3.1. The energy spectra

We begin our study by investigating how energy is distributed across scales in a statistically stationary state in flows of polymeric fluids. The distribution of energy is given by its spectrum $E(k)$ which is a measure of the energy content at scale (mode) $k$ such that the total energy $\mathcal{E} = \langle |\boldsymbol{u}|^2 \rangle /2 = \int E(k) {\rm d}k$ . The energy spectrum is one of the most commonly studied quantities in turbulence and was shown by Kolmogorov (Reference Kolmogorov1941) to have a self-similar distribution across scales in (the inertial range of) HIT as $E(k) \sim k^{-5/3}$ (see also Frisch Reference Frisch1996). We now show in figure 2 how this self-similar distribution of fluid kinetic energy is modified in PHIT, and its dependence on polymer elasticity $\textit{De}$ in figure 2(a) and the intensity of turbulence ${Re}_\lambda$ in figure 2(b).

Figure 2. The non-unique scaling of the fluid energy spectrum $E(k)$ in polymeric flows at different (a) $\textit{De}$ and (b) $\textit{Re}$ numbers. The spectra have been shifted vertically for visual clarity by factors of powers of 10. Three distinct scaling regimes (in different shades) have been shown in dashed ( $k^{-5/3}$ , Newtonian), dash–dotted ( $k^{-2.3}$ , polymeric) and dotted ( $k^{-\gamma }$ ; $\gamma \geqslant 4$ , smooth) lines. All three regimes coexist when De $\approx 1$ . (a) The smallest $\textit{De}$ has a close to Newtonian behaviour as the elastic effects are minimal. (b) The triple scaling at largest $\textit{Re}$ gives way to a unique, smooth $k^{-4}$ regime at the smallest ${Re}_\lambda$ .

Figure 2(a) shows how the spectrum $E(k)$ changes with $\textit{De}$ at large ${Re}_\lambda \approx 450$ . When $\textit{De}$ is small, i.e. in the limit $\tau _{{p}} \to 0$ , polymers are stretch minimally, and remain close to their equilibrium unstretched configuration $\boldsymbol{C} \approx \boldsymbol{I}$ . This is easily seen by multiplying (2.1b ) by $\tau _{{p}}$ , and taking the limit $\tau _{{p}} \to 0$ ,

(3.1) \begin{align} \tau _{{p}} \left ( \partial _t \boldsymbol{C} + \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{C} \right ) = \tau _{{p}} \big( \boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} + (\boldsymbol{\nabla }\boldsymbol{u})^T \boldsymbol{C} \big) - (\boldsymbol{C} -\boldsymbol{I}) \overset {\tau _{{p}} \to 0}{\implies } \boldsymbol{C} -\boldsymbol{I} = 0. \end{align}

As the fluctuations in $\boldsymbol{C}$ are small, the polymer stresses in (2.1a ) have a minimal contribution: $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{C} \approx 0$ . Thus, elastic effects do not play a significant role at small $\textit{De}$ , and the behaviour remains largely Newtonian. This results in the classical K41, Newtonian-like scaling $E(k) \sim k^{-5/3}$ in figure 2(a). (Henceforth, we refer to this classical scaling range as the Newtonian regime.) This intermediate, Newtonian scaling range is of course followed by the dissipation range where the spectrum shows an exponential decay, a consequence of velocity fields being analytic.

As $\textit{De}$ is increased, polymers begin to stretch to longer lengths so that $\boldsymbol{C}$ now ventures far from $\boldsymbol{I}$ and has rather large spatiotemporal fluctuations. The resulting stretching and relaxation of polymers generates large non-Newtonian stresses (given by $ ( \mu _{{p}}/\tau _{{p}} ) \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{C}$ ) on the carrier fluid, especially for a large enough elasticity as $De \approx 1$ . The resulting dominant elastic effects along with a suppressed fluid nonlinearity (see § 3.2 for detailed discussion) means that the energy spectrum is modified to $E(k) \sim k^{-2.3}$ at large ${Re}_\lambda$ , as was also observed by Zhang et al. (Reference Zhang, Zhang, Li, Yu and Li2021a ) and Rosti et al. (Reference Rosti, Perlekar and Mitra2023). (We refer to this as the polymeric range, henceforth.) Viscous effects begin to dominate beyond this polymeric range and the fluid nonlinearity ceases to have any contribution. The polymeric stresses, however, are still active in this range due to the small scale fluctuations in polymer lengths which, in turn, create fluctuations in the fluid velocity. Consequently, the spectrum exhibits a steep power-law decay $E(k) \sim k^{-\delta }$ with $\delta \geqslant 4$ , rather than an exponential fall-off in the smooth dissipative range. Thus, the elasticity of polymers results in a dual scaling behaviour of the energy spectrum $E(k) \sim k^{-\delta }$ in PHIT, especially for $De \approx 1$ (see figure 2 a), with the exponent $\delta \approx 2.3$ in the intermediate, polymeric range of scales ( $6 \lesssim k \lesssim 60$ ) and $\delta \gtrsim 4$ for $k \gtrsim 100$ in the smooth but steep dissipative range.

On a further increase in polymer elasticity, the back reaction of polymers on the Newtonian carrier flow begins to diminish. This is because polymers with very large relaxation times respond to the background flow after a large time lag, thereby staying in a stretched configuration for long times. This means fluctuations in polymer lengths are suppressed at large $\textit{De}$ as polymer lengths do not change appreciably in space. As a result, their gradients also remain typically small thereby weakening the elastic stresses $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{C}$ on an average. This is coupled with the fact that a large factor of $\tau _{{p}}$ further weakens the feedback of polymer stresses on the carrier flow. Thus, Newtonian scaling is recovered for large $\textit{De}$ , and is seen in the re-emergence of $E(k) \sim k^{-5/3}$ regime at $De \approx 9$ . Indeed, at even larger $\textit{De}$ we have an intermediate range of scales with completely Newtonian scaling behaviour Rosti et al. (Reference Rosti, Perlekar and Mitra2023). Again in the deep dissipation range, small fluctuations in the polymer lengths result in fluctuations of the flow field, which are rough at the subleading order with a steep power-law decay of the spectrum as $E(k) \sim k^{-4}$ .

Figure 2(b) shows the dependence of the energy spectrum on ${Re}_\lambda$ , where we plot $E(k)$ for a fixed polymer elasticity of $De \approx 1$ . The top curve is just the spectrum at ${Re}_\lambda \approx 450$ and $De \approx 1$ . Now, it is well known that with decreasing ${Re}_\lambda$ in HIT the fluid nonlinearity becomes increasingly less important (with respect to viscous dissipation) causing the inertial range to shrink (and the viscous dissipation range to widen). In PHIT, a decreasing ${Re}_\lambda$ entails that now the intermediate polymeric range shrinks as seen from figure 2(b). This is a consequence of the fluid inertia weakening even further in the presence of polymers (see § 3.2 for detailed discussion). A further weakened fluid inertia means that viscosity becomes important at even larger scales. The dissipative range thus widens, and the steep power-law fall-off begins at even smaller $k$ as ${Re}_\lambda$ decreases. In fact, at ${Re}_\lambda \approx 40$ , the polymeric regime is completely absent and the smooth dissipation range spans the entire range of scales (see also Singh et al. Reference Singh, Perlekar, Mitra and Rosti2024). This is indeed very different from Newtonian turbulence, where a decreasing ${Re}_\lambda$ results in a shrinking inertial range that completely disappears as ${Re}_\lambda \to 0$ and the spectrum decays exponentially. Polymeric turbulence, on the other hand, shows a transition from $E(k) \sim k^{-2.3}$ , as ${Re}_\lambda$ decreases, to a steeper power-law spectrum of $E(k) \sim k^{-4}$ as ${Re}_\lambda \to 0$ for large $\textit{De}$ .

Next, we discuss how different dominant contributions lead to this non-unique scaling behaviour in polymeric flows, as a function of ${Re}_\lambda$ and $\textit{De}$ .

3.2. The flux contributions

The coexistence of various scaling regimes of the energy spectrum, as discussed in § 3.1, can be better understood by looking at the dominant energy transfer mechanisms across scales (see also Casciola & De Angelis (Reference Casciola and De Angelis2007) for a similar discussion on spectral decomposition with a slightly different approach). We obtain these dominant contributions from (2.1a ) by computing its Fourier transform and its complex conjugate,

(3.2a) \begin{align}& \partial _t \hat {u}_i(\boldsymbol{k}) + \mathcal{N}_i(\boldsymbol{k}) = i k_i \hat {p} - \nu k^2 \hat {u}_i (\boldsymbol{k}) + \frac {\mu _{{p}}}{\rho \tau _{{p}}} ik_j \hat {C}_{\textit{ij}} + \hat {F}_i(\boldsymbol{k}) , \quad k_i \hat {u}_i = 0; \end{align}

(3.2b) \begin{align}&\quad\,\,\, \partial _t \hat {u}^\dagger _i(\boldsymbol{k}) + \mathcal{N}^\dagger _i(\boldsymbol{k}) = -i k_i \hat {p}^\dagger - \nu k^2 \hat {u}^\dagger _i (\boldsymbol{k}) - \frac {\mu _{{p}}}{\rho \tau _{{p}}} ik_j \hat {C}^\dagger _{\textit{ij}} + \hat {F}^\dagger _i(\boldsymbol{k}), \end{align}

where the hat $\hat {\boldsymbol{\cdot }}$ denotes the Fourier transformed variables, $\boldsymbol{\cdot }^\dagger$ denotes the complex conjugate and $\mathcal{N}_i(\boldsymbol{k})$ is the Fourier transform of the fluid nonlinear term. Summation over repeated indices is implied. The energy equation for any $k$ th mode can be obtained by multiplying (3.2a ) by $u^\dagger _i(\boldsymbol{k})$ , and (3.2b ) by $u_i(\boldsymbol{k})$ , and adding the two together,

(3.3) \begin{align} \partial _t \mathcal{E}(\boldsymbol{k}) + \ \textrm{Re}\ \big( \hat {u}^\dagger _i(\boldsymbol{k}) \mathcal{N}_i (\boldsymbol{k}) \big) = - 2 \nu k^2 \mathcal{E}(\boldsymbol{k}) + \frac {\mu _{{p}}}{\rho \tau _{{p}}}\ \textrm{Im}\ \big( \hat {u}^\dagger _i(\boldsymbol{k}) k_j \hat {C}_{\textit{ij}} \big) +\ \textrm{Re}\ \big( \hat {u}^\dagger _i(\boldsymbol{k}) \hat {F}_i(\boldsymbol{k}) \big), \end{align}

where $\mathcal{E}(\boldsymbol{k}) = \lvert \hat {u}_i (\boldsymbol{k}) \rvert ^2 /2$ is the energy in mode $\boldsymbol{k}$ , the pressure term vanishes away due to incompressibility, and $\textrm{Re} ( \boldsymbol{\cdot })$ and $\textrm{Im} ( \boldsymbol{\cdot })$ denote the real and imaginary parts, respectively. The total energy flux from modes $k \lt K$ to those $k \gt K$ can be obtained by integrating from $0$ to $K$ ,

(3.4) \begin{align} & \partial _t \int _{0}^{K} {\rm d}\boldsymbol{k} \ \mathcal{E}(\boldsymbol{k}) + \int _{0}^{K} {\rm d}\boldsymbol{k} \ \textrm{Re}\ \big( \hat {u}^\dagger _i(\boldsymbol{k}) \mathcal{N}_i (\boldsymbol{k}) \big) = \nonumber \\ & - \nu \int _{0}^{K} {\rm d}\boldsymbol{k} \ k^2 \lvert \hat {u}_i(\boldsymbol{k}) \rvert ^2 + \frac {\mu _{{p}}}{\rho \tau _{{p}}} \int _{0}^{K} {\rm d}\boldsymbol{k} \ \textrm{Im}\ \big( \hat {u}^\dagger _i(\boldsymbol{k}) k_j \hat {C}_{\textit{ij}} \big) + \int _{0}^{K} {\rm d}\boldsymbol{k} \ \textrm{Re}\ \big( \hat {u}^\dagger _i(\boldsymbol{k}) \hat {F}_i(\boldsymbol{k}) \big). \end{align}

Figure 3. Normalised flux contributions at (a) Re $_\lambda \approx 450$ , (b) $240$ and (c) $40$ for flows with De $\approx 1$ . The polymeric contribution $\mathcal{P}$ is split into flux $\varPi _{{p}}$ and dissipation $\mathcal{D}_{\!{p}}$ contributions as $ \mathcal{P} = \varPi _{{p}} + \mathcal{D}_{\!{p}}$ as in Rosti et al. (Reference Rosti, Perlekar and Mitra2023). (a) At large ${Re}_\lambda$ , three distinct regimes are determined by different dominant contributions: large scales are dominated by the fluid nonlinear flux $\varPi _{{f}}$ ; intermediate scales by the polymer flux $\varPi _{{p}}$ ; small scales by the polymer dissipation $\mathcal{D}_{\!{p}}$ . (b) At moderate ${Re}_\lambda$ , the fluid-nonlinearity $\varPi _{{f}}$ is weakened and comprises two distinct regimes dominated by $\varPi _{{p}}$ and $\mathcal{D}_{\!{p}}$ . (c) At extremely small ${Re}_\lambda$ , only the dissipative terms $\mathcal{D}_{\!{f}}$ , $\mathcal{D}_{\!{p}}$ remain important. All terms are normalised by $\epsilon _{\textit {t}}$ .

In a statistically stationary state, the total energy in any set of modes is a constant, so that $\partial _t \int _{0}^{K} {\rm d}\boldsymbol{k} \ \mathcal{E}(\boldsymbol{k}) = 0$ . Choosing suitable symbols for the remaining integrals, and using isotropy to argue that their dependence is only on $K = \lvert \boldsymbol{K} \rvert$ , we have

(3.5) \begin{align} \varPi ^{\prime}_{{f}}(K) = - \mathcal{D}^{\prime}_{{f}}(K) - \mathcal{P}'(K) + \mathcal{F}(K), \end{align}

where $\varPi ^{\prime}_{{f}}$ , $\mathcal{D}^{\prime}_{{f}}$ , $\mathcal{P}'$ and $\mathcal{F}$ are the contributions from fluid nonlinearity, fluid dissipation, polymer stresses and the external forcing, respectively. Since the forcing $\hat {F}_i$ is applied at only $K = 1$ , we have that $\mathcal{F}(K)$ is a constant for any $K \gt 1$ ,

(3.6) \begin{align} \varPi ^{\prime}_{{f}}(K) + \mathcal{D}^{\prime}_{{f}}(K) + \mathcal{P}'(K) = \mathcal{F}(K) = \epsilon _{\textit {t}}, \end{align}

where $\epsilon _t$ is the rate of total energy injection into the system by the forcing $\boldsymbol{F}$ . Thus, upon normalisation by $\epsilon _{\textit {t}}$ , we obtain

(3.7) \begin{align} \varPi _{{f}}(K) + \mathcal{D}_{\!{f}}(K) + \mathcal{P}(K) = 1, \end{align}

where $\varPi _{{f}} = \varPi ^{\prime}_{{f}}/\epsilon _{\textit {t}}$ , $\mathcal{D}_{\!{f}} = \mathcal{D}^{\prime}_{{f}}/\epsilon _{\textit {t}}$ , $\mathcal{P} = \mathcal{P}'/\epsilon _{\textit {t}}$ (see also Abdelgawad, Cannon & Rosti Reference Abdelgawad, Cannon and Rosti2023). We show the curves for all the three contributions, $\varPi _{{f}}$ , $\mathcal{D}_{\!{f}}$ and $\mathcal{P}$ , as a function of ${Re}_\lambda$ in figure 3, for a polymer elasticity of $De \approx 1$ . Following Rosti et al. (Reference Rosti, Perlekar and Mitra2023), we partition the polymeric contribution into flux ( $\varPi _{{p}}$ ) and dissipative ( $\mathcal{D}_{\!{p}}$ ) terms as

(3.8) \begin{align} \mathcal{P}(K) = \varPi _{{p}}(K) + \mathcal{D}_{\!{p}}(K) , \quad \textrm {where} \quad \mathcal{D}_{\!{p}}(K) \equiv (\epsilon _{{p}}/\epsilon _{{f}}) \mathcal{D}_{\!{f}}(K). \end{align}

Such a definition is motivated by the requirement that at small scales (or large $k$ ) the polymeric term must get all its contribution from dissipative effects. Indeed, the full polymeric contribution $\mathcal{P}$ captures both the dissipative and flux contributions. We note that a purely dissipative part $\mathcal{D}_{\!{p}}$ of $\mathcal{P}$ must be monotonically growing in $k$ , thus we choose to model $\mathcal{D}_{\!{p}}$ in the very simple and naïve way by prescribing the same scale dependence as the fluid dissipation $\mathcal{D}_{\!{f}}$ , which we already know grows monotonically in $k$ . We tried different prescriptions for $\mathcal{D}_p$ under the constraint of $\mathcal{D}_{\!{p}} = 0$ at $k=0$ , $\mathcal{D}_{\!{p}} = \epsilon _{{p}}$ at $k=k_{\textit{max}}$ with a monotonic growth. The results do not change qualitatively but only quantitatively, in the sense that the boundaries of the different scaling ranges shift only marginally. Since these changes remain limited, the estimates for where different flux contributions crossover is reasonably robust. We plot the contributions $\varPi _{{p}}$ and $\mathcal{D}_{\!{p}}$ thus obtained in figure 3.

At large ${Re}_\lambda$ , figure 3(a) shows that the fluid nonlinearity $\varPi _{{f}}$ has the dominant contribution to the total energy balance at large scales (i.e. for $k \leqslant 4$ ). This gives the Newtonian $k^{-5/3}$ scaling at the large but slim band of scales in figure 2. Away from this range, i.e. for $k \gtrsim 10$ , the polymeric flux contribution $\varPi _{{p}}$ begins to dominate. This dominant non-Newtonian flux results in a polymeric scaling regime with $E(k) \sim k^{-2.3}$ up to $k \approx 70$ in figure 2 (see also Rosti et al. Reference Rosti, Perlekar and Mitra2023). At yet smaller scales, i.e. for $k \gtrsim 200$ , fluid ( $\mathcal{D}_{\!{f}}$ ) and polymer ( $\mathcal{D}_{\!{p}}$ ) dissipation remove energy from the flow rapidly. The polymer contribution $\varPi _{{p}}$ , has a small, subdominant yet non-zero contribution in this range. (This aligns with our assertion of small-scale fluctuations in polymer lengths in § 3.1.) This subdominant contribution results in a fluid velocity spectrum given by a steep power-law decay at large $k$ , in a manner similar to that in ET at very small ${Re}_\lambda$ (see Singh et al. Reference Singh, Perlekar, Mitra and Rosti2024). Now, as ${Re}_\lambda$ is decreased to a moderate value of ${Re}_\lambda \approx 240$ , the fluid nonlinearity is weakened evidently further as seen in figure 3(b), and the flux of energy is primarily via the fluid–polymer interactions $\varPi _{{p}}$ . However, this range is rather limited as a smaller ${Re}_\lambda$ also means that viscous effects become important at relatively smaller $k$ . Thus, we have a restricted polymeric and a wider $k^{-4}$ regime at ${Re}_\lambda \approx 240$ in figure 2(b). Finally, for very small ${Re}_\lambda \approx 40$ in figure 3(c), $\mathcal{D}_{\!{p}}$ and $\mathcal{D}_{\!{f}}$ are always large while $\varPi _{{f}}$ and $\varPi _{{p}}$ remain subdominant. In fact, $\varPi _{{f}}$ at such small ${Re}_\lambda$ is expected to be dormant and indeed shows an exponential fall off in wavenumbers $k$ in figure 4(c). Here $\varPi _{{p}}$ instead decays only as a steep power-law, and forces fluctuations in the velocity field that result in the steep, and extended $k^{-4}$ range in figure 2(d) .

3.3. Slowing down the energy cascade

While the above discussion in § 3.1 on flux contributions sheds light on the non-unique scaling nature of the energy spectra, there is more that remains in hiding in figure 3. We uncover this by plotting the flux contributions on a log–log scale in figure 4 which clearly reveals the $k$ -dependence of the flux contributions. We are particularly interested in the fluid $\varPi _{{f}}$ and polymeric $\varPi _{{p}}$ fluxes.

Figure 4(a) shows that at large ${Re}_\lambda$ our polymeric regime $E(k) \sim k^{-2.3}$ is marked by a fluid nonlinear flux decaying as $\varPi _{{f}} \sim k^{-1.2}$ , while $\varPi _{{p}}$ remains almost constant. Now, $E(k)$ and $\varPi _{{f}}$ can be related to each other using the following simple arguments. Here $\varPi _{{f}}$ is the average rate of net energy transfer through a scale $k$ . This means velocity fluctuations $u_k$ upto the scale $k$ transfer their energy to smaller scales in some characteristic time $\tau _k \sim (k u_k)^{-1}$ . In HIT, this characteristic time scale is equal to the typical lifetime of an eddy within the K41 phenomenology. This is because energy flux $\varPi _{{f},\textit {N}}$ in the inertial range is a constant, i.e. $\varPi _{{f},\textit {N}} = \epsilon$ , with $\epsilon$ being the average rate of energy dissipation. Therefore, we have

(3.9) \begin{align} \varPi _{{f},\textit {N}} \sim \frac {u_k^2}{\tau _k} \sim k u_k^3 \sim \epsilon \,\implies\, u_k \sim k^{-1/3} \end{align}

in the inertial range of scales. This means that the time to transfer energy to smaller scales and the typical lifetime of eddies are, respectively, estimated as

(3.10) \begin{align} \tau _{k,\textit {N}} \sim \frac {u_k^2}{\varPi _{{f},\textit {N}}} \sim \frac {k E_k}{k^0} \sim k^{-2/3} ; \qquad \tau _{k,\textit {N}} \sim (k u_k)^{-1} \sim (k k^{-1/3})^{-1} \sim k^{-2/3}. \end{align}

Figure 4. The scaling behaviour of flux contributions obtained by replotting figure 3 on a log–log scale. The approximate scaling forms are shown as dash–dotted lines. The fluid flux $\varPi _{{f}}$ decays as a power-law in the presence of polymers at large to moderate ${Re}_\lambda$ , implying a slow and weak fluid nonlinear cascade.

In HIT, these time scales are identical as they both are a consequence of a constant inertial range energy flux $\varPi _{{f},\textit {N}}$ . This, however, is no longer true in PHIT where the fluid nonlinear flux $\varPi _{{f}}$ is no longer a constant, but is rather a function of the scale $k$ (and therefore is no longer the invariant that determines the scaling form of the velocity fluctuations). For the remainder of this section, we refer to $De = 1$ as it shows maximal deviation from a Newtonian behaviour. We can still estimate the eddy turnover time scale in PHIT as

(3.11) \begin{align} \tau _k \sim (k u_k)^{-1} \sim \left ( k \sqrt {kE_k} \right )^{-1} \sim k^{-0.3}. \end{align}

So, the typical lifetimes of fluctuations in polymeric turbulence are larger compared with those in Newtonian turbulent flows as $\tau _k$ now falls off slower with $k$ . Thus, energy is transferred at a slower rate to smaller scales of the fluid in the presence of polymers. With this knowledge, we can now determine the fate of the fluid nonlinear flux $\varPi _{{f}}$ via the relation $\tau _k \sim u_k^2/\varPi _{{f}}$ as

(3.12) \begin{align} \varPi _{{f}} \sim \frac {u_k^2}{\tau _k} \sim \frac {k E_k}{k^{-0.3}} \sim k^{-1.0}. \end{align}

Clearly, the scale-by-scale energy transfer via the nonlinear cascade is weakened in polymeric turbulence by the virtue of a smaller proportion of the (total averaged) fluid kinetic energy $u_k^2/u^2 \sim k^{-1.3}$ being passed onto the larger $k$ at a smaller rate $\tau _k/\tau _L \sim k^{-0.3}$ , compared with a Newtonian fluid. The estimated scaling behaviour of $\varPi _{{f}} \sim k^{-1}$ is indeed found to hold in figure 4 for the range $k \in [8,40]$ which is separated from both the forcing effects (acting at $k=1$ ) and fluid dissipation (which becomes dominant around $k \approx 40$ ). (Recall, however, that the energy spectrum scaling $E(k) \sim k^{-2.3}$ holds for the range $k \in [4,40]$ .) This depleted fluid nonlinear flux is accompanied by a finite rate of energy transfer to the dissolved polymers. The typical time scale of this transfer can be estimated by making a crude assumption of a constant polymeric flux, i.e. $\varPi _{{p}} \approx k^0$ , in $k \in [10,70]$ from figure 4(a). This assumption then yields the scale-by-scale energy transfer rate to polymers as

(3.13) \begin{align} \tau _k \sim \frac {u_k^2}{\varPi _{{p}}} \sim \frac {k E_k}{k^0} \sim k^{-1.3}. \end{align}

This shows that the remaining energy of fluctuations, that was not handed down to the smaller active scales of motion of the carrier fluid by the nonlinear flux, is transferred to the polymers, on an average, at a much faster rate $\tau _k \sim k^{-1.3}$ . Now, we expect this relation to hold only approximately for a restricted range of $k \in [10,40]$ as $\varPi _{{p}}$ is not exactly constant in this range. With these caveats in mind, we now plot the compensated time scales in figure 5(a) for Re $_\lambda = 450$ . The relevant ranges are marked by the flattening of the curves which are computed using the relations (3.11), (3.12) and (3.13). Of these, the first two are the estimates of eddy-turnover times/lifetime of velocity fluctuations in polymeric turbulence. Now, the relations (3.11) and (3.12) must yield the same estimates of $\tau _k$ up to some constant of ${O}(1)$ . (To obtain $\tau _k$ from the second relation (3.12) we use the estimate $\varPi _{{f}} \sim k^{-1}$ ). This is indeed found to be consistent in figure 5(a) where we show these two estimates as curves with triangle and square markers, respectively. And while these curves do not stay flat over a wide range, it is certainly true for the expected range $k \in [8,40]$ . Lastly, the typical time scales of energy transfer to the polymeric mode is given by the compensated curve in diamond markers which corresponds to the relation (3.13). In confirmation to our expectations, we indeed find that this curve is (approximately) flat in the range $k \in [10,40]$ in figure 4(a). So, it is indeed the case that polymers extract energy at a faster rate from turbulence than the fluid nonlinearity transfers the remaining to smaller scales, on an average resulting in a weakened cascade/fluid nonlinearity.

Figure 5. The plots of various time scales discussed in § 3.3 for (a) Re $_\lambda$ = 450 and (b) 240, at $De = 1$ . We show the compensated plots for the eddy turnover time (triangles) and the time obtained from the nonlinear (squares) and polymer fluxes.

We now turn our attention to when Re $_\lambda = 240$ . At the outset, we notice from figure 4(b) that there is a rather very small set of scales that lies between the forcing scale ( $k=1$ ) and the scale where fluid dissipation becomes important ( $k = 10$ ). Naturally, it cannot be expected that the lifetime of fluctuations $\tau _k$ is reliably estimated by relating it to the nonlinear flux via $\varPi _{{f}} \sim u_k^2 /\tau _k$ , as this relation does not account for losses due to viscous dissipation. However, it still estimates the rate at which energy is transferred to smaller scales. Noting that $\varPi _{{f}}$ still has a power-law behaviour of $\varPi _{{f}} \sim k^{-1.4}$ in $k \in [3,12]$ and $\varPi _{{p}} \approx {\rm constant}$ in this range as well, we can now estimate the various time scales at Re $_\lambda = 240$ as

(3.14) \begin{align} \tau _k \sim (k u_k)^{-1} \sim k^{-0.3} \ ; \quad \tau _k \sim \frac {kE_k}{\varPi _{{f}}} \sim k^{0.1} \ ; \quad \tau _k \sim \frac {kE_k}{\varPi _{{p}}} \sim k^{-1.3}. \end{align}

We plot the corresponding compensated curves of these transfer time scales in figure 5(b). Consistent with our expectations, all of these curves are fairly constant in the rather small desired range $k \in [4,10]$ .

In a nutshell, the effect of the addition of polymers is to slow down the cascade process which we quantify using scaling arguments the knowledge of flux contributions, and the fluid energy spectra. In the next section, we discuss how energy is distributed in the polymeric scales of motion via the polymer energy spectrum using data as well as scaling arguments.

3.4. The polymer spectra

The distribution of energy across polymeric scales, in a statistically stationary state, can be obtained by computing the polymer energy spectrum $E_{{p}} (k)$ in analogy with the fluid energy spectrum $E (k)$ . To do so, we start by defining the polymer energy spectrum in a sense similar to that of Casciola & De Angelis (Reference Casciola and De Angelis2007), Balci et al. (Reference Balci, Thomases, Renardy and Doering2011) and Nguyen et al. (Reference Nguyen, Delache, Simoëns, Bos and El Hajem2016). The measure of average total polymer energy $ \mathcal{E}_{{p}}$ is given by the ${\textit{Tr}}(\boldsymbol{C})$ as

(3.15) \begin{align} \mathcal{E}_{{p}} = \frac {\mu _{{p}}}{2 \tau _{{p}}} \left \langle {\textit{Tr}}(\boldsymbol{C}) \right \rangle = \frac {\mu _{{p}}}{2 \tau _{{p}} V} \int _{V} C_{ii}(\boldsymbol{x}) {\rm d}\boldsymbol{x} = \frac {\mu _{{p}}}{2 \tau _{{p}} V} \int _{V} B_{\textit{ij}}(\boldsymbol{x}) B_{ji}(\boldsymbol{x}) {\rm d}\boldsymbol{x} , \end{align}

where $B$ (with components $B_{\textit{ij}}$ ) is the positive-definite square root of the conformation tensor $\textit {C}$ , i.e. $C_{\textit{ij}} (\boldsymbol{x},t) = B_{ik} (\boldsymbol{x},t) B_{kj} (\boldsymbol{x},t)$ . One can now define the polymer energy spectrum $E_{{p}}(k)$ by employing Parseval’s relation

(3.16) \begin{align} \mathcal{E}_{{p}} = \frac {\mu _{{p}}}{2 \tau _{{p}} V} \int _{V} B_{\textit{ij}}(\boldsymbol{x}) B_{ji}(\boldsymbol{x}) {\rm d}\boldsymbol{x} &= \frac {\mu _{{p}}}{2 \tau _{{p}} V} \int _{\mathbb{R}^3} B_{\textit{ij}}(\boldsymbol{k}) B_{ji}(-\boldsymbol{k}) {\rm d}\boldsymbol{k} = \int _0^{\infty } E_{{p}}(k) \ {\rm d}k \nonumber \\ \text{where} \quad E_{{p}}(k) &= \frac {\mu _{{p}}}{2 \tau _{{p}} V} \int _{|\boldsymbol{k}| = k} B_{\textit{ij}}(\boldsymbol{k}) B_{ji}(-\boldsymbol{k}) \ {\rm d}\varOmega _k, \end{align}

where $V$ is volume of integration and the last integral is defined over a spherical shell of radius $k$ .

Now, the scale-by-scale energy $(\mu _{{p}}/2\tau _{{p}})C_k$ can then be estimated as the energy contained in the modes $\in [ k, k+{\rm d}k ]$ ,

(3.17) \begin{align} \frac {\mu _{{p}}}{2 \tau _{{p}}} C_k = E_{{p}}(k) {\rm d}k, \end{align}

in analogy with the classical

(3.18) \begin{align} \frac {1}{2} u_k^2 = E_k (k) {\rm d}k. \end{align}

Thus, it is easy to see using dimensional arguments that $C_k \sim k E_{{p}}(k)$ , similar to $u_k^2 \sim k E_k(k)$ . We now use this relation to estimate $E_{{p}}(k)$ by relating it with $E(k)$ .

Figure 6. The dependence of the polymer energy spectrum $E_{{p}}(k)$ on (a) ${Re}_\lambda$ (shown for $De \approx 1$ ), and on (b) De (shown for Re $_\lambda \approx$ 450). The two different scaling regimes with exponents $-1.35$ (at large Re) and $-1.5$ (at small Re) are shown in dash–dotted and dotted lines, respectively. (b) Small De flows at large Re show a close to Newtonian behaviour and $E_{{p}}(k)$ remains devoid of any scaling form. As De $\gtrsim 1$ , the expected scaling with exponent $-1.35$ begins to appear. The spectra are shifted vertically for visual clarity by factors of powers of 10.

Let us begin by looking at the $\varPi _{{p}}$ curves in figure 3 (or figure 4). For large ${Re}_\lambda$ , the flux contribution $\varPi _{{p}}$ remains approximately constant for $k \in [ 20,60 ]$ . Therefore,

(3.19) \begin{align} \frac {\mu _{{p}}}{\tau _{{p}}} k u_k C_k \sim k^0 \,\implies C_k \sim \left ( ku_k \right )^{-1} \sim k^{-0.35}. \end{align}

This immediately implies, using (3.17), that $E_{{p}}(k) \sim k^{-1} C_k \sim k^{-1.35}$ for $k \in [ 20,60 ]$ . Indeed, we find this to be consistent with the data from simulations plotted in figure 6(a). At ${Re}_\lambda \approx 240$ , this scaling range of $E_{{p}}(k)$ shows a slight shift to larger scales, in correspondence to a similar shift of constant $\varPi _{{p}}$ in figure 4. We point out here that in contrast to the behaviour of $E(k)$ , the scaling range for $E_{{p}} (k)$ appear cleaner with decreasing Re $_\lambda$ . This is because the assumption of $\varPi _{{p}} \sim k^0$ is not exactly true, especially for the largest Re $_\lambda$ , and only provides a rough estimate of the self-similarity of $E_{{p}} (k)$ . However, the scaling ranges get better with decreasing Re $_\lambda$ also because the fluid nonlinearity becomes weaker and elastic effects begin to dominate the flux contributions. This is clearly brought out comparing figures 4(a) and 4(b). More importantly, these estimates are obtained by relating the polymer statistics to that of the fluid velocity field, i.e. the knowledge of one is required to obtain the other. However, the relation between these statistics changes at the smallest Re $_\lambda = 40$ where the fluid nonlinearity remains dormant and falls-off exponentially as shown in figure 4(c).

At the smallest ${Re}_\lambda \approx 40$ , dissipative effects become dominant so that polymeric contribution $\mathcal{P}$ to the total flux is of the same order as $\mathcal{D}_{\!{f}}$ for $k \gtrsim 10$ in figure 4(c). Hence, a simple comparison of the fluid dissipation and polymer terms gives (see also Singh et al. Reference Singh, Perlekar, Mitra and Rosti2024)

(3.20) \begin{align} k^2 u_k \sim k C_k \implies E_{{p}} (k) \sim u_k \sim \sqrt {k E_k} \sim k^{-3/2}, \end{align}

which is also consistent with the data shown in figure 6(a).

We now show the variation of the polymer spectrum with $\textit{De}$ in figure 6(b) for large ${Re}_\lambda$ . At very small $\textit{De}$ , the polymers barely stretch and they quickly relax back to their equilibrium lengths. This means fluctuations in their lengths are largely limited to only small scales and minimal large-scale fluctuations. Thus, at relatively large scales and small $k$ , for small $\textit{De}$ polymers, one expects a minimal growth in the spectrum of fluctuations making it appear rather flat. This can otherwise also be understood in terms of real space fluctuations as follows. In real space, as the polymers remain small, their fluctuations are localised to small scales as well such that their fluctuations are correlated over only small scales. So, at large separations, these fluctuations are decorrelated and appear as white noise. This gives the spectrum its flat shape over a wide range of $k$ . However, at very large $k$ , the spectrum $E_{{p}} (k)$ falls off much rapidly which is just a consequence of the small scales being well resolved and the polymer fields being analytic.

Now, as $\textit{De}$ increases, polymers begin to stretch to longer lengths, so that their fluctuations are now correlated over longer lengths whose effect begins to be seen in the spectrum being more pronounced at small $k$ . At largest $\textit{De}$ , in a close agreement with the scaling arguments (3.19), we find that the spectrum $E_{{p}} (k) \sim k^{-1.35}$ for a wide range. The small scale fall-off of the spectra still confirms the analyticity of the polymer stress field.

We now move on to real space statistics, by first discussing the non-unique scaling nature using structure functions, complementing the discussion in § 3.1.

3.5. Velocity differences and structure functions

In this section, we discuss how a non-unique scaling behaviour is also manifested in the real-space statistics of PHIT. This complements the Fourier space discussion of § 3.1. One of the most common real-space measures that admit a scaling behaviour in turbulence are the well-known structure functions $S_{{p}}(r)$ , defined as the ${p}$ th moments of the longitudinal velocity increments $\delta _{\boldsymbol{r}} u$ over a separation $\boldsymbol{r}$ ,

(3.21) \begin{align} \delta _{\boldsymbol{r}} u &\equiv \left [ \boldsymbol{u}(\boldsymbol{x} + \boldsymbol{r}) - \boldsymbol{u}(\boldsymbol{x}) \right ] \boldsymbol{\cdot }\hat {\boldsymbol{r}}, \end{align}

(3.22) \begin{align} S_{{p}}(r) = \langle \left ( \delta _{\boldsymbol{r}} u \right )^{{p}} \rangle &= \frac {1}{3} \sum _{i=1}^{3} \left \langle \left [ u_i \left ( x_i + r_i \right ) - u_i\left ( x_i \right ) \right ]^{{p}} \right \rangle\! . \end{align}

In particular, $S_2 (r)$ is related to the energy spectrum $E(k)$ via a Fourier transform. For ease of arguments, we illustrate this using the vector second-order structure functions defined as (see (6.29), (6.40a) in Davidson (Reference Davidson2015))

(3.23) \begin{align} \big\langle \left [ \boldsymbol{u} \left ( \boldsymbol{x} + \boldsymbol{r} \right ) - \boldsymbol{u} \left ( \boldsymbol{x} \right ) \right ]^2 \big\rangle \equiv \big\langle \left [ \Delta \boldsymbol{u} \right ]^2 \big\rangle = \frac {1}{r^2} \frac {\partial }{\partial r}\big[ r^3 S_2 (r) \big]. \end{align}

The term on the left-hand side can be expanded as

(3.24) \begin{align} \big\langle \left [ \boldsymbol{u} \left ( \boldsymbol{x} + \boldsymbol{r} \right ) - \boldsymbol{u} \left ( \boldsymbol{x} \right ) \right ]^2 \big\rangle &= 2 \langle \boldsymbol{u}^2 \rangle - 2 \left \langle \boldsymbol{u} \left ( \boldsymbol{x}+\boldsymbol{r} \right ) \boldsymbol{\cdot }\boldsymbol{u} \left ( \boldsymbol{x} \right ) \right \rangle \nonumber \\ &= 4 \int {\rm d}\boldsymbol{k} \ |\hat {u}(\boldsymbol{k})|^2 - 4 \int {\rm d}\boldsymbol{k} |\hat {u}(\boldsymbol{k})|^2 e^{-i \boldsymbol{k} \boldsymbol{\cdot }\boldsymbol{r}} \nonumber \\ & = 4 \int {\rm d}k \ E(k) \left ( 1-\frac {\sin {kr}}{kr} \right ) \end{align}

where we have the definition of Fourier transforms to obtain the second line and isotropy to obtain the last line. The previous relation can then be easily used to relate the power-law behaviour of $S_2$ and $E(k)$ . Consider that the separation $r$ in the real-space is scaled by a factor $b$ , i.e. $r \to b r$ , so that the Fourier space wavenumber scales as $k \to b^{-1}k$ . Consequently, if the spectrum scales with an exponent $\alpha$ , i.e. $E(k) \sim k^{-\alpha }$ , then $E(k) \to E(b^{-1}k) = b^{\alpha } E(k)$ . Therefore, the right-hand side of (3.24) can be rewritten under this rescaling as

(3.25) \begin{align} \big\langle \left [ \Delta \boldsymbol{u} \right ]^2 \big\rangle _{\textit {rescaled}} &= 4 \int \big( b^{-1} {\rm d}k \big)\ E(b^{-1}k) \left ( 1-\frac {\sin {kr}}{kr} \right ) \nonumber \\ &= 4 b^{\alpha -1} \int {\rm d}k \ E(k) \left ( 1-\frac {\sin {kr}}{kr} \right ) = b^{\alpha -1} \big\langle \left [ \Delta \boldsymbol{u} \right ]^2 \big\rangle. \end{align}

Thus, it is easy to see that the vector and longitudinal structure functions transform identically under rescaling since

(3.26) \begin{align} \big \langle \left [ \Delta \boldsymbol{u} \right ]^2 \big\rangle _{\textit {rescaled}} &= \frac {1}{ (br)^2} \frac {\partial }{\partial (br)}\big[ (br)^3 S_2 (br) \big] = \frac {1}{ r^2} \frac {\partial }{\partial r}\big[ r^3 S_2 (br) \big] \nonumber \\ & = b^{\alpha -1} \big\langle \left [ \Delta \boldsymbol{u} \right ]^2 \big\rangle = \frac {1}{r^2} \frac {\partial }{\partial r}\big[ r^3 b^{\alpha -1} S_2 (r) \big] \implies S_2 (br) = b^{\alpha -1} S_2 (r). \end{align}

Figure 7. Manifestation of the non-unique scaling behaviour in the structure functions of second ( $S_2$ : panels (a) and (c)) and fourth ( $S_4$ : panels (b) and (d)) orders. Panels (a) and (b) show the dependence on ${Re}_\lambda$ , while panels (c) and (d) show the dependence on the polymer elasticity $\textit{De}$ . Note that the (c) $S_2(r)$ and (d) $S_4(r)$ curves fall on top of the Newtonian for small $\textit{De}$ . At large $\textit{De}$ , the elastic scaling regime becomes clear as elasticity begins to play a significant role showing a clear departure from the Newtonian curve.

Hence, we have that $E(k) \sim k^{-\alpha } \iff S_2(r) \sim r^{\alpha -1}$ . Note that this relation holds only for $\alpha \leqslant 3$ . For $\alpha \gt 3$ , the real-space exponent is constrained by the leading order in Taylor expansion as $S_2 \sim r^{2}$ . In HIT, these arguments imply that $S_2 \sim r^{2/3}$ given $E(k) \sim k^{-5/3}$ whereas in PHIT we now expect $S_2 \sim r^{1.3}$ using $E(k) \sim k^{-2.3}$ . Additionally, using dimensional arguments, one also expects within the K41 phenomenology that $S_4 \sim S_2^2$ . With these expected results in mind, we show in figure 7(a,b) the plots of $S_2(r), S_4(r)$ for PHIT at different ${Re}_\lambda$ and $De \approx 1$ alongside the HIT result. The variation of $S_2(r)$ and $S_4(r)$ with $\textit{De}$ is shown in figure 7(c,d). We of course have that the HIT structure functions follow the Kolmogorov scaling $r^{{p}/3}$ closely in figure 7(a,b), while those for PHIT show a very clear departure from this behaviour. Indeed, and in consistency with the discussion above, the polymeric scaling is given by $S_2 \sim r^{1.3} (S_4 \sim r^{2.6})$ and is seen to have the widest span at the largest ${Re}_\lambda \approx 450$ . This is also consistent with the discussion in § 3.1 where we show that the polymeric regime shrinks with decreasing ${Re}_\lambda$ , while the smooth dissipation range marked by $S_2 \sim r^2$ expands (corresponding to $E(k) \sim k^{-4}$ in figure 2). At the smallest ${Re}_\lambda \approx 40$ , the smooth scaling $S_2 \sim r^2$ spans almost a decade.

We show the variation of the structure functions with $\textit{De}$ in figure 7(c,d). As expected from figure 2(a), this dependence is rather non-monotonic. The small $\textit{De}$ cases show an approximately Newtonian behaviour, owing to the minimal non-Newtonian contributions as the polymers strongly resist any stretching (see § 3.1 for detailed discussion). As $De \to 1$ , we have a clear polymeric scaling $S_2 \sim r^{1.3} (S_4 \sim r^{2.6})$ . This steeper than Kolmogorov slope starts to vanish when polymer elasticity is further increased, and the $S_2/ S_4$ curves now have a shallower slope. At much larger polymer elasticity one indeed recovers the Kolmogorov scaling (see Rosti et al. (Reference Rosti, Perlekar and Mitra2023)). This behaviour is again in consistency with that for the spectrum in § 3.1.

We also make an important note about figure 7(b,d). The $S_4(r)$ curves evidently show smaller scaling ranges compared with those of $S_2(r)$ in figure 7(a,c). This means that the exponents quickly deviate from the naïve expectation of $S_4(r) \sim S_2^2(r)$ . This is not only an artefact of the deviations from a K4-like behaviour but also captures the non-Gaussianity of $\delta _{\boldsymbol{r}} u$ . We quantify these deviations by measuring the kurtosis of the velocity increment distributions in the next section.

3.6. Kurtosis of velocity differences

Consider the distribution of (longitudinal) velocity differences $\delta _{\boldsymbol{r}} u$ , defined by (3.21), as a function of the scale $r$ . The kurtosis (or flatness) $\mathcal{K}(r)$ of such a distribution is defined as

(3.27) \begin{align} \mathcal{K} (r) \equiv \frac {\left \langle \left ( \delta _{\boldsymbol{r}} u \right )^4 \right \rangle }{\left \langle \left ( \delta _{\boldsymbol{r}} u \right )^2 \right \rangle ^2}. \end{align}

The kurtosis $\mathcal{K}$ captures the relative importance of the tails of a distribution and quantifies their contribution to the overall statistics of $\delta _{\boldsymbol{r}} u$ . For a Gaussian distribution, $\mathcal{K} = 3$ . We expect the velocity increments to be uncorrelated over very large separations, so that the distribution of $\delta _{\boldsymbol{r}} u$ is close to a Gaussian for large $r$ . This is indeed confirmed in our data in figure 8 where we find $\mathcal{K}(r) \approx 3$ for large $r$ , irrespective of ${Re}_\lambda$ and $\textit{De}$ . More importantly, $\mathcal{K}$ is not a constant but a function of the scale $r$ . The distributions of $\delta _{\boldsymbol{r}} u$ are, evidently, devoid of scale-invariance and their departures from Gaussianity become increasingly stronger at small $r$ . This is a result of more important contributions from the tails of the distribution of $\delta _{\boldsymbol{r}} u$ , a phenomenon referred to as intermittency, and causes deviations from the dimensional expectation of $S_4 \sim S_2^2$ (this would have meant $\mathcal{K} \sim r^0$ from (3.27)). The log–log plots of $\mathcal{K}$ in figure 8(a) clearly show a power-law dependence on $r$ at large ${Re}_\lambda \approx 450$ . Crucially, this power-law is very weakly dependent on $\textit{De}$ as the curves for different $\textit{De}$ follow each other closely. Thus, the intermittent nature of the velocity fluctuations is still largely determined by the fluid nonlinearity. This was also observed by Rosti et al. (Reference Rosti, Perlekar and Mitra2023) via the multifractal spectrum for energy dissipation. The influence of polymers on velocity fluctuations, however, is more apparent at small ${Re}_\lambda$ , as shown in figure 8(b). At small ${Re}_\lambda$ , although the fluid nonlinearity becomes considerably weaker (see figure 4 b), the polymer stresses feeding back on the flow result in the velocity fluctuations $\delta _{\boldsymbol{r}} u$ being more intermittent, and the departure from HIT curve becomes significant. This is even more pronounced at the smallest ${Re}_\lambda$ (we considered) where the presence of polymers results in much large deviations of $\delta _{\boldsymbol{r}} u$ at small $r$ which is manifested in a much steeper slope of the kurtosis $\mathcal{K} (r) \sim r^{-0.3}$ compared with the Newtonian.

Figure 8. The kurtosis of velocity differences as a function of scale $r$ . Here $\mathcal{K} \to 3$ as $r \to L$ as velocity differences decorrelate at very large separations. (a) Here $\mathcal{K}$ at large ${Re}_\lambda \approx 450$ and different $\textit{De}$ are almost coincident implying a very weak dependence of intermittency on $\textit{De}$ . (b,c) Intermittency in polymeric flows increases as ${Re}_\lambda$ is decreased, especially when $\textit{De}$ is large.

So, the departure from a Kolmogorov-like behaviour in polymeric flows is not only due to a modified energy flux – resulting in a different scaling of the spectrum – but also due to the modified nature of extreme fluctuations that result from the action of the polymer stresses. This also underlines the fact that intermittency is due not only to fluid nonlinearity but even polymer stresses can drive highly intermittent velocity fluctuations. Such intermittent behaviour is also discernible in the statistics of velocity gradients. In the following sections, we discuss the intermittent nature of velocity gradients, and how they influence the stretching of polymers. This also forms the basis for the discussion of the two energy dissipation rates: fluid $\epsilon _{{f}}$ and polymer $\epsilon _{{p}}$ .

3.7. Polymer stretching, velocity gradients and their relation to the energy dissipation

We now discuss in this section how polymers of different $\textit{De}$ are themselves affected by the fluid flow at different ${Re}_\lambda$ . To this end, we begin by simply looking at the stretching statistics of the polymers, the measure of the stretching of the polymers is their end-to-end lengths whose instantaneous squared value is given by ${\textit{Tr}}(\boldsymbol{C})$ . We naturally expect polymers to stretch more with increasing $\textit{De}$ at any given ${Re}_\lambda$ . As stated earlier, in the limit of $De \to 0$ the polymers are almost inextensible and $\boldsymbol{C}$ does not stray far away from its equilibrium configuration $\boldsymbol{I}$ . On the contrary, the other limit of $De \to \infty$ implies a weak restoring force, allowing $\boldsymbol{C}$ to grow to large values. That is, polymers take a very long time to relax back to their equilibrium lengths for a very large $\tau _{{p}}$ . Or in other words their tendency to relax back, when extended, is much less. Evidently, polymers are stretched by the local velocity gradients given by the first two terms on the right-hand side of (2.1b ).

Figure 9. Probability distribution functions (p.d.f.s) of the end-to-end polymer lengths ${\textit{Tr}}(\boldsymbol{C})$ for different ${Re}_\lambda$ : (a) ${Re}_\lambda \approx 450$ , (b) ${Re}_\lambda \approx 240$ and (c) ${Re}_\lambda \approx 40$ and all $\textit{De}$ . A large $\textit{De}$ polymer has a larger average end-to-end length, as seen in the right shift of the p.d.f.s.

Figure 10. (a) Mean $\mu _{{\textit{Tr}}(\boldsymbol{C})}$ and (b) standard deviation $\sigma _{{\textit{Tr}}(\boldsymbol{C})}$ of the distributions in figure 9, shown as a function of ${Re}_\lambda$ and $\textit{De}$ .

We plot the p.d.f.s of ${\textit{Tr}}(\boldsymbol{C})$ for different ${Re}_\lambda$ in figure 9. We first note that although polymers described by the Oldroyd-B model are infinitely stretchable, they can only do so in the presence of very large and sustained velocity gradients. This is confirmed by the very fast decay of the tails of the ${\textit{Tr}}(\boldsymbol{C})$ -p.d.f.s in figure 9, clearly suggesting that the polymer lengths always remain bounded in our simulations. Figure 9(a) shows that, at large ${Re}_\lambda$ , the p.d.f.s shift towards the right with increasing $\textit{De}$ indicating a larger averaged polymer length. Moreover, the width of the distributions also increases with $\textit{De}$ , meaning that polymer lengths also fluctuate over a wider range. This is more accurately shown by figure 10 where we plot the mean ( $\mu _{{\textit{Tr}}(\boldsymbol{C})}$ ) and standard deviation ( $\sigma _{{\textit{Tr}}(\boldsymbol{C})}$ ) of the distributions as a function of ${Re}_\lambda$ and $\textit{De}$ . Indeed, the polymer lengths and their fluctuations increase with both ${Re}_\lambda$ and $\textit{De}$ .

Figures 9(b) and 9(c) show the ${\textit{Tr}}(\boldsymbol{C})$ distributions for smaller ${Re}_\lambda$ . With decreasing ${Re}_\lambda$ , turbulence intensity decreases and the velocity gradients become weaker on an average and stretch the polymers less (as can be seen from (2.1b )). Therefore, the mean polymer lengths decrease with decreasing ${Re}_\lambda$ as the p.d.f.s shift towards smaller values of ${\textit{Tr}}(\boldsymbol{C})$ . Moreover, polymers are also limited in their maximum extensions as ${Re}_\lambda$ decreases. At the smallest ${Re}_\lambda$ , the velocity gradients are unable to stretch the less extensible, small $\textit{De}$ polymers considerably. This is captured in figure 9(c), where the ${\textit{Tr}}(\boldsymbol{C})$ -p.d.f.s fall-off very fast at small $\textit{De}$ showing that polymer lengths have rather small fluctuations. (This also implies that polymer stresses remain very small so that the flow remains laminar for $De \lt 1$ ). However, for a large enough $De (\approx 1)$ , polymers stretch to considerably long lengths, and the resulting large fluctuations are enough to excite subdominant velocity fluctuations which gives rise to ET with $E(k) \sim k^{-4}$ (see figure 2).

Figure 11. The plot of the joint distributions of the polymer lengths on log–log scale with the (a–c) dissipation rate $\epsilon _{{f}}$ and (d–f) enstrophy $\varOmega$ , for different ${Re}_\lambda$ with $De \approx 1$ . The colour bar shows the colour coding of the logarithm of probability $\log _{10} p$ .

The above discussion is largely based on how polymers with different $\textit{De}$ are stretched by carrier flows at different ${Re}_\lambda$ . To understand this better, we now discuss how the polymer stretching ${\textit{Tr}}(\boldsymbol{C})$ is correlated with the velocity gradients $A_{\textit{ij}} (\boldsymbol{x},t) \equiv \partial _i u_j (\boldsymbol{x},t)$ in the flow. To this end, we first decompose $A_{\textit{ij}}$ into its symmetric $S_{\textit{ij}}$ and antisymmetric $\omega _{\textit{ij}}$ components that, respectively, capture the purely extensional (compressional) and the purely rotational contributions. The squared magnitude of these components are given by the fluid dissipation rate $\epsilon _{{f}}$ and enstrophy $\varOmega$ ,

(3.28) \begin{align} \varOmega = \omega _{\textit{ij}} \omega _{\textit{ij}} &\ ; \quad \epsilon _{{f}} = 2 \nu S_{\textit{ij}}S_{\textit{ij}}, \nonumber \\ \text{where} \quad \omega _{\textit{ij}} = ( \partial _i u_j - \partial _j u_i )/2 &\ ; \quad S_{\textit{ij}} = ( \partial _i u_j + \partial _j u_i )/2. \end{align}

The presence of the symmetric term $\boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} + (\boldsymbol{\nabla }\boldsymbol{u})^T \boldsymbol{C} = 2 C_{ik} S_{kj}$ in (2.1b ) means that the extensional (compressional) regions of the flow are directly responsible for the stretching (relaxation) of the polymers. On the other hand, the absence of any $\omega _{\textit{ij}}$ terms means that there is no direct effect of the rotation in changing the lengths of polymers. However, turbulence in three dimensions is marked by the stretching and intensification of vorticity in the direction of local velocity gradients. So, the vortical regions can be expected to correlate with large polymer lengths, although to a lesser degree than the straining regions. (A direct computation of Pearson correlation coefficients of the joint distributions of $({\textit{Tr}}(\boldsymbol{C}),\epsilon _{{f}})$ as well as those of $({\textit{Tr}}(\boldsymbol{C}),\varOmega )$ yielded values $\approx 0.3$ for all cases, while those of the $\log$ of normalised variables which are actually studied, e.g. of $(\log [{\textit{Tr}}(\boldsymbol{C})/ \langle {\textit{Tr}}(\boldsymbol{C}) \rangle ], \log [ \epsilon _{{f}}/ \langle \epsilon _{{f}} \rangle ])$ , yielded values $\approx 0.4$ showing that indeed these variables have a non-zero, finite correlation.)

To illustrate the above, we show in figure 11 the joint p.d.f.s (j.p.d.f.s) of the polymer lengths (given by ${\textit{Tr}}(\boldsymbol{C})$ ) with the local stretching rates (given by $\epsilon _{{f}}$ ) as a function of ${Re}_\lambda$ for one instance of polymer elasticity of $De = 1$ in figure 11(a–c). We observe that the longest end-to-end lengths of the polymers indeed coincide with the largest stretching rates, while the smallest polymers are found in regions of the flow with small extensional rates. This results in a perceivable tilt of the joint distribution in figure 11(a). With decreasing ${Re}_\lambda$ , the mean lengths of the polymers decreases (see figure 10) and their preference for regions with small-scale fluctuations increases. This is evident from figure 11(b,c) where elongated bottom tails of the j.p.d.f.s imply small polymers lengths are more probable at smaller ${Re}_\lambda$ . While we still have that the largest polymer extensions coincide with the largest velocity gradients, the core of the j.p.d.f.s is now more vertical which indicates that much larger extensions are now possible at even the average stretching rates. This hints towards a decreasing degree of correlation between extreme stretching rates and polymer stretching as ${Re}_\lambda$ is reduced.

Figure 12. Joint distributions of the polymer lengths with the (a–c) $\epsilon _{{f}}$ and (d–f) $\varOmega$ , for different $\textit{De}$ at ${Re}_\lambda \approx 450$ . Different colours correspond to the logarithm of probability $\log _{10} p$ as coded by the colourbar.

Figure 11(d–f) show the j.p.d.f.s of ${\textit{Tr}}(\boldsymbol{C})$ and enstrophy $\varOmega$ . At large ${Re}_\lambda$ , large polymer lengths also coincide with extreme vorticity regions. This means polymers are also stretched in regions of large vorticity (similar to what observed for long, polymer-like fibres in Picardo et al. (Reference Picardo, Singh, Ray and Vincenzi2020)). The maximum enstrophy regions, however, are more likely to have polymers with close to average lengths, implied by figure 11(d), unlike the straining regions whose extremes see maximally extended polymers. More importantly, polymers are likely to be stretched even in regions with very small $\varOmega$ as suggested by the left bulge of the j.p.d.f.s. This shows that the polymer stretching–enstrophy correlation is not as strong as the stretching–straining correlation. This is even more evident with decreasing ${Re}_\lambda$ as vortex stretching becomes progressively weaker. Indeed, figure 11( f) shows that polymer lengths are rather uniformly distributed, especially for small enstrophy, indicating an even less correlation with polymer stretching.

Figure 12 shows instead how polymer stretching correlates with the local stretching (figure 12 a–c) and rotation rates (figure 12 d–f) of the flow, as a function of polymer elasticity. It was already shown that large $\textit{De}$ polymers stretch more and is clearly captured by their mean lengths in figure 10. At small $\textit{De}$ , polymers stretch minimally and have small relaxation times. Therefore, while they are stretched by regions with very large stretching rates, those regions have rather small lifetimes (a quick estimate gives $\tau \sim S_{\textit{ij}}^{-1}$ ). So, maximally stretched polymers are more likely to persist for longer in regions with moderate stretching rates. This means that the maximum of ${\textit{Tr}}(\boldsymbol{C})$ coincides with only slightly larger than the average values of $ \langle \epsilon _{{f}} \rangle$ . However, at larger $\textit{De}$ , polymers are more stretchable and easily acquire their maximal lengths upon encountering strong straining regions (similar to the observation in Singh (Reference Singh2024) that long, elastic, polymer-like fibres preferentially sample straining regions). At these large $\textit{De}$ , polymers remain stretched for longer times so that the fluctuations about the mean become smaller, across all values of $\epsilon _{{f}}$ , thus giving the j.p.d.f.s more rounded contours. Figure 12(d–f) plots the ${\textit{Tr}}(\boldsymbol{C})-\varOmega$ joint distributions. We, of course, again have a much wider spread of the distributions compared with ${\textit{Tr}}(\boldsymbol{C})-\epsilon _{{f}}$ , yet again suggesting that enstrophy and polymer stretching are less correlated compared with $\epsilon _{{f}}-{\textit{Tr}}(\boldsymbol{C})$ , as polymers can also be stretched when enstrophy is very small. Thus, while the maximally stretched polymers are most likely found in regions with maximal enstrophy at large $\textit{De}$ , they also tend to be stretched in regions with very small $\varOmega$ when $\textit{De}$ is large.

3.8. Energy dissipation

In this last section, we discuss how the nature of energy dissipation in turbulence is modified in the presence of polymers. In polymeric flows, energy is dissipated away by both the carrier fluid $\epsilon _{{f}}$ and the dissolved polymers $\epsilon _{{p}}$ . The positive-definite fluid dissipation is given by $\epsilon _{{f}} = 2\nu S_{\textit{ij}}S_{\textit{ij}}$ , whereas the positive-definite polymeric dissipation is related to the average end-to-end polymer length by $\epsilon _{{p}} = (\mu _{{p}}/2\tau _{{p}}^2) \left \langle {\textit{Tr}}(\boldsymbol{C}) -3 \right \rangle$ . Both these relations can be readily obtained by multiplying (2.1a ) with $\boldsymbol{u}$ , and using (2.1b ) as follows:

(3.29) \begin{align} \partial _t \langle \boldsymbol{u}^2 \rangle /2 &= \nu \langle \boldsymbol{u} {\nabla}^2 \boldsymbol{u} \rangle + \frac {\mu _{{p}}}{\tau _{{p}}} \left \langle \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{C} \right \rangle = - \nu \big\langle \left ( \boldsymbol{\nabla }\boldsymbol{u} \right )^2 \big\rangle - \frac {\mu _{{p}}}{\tau _{{p}}} \left \langle {\textit{Tr}} \left [ \boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} \right ] \right \rangle + \left \langle \boldsymbol{F} \boldsymbol{\cdot }\boldsymbol{u} \right \rangle \nonumber \\ \implies 0 &= -2 \nu \left \langle S_{\textit{ij}}S_{\textit{ij}} \right \rangle - \frac {\mu _{{p}}}{\tau _{{p}}} \left \langle {\textit{Tr}} \left [ \boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} \right ] \right \rangle + \left \langle \boldsymbol{F} \boldsymbol{\cdot }\boldsymbol{u} \right \rangle = - \epsilon _{{f}} - \epsilon _{{p}} + \epsilon _{\textrm {I}}. \end{align}

Similarly,

(3.30) \begin{align} \mu _{{p}} \partial _t \left \langle \boldsymbol{C} \right \rangle &= \mu _{{p}} \left \langle \boldsymbol{C} \left ( \boldsymbol{\nabla }\boldsymbol{u} \right ) \right \rangle + \mu _{{p}} \big\langle \left ( \boldsymbol{\nabla }\boldsymbol{u} \right )^T \boldsymbol{C} \big\rangle - \frac {\mu _{{p}}}{\tau _{{p}}} \left \langle \boldsymbol{C} - \boldsymbol{I} \right \rangle \nonumber \\ \implies 0 &= \frac {2 \mu _{{p}}}{ \tau _{{p}}}\left \langle {\textit{Tr}} \left [ \boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} \right ] \right \rangle - \frac {\mu _{{p}}}{\tau _{{p}}} \left \langle {\textit{Tr}}(\boldsymbol{C}) - 3 \right \rangle \implies \epsilon _{{p}} = \frac {\mu _{{p}}}{2 \tau _{{p}}^2} \left \langle {\textit{Tr}}(\boldsymbol{C}) - 3 \right \rangle\! , \end{align}

where we have used stationarity, periodicity and incompressibility to set time derivatives, fluid nonlinearity, pressure contribution and polymer advection terms to zero. Note that, the previous relation $\epsilon _{{p}} = ({\mu _{{p}}}/{\tau _{{p}}}) \left \langle {\textit{Tr}} [ \boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} ] \right \rangle = ({\mu _{{p}}}/{2 \tau _{{p}}^2}) \left \langle {\textit{Tr}}(\boldsymbol{C}) - 3 \right \rangle$ holds true only on average, and thus the two processes are equal only in a statistically stationary state, but very different instantaneously. The $ {\textit{Tr}} [ \boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} ]$ term captures the exchange of energy between fluid and the polymers, and can take both positive and negative values, while $\left \langle {\textit{Tr}}(\boldsymbol{C}) - 3 \right \rangle$ is a positive-definite quantity and captures purely dissipative effects. The interaction term $ ({\mu _{{p}}}/{\tau _{{p}}}){\textit{Tr}} [ \boldsymbol{C} \boldsymbol{\nabla }\boldsymbol{u} ]$ leads to a net transfer of energy to polymers, which is dissipated away as $({\mu _{{p}}}/{2 \tau _{{p}}^2}) \left \langle {\textit{Tr}}(\boldsymbol{C}) - 3 \right \rangle$ . This is seen from figure 13, where we plot the normalised distributions of the two terms using data from our simulations. The positive-definiteness of $\epsilon _{{f}}$ and $\epsilon _{{p}}$ means that the total energy injected by the forcing $\epsilon _{\textrm {I}}$ is dissipated away by both polymers and the fluid in a stationary state. The internal energy of the polymers is proportional to their instantaneous extension so that elongated polymers dissipate away energy as they relax. We now discuss the nature of these measures of dissipation in our polymeric flows.

Figure 13. The p.d.f.s comparing the local dissipation and transfer processes associated with polymers. They equal each other only on an average, while locally the processes are very different.

Figure 14. The p.d.f. of the fluid energy dissipation rate $\epsilon _{{f}}$ , compared with a log-normal distribution, for different $\textit{De}$ and ${Re}_\lambda$ . At (a) ${Re}_\lambda \approx 450$ , polymer elasticity has no influence on the intermittent nature of the distributions. However, at smaller (b) ${Re}_\lambda \approx 240$ , the presence of large $\textit{De}$ polymers leads to more large deviations, which are even more prominent at (c) ${Re}_\lambda \approx 40$ .

We first recall that $\epsilon _{{f}}$ in HIT has a distribution whose tails deviate from a log-normal behaviour, in contradiction to the Kolmogorov (Reference Kolmogorov1962) prediction. We now show in figure 14(a) that the normalised $\log ( \epsilon _{{f}})$ -p.d.f.s remain largely unaffected by the addition of polymers in flows at large ${Re}_\lambda$ . The curves for different $\textit{De}$ all collapse over the Newtonian curve in figure 14(a) which shows that spatiotemporal fluctuations in energy dissipation are unaffected by the addition of polymers. Polymer stresses, though present, do not result in yet larger deviations which are still dominated by the fluid nonlinearity. (Note that the mean fluid dissipation $ \langle \log \epsilon _{{f}} \rangle$ is still a function of $\textit{De}$ .) At a smaller ${Re}_\lambda \approx 240$ , the Newtonian curve shows an even larger deviation from log-normality, especially at the left tails, suggesting that dissipation field now has relatively more quiescent regions. There is no appreciable change in the distributions upon the addition of small $\textit{De}$ polymers as the polymer stresses remain small. However, when $\textit{De}$ is large, elastic effects become appreciable compared with the weakened fluid nonlinearity. The polymer stresses are now large enough to create significant fluctuations in the flow so that now there are less quiescent regions and more extreme events. This is apparent from the shrinking left tails and the widening right tails at $De = 1,3$ in figure 14(b). Curiously, at the largest $De = 9$ , when the elastic effects begin to weaken again, the large deviations are reduced (this is in parallel to the weakened elastic effects resulting in the shrinking polymeric range in figure 2 b). These effects are most clearly seen at the smallest ${Re}_\lambda = 40$ where fluid nonlinearity does not play a role (and decays exponentially in scales, see figure 4 c). The fluctuations in velocity gradients, therefore, only result from the strong polymer stresses at large $\textit{De}$ and are manifested in the wide-tailed, super log-normal distributions of $\epsilon _{{p}}$ in figure 14(c). However, for a large polymer elasticity such as $De = 9$ , a weakening of elastic effects results in a mild shrinking of the tails of the p.d.f. Figure 15 shows the normalised p.d.f.s of the bare $\epsilon _{{f}}$ for three different Re $_\lambda$ and all $\textit{De}$ . It is clear from these plots that at large Re the presence of polymers does not affect the nature of extreme events in polymer dissipation, which overlap to the Newtonian curve. This is to say the intermittency corrections are the same in Newtonian and polymeric turbulence at large Re, consistently with the collapse of the multifractal spectra presented by (Rosti et al. Reference Rosti, Perlekar and Mitra2023). The differences begin to surface at smaller Re where extreme events become more prominent, especially at large De. This is again consistent with the kurtosis picture in figures 8(b) and 8(c), where the curves steepen at large De and small Re. A crucial difference with figure 14 is seen at Re $_\lambda$ = 40, where the log-normal plot shows that the dissipation distributions vary non-monotonically with De, whereas figure 15 shows that the dependence is monotonic.

Figure 15. The p.d.f. of the fluid energy dissipation rate $\epsilon _{{f}}$ for different $\textit{De}$ and ${Re}_\lambda$ . Each panel corresponds to a decreasing ${Re}_\lambda$ from (a) to (c), for all the investigated $\textit{De}$ .

Figure 16. The p.d.f. of the logarithm of polymer dissipation $\epsilon _{{p}} \propto {\textit{Tr}}(\boldsymbol{C})$ , as a function of $\textit{De}$ for different ${Re}_\lambda$ given in the panels. The black dashed curve shows a log-normal distribution for reference. The polymer dissipation becomes less intermittent with increasing $\textit{De}$ .

We show the distributions of the polymer dissipation $\epsilon _{{p}}$ in figure 16, which are sublognormal, unlike $\epsilon _{{f}}$ , thus indicating a significantly less intermittent behaviour. In particular, at small $\textit{De}$ , $\epsilon _{{p}}$ shows more large deviations. These polymers rarely stretch and even when stretched, they quickly relax back to their small, equilibrium lengths. This means any deviations about a small mean are more significant. Thus, the polymer dissipation $\epsilon _{{p}} \propto {\textit{Tr}}(\boldsymbol{C})$ shows the widest tails at the smallest $De = 1/9$ , which are wider than even a log-normal distribution, especially at larger ${Re}_\lambda$ in figure 16(a,b). With increasing $\textit{De}$ polymers tend to stretch more and for longer duration owing to the larger relaxation times. Thus, fluctuations in lengths become less important at large $\textit{De}$ , and consequently, the stored energy is dissipated away in a more uniform and less intermittent manner, with fewer large deviations. This results in the $\log \epsilon _{{p}}$ distributions becoming progressively sublognormal. This effect is rather persistent across all ${Re}_\lambda$ , even into the ET regime at ${Re}_\lambda \approx 40$ , as seen from figure 16.