In panels (d–f), the insets provide a zoom of the normalised distributions.

Figure 8.

Dependence of the distribution of the non-Newtonian flux $\varPhi _{p}^{\delta q^2}$ on the Deborah number, for (a,d) $r=0.1$ , (b,e) $r=0.3$ and (c,f) $r=0.9$ . (a–c) Distribution of $\varPhi _p^{\delta q^2}$ . (d–f) Distribution of $\varPhi _p^{\delta q^2}/\varPhi _{p,rms}^{\delta q^2}$ . In panels (d–f), the insets provide a zoom of the normalised distributions.

We now examine how polymers affect the tails of the distributions, focusing on extreme events. Starting with the nonlinear flux $\varPhi _{\!f}^{\delta q^2}$ shown in figure 7, we observe that across all three scales, the presence of polymers narrows the tails of the distribution. This indicates a suppression of extreme forward and backward energy transfer events along the classical cascade route. This attenuation becomes more pronounced as ${\textit{De}}$ increases, consistent with a smoother velocity field and the diversion of energy towards the polymer-driven transfer mechanism. When examining the fluxes normalised by their root-mean-square values (panels d–f), we find that the distributions at $r=0.3$ and $r=0.9$ collapse reasonably well across all ${\textit{De}}$ . Some deviations appear at the largest values, likely due to insufficient statistical convergence. This suggests that, in the $\varPhi _{\!f}^{\delta q^2}$ -dominated range, polymers modify the intensity of extreme nonlinear cascade events while having little effect on their intermittency. In other words, in the inertial range of scales where the polymeric contribution is weak, the self-similarity of the classical cascade process appears to be largely maintained. In contrast, the normalised distributions do not overlap at the small scales $r=0.1$ . In the elastic range of scales, the nonlinear flux $\varPhi _{\!f}^{\delta q^2}$ has a small average contribution and its intermittency increases with ${\textit{De}}$ . We now move to the polymeric flux $\varPhi _p^{\delta q^2}$ , shown in figure 8. As ${\textit{De}}$ increases, the distribution tails widen and the asymmetry becomes more pronounced consistent with the rise in the average value. When examining the normalised fluxes (panels d–f), the behaviour differs from that of $\varPhi _{\!f}^{\delta q^2}$ . Specifically, the distributions collapse reasonably well at the small scale $r=0.1$ (with some deviations at extreme values), but not at the larger scales $r=0.3$ and $r=0.9$ . An increase in ${\textit{De}}$ appears to alter the space-time intermittency of the interscale energy transfer driven by the polymeric microstructure primarily at larger scales, while self-similarity with respect to ${\textit{De}}$ appears to hold at smaller scales within the elastic range of scales. Consistent with the increased asymmetry, at large scales, higher ${\textit{De}}$ corresponds to narrower positive tails and slightly heavier negative tails.

We now provide a further insight on the influence of the polymers on the inertia-driven energy cascade process by looking at the simple relation introduced by Baj et al. (Reference Baj, Portela and Carter2022) that links the nonlinear flux with the Lamb vector $\delta \boldsymbol{\omega } \times \delta \boldsymbol{u}$ . Exploiting the Lamb decomposition and manipulating the equations, it is indeed possible to show that

(4.2) \begin{equation} -\frac {\partial }{\partial r_{\!j}} \left \langle { \delta u_{\!j} \delta q^2}\right \rangle = 2 \left \langle {\boldsymbol{u}^* \boldsymbol{\cdot }( \delta \boldsymbol{\omega } \times \delta \boldsymbol{u}) }\right \rangle . \end{equation}

Since $|| \delta \boldsymbol{u} \times \delta \boldsymbol{\omega }||^2 = || \delta \boldsymbol{u} ||^2 || \delta \boldsymbol{\omega } ||^2 - || \delta h ||^2$ , it is clear that there is an implicit connection between the scale-by-scale helicity $\delta h$ and the interscale energy transfer. Indeed, a large $\delta h = \delta \boldsymbol{\omega } \boldsymbol{\cdot }\delta \boldsymbol{u}$ corresponds to a small $\delta \boldsymbol{\omega } \times \delta \boldsymbol{u}$ and, therefore, to a weaker local transfer of energy. This agrees with the observation of previous authors who found that the magnitude of the helicity has an impact on the local transfer of energy (see for example Pelz et al. Reference Pelz, Yakhot, Orszag, Shtilman and Levich1985; Stepanov et al. Reference Stepanov, Golbraikh, Frick and Shestakov2015). To gain further insights on how polymers influence the interscale energy transfer, we look at the angle $\theta$ between $\boldsymbol{u}^*$ and $\delta \boldsymbol{\omega } \times \delta \boldsymbol{u}$ such that

(4.3) \begin{equation} \cos (\theta) = \frac { \boldsymbol{u}^* \boldsymbol{\cdot }\left ( \delta \boldsymbol{\omega } \times \delta \boldsymbol{u} \right) } { || \boldsymbol{u}^* || \ || \delta \boldsymbol{\omega } \times \delta \boldsymbol{u}|| }. \end{equation}

Figure 9 plots the distributions of $\cos (\theta)$ at different scales $r$ for $0 \leqslant De \leqslant 9$ . We start our discussion with the Newtonian case, where the distribution is positively skewed for all $r$ . This is consistent with (4.2), which implies that, on average, the left-hand side ( $|| \boldsymbol{u}^* || || \delta \boldsymbol{\omega } \times \delta \boldsymbol{u} || \cos (\theta)$ ) is positive. Similar to what was found by Baj et al. (Reference Baj, Portela and Carter2022), as the separation increases, the probability of parallel ( $\cos (\theta) = 1$ ) and antiparallel ( $\cos (\theta) = -1$ ) alignment increases, with a larger probability of the parallel events. For PHIT ( ${\textit{De}}\gt 0$ ), the picture changes in a scale-dependent manner. At small ${\textit{De}} (\ll 1)$ , as expected, the distributions remain similar to Newtonian HIT. For large ${\textit{De}} (\gtrsim 1)$ , instead, at small scales $r$ where the polymeric flux $\varPhi _p^{\delta q^2}$ dominates, the Lamb vector has a larger likelihood of being normal to the local advecting velocity, with the distribution still being right skewed. This agrees with the fact that, in this range of scales, only a small fraction of energy is transferred to smaller scales by the inertia-driven transfer route. The weakening of the inertia-driven interscale-energy transfer within the elastic range of scales is thus accompanied by an increase of the events where $\boldsymbol{u}^*(r)$ and $\delta \boldsymbol{\omega }(r) \times \delta \boldsymbol{u}(r)$ are perpendicular. When considering large $r$ , where the influence of the polymers is weaker and $\varPhi _{\!f}^{\delta q^2}$ dominates, the distribution of $\cos (\theta)$ flattens and progressively recovers a distribution similar to that in Newtonian HIT.

Figure 9.

Distribution of $\cos (\theta)= \boldsymbol{u}^* \boldsymbol{\cdot }( \delta \boldsymbol{\omega } \times \delta \boldsymbol{u})/( ||\boldsymbol{u}^*|| || \delta \boldsymbol{u} \times \delta \boldsymbol{\omega } ||)$ at different scales for (a) ${\textit{De}}=0$ , (b) ${\textit{De}}=1/3$ , (c) ${\textit{De}}=1$ and (d) ${\textit{De}}=9$ .

4.2.1. Scale-by-scale budget

This section considers the enstrophy structure function. Unlike energy and helicity, enstrophy is not an inviscid invariant of the Navier–Stokes equations. The budget equation for $\langle {\delta \omega ^2}\rangle$ , indeed, features a source term $\langle V_{st} \rangle$ , which quantifies the amount of enstrophy that is created up to the scale $r$ by means of vortex stretching-like processes. This means that even in Newtonian HIT, the notion of cascade used for energy transfer does not apply here, and most of the enstrophy is generated directly at the small scales. However, as stated by several authors (see for example Davidson et al. Reference Davidson, Morishita and Kaneda2008), vortex stretching does occur in the inertial range of HIT and hence a net transfer of enstrophy to smaller scales is expected at these $r$ .

We plot the terms in the budget equation for $\langle {\delta \omega ^2}\rangle$ in figure 10. We start by looking at the Newtonian case ( ${\textit{De}}=0$ ). At intermediate and large scales, vortex stretching substantially matches the dissipation of enstrophy, i.e. $\langle V_{st} \rangle (r) \approx (4/3) \langle \varepsilon _{\!f}^{\delta \omega ^2} \rangle r$ , resulting in a $\langle V_{st} \rangle (r) \sim r^{1}$ scaling. In this intermediate range of scales, the forcing $\langle F^{\delta \omega ^2} \rangle$ and dissipation $\langle D_{\!f}^{\delta \omega ^2} \rangle$ contributions are negligible, and the flux is subdominant $\langle \varPhi _{\!f}^{\delta \omega ^2} \rangle \ll \langle V_{st}^{\delta \omega ^2} \rangle$ .

Figure 10.

All contributions in the budget equation for $\langle {\delta \omega ^2}\rangle$ for (a) ${\textit{De}}=0$ , (b) ${\textit{De}}=1/3$ , (c) ${\textit{De}}=1$ and (e) ${\textit{De}}=9$ . For ${\textit{De}}=0$ , the inset shows the budget with the $y$ axis as logarithmic scale. The black dashed line denotes $r^{-2}$ .

An insightful interpretation of the linear scaling $\langle V_{st}^{\delta \omega ^2} \rangle \sim r^1$ arises from the observation that vortex stretching is predominantly contributed by the smallest scales. This can be seen by the dimensional estimate of the scale-by-scale vorticity $\omega _r$ ,

(4.4) \begin{align} \omega _r \sim \frac {u_r}{r} \sim \left\langle \varepsilon _{\!f}^{\delta q^2} \right\rangle ^{1/3} r^{1/3} r^{-1} = \left\langle \varepsilon _{\!f}^{\delta q^2} \right\rangle ^{1/3} r^{-2/3}, \end{align}

which becomes maximum as $r \to \eta$ , so that $\omega _\eta \sim \langle \varepsilon _{\!f}^{\delta q^2} \rangle ^{1/3} \eta ^{-2/3}$ ; here, we have used the $u_r \sim \langle \varepsilon _{\!f}^{\delta q^2} \rangle ^{1/3} r^{1/3}$ Kolmogorov prediction. Now, to estimate the cumulative vortex stretching $\langle {V_{st}}\rangle$ up to scale $r$ ,

(4.5) \begin{align} \left \langle {V_{st}}\right \rangle (r) &= \frac {1}{\mathcal{S}(r)} \int _{\mathcal{V}(r)} \ \underbrace { 2 \left ( \left \langle { \delta \omega _i \omega _{\!j}^* \delta \left ( \frac {\partial u_i}{\partial x_{\!j}} \right) }\right \rangle + \left \langle { \delta \omega _i \delta \omega _{\!j} \left ( \frac {\partial u_i}{\partial x_{\!j}} \right)^* }\right \rangle \right) }_{v_{st,r}}\, \text{d}\varOmega , \end{align}

we begin by estimating the scale-by-scale vortex stretching $v_{st,r}$ , as

(4.6) \begin{align} v_{st,r} \sim \omega _r \omega _r \frac {u_\eta }{\eta }. \end{align}

We now use the notion that $v_{st,r}$ gets a maximal contribution from the smallest scales (Davidson et al. Reference Davidson, Morishita and Kaneda2008) and write

(4.7) \begin{align} v_{st,\eta } \sim \omega ^2_{\eta } \ \frac {u_\eta }{\eta } \sim \left\langle \varepsilon _{\!f}^{\delta q^2} \right\rangle ^{2/3} \eta ^{-4/3} \left\langle \varepsilon _{\!f}^{\delta q^2} \right\rangle ^{1/3} \eta ^{1/3} \eta ^{-1} = \left\langle \varepsilon _{\!f}^{\delta q^2} \right\rangle \eta ^{-2} \sim \left\langle \varepsilon _{\!f}^{\delta \omega ^2} \right\rangle , \end{align}

where we have used the $\langle \varepsilon _{\!f}^{\delta \omega ^2} \rangle \sim \langle \varepsilon _{\!f}^{\delta q^2} \rangle \eta ^{-2}$ dimensional estimate. By neglecting the contribution to vortex stretching from larger scales, we thus estimate the cumulative vortex stretching up to a scale $r$ as $\langle V_{st} \rangle (r) \approx 1/\mathcal{S}(r)\int _{\mathcal{V}(r)} v^{st}_\eta \text{d}\varOmega \sim r^{-2}r^{3} \langle \varepsilon _{\!f}^{\delta \omega ^2} \rangle \sim \langle \varepsilon _{\!f}^{\delta \omega ^2} \rangle r$ , which is consistent with the data plotted in figure 10.

Moreover, figure 10 shows that the nonlinear flux exhibits a $\langle \varPhi _{\!f}^{\delta \omega ^2} \rangle (r) \sim r^{-1}$ scaling behaviour at the intermediate scales, with its magnitude increasing as $r$ decreases. Since enstrophy is predominantly generated at the smallest scales, the inter-scale transfer rate intensifies with decreasing $r$ , reaching a maximum near the transition between the inertial and dissipative ranges. Following the dimensional analysis approach of Davidson et al. (Reference Davidson, Morishita and Kaneda2008), this scaling can be interpreted by analogy with the vortex stretching mechanism. Consider, for example, a vortex tube of length $\ell$ and cross-sectional area $A \sim r^2$ . When subjected to external shear, the tube is stretched, reducing its cross-section and thereby increasing its local vorticity. This illustrates how enstrophy at a larger scale is transferred to more intense enstrophy at smaller scales. Based on this picture, the enstrophy transfer in the inertial range is driven by eddies of size $r$ , with characteristic velocity $\delta u(r) \sim u_r \sim \langle \varepsilon _{\!f}^{\delta q^2} \rangle ^{1/3} r^{1/3}$ and vorticity $\delta \omega (r) \sim \omega _r \sim \langle \varepsilon _{\!f}^{\delta q^2} \rangle ^{1/3} r^{-2/3}$ . These scalings naturally lead to the observed behaviour:

(4.8) \begin{align} \left \langle {\varPhi _{\!f}^{\delta \omega ^2}}\right \rangle (r) \sim \frac {1}{\mathcal{S}(r)} \int _{\mathcal{S}(r)} \ u_r \omega ^2_r \,\text{d} \varSigma \sim r^{-2} r^2 \ \left \langle {\varepsilon _{\!f}^{\delta q^2}}\right \rangle r^{1/3} r^{-4/3} = \left \langle {\varepsilon _{\!f}^{\delta q^2}}\right \rangle r^{-1}. \end{align}

We now examine the influence of polymers on the budget of $\langle \delta \omega ^2 \rangle$ . Their presence introduces two additional terms into the enstrophy budget: a polymeric source/sink term, $\pi ^{\delta \omega ^2}$ , and the associated flux term, $\varPhi _p^{\delta \omega ^2}$ . In other words, analogous to the case of $\langle \delta q^2 \rangle$ , polymers contribute with a source/sink mechanism and an alternative pathway for interscale enstrophy transfer. Recall that for enstrophy, we have not split $\varPhi _p^{\delta \omega ^2}$ into the inertial and dissipative contributions.

For small Deborah numbers ( ${\textit{De}} \lt 1$ ), the polymeric flux $\langle \varPhi _{p}^{\delta \omega ^2} \rangle$ and source term $\langle \pi ^{\delta \omega ^2} \rangle$ remain weak, and the classical enstrophy production $\langle V_{st} \rangle$ and transfer $\langle \varPhi _{\!f}^{\delta \omega ^2} \rangle$ are only marginally affected by the polymers; see figure 10(b). In this regime, $\langle \pi ^{\delta \omega ^2} \rangle \lt 0$ , indicating that polymers act, on average, as a sink of enstrophy for the fluid phase (see also figure 11). Moreover, $\langle \varPhi _p^{\delta \omega ^2} \rangle \lt 0$ at all scales, implying that enstrophy is transferred from larger to smaller scales even along the polymer-induced interscale transfer.

Figure 11.

Dependence of $\langle v_{st} \rangle$ and $\langle \pi ^{\delta \omega ^2} \rangle$ on ${\textit{De}}$ . All quantities are made dimensionless with $\langle \varepsilon _{\!f}^{\delta \omega ^2} \rangle$ . Here, $v_{st} = \omega _i \omega _{\!j} S_{\textit{ij}}$ is the vortex-stretching production term that appears in the budget equation for $\langle \omega ^2 \rangle$ . The bars show the standard deviation computed from six different snapshots.

At large Deborah numbers, the dynamics of the enstrophy transfer undergoes a marked transformation. As ${\textit{De}}$ increases, the average fluid contribution $\langle \varPhi _{\!f}^{\delta \omega ^2} \rangle$ remains nearly unchanged, indicating that polymers exert limited influence on the classical inertia-driven enstrophy interscale transfer. However, once ${\textit{De}} \geqslant 1$ , the polymeric contribution $\langle \varPhi _p^{\delta \omega ^2} \rangle$ becomes comparable in magnitude to $\langle \varPhi _{\!f}^{\delta \omega ^2} \rangle$ and displays a different behaviour across scales. At intermediate scales, both fluid- and polymer-driven transfer processes contribute to the forward enstrophy transfer, from larger to smaller scales, though both remain subdominant compared with the vortex-stretching term. At sufficiently small scales, the scenario changes. For ${\textit{De}} \geqslant 1$ , the polymeric flux $\langle \varPhi _p^{\delta \omega ^2} \rangle$ becomes negative as $r \rightarrow 0$ , suggesting that the fluid–polymer interaction facilitates an inverse enstrophy transfer from smaller to larger scales. Notably, this shift is accompanied by a sign reversal in $\langle \pi ^{\delta \omega ^2} \rangle$ (see §§ 3 and 5). The data suggest that, at these values of ${\textit{De}}$ , polymers generate enstrophy at the smallest scales, which is then transferred towards slightly larger scales before being ultimately dissipated by viscosity. This behaviour stands in contrast to that observed for $\langle \delta q^2 \rangle$ , where the nonlinear flux dominates the energy cascade at large scales and the polymeric flux becomes dominant at small scales, but both consistently drive a forward energy transfer across all $r$ .

A further key effect at high ${\textit{De}}$ is the strong modulation of the vortex-stretching term. Figure 11 illustrates this by showing the dependence of $\langle v_{st} \rangle = \langle \omega _i \omega _{\!j} S_{\textit{ij}} \rangle$ on ${\textit{De}}$ , where $S_{\textit{ij}} = ( \partial u_i/\partial x_{\!j} + \partial u_{\!j}/\partial x_i)/2$ is the rate-of-strain tensor. The quantity $v_{st}$ represents the one-point vortex-stretching production of enstrophy $\omega ^2$ (Tsinober Reference Tsinober2001) and satisfies $\langle V_{st}^{\delta \omega ^2} \rangle \rightarrow 2 \langle v_{st} \rangle$ as $r \rightarrow \infty$ . As shown in figure 11, $\langle v_{st} \rangle$ decreases significantly with increasing ${\textit{De}}$ , in agreement with earlier findings by Liberzon et al. (Reference Liberzon, Guala, Luthi, Kinzelbach and Tsinober2005, Reference Liberzon, Guala, Luthi, Kinzelbach and Tsinober2006), Cai, Li & Zhang (Reference Cai, Li and Zhang2010) and ur Rehman, Lee & Lee (Reference ur Rehman, Lee and Lee2022). Remarkably, this attenuation of vortex stretching is closely accompanied by the striking sign reversal in $\langle \pi ^{\delta \omega ^2} \rangle$ , which becomes negative and approximately balances the viscous dissipation $\langle \pi ^{\delta \omega ^2} \rangle \approx \langle \varepsilon _{\!f}^{\delta \omega ^2} \rangle$ for ${\textit{De}} \geqslant 1$ . In other words, at high elasticity, polymers not only suppress classical enstrophy production via vortex stretching, but also introduce a different dynamical regime, in which polymer stresses act as a net source of enstrophy.

As discussed in § 5, the suppression of vortex stretching and the sign reversal in $\langle \pi ^{\delta \omega ^2} \rangle$ with increasing ${\textit{De}}$ are closely linked to changes in the local flow topology and alignments, a transformation that redefines enstrophy dynamics in polymer-laden turbulence.

4.3.1. Scale-by-scale budget

We now move to the helicity structure function $\delta h$ . In the present set up, turbulence is sustained with the ABC forcing that injects both energy and helicity at the largest scales. As such, like energy, helicity is expected to cascade from large to small scales in the inertial range of scales and to eventually vanish at the small scales where the mirror symmetry $\boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\omega } = 0$ is restored.

Our results, presented in figure 12, provide clear evidence of a dual direct cascade of energy and helicity across all values of ${\textit{De}}$ , consistent with the theoretical predictions and numerical observations by Kraichnan (Reference Kraichnan1973), Polifke & Shtilman (Reference Polifke and Shtilman1989) and Borue & Orszag (Reference Borue and Orszag1997).

Figure 12.

All contributions in the budget equation for $\langle {\delta h}\rangle$ for (a) ${\textit{De}}=0$ , (b) ${\textit{De}}=1/3$ , (c) ${\textit{De}}=1$ and (d) ${\textit{De}}=9$ . The yellow and blue shaded regions identify the inertial and elastic ranges of scales. Here, $\varPhi _t = \varPhi _{\!f} + \varPhi _p$ and $D_t = D_{\!f} + D_p$ .

In the purely Newtonian case, helicity is injected at the largest scales ( $r \gt r_{\textit{IE}}$ ) at a rate that matches its dissipation, i.e. $ \langle F^{\delta h} \rangle = (4/3) \langle \varepsilon _{\!f}^{\delta h} \rangle r$ . It is then transferred from large to small scales within the inertial range ( $r_{\textit{DI}} \lt r \lt r_{\textit{IE}}$ ), where $\langle \varPhi _{\!f}^{\delta h} \rangle \sim - (4/3) \langle \varepsilon _{\!f}^{\delta h} \rangle r$ , and is ultimately dissipated by viscous effects at the smallest scales.

As with the energy $\langle \delta q^2 \rangle$ , polymers introduce additional helicity transfer and dissipation mechanisms, whose significance increases with ${\textit{De}}$ . Specifically, the helicity cascade is governed predominantly by the nonlinear flux $\langle \varPhi _{\!f}^{\delta h} \rangle$ in the range $r_p^* \lessapprox r \lt r_{\textit{IE}}$ and by the polymeric flux $\langle \varPhi _p^{\delta h} \rangle$ in the smaller-scale range $r_{\textit{DI}} \lt r \lessapprox r_p^*$ . As ${\textit{De}}$ increases and the polymers stretch more, the inertia-driven helicity cascade $\langle \varPhi _{\!f}^{\delta h} \rangle$ is progressively suppressed, mirroring the behaviour observed for $\langle \varPhi _{\!f}^{\delta q^2} \rangle$ .

The emergence of two distinct cascade mechanisms dominating in separate scale ranges is consistent with the multiscaling behaviour of $\langle \delta h \rangle$ reported in figure 2.

Figure 13 offers a more detailed view of how polymer additives influence the nonlinear helicity flux $\langle \varPhi _{f}^{\delta h} \rangle$ . As discussed in § 2, this flux comprises two distinct contributions. The first is the classical convective term, $\langle \varPhi _{f,a}^{\delta h} \rangle$ , associated with the correlation $\langle \delta h \delta u_{\!j} \rangle$ . The second, $\langle \varPhi _{f,b}^{\delta h} \rangle$ , captures a transfer mechanism involving the correlation $\langle \delta \omega _{\!j} \delta q^2 \rangle$ . In the Newtonian case, figure 13 shows that $\langle \varPhi _{f,a}^{\delta h} \rangle$ clearly dominates, being nearly twice as large as $\langle \varPhi _{f,b}^{\delta h} \rangle$ across all scales. This indicates that, in Newtonian HIT, helicity is primarily transferred through the convective mechanism. As ${\textit{De}}$ increases, however, both components of the flux are progressively suppressed in the elastic range of scales, reflecting the polymers’ ability to attenuate both transfer pathways while redirecting a portion of the helicity flux through the polymeric channel. Notably, polymers have a stronger damping effect on $\langle \varPhi _{f,a}^{\delta h} \rangle$ than on $\langle \varPhi _{f,b}^{\delta h} \rangle$ . As a result, at ${\textit{De}} = 9$ , the two contributions become comparable on average, with $\langle \varPhi _{f,a}^{\delta h} \rangle \approx \langle \varPhi _{f,b}^{\delta h} \rangle$ .

Figure 13.

Dependence of (a) $\langle \varPhi _{f,a}^{\delta h} \rangle$ and (b) $\langle \varPhi _{f,b}^{\delta h} \rangle$ on $r$ for different ${\textit{De}}$ .

4.3.2. Relative helicity

We now examine the relative helicity, defined as $h_r \equiv r \langle \delta h \rangle / (2 \langle \delta q^2 \rangle)$ following Borue & Orszag (Reference Borue and Orszag1997), to assess how polymers influence the flow’s tendency to restore mirror symmetry at small scales. As shown in figure 14, the flow exhibits maximal helicity at large scales, with $h_r$ decaying monotonically as $r$ decreases, indicating that helicity becomes dynamically negligible at small scales.

Figure 14.

(a) Dependence of the relative helicity $r\langle {\delta h}\rangle /(2 \langle {\delta q^2}\rangle)$ on $r$ for different ${\textit{De}}$ . (b) Dependence of $\langle {\delta h}\rangle /\langle {\delta q^2}\rangle$ on $r$ for different ${\textit{De}}$ .

In the Newtonian case, we recover the classical scaling $h_r \sim r^1$ within the inertial range, consistent with the findings of Borue & Orszag (Reference Borue and Orszag1997). This behaviour is illustrated in figure 14(b), where the ratio $\langle \delta h \rangle / \langle \delta q^2 \rangle \approx 0.75$ is observed, comparable to that found for passive scalar transport in isotropic turbulence (Borue & Orszag Reference Borue and Orszag1997).

In PHIT, however, the decay of $h_r$ with $r$ is noticeably slower in the intermediate range, and the Newtonian $r^1$ scaling in the inertial range is lost. This reflects the fact that $\langle \delta h \rangle$ and $\langle \delta q^2 \rangle$ scale differently for ${\textit{De}} \gt 0$ , as previously shown in figure 2. Furthermore, we find that $h_r$ increases systematically with ${\textit{De}}$ across all scales, suggesting a growing asymmetry with increasing elasticity.

At the smallest scales, the ratio $\langle \delta h \rangle / \langle \delta q^2 \rangle$ tends to a plateau for all cases, which reflects the smoothness of the velocity and vorticity fields and is consistent with a Taylor expansion representation (see also figure 2). Overall, the results in figure 14 indicate that polymers act to break mirror symmetry and that this symmetry-breaking effect becomes more pronounced with increasing elasticity.