2.1. Uncertainty in the flow

We define a positive scalar metric to quantify the uncertainty in the flow as the kinetic energy of the velocity difference: $E_{\varDelta }=({1}/{2})\Delta {u}_{i}\Delta {u}_{i}$ . An evolution equation for $E_{\varDelta }$ can be obtained by multiplying (2.5) by $\Delta {u}_{i}$ , resulting in

(2.7) \begin{align} &\frac {\partial {E}_\Delta }{\partial {t}}+\frac {\partial \left (u_{\!j}^{\left (1\right )}+\Delta {u}_{\!j}\right )E_{\varDelta }}{\partial {x}_{\!j}}+\Delta {u}_{i}\Delta {u}_{\!j}\frac {\partial {u}^{\left (1\right )}_{i}}{\partial {x}_{\!j}} \nonumber \\&\quad =-\frac {\partial \Delta {u}_{i}\Delta {p}}{\partial {x_{i}}} +\frac {\beta }{{\textit{Re}}}\frac {\partial }{\partial {x}_{\!j}}\left [\frac {\partial {E}_{\varDelta }}{\partial {x}_{\!j}}\right ] -\frac {\beta }{{\textit{Re}}}\frac {\partial \Delta {u}_{i}}{\partial {x}_{\!j}}\frac {\partial \Delta {u}_{i}}{\partial {x}_{\!j}}+\frac {1-\beta }{{\textit{Re}}\textit{Wi}}\Delta {u}_{i}\frac {\partial \Delta {c}_{\textit{ji}}}{\partial {x}_{\!j}}. \end{align}

We take the average of (2.7) over the periodic domain and, denoting the spatial average $\langle \boldsymbol{\cdot }\rangle$ , we obtain

(2.8) \begin{equation} \frac {{\rm d}\left \langle {E}_\Delta \right \rangle }{{\rm d}{t}}=\left \langle -\Delta {u}_{i}S^{\left (1\right )}_{\textit{ij}}\Delta {u}_{\!j}\right \rangle -\left \langle \frac {\beta }{{\textit{Re}}}\frac {\partial \Delta {u}_{i}}{\partial {x}_{\!j}}\frac {\partial \Delta {u}_{i}}{\partial {x}_{\!j}}\right \rangle +\left \langle -\frac {1-\beta }{{\textit{Re}}\textit{Wi}}\Delta {c}_{\textit{ij}}\Delta {S}_{\textit{ij}}\right \rangle , \end{equation}

where $S^{(1)}_{\textit{ij}}=({1}/{2})(({\partial {u}_{i}^{(1)}}/{\partial {x}_{\!j}})+({\partial {u}_{\!j}^{(1)}}/{\partial {x}_{i}}))$ , and we have exploited the fact that $\langle \partial (\boldsymbol{\cdot })_{k}/\partial {x}_{k}\rangle =0$ on the periodic domain. Note that (2.8) is only valid on periodic domains; in more general settings, the evolution of $E_{\varDelta }$ is given by (2.7). The first two terms on the right-hand side of (2.8), which (respectively) represent inertial production of uncertainty (denoted $I_{\varDelta }$ ) and viscous dissipation of uncertainty (denoted $D_{\varDelta }$ ), match those derived by Ge et al. (Reference Ge, Rolland and Vassilicos2023) for Newtonian flows. The final term represents the polymeric propagation of uncertainty (denoted $P_{\varDelta }$ ), and governs the propagation of uncertainty from the polymer deformation to the flow. We express (2.8) in these terms as

(2.9) \begin{equation} \frac {{\rm d}\left \langle {E}_\Delta \right \rangle }{{\rm d}{t}}=\left \langle {I}_{\varDelta }\right \rangle -\left \langle {D}_{\varDelta }\right \rangle +\left \langle {P}_{\varDelta }\right \rangle . \end{equation}

Viscous dissipation $D_{\varDelta }$ is positive by construction; Newtonian viscosity always reduces uncertainty. Inertial production may be positive or negative, and its sign is dependent on the alignment of the uncertainty velocity field $\Delta {u}_{i}$ with the reference strain rate, as discussed in detail in Ge et al. (Reference Ge, Rolland and Vassilicos2023) in the context of three-dimensional inertial turbulence. Considering for now the two-dimensional case, we denote the eigenvalues of ${S}_{\textit{ij}}^{(1)}$ as $\varLambda _{i}^{(1)}$ (following the convention that $\varLambda _{1}\leqslant \varLambda _{2}$ ), and note that due to incompressibility the two eigenvalues are real, equal and opposite ( $\varLambda _{2}^{(1)}=-\varLambda _{1}^{(1)}=\varLambda ^{(1)}\gt 0$ ). We define the components of $\Delta {u}_{i}$ projected onto the corresponding principal axes as $\Delta {u}_{i}^{\prime }$ , and the inertial production term can be expressed as

(2.10) \begin{equation} \left \langle {I}_{\varDelta }\right \rangle =\left \langle \varLambda ^{\left (1\right )}\left (\Delta {u}_{1}^{\prime {2}}-\Delta {u}_{2}^{\prime {2}}\right )\right \rangle , \end{equation}

in which $\Delta {u}_{1}^{\prime }$ corresponds to a velocity difference aligned with the compressive flow direction in the reference field and $\Delta {u}_{2}^{\prime }$ corresponds to a velocity difference aligned with the stretching direction. This is the two-dimensional equivalent of the expression given by Ge et al. (Reference Ge, Rolland and Vassilicos2023) for three-dimensional flows: the inertial production of uncertainty is increased by uncertainty aligned with compressive flows, and decreased by uncertainty aligned with flow stretching. A similar approach may be taken to provide insight into the polymeric propagation of uncertainty $\langle {P}_{\varDelta }\rangle$ , which depends on the alignment of the uncertainty in the conformation tensor with the uncertainty in the strain rate. We again consider the two-dimensional case, and now take the principal axes of $\Delta {S}_{\textit{ij}}$ as a local orthonormal reference frame. Denoting the eigenvalues of $\Delta {S}_{\textit{ij}}$ as $\pm \varLambda _{\varDelta }$ ( $\varLambda _{\varDelta }\gt 0$ ), and the components of $\Delta {c}$ in this basis as $\Delta {c}_{\textit{ij}}^{\star }$ , we obtain

(2.11) \begin{equation} \left \langle {P}^{2D}_{\varDelta }\right \rangle =\left \langle \frac {1-\beta }{{\textit{Re}}\textit{Wi}}\left [\left (\Delta {c}^{\star }_{11}-\Delta {c}^{\star }_{22}\right )\varLambda _{\varDelta }\right ]\right \rangle . \end{equation}

Here $\Delta {c}_{11}^{\star }$ corresponds to a difference in the polymer deformation aligned with compression in the velocity difference field, whilst $\Delta {c}_{22}^{\star }$ corresponds to a difference in the polymer deformation aligned with extension in the velocity difference field. We denote by $\Delta {N}_{1}^{\star }$ the term in parentheses, which is proportional to the difference in the first normal stress difference in a basis aligned with a compressive flow difference. The polymeric propagation of uncertainty depends on the difference in the first normal stress difference in a basis aligned with the principal axes of the velocity difference. Little can be said a priori of the likely sign or magnitude of $\Delta {N}_{1}^{\star }$ , as $\Delta {c}_{\textit{ij}}$ is not constrained to remain positive definite, and even if both realisations exhibit a positive first normal stress the difference, $\Delta {N}_{1}^{\star }$ , may be negative.

Considering the viscous dissipation of uncertainty $\langle {D}_{\varDelta }\rangle$ , in two dimensions we may write

(2.12) \begin{equation} D_{\varDelta }=\frac {\beta }{{\textit{Re}}}\frac {\partial \Delta {u}_{i}}{\partial {x}_{\!j}}\frac {\partial \Delta {u}_{i}}{\partial {x}_{\!j}}=\frac {\beta }{{\textit{Re}}}\left (\Delta {S}_{\textit{ij}}\Delta {S}_{\textit{ij}}+\frac {1}{2}\Delta \omega ^{2}\right ), \end{equation}

where $\Delta \omega$ is the vorticity difference. As before, we can take the principal axes of $\Delta {S}_{\textit{ij}}$ as a local orthonormal basis, and we obtain

(2.13) \begin{equation} D_{\varDelta }=\frac {\beta }{{\textit{Re}}}\left (2\varLambda _{\varDelta }^{2}+\frac {1}{2}\Delta \omega ^{2}\right ), \end{equation}

highlighting that viscous dissipation occurs due to differences in both extension ( $\varLambda _{\varDelta }$ ) and vorticity ( $\Delta \omega$ ).

As a consequence of the elliptic nature of the incompressibility constraint in (2.1) and (2.2), a perturbation to the flow which is local in space can generate a non-zero uncertainty over the entire flow field instantly. For uncertainty in the flow this non-local behaviour is via the pressure-gradient term in (2.5). The pressure-gradient term in (2.7) disappears on spatial averaging (i.e. it does not appear in (2.8)). Whilst it therefore does not explicitly contribute to the evolution of $\langle {E}_{\varDelta }\rangle$ as a global measure of uncertainty, it does still influence the evolution of $\langle {E}_{\varDelta }\rangle$ , through its influence on both $\Delta {u}_{i}$ and $\Delta {c}_{\textit{ii}}$ . This is verified by numerical experiments, and discussed further in § 3.

2.2. Uncertainty in polymer deformation

Identifying a suitable scalar measure of the uncertainty in the conformation tensor is non-trivial. Two requirements for such a measure are (i) that it is strictly positive and (ii) that it is independent of the conformation tensor basis. Although $c_{\textit{ij}}^{(1)}$ and $c_{\textit{ij}}^{(2)}$ are (symmetric) positive definite, the invariants of $\Delta {c}_{\textit{ij}}$ are not necessarily positive. Several approaches are offered by Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018), including the logarithmic volume ratio and the squared distance between $c_{\textit{ij}}^{(1)}$ and $c_{\textit{ij}}^{(2)}$ along a geodesic curve on a Riemannian manifold. Whilst both approaches provide a strictly positive scalar measure, the derivation of evolution equations for these measures becomes prohibitively complex. For the Oldroyd B model, the elastic energy stored in the polymers is proportional to the conformation tensor trace, and a measure which can be related to the uncertainty in the stored elastic energy is desirable. We opt to take the square of the conformation tensor trace difference as a scalar measure of uncertainty on the polymeric deformation, which satisfies both the above requirements. We define $\varGamma _{\varDelta }=(\Delta {c}_{\textit{ii}})^{2}$ and multiply the trace of (2.6) by $2\Delta {c}_{\textit{jj}}$ to obtain an evolution equation for $\varGamma _{\varDelta }$ :

(2.14) \begin{align} \frac {\partial \varGamma _{\varDelta }}{\partial {t}}+u^{\left (1\right )}_{k}\frac {\partial \varGamma _{\varDelta }}{\partial {x}_{k}}+\Delta {u}_{k}\frac {\partial \varGamma _{\varDelta }}{\partial {x}_{k}}+2\Delta {c}_{\textit{jj}}\Delta {u}_{k}\frac {\partial {c}^{\left (1\right )}_{\textit{ii}}}{\partial {x}_{k}} -4\Delta {c}_{\textit{jj}}\Delta {c}_{\textit{ik}}S_{\textit{ik}}^{\left (1\right )}-4\Delta {c}_{\textit{jj}}{c}^{\left (1\right )}_{\textit{ik}}\Delta {S}_{\textit{ik}}\nonumber \\-\,4\Delta {c}_{\textit{jj}}\Delta {c}_{\textit{ik}}\Delta {S}_{\textit{ik}} =-\frac {2\varGamma _{\varDelta }}{\textit{Wi}}\big [1+2\varepsilon {c}_{\textit{ii}}^{\left (1\right )}+\varepsilon \Delta {c}_{\textit{ii}}\big ]+\kappa \frac {\partial ^{2}\varGamma _{\varDelta }}{\partial {x}_{k}\partial {x}_{k}}-{2}\kappa \frac {\partial \Delta {c}_{\textit{ii}}}{\partial {x}_{k}}\frac {\partial \Delta {c}_{\textit{jj}}}{\partial {x}_{k}}. \end{align}

We take the spatial average, and again exploiting the fact that any terms expressed in flux form average to zero over a periodic domain, we obtain

(2.15) \begin{align} \frac {{\rm d}\left \langle \varGamma _{\varDelta }\right \rangle }{{\rm d}t}&=\left \langle -2\Delta {c}_{\textit{ii}}\Delta {u}_{k}\frac {\partial {c}^{\left (1\right )}_{\textit{jj}}}{\partial {x}_{k}}\right \rangle +\big \langle 4\Delta {c}_{\textit{jj}}\Delta {c}_{\textit{ik}}S_{\textit{ik}}^{\left (1\right )}\big \rangle +\big \langle 4\Delta {c}_{\textit{jj}}{c}^{\left (1\right )}_{\textit{ik}}\Delta {S}_{\textit{ik}}\big \rangle \nonumber \\ &\quad +\left \langle 4\Delta {c}_{\textit{jj}}\Delta {c}_{\textit{ik}}\Delta {S}_{\textit{ik}}\right \rangle -\left \langle \frac {2\varGamma _{\varDelta }}{\textit{Wi}}\big [1+2\varepsilon {c}_{\textit{ii}}^{\left (1\right )}+\varepsilon \Delta {c}_{\textit{ii}}\big ]\right \rangle -\left \langle {2}\kappa \frac {\partial \Delta {c}_{\textit{ii}}}{\partial {x}_{k}}\frac {\partial \Delta {c}_{\textit{jj}}}{\partial {x}_{k}}\right \rangle . \end{align}

The first term on the right-hand side relates to the production of uncertainty due to advection. The next three terms originate from the upper convected derivative, and relate to the production and propagation of uncertainty due to stretching and rotation of the polymers. The penultimate term corresponds to the destruction of uncertainty due to polymer relaxation. The final term is the dissipation of uncertainty due to polymeric diffusion. For convenience, we rewrite (2.15) with the terms denoted

(2.16) \begin{equation} \frac {{\rm d}\left \langle \varGamma _{\varDelta }\right \rangle }{{\rm d}t}=\left \langle {A}_{\varDelta }\right \rangle +\left \langle {\textit{UC}1}_{\varDelta }\right \rangle +\left \langle {\textit{UC}2}_{\varDelta }\right \rangle +\left \langle {\textit{UC}3}_{\varDelta }\right \rangle -\left \langle {R}_{\varDelta }\right \rangle -\left \langle {\textit{PD}}_{\varDelta }\right \rangle . \end{equation}

We first comment on the effect of polymer relaxation on uncertainty. In the limit of $\varepsilon \to 0$ (the Oldroyd B limit) $R_{\varDelta }$ is positive by construction; in the Oldroyd B limit, polymer relaxation always reduces uncertainty in the polymer deformation. For finite $\varepsilon$ , the nonlinearity in the polymer relaxation may result in (locally) negative $R_{\varDelta }$ , when and where the inequality $c_{\textit{ii}}^{(2)}\gt 3c_{\textit{ii}}^{(1)}+1/\varepsilon$ is satisfied. Certainly in the early stages of uncertainty evolution, when the two fields are closely correlated, we would not expect this to occur. With the two realisations following the same evolution equations, and expected to have the same statistics, we would not expect $\langle {R}_{\varDelta }\rangle \lt 0$ either. Indeed, $\langle {R}_{\varDelta }\rangle$ remains positive throughout all our numerical simulations, and in fact we see a trend that $\langle {R}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle$ increases with increasing $\varepsilon$ . The term $\langle {\textit{PD}}_{\varDelta }\rangle$ is positive by construction; polymeric diffusion always reduces uncertainty in the polymer deformation. The remaining terms $\langle {A}_{\varDelta }\rangle$ , $\langle {\textit{UC}1}_{\varDelta }\rangle$ , $\langle {\textit{UC}2}_{\varDelta }\rangle$ and $\langle {\textit{UC}3}_{\varDelta }\rangle$ may be either positive or negative. We note that an alternative positive scalar measure of the uncertainty in the polymer deformation could be defined as $\varPi _{\varDelta }=\Delta {c}_{\textit{ij}}\Delta {c}_{\textit{ij}}$ . This measure is frame-invariant, and includes information on the off-diagonal elements of $\Delta {c}_{\textit{ij}}$ ; note that, in two dimensions, we can write $\varPi _{\varDelta }=\varGamma _{\varDelta }-2\det (\Delta {c}_{\textit{ij}})$ . Although we do not explore this measure in detail, we include it in our investigation of the orientation of uncertainty in § 3.1.1, and an evolution equation for $\varPi _{\varDelta }$ is provided in Appendix C.

Returning to the evolution of $\langle \varGamma _{\varDelta }\rangle$ and considering the production term $\langle {\textit{UC}1}_{\varDelta }\rangle$ , we take the principal axes of ${S}_{\textit{ij}}^{(1)}$ as a local orthonormal basis, and we denote the components of $\Delta {c}$ in this basis as $\Delta {c}_{\textit{ij}}^{\prime }$ . Expressed in this basis, we can write

(2.17) \begin{equation} \textit{UC}1_{\varDelta }=4\varLambda ^{\left (1\right )}\left (\Delta {c}_{22}^{\prime {2}}-\Delta {c}_{11}^{\prime {2}}\right ), \end{equation}

where $\Delta {c}_{11}^{\prime }$ represents deformation in the conformation tensor difference aligned with compressive flow in the reference field and $\Delta {c}_{22}^{\prime }$ represents that aligned with stretching flow. From (2.17), we see that regions of extensional deformation in the conformation tensor difference aligned with extensional flow produce uncertainty, whilst extensional deformation in the conformation tensor difference aligned with compressive flow reduce uncertainty. Next we consider the term $\langle {UC2}_{\varDelta }\rangle$ , which is a propagation or transfer term, through which uncertainty in the flow (via $\varLambda _{\varDelta }$ ) propagates into the conformation tensor uncertainty (but only where some uncertainty in the polymeric deformation already exists, i.e. $\Delta {c}_{\textit{ii}}\ne 0$ ). We take the principal axes of $\Delta {S}_{\textit{ij}}$ as a local orthornormal basis, and write

(2.18) \begin{equation} \textit{UC}2_{\varDelta }=4\Delta {c}_{\textit{ii}}\varLambda _{\varDelta }\left (c_{22}^{\left (1\right )\star }-c_{11}^{\left (1\right )\star }\right ), \end{equation}

in which $c_{22}^{(1)\star }$ and $c_{11}^{(1)\star }$ correspond to polymeric deformation in the reference field aligned with stretching and compression (respectively) in the velocity difference field. Whether or not this term is positive or negative depends on the sign of the conformation tensor trace difference, and the normal stress difference in the reference field aligned with the principal stretching direction of the velocity difference. The term $\langle {\textit{UC}3}_{\varDelta }\rangle$ , also describing uncertainty propagation, is constructed from the product of three differences, whilst the terms $\langle {\textit{UC}1}_{\varDelta }\rangle$ and $\langle {\textit{UC}2}_{\varDelta }\rangle$ are each formed of the product of two differences and a reference field. Hence, whilst the two realisations are largely correlated and uncertainty is small, we expect the term $\langle {\textit{UC}3}_{\varDelta }\rangle$ to be small relative to the other two upper convected terms. The term in (2.15) relating to the advection of polymers contains only invariants of the conformation tensor and its difference. If we take the principal axes of $c_{\textit{ij}}^{(1)}$ as a local orthonormal basis, and denote tensors expressed in this basis, and vectors projected onto this basis, with a double prime, we can express the advection term in (2.15) as

(2.19) \begin{equation} {A}_{\varDelta }=-2\Delta {c}_{\textit{ii}}\Delta {u}^{\prime \prime }_{k}\frac {\partial {c}_{\textit{jj}}^{\left (1\right )}}{\partial {x}^{\prime \prime }_{k}}. \end{equation}

A characteristic of elastic turbulence is thin regions of highly deformed polymers, and here the gradient of the conformation tensor trace $\partial {c}^{(1)}_{\textit{ii}}/\partial {x}_{k}$ will be predominantly aligned normal to the principal stretching direction of the polymers. Hence, we expect $\partial {c}_{\textit{jj}}^{(1)}/\partial {x}^{\prime \prime }_{1}\gg \partial {c}_{\textit{jj}}^{(1)}/\partial {x}^{\prime \prime }_{2}$ in regions of large polymer deformation. Noting that $\Delta {c}_{\textit{ii}}$ may be negative, but whilst the two realisations are still predominantly correlated, $\lvert \Delta {c}_{\textit{ii}}\rvert$ is likely to be larger in regions where $c_{\textit{ii}}^{(1)}$ is large, we may expect $\langle {A}_{\varDelta }\rangle$ to depend on the relative orientation of the uncertainty in the velocity, and the reference polymer deformation. Such analysis can only provide a certain amount of insight, and to investigate the dynamics of uncertainty further, we turn to numerical simulations.

3.1. The evolution of uncertainty

Figure 2(a) shows the evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ with $\tau$ , for individual instances (red lines) and the ensemble average (blue lines). There is a clear period of exponential growth, over approximately six orders of magnitude, with the same exponent in both $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ . This exponential growth starts at $\tau \approx {1}$ for the flow uncertainty and $\tau \approx 3$ for $\langle {\varGamma }_{\varDelta }\rangle$ . As observed by Berti et al. (Reference Berti, Bistagnino, Boffetta, Celani and Musacchio2008) and Berti & Boffetta (Reference Berti and Boffetta2010) for elastic turbulence driven by a Kolmogorov forcing, there is a strong imprint of the laminar fixed point in the chaotic flow field, and as a consequence there is a limit to the uncertainty (the extent to which the two realisations can decorrelate). This is clear in figure 2(a) as both $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ reach limiting values of approximately $0.3$ and $2000$ (respectively) after approximately $30$ time units. Note that $\langle ({c}_{\textit{ii}}^{(1)})^{2}+({c}_{\textit{ii}}^{(2)})^{2}\rangle /\textit{Wi}^{2}\approx 6645$ , and the saturation value of $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ is about $0.3$ times this. From these results we can identify four regimes: (I) very short time growth $\langle {E}_{\varDelta }\rangle$ and $\langle \varGamma _{\varDelta }\rangle$ until $\tau \approx {0.1}$ ; (II) a reduction in uncertainty, initially in $\langle \varGamma _{\varDelta }\rangle$ , over time scales of the order of unity; (III) exponential growth of uncertainty, initially in $\langle {E}_{\varDelta }\rangle$ from $\tau \approx {1}$ , then also $\langle \varGamma _{\varDelta }\rangle$ from $\tau \approx {4}$ ; and (IV) saturation of uncertainty from $\tau \approx {30}$ . We discuss the mechanisms controlling these regimes further below. Note that we have run simulations for an additional $150$ time units, up to $\tau =200$ , and do not observe any further growth in uncertainty. During the exponential regime (III), the evolution of the quantities $\langle {E}_{\varDelta }\rangle$ and $\langle \varGamma _{\varDelta }\rangle$ can be modelled by

(3.3) \begin{equation} \frac {{\rm d}\left \langle {E}_{\varDelta }\right \rangle }{{\rm d}t}=\lambda \left \langle {E}_{\varDelta }\right \rangle ;\qquad \frac {{\rm d}\left \langle {\varGamma }_{\varDelta }\right \rangle }{{\rm d}t}=\lambda \left \langle {\varGamma }_{\varDelta }\right \rangle \end{equation}

in which $\lambda$ is a characteristic growth rate, equal to twice the maximal Lyapunov exponent. Letting $\{\boldsymbol{\cdot }\}_{(\textit{III})}$ represent the temporal mean of a quantity over regime (III), we calculate $\{\lambda \}_{(\textit{III})}=0.68$ . This growth rate is shown by the dashed black lines in figure 2(a).

Figure 2(b) shows the evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ with $\tau$ for different values of $A_{0}$ , with results scaled by $A_{0}^{-2}$ , for a single simulation. Prior to the saturation of uncertainty, the lines collapse, as the magnitude of the uncertainty scales with $A_{0}^{2}$ , the square of the perturbation amplitude, by definition of $E_{\varDelta }$ and $\varGamma _{\varDelta }$ . Just as the evolution of uncertainty is independent of $\delta _{t0}$ it is unchanged by the magnitude of $A_{0}$ , despite the magnitude of the perturbation changing by over three orders of magnitude. The imprint of the reference flow on the uncertainty is visible in figure 2(b), where for all values of $A_{0}$ the same reference flow is used.

3.1.1. The orientation of uncertainty

As before, we take the principal axes of the reference strain rate tensor $S_{\textit{ij}}^{(1)}$ to define a local orthonormal basis (denoted with a prime). We also use the principal axes of the reference conformation tensor $c_{\textit{ij}}^{(1)}$ to define another basis (denoted with a double prime). We use each basis to obtain a projection of the flow uncertainty $\Delta {u}_{i}$ , which we denote $\Delta {u}_{i}^{\prime }$ and $\Delta {u}_{i}^{\prime \prime }$ . We also express the uncertainty in the conformation tensor in these bases (denoted $\Delta {c}_{\textit{ij}}^{\prime }$ and $\Delta {c}_{\textit{ij}}^{\prime \prime }$ ). We denote the proportion of the uncertainty energy aligned with each principal direction as

(3.4) \begin{equation} \theta ^{\prime }_{i}=\frac {\Delta {u}_{i}^{\prime {2}}}{2\left \langle {E}_{\varDelta }\right \rangle }, \end{equation}

where an equivalent definition follows for $\theta _{i}^{\prime \prime }$ . Note that $\theta _{1}^{\prime }+\theta _{2}^{\prime }=\theta _{1}^{\prime \prime }+\theta _{2}^{\prime \prime }=1$ . The quantity $\theta _{1}^{\prime }$ represents the proportion of uncertainty energy aligned with compressive flow and $\theta _{2}^{\prime }$ represents the proportion aligned with stretching flow. The quantity $\theta _{2}^{\prime \prime }$ represents the proportion of uncertainty energy aligned with the principal stretching direction of the conformation tensor and $\theta _{1}^{\prime \prime }$ is the proportion of uncertainty energy orthogonal to this.

Figure 3. The evolution of the orientation of uncertainty for the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ). (a) The evolution of $\langle \theta ^{\prime }\rangle$ and $\langle \theta ^{\prime \prime }\rangle$ . (b) The evolution of components of $\langle \psi ^{\prime }\rangle$ and $\langle \psi ^{\prime \prime }\rangle$ . Note that the components of $\langle \psi ^{\prime \prime }\rangle$ may be greater than unity, and where we have plotted $\langle 1-\psi _{22}^{\prime \prime }\rangle$ and this quantity is negative, it is plotted with a dotted line. The black line shows the evolution of $-(\langle \varGamma _{\varDelta }\rangle -\langle \varPi _{\varDelta }\rangle )/2\langle \varGamma _{\varDelta }\rangle$ . The different regimes of uncertainty evolution are indicated separated by dotted red vertical lines.

In Ge et al. (Reference Ge, Rolland and Vassilicos2023), they observed an uneven distribution of the orientation of the uncertainty energy for three-dimensional turbulence in a Newtonian fluid: during the similarity regime characterised by exponential growth of uncertainty, the majority of the uncertainty energy was aligned with the compressive flow direction. The picture is different in the two-dimensional low- ${\textit{Re}}$ elastic turbulence case. Figure 3(a) shows the evolution of $\langle \theta ^{\prime }\rangle$ and $\langle \theta ^{\prime \prime }\rangle$ . The variation of $\langle \theta _{1}^{\prime }\rangle$ is small: at all times $0.467\leqslant \langle \theta _{1}^{\prime }\rangle \leqslant 0.542$ ; the uncertainty energy is approximately evenly split between regions of compression and extension. There does not appear to be any preferential orientation of uncertainty energy due to the reference flow. The relative orientation of the velocity difference field and the reference flow field only appears in (2.8) in the inertial production term, and for small ${\textit{Re}}$ , this is small. At early times, the quantity $\langle \theta _{1}^{\prime \prime }\rangle$ is very small, and $\langle \theta _{2}^{\prime \prime }\rangle \approx {1}$ ; i.e. the perturbation we impose in the conformation tensor is predominantly aligned with the direction of polymer extension. During regime (I) ( $\tau \lesssim {0.1}$ ), the orientation changes, and $\langle \theta _{1}^{\prime \prime }\rangle$ , the proportion of the uncertainty energy $\langle {E}_{\varDelta }\rangle$ perpendicular to the extensional direction of the polymers, increases to approximately $0.15$ . There is a further increase in $\langle \theta _{1}^{\prime \prime }\rangle$ during regime (II) ( $0.1\lesssim \tau \lesssim {1}$ ), and during regime (III) ( $1\lesssim \tau \lesssim 30$ ), $\langle \theta _{1}^{\prime \prime }\rangle \approx 0.25$ . During the exponential growth regime approximately a quarter of the uncertainty energy is aligned perpendicular to the direction of polymeric extension. The increase in the proportion of the uncertainty energy aligned normal to the principal direction of reference polymer deformation prior to regime (III) is consistent with the analysis in § 2; we expect the production of uncertainty due to polymer advection $\langle {A}_{\varDelta }\rangle$ to be larger when the uncertainty energy is aligned with the gradient of the polymer deformation trace, which we expect to be normal to the primary polymer deformation. In regime (IV) ( $\tau \gtrsim 30$ ), there is a slight further increase in $\langle \theta _{1}^{\prime \prime }\rangle$ . That $\langle \theta _{1}^{\prime \prime }\rangle$ always remains below approximately $0.3$ is further indication that there is a limit to complete decorrelation: both realisations are subject to the same forcing, and even when their evolutions are fully diverged, the imprint of the forcing in each flow results in a residual correlation.

As a measure of the relative orientation of the uncertainty in the conformation tensor we define

(3.5) \begin{equation} \psi ^{\prime }_{\textit{ij}}=\frac {\Delta {c}^{\prime {2}}_{\textit{ij}}}{\left \langle \varPi _{\varDelta }\right \rangle }, \end{equation}

with an equivalent definition for $\psi ^{\prime \prime }_{\textit{ij}}$ . We note here that the normalisation of $\psi ^{\prime }_{\textit{ij}}$ by $\langle \varPi _{\varDelta }\rangle$ (as opposed to $\langle \varGamma _{\varDelta }\rangle$ ) ensures the sum over all components of $\psi _{\textit{ij}}^{\prime }$ equals one. Figure 3(b) shows the evolution of $\langle \psi ^{\prime }\rangle$ and $\langle \psi ^{\prime \prime }\rangle$ .

At all times the largest component of $\langle \psi ^{\prime }\rangle$ is $\langle \psi _{22}^{\prime }\rangle$ and the largest component of $\langle \psi ^{\prime \prime }\rangle$ is $\langle \psi _{22}^{\prime \prime }\rangle$ – the largest component of the uncertainty in the polymer deformation is aligned with stretching and polymeric extension in the reference flow. During regime (II) this proportion decreases, from approximately $0.6$ to approximately $0.35$ at $\tau =1$ , before increasing again during regime (III) to approximately $0.5$ . Following the expression for $UC1_{\varDelta }$ in (2.17), this implies that $\langle {\textit{UC}1}_{\varDelta }\rangle$ will be positive at all times. We next inspect the evolution of $\langle \psi ^{\prime \prime }\rangle$ . Note that in figure 3(b) we have plotted $\langle \psi _{11}^{\prime \prime }\rangle$ , $\langle \psi _{12}^{\prime \prime }\rangle$ and $\langle 1-\psi _{22}^{\prime \prime }\rangle$ . At short times the orientation of the uncertainty in the polymer deformation relative to the reference polymeric deformation is roughly constant, with $\langle \psi _{22}^{\prime \prime }\rangle \approx 0.997$ . At the start of regime (III), from $\tau \approx {1}$ to $\tau \approx {4}$ , $\langle \psi _{22}^{\prime \prime }\rangle$ decreases. This period coincides with exponential growth starting in $\langle {E}_{\varDelta }\rangle$ whilst $\langle \varGamma _{\varDelta }\rangle$ continues to decay. As we discuss later, during this period, uncertainty is growing exponentially at large scales, but is reducing due to polymeric dissipation and relaxation at small scales. During this period at the start of regime (III), there is also a decrease in $\langle \psi _{11}^{\prime \prime }\rangle$ and an increase in $\langle \psi _{12}^{\prime \prime }\rangle$ . During the transition to exponential growth, a greater proportion of the uncertainty in the polymeric deformation is deviatoric in an orthornormal basis aligned with the reference polymeric deformation, indicating a rotation of the uncertainty relative to the polymeric deformation of the reference field as the two realisations decorrelate. However, we note that for the remainder of regime (III) ( $4\lesssim \tau \lesssim 30$ ) the orientation of uncertainty in the polymeric deformation is roughly constant with respect to the reference polymeric deformation. At the end of regime (III), there is a sharp decrease in $\langle \psi _{22}^{\prime \prime }\rangle$ ( $\langle 1-\psi _{22}^{\prime \prime }\rangle$ increases by two orders of magnitude) and a corresponding increase in $\langle \psi _{11}^{\prime \prime }\rangle$ and $\langle \psi _{12}^{\prime \prime }\rangle$ . Figure 3(b) shows the evolution of $-(\langle \varGamma _{\varDelta }\rangle -\langle \varPi _{\varDelta }\rangle )/2\langle \varGamma _{\varDelta }\rangle =-\langle \det (\Delta {c}_{\textit{ij}})\rangle /\langle \varGamma _{\varDelta }\rangle$ , and we note this quantity is always positive, indicating the spatial average of the determinant of $\Delta {c}_{\textit{ij}}$ is negative. The relative differences between $\langle \varGamma _{\varDelta }\rangle$ and $\langle \varPi _{\varDelta }\rangle$ are small: $-(\langle \varGamma _{\varDelta }\rangle -\langle \varPi _{\varDelta }\rangle )/2\langle \varGamma _{\varDelta }\rangle$ remains below $0.01$ during regimes (I) to (III), increasing to approximately $0.1$ in regime (IV). The evolution of $-(\langle \varGamma _{\varDelta }\rangle -\langle \varPi _{\varDelta }\rangle )/2\langle \varGamma _{\varDelta }\rangle$ closely follows the evolution of $\langle \psi _{12}^{\prime \prime }\rangle$ , suggesting that the exponential growth in regime (III) is characterised by not only a relative rotation of the uncertainty in polymer deformation, but also a relative increase in $\lvert \det (\Delta {c}_{\textit{ij}})\rvert$ .

Figure 4. The evolution of ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t$ and terms in (2.8) for the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ). (a) The sum of terms in (2.8) and the calculated values of ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t$ . Where ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t\gt 0$ , a solid red line is used, and where ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t\lt 0$ , a dotted red line. (b) Ratios of the terms in (2.8). The lower inset highlights the early-time evolution. The upper inset shows the evolution of the individual terms $\langle {I}_{\varDelta }\rangle$ , $\langle {D}_{\varDelta }\rangle$ and $\langle {P}_{\varDelta }\rangle$ .

3.1.2. Terms contributing to the evolution of $\langle {E}_{\varDelta }\rangle$

We now consider how the terms in (2.8) evolve. Figure 4(a) shows the evolution of ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t$ , and the terms contributing to ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t$ in (2.8). Note that we have calculated ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t$ with a first-order forward difference approximation based on successive values of $\langle {E}_{\varDelta }\rangle$ separated by $\delta \tau =10^{-3}$ . There is very close agreement between the calculated value of ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t$ and $\langle {I}_{\varDelta }\rangle -\langle {D}_{\varDelta }\rangle +\langle {P}_{\varDelta }\rangle$ , providing confirmation that (2.8) holds. Figure 4(b) shows the evolution of the ratios $-\langle {I}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle$ and $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ (scaled by $10^{4}$ ). The upper inset shows the individual terms $\langle {I}_{\varDelta }\rangle$ , $\langle {P}_{\varDelta }\rangle$ and $\langle {D}_{\varDelta }\rangle$ . Firstly, we note that $\langle {I}_{\varDelta }\rangle$ is almost always negative (we have plotted $\lvert \langle {I}_{\varDelta }\rangle \rvert$ in the inset), indicating that in this regime the (small) net effect of the inertial terms is to reduce uncertainty. The quantities $\langle {P}_{\varDelta }\rangle$ and $\langle {D}_{\varDelta }\rangle$ are closely matched, and approximately four orders of magnitude larger than $\langle {I}_{\varDelta }\rangle$ . The quantity $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ provides a measure of this match: for $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1\gt 0$ , polymeric propagation of uncertainty outweighs viscous dissipation of uncertainty. For $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1\lt 0$ , viscous dissipation dominates. The evolution of $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ shows several distinct behaviours. At very early times (lower inset) the quantity $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ is positive, indicating the early-stage increase in uncertainty is driven by polymeric propagation. Subsequently, $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ becomes negative on time scales of the order of $10^{-1}$ . In this period, there is a net reduction in uncertainty as viscous dissipation dominates polymeric propagation. From $\tau \approx {1}$ to $\tau \approx {25}$ , $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ fluctuates, but is predominantly positive. This period corresponds to the regime of exponential growth of uncertainty. As uncertainty saturates, the quantity $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ increases, with a trend matching that of $-\langle {I}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle$ .

Figure 5. The evolution of ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t$ and terms in (2.15) for the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ). (a) The sum of terms in (2.15) and the calculated values of ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t$ . Where ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t\gt 0$ , a solid red line is used, and where ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t\lt 0$ , a dotted red line. (b) The individual terms in (2.15). The inset shows these same terms normalised by $\langle \varGamma _{\varDelta }\rangle$ . Where $\langle {A}_{\varDelta }\rangle$ is negative, it is plotted with a dotted red line, and with a solid red line where positive.

3.1.3. Terms contributing to the evolution of $\langle {\varGamma }_{\varDelta }\rangle$

Figure 5 shows the evolution of ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t$ and the contributing terms in (2.15). Figure 5(a) shows the evolution of ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t$ (evaluated from successive values of $\langle \varGamma _{\varDelta }\rangle$ at intervals of $\delta {\tau }=10^{-3}$ ), and the sum of the terms in the right-hand side of (2.15). The two quantities match very closely, with discrepancies only arising when ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t$ is rapidly changing, due to the first-order finite-difference approximation used to evaluate ${\rm d}\langle {\varGamma }_{\varDelta }\rangle /{\rm d}t$ .

Figure 5(b) shows the evolution of the individual terms in (2.15). For this configuration with $\varepsilon =0$ (i.e. Oldroyd B), $\langle {R}_{\varDelta }\rangle =\langle 2\varGamma _{\varDelta }/\textit{Wi}\rangle =\langle \varGamma _{\varDelta }\rangle$ . We first consider the upper convected terms. Term $\langle {\textit{UC}1}_{\varDelta }\rangle$ is at all times positive. This observation is consistent with the finding in § 3.1.1 that $\langle \psi _{22}^{\prime }\rangle$ is always the largest component of $\langle \psi ^{\prime }\rangle$ . Stretching in the conformation tensor difference field is predominantly aligned with stretching in the reference flow, and this results in a production of uncertainty. Term $\langle {\textit{UC}2}_{\varDelta }\rangle$ is at all times negative (note that we have plotted $-\langle {\textit{UC}2}_{\varDelta }\rangle$ ), and always approximately an order of magnitude smaller than $\langle {\textit{UC}1}_{\varDelta }\rangle$ . Term $\langle {\textit{UC}3}_{\varDelta }\rangle$ is sometimes positive and sometimes negative, and with magnitude several orders smaller than $\langle {\textit{UC}1}_{\varDelta }\rangle$ , as predicted in § 2. The relative importance of $\langle {\textit{UC}3}_{\varDelta }\rangle$ increases with time as the uncertainty increases, but even in regime (IV) it remains an order of magnitude smaller than the other two upper convected terms. Of the upper convected terms, it is $\langle {\textit{UC}1}_{\varDelta }\rangle$ which dominates the dynamics of uncertainty evolution, with $\langle {\textit{UC}2}_{\varDelta }\rangle \sim -0.1\langle {\textit{UC}1}_{\varDelta }\rangle$ through regime (III), and $\langle {\textit{UC}1}_{\varDelta }\rangle \gt 0$ at all times. The net contribution of the upper convected terms is to amplify uncertainty.

The inset of figure 5(b) shows the evolution of the terms in (2.15) normalised by $\langle \varGamma _{\varDelta }\rangle$ . At early times, $\langle {\textit{PD}}_{\varDelta }\rangle$ dominates, as the perturbation is focused at small scales. Over the first few time units, the relative magnitude of $\langle {\textit{PD}}_{\varDelta }\rangle$ reduces, and during the exponential regime ( $\tau \in [1,30]$ ), we see $\langle {\textit{PD}}_{\varDelta }\rangle \approx \langle {R}_{\varDelta }\rangle =\langle \varGamma _{\varDelta }\rangle$ . During the period of exponential growth of $\langle \varGamma _{\varDelta }\rangle$ , the ratio $\langle {\textit{UC}1}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle$ is always greater than unity and the ratio $\langle {\textit{PD}}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle$ is always less than unity. The growth of uncertainty is approximately determined by the balance of the quantity

(3.6) \begin{equation} \frac {\left \langle {\textit{UC}1}_{\varDelta }\right \rangle +\left \langle {A}_{\varDelta }\right \rangle -\left \langle {R}_{\varDelta }\right \rangle -\left \langle {\textit{PD}}_{\varDelta }\right \rangle }{\left \langle \varGamma _{\varDelta }\right \rangle }, \end{equation}

which is equivalent (neglecting the contributions of $\left \langle {\textit{UC}2}_{\varDelta }\right \rangle$ and $\langle {\textit{UC}3}_{\varDelta }\rangle$ ) to twice the maximal Lyapunov exponent of $\langle \varGamma _{\varDelta }\rangle$ . With $\langle {R}_{\varDelta }\rangle =\langle {\varGamma }_{\varDelta }\rangle$ , and $\langle {\textit{UC}1}_{\varDelta }\rangle$ and $\langle {\textit{PD}}_{\varDelta }\rangle$ closely correlated to $\langle {\varGamma }_{\varDelta }\rangle$ , it is the advective term which largely determines whether uncertainty increases or reduces (note that this advective term, originating from (2.3), remains relevant even at very low ${\textit{Re}}$ ). This can be seen in figure 5, where regions of negative $\langle {A}_{\varDelta }\rangle$ (shown with a dotted red line), correspond to short-term decreases in $\langle {R}_{\varDelta }\rangle$ .

3.1.4. The spectra of uncertainty

We next investigate how the spectra of the uncertainty evolve. We calculate the energy spectra of $\Delta {u}_{i}$ (denoted $\hat {E}_{\varDelta }$ ) and the spectra of $\Delta {c}_{\textit{ii}}$ (denoted $\Delta \hat{c}_{\textit{ii}}$ ). Figure 6 shows $\hat {E}_{\varDelta }$ and ${\Delta \hat{c}}_{\textit{ii}}$ at very early times $\tau \leqslant 0.05$ , covering until nearly the end of regime (I). Figure 7 shows the same for $\tau \in [0.1,0.6]$ , covering regime (II). Figure 8 shows the same at later times $\tau \in [1,32]$ , covering regime (III). In all three figures, the forcing wavenumber is indicated by a vertical dotted red line, and the spectra of the reference field (averaged over $50$ time units) are shown with a black dashed line.

Figure 6. Evolution of the uncertainty energy spectrum $\hat {E}_{\varDelta }(k)$ (a) and the uncertainty in the conformation tensor trace ${\Delta \hat {c}}_{\textit{ii}}(k)$ (b) for the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ) at very short times (regime (I)). The dashed black lines show the reference energy spectrum $\hat {E}^{(1)}$ and conformation tensor trace $\hat {c}^{(1)}(k)$ . The dotted red line indicates the forcing wavenumber $k=2\pi$ .

At early times (figure 6) the uncertainty energy and uncertainty in the conformation tensor trace are predominantly at high wavenumbers. This is due to the nature of the perturbation, which is imposed via (3.1). At the earliest times, both $\hat{E}_{\varDelta }$ and ${\Delta \hat {c}}_{\textit{ii}}$ have a slope of $k^{4}$ , consistent with the $\boldsymbol{\nabla} ^{4}$ operator used to impose the perturbation. For $\tau \leqslant 10^{-3}$ , there is an increase in $\hat {E}_{\varDelta }$ and ${\Delta \hat {c}}_{\textit{ii}}$ across all wavenumbers, as the perturbation is applied over a finite time – the characteristic time scale of the perturbation in (3.1) is $10^{-3}$ . From the earliest times, there is an almost immediate increase in uncertainty at large scales, and this increase continues after the perturbation has ceased, and spreads across a greater range of wavenumbers with increasing $\tau$ . This transfer of uncertainty from small scales to large scales is due to the elliptic constraint of incompressibility: small-scale uncertainty localised at one point in space creates uncertainty everywhere, instantly.

At slightly longer times – $\tau \in [0.1,1]$ (figure 7) – there is a reduction in uncertainty at small scales, and the uncertainty at large scales remains roughly constant. This period corresponds to regime (II) as defined above, and the time during which the viscous dissipation of uncertainty dominates polymeric propagation. We see a slight reduction in $\hat {E}_{\varDelta }$ across all scales and a decrease in ${\Delta \hat {c}}_{\textit{ii}}$ at small scales. In this regime the spectrum $\hat {E}_{\varDelta }$ has a slope of approximately $k^{-2}$ and ${\Delta \hat {c}}_{\textit{ii}}$ has a slope (which matches that of $\hat {{c}}^{(1)}_{\textit{ii}}$ ) of $k^{1}$ . After this (regime (III); figure 8), there is exponential growth of $\hat {E}_{\varDelta }$ and ${\Delta \hat {c}}_{\textit{ii}}$ . This exponential growth occurs initially at large scales, whilst diffusion still dominates small scales until $\tau \approx 4$ , after which we see exponential growth across all scales. With increasing $\tau$ , the slope of $\hat {E}_{\varDelta }$ increases towards $k^{-3.7}$ , the slope of the reference energy spectra.

Figure 7. Evolution of the uncertainty energy spectrum $\hat {E}_{\varDelta }(k)$ (a) and the uncertainty in the conformation tensor trace ${\Delta \hat {c}}_{\textit{ii}}(k)$ (b) for the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ) at short times (regime (II)). The dashed black lines show the reference energy spectrum $\hat {E}^{(1)}$ and conformation tensor trace $\hat {c}^{(1)}(k)$ . The dotted red line indicates the forcing wavenumber $k=2\pi$ .

Figure 8. The evolution of the uncertainty energy spectrum $\hat {E}_{\varDelta }(k)$ (a) and the uncertainty in the conformation tensor trace ${\Delta \hat {c}}_{\textit{ii}}(k)$ (b) for the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ) at longer times (regime (III)). The dashed black lines show the reference energy spectrum $\hat {E}^{(1)}$ and conformation tensor trace $\hat {c}^{(1)}(k)$ . The dotted red line indicates the forcing wavenumber $k=2\pi$ .

In figure 9 we plot the time evolution of individual components of $\hat {E}_{\varDelta }$ and ${\Delta \hat {c}}_{\textit{ii}}$ . The different regimes of uncertainty evolution are marked with vertical dotted red lines. We denote regime (0) as the time scale over which the perturbation is imposed. For $\hat {E}_{\varDelta }$ , the uncertainty at small scales initially grows at a rate proportional to $\tau ^{2}$ , which is consistent with the injection of uncertainty during the imposition of the perturbation for $\tau \lt \delta _{t0}$ . For $\tau \gt \delta _{t0}$ , this growth at small scales ceases. For ${\Delta \hat {c}}_{\textit{ii}}$ , this early growth at small scales is at a rate proportional to $\tau$ . The picture is different at large scales, with $\hat {E}_{\varDelta }$ growing proportional to $\tau ^{6}$ and ${\Delta \hat {c}}_{\textit{ii}}$ growing proportional to $\tau ^{2}$ . This growth at large scales continues for $\tau \gt \delta _{t0}$ . We have also performed simulations with $\delta _{t0}=10^{-5}\ll \delta {t}=4\times 10^{-4}$ , where $\delta {t}$ is the computational time step. Even in this limit, the same short-time growth rate of $\tau ^{6}$ is observed, and this growth rate persists for the same duration (i.e. it persists until $\tau \gg \delta _{t0}$ ).

Through regime (I), the uncertainty in both flow and conformation tensor continues to grow at large scales, whilst the uncertainty at small scales is roughly constant. The growth rate decreases with time, as the balance between viscous dissipation and polymeric propagation changes. By $\tau \approx 0.1$ , viscous dissipation of uncertainty becomes greater than polymeric propagation, and the uncertainty in both flow and conformation tensor decreases at small scales through regime (II). For $\tau \gt 1$ , in regime (III), there is exponential growth of uncertainty in both the flow and conformation tensor, again initially at large scales, and followed by small scales by $\tau \approx {4}$ . For $\tau \gt 30$ , in regime (IV), the uncertainty saturates at all scales.

Figure 9. The time evolution of components of the uncertainty energy spectra $\hat {E}_{\varDelta }(k)$ (a) and the spectra of the uncertainty in conformation tensor trace ${\Delta \hat {c}}_{\textit{ii}}(k)$ (b), normalised by the reference spectra, for the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $Sc=4\times {10}^{6}$ , $n=4$ , $\textit{Wi}=2$ ).

We comment here on the short-time evolution of uncertainty. If we impose a perturbation on $c_{\textit{ij}}^{ (2 )}$ using a Laplacian operator (rather than a hyperviscous operator), we observe qualitatively similar behaviour during regimes (I) and (II). We note that for all perturbations, regimes (III) and (IV) are unchanged; the growth rate in regime (III) and the saturation values in regime (IV) are independent of the perturbation imposed.

3.2. The effect of polymeric diffusivity $\kappa$

Polymeric diffusivity acts to limit the smallest length scales of the polymer deformation field, and we see from (2.15) that the effect of polymeric diffusivity on uncertainty is to reduce it. Figure 10(a) shows the evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle {\varGamma }_{\varDelta }\rangle /\textit{Wi}^{2}$ for $\kappa \in [10^{-4},5\times {10}^{-5},2.5\times {10}^{-5},10^{-5}]$ . The growth rate of uncertainty during regime (III) increases with decreasing $\kappa$ . At short times, the polymeric diffusivity has a significant influence on the uncertainty. The inset shows the short-time evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle {\varGamma }_{\varDelta }\rangle \textit{Wi}^{2}$ , collapsing when scaled by $\kappa ^{3}$ . This dependence on $\kappa$ is a result of the form of the perturbation: the perturbation imposed generates polymeric deformation difference spectra which scales with $k^{4}$ (as in figure 6). Polymeric diffusivity limits the smallest length scales of the polymer deformation field, and consequently the imposed perturbation is smaller for larger $\kappa$ . Figure 10(b) shows the evolution of the terms in (2.15), normalised by $\langle \varGamma _{\varDelta }\rangle$ , for different $\kappa$ . The inset shows the early time evolution. For $\kappa \geqslant 2.5\times {10}^{-5}$ , the relative magnitude of $\langle {\textit{UC}1}_{\varDelta }\rangle$ and $\langle {A}_{\varDelta }\rangle$ is little changed by $\kappa$ . At very short times, $\langle {\textit{PD}}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle$ is independent of $\kappa$ . During regime (III), $\langle {\textit{PD}}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle$ is approximately constant, with a value which decreases with decreasing $\kappa$ . One could extrapolate this trend to estimate the maximum permissible value of $\kappa$ such that the effect of polymeric diffusivity on the chaotic dynamics is negligible (e.g. the value of $\kappa$ such that $\langle {\textit{PD}}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle \lt 0.1$ during regime (III)), but in the absence of extremely high-fidelity simulations at very small $\kappa$ to confirm such an extrapolation, we leave this as a suggestion for future studies. For all four values of $\kappa$ , the terms $\langle {\textit{UC}1}_{\varDelta }\rangle$ and $\langle {A}_{\varDelta }\rangle$ are larger than $\langle {\textit{PD}}_{\varDelta }\rangle$ during regime (III).

Figure 10. (a) The evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle {\varGamma }_{\varDelta }\rangle /\textit{Wi}^{2}$ for different values of $\kappa$ . (b) The evolution of terms in (2.15), normalised by $\langle \varGamma _{\varDelta }\rangle$ , for different values of $\kappa$ . All other parameters match the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $n=4$ , $\textit{Wi}=2$ ). For $\kappa =10^{-5}$ , the resolution is $(192n)^{2}$ ; for all larger $\kappa$ , the resolution is $(128n)^{2}$ .

3.3. The influence of ${\textit{Re}}$

We next investigate the influence of ${\textit{Re}}$ , keeping all other parameters fixed. Figure 11(a) shows the evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ for a range ${\textit{Re}}\in [10^{-4},10^{-3},10^{-2}, 10^{-1},1]$ . The growth rate of uncertainty during regime (III) is shown in the inset, and appears weakly dependent on ${\textit{Re}}$ for ${\textit{Re}}\lt 0.1$ . For ${\textit{Re}}={1}$ , the growth rate decreases. Figure 11(b) shows the evolution of $\langle {I}_{\varDelta }\rangle$ , $\langle {D}_{\varDelta }\rangle$ and $\langle {P}_{\varDelta }\rangle$ . The evolution of $\lvert \langle {I}_{\varDelta }\rangle \rvert$ grows in approximate proportion to $\langle {E}_{\varDelta }\rangle$ , with the proportionality independent of ${\textit{Re}}$ . The magnitudes of $\langle {D}_{\varDelta }\rangle$ and $\langle {P}_{\varDelta }\rangle$ scale with ${\textit{Re}}^{-1}$ . The quantities $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ and $-\langle {I}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle$ also scale with ${\textit{Re}}^{-1}$ ; this collapse is anticipated provided the average flow is independent of ${\textit{Re}}$ , as both $\langle {P}_{\varDelta }\rangle$ and $\langle {D}_{\varDelta }\rangle$ contain ${\textit{Re}}$ in the denominator. What is remarkable is that the average ratio of these quantities during regime (III) is only weakly dependent on ${\textit{Re}}$ . Defining

(3.7) \begin{equation} C({\textit{Re}})=\frac {\{\langle {I}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle \}_{(\textit{III})}}{[\{\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle \}_{(\textit{III})}-1]}, \end{equation}

in which $C({\textit{Re}})$ is a function of ${\textit{Re}}$ , we find $C\in [0.113,0.117,0.121,0.143,0.153]$ for ${\textit{Re}}\in [10^{-4},10^{-3},10^{-2},10^{-1},1]$ . The relative importance of the inertial production of uncertainty is still of the order of $10\,\%$ even for ${\textit{Re}}=10^{-4}$ . How $C({\textit{Re}})$ approaches zero in the limit ${\textit{Re}}\to 0$ is an interesting question, but one which we leave for future studies. We have also calculated the ratios $\{\langle {A}_{\varDelta }\rangle /\langle {R}_{\varDelta }\rangle \}_{(\textit{III})}$ and $\{\langle {\textit{UC}1}_{\varDelta }\rangle /\langle {R}_{\varDelta }\rangle \}_{(\textit{III})}$ and find that both ratios are independent of ${\textit{Re}}$ .

Figure 11. (a) The evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ for a range of ${\textit{Re}}$ . All other parameters match the reference configuration ( $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ). Inset shows the average growth rate over regime (III). (b) The evolution of $\langle {I}_{\varDelta }\rangle$ , $\langle {D}_{\varDelta }\rangle$ and $\langle {P}_{\varDelta }\rangle$ for a range of ${\textit{Re}}$ .

Figure 12. Snapshots of the conformation tensor trace field for ${\textit{Re}}\in [10^{-4},10^{-2},1]$ . All other parameters match the reference configuration ( $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ).

Figure 12 shows snapshots of the conformation tensor trace field for three values of ${\textit{Re}}$ . Note that in these snapshots, structures similar to the ‘narwhal’ or ‘arrowhead’ structures recently identified as exact travelling-wave solutions (Page et al. Reference Page, Dubief and Kerswell2020; Morozov Reference Morozov2022) are visible for both ${\textit{Re}}=10^{-4}$ and ${\textit{Re}}=1$ . These larger-scale structures spanning the domain and breaking the cellular structure of the flow are present intermittently at all ${\textit{Re}}$ investigated, but are present for a greater proportion of the time at larger ${\textit{Re}}$ . We also note that the occurrence of these structures is less frequent at larger $\varepsilon$ : we do not observe them at all for simulations with $\varepsilon =10^{-2}$ .

We next assess the influence of ${\textit{Re}}$ on the spectra of uncertainty, following the same analysis as above for the reference configuration. The results are shown in figure 13. From figure 11, we have observed that the evolution of uncertainty in regime (III) is only weakly dependent on ${\textit{Re}}$ , and here we are interested in how the inertial terms influence the early stages of uncertainty growth. At short times, for all ${\textit{Re}}\in [10^{-4},1]$ we see the same growth rates: at the largest scales, $\hat {E}_{\varDelta }$ grows with $\tau ^{6}$ , and this growth rate persists for some time after the imposition of the perturbation. At the smallest scales, $\hat {E}_{\varDelta }$ grows with $\tau ^{2}$ during the imposition of the perturbation, after which it plateaus, before diffusive effects begin to reduce uncertainty. Although the growth rate is independent of ${\textit{Re}}$ , the magnitude of $\hat {E}_{\varDelta }$ at small $k$ scales approximately with ${\textit{Re}}^{-2}$ . This can be seen in the inset of figure 13(a), which shows the early stages of the evolution of ${\textit{Re}}^{2}\hat {E}_{\varDelta }$ . There appears to be a maximum uncertainty in the large scales during regime (II). For smaller ${\textit{Re}}$ , the growth proportional to $\tau ^{6}$ persists for a shorter time. For all ${\textit{Re}}\le {1}$ , the low-wavenumber components of $\hat {E}_{\varDelta }$ reach a similar magnitude during the transition from regime (I) to regime (II), after which the evolution during the exponential growth regime (III) tracks approximately the same course. For ${\textit{Re}}=1$ , the evolution of the large scales of uncertainty is qualitatively similar, but with smaller magnitudes and lower growth rates during the exponential regime (III).

Figure 13. The time evolution of components of the uncertainty energy spectra $\hat {E}_{\varDelta }(k)$ (a) and the spectra of the uncertainty in conformation tensor trace ${\Delta \hat {c}}_{\textit{ii}}(k)$ (b), normalised by the reference spectra, for different values of ${\textit{Re}}$ . The inset of (a) shows components of $\hat {E}_{\varDelta }(k)$ scaled by ${\textit{Re}}^{2}$ . The line styles correspond to different wavenumbers: $k=1/n$ , solid lines; $k=10$ , dashed lines; $k=50$ , dash-dot lines. All other parameters match the reference configuration ( $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ , $\textit{Wi}=2$ ).

3.4. The effect of increasing $\textit{Wi}$

Figure 14. Snapshots of the vorticity field (a) and normalised conformation tensor trace ( $c_{\textit{ii}}^{(1)}/\textit{Wi}$ ) (b) for increasing $\textit{Wi}$ . Other parameters are ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ ).

Figure 15. Time evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ with increasing $\textit{Wi}$ . (a) Other parameters are ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ ). (b) The polymeric diffusivity is increased to $\kappa =10^{-4}$ .

We next investigate the influence of elasticity on the evolution of uncertainty. Following the observation in § 3.2 that, for a given configuration, increased nonlinearity reduces the relative influence of polymeric diffusivity on the evolution of uncertainty, in this section, we take a base configuration with $\varepsilon =10^{-2}$ ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $n=4$ ) and vary $\textit{Wi}$ . Relative to the case with $\varepsilon =0$ , we expect this to allow us to reach larger $\textit{Wi}$ without the effects of polymeric diffusivity masking the nature of how the uncertainty evolution varies with $\textit{Wi}$ . Figure 14 shows snapshots of the vorticity field (upper panel) and normalised conformation tensor trace $c_{\textit{ii}}^{(1)}/\textit{Wi}$ (lower panel) for different values of $\textit{Wi}$ . For $\textit{Wi}=0.5$ , the flow is steady, the flow pattern is periodic on the scale of the forcing and in a laminar regime. With increasing $\textit{Wi}$ there is a symmetry breaking as the thin regions of high polymer deformation between cells interact with stagnation points and are swept laterally. Figure 15 shows the evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ for a range of $\textit{Wi}\in [0.5,3]$ . In our simulations, at larger $\textit{Wi}$ we obtain finer flow features, and hence to reach larger $\textit{Wi}$ , we require either a finer resolution or larger values of polymeric diffusivity. The former will increase computational costs to a level beyond our resources for this work (doubling resolution increases costs by a factor of ${\gt } 8$ ) whilst for the latter option the chaotic dynamics will be altered by the polymeric diffusivity. We show in figure 15(b) the evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ for a range of $\textit{Wi}\in [1,20]$ , with polymeric diffusivity increased to $\kappa =10^{-4}$ . This increase in $\kappa$ permits stable simulations at larger $\textit{Wi}$ , but it is likely that the dynamics of the flow is influenced by the polymeric diffusivity, particularly at larger elasticities. For smaller $\textit{Wi}$ we see a sublinear increase in the growth rate of uncertainty during the exponential regime (insets of figure 15). The dashed black lines in both insets are linear in $\textit{Wi}^{0.7}$ , and for $\kappa =2.5\times 10^{-5}$ in figure 15(a), there is a reasonable fit, suggesting the growth rate of uncertainty scales approximately with $\textit{Wi}^{0.7}$ . Acknowledging that the Lyapunov dimension and Lyapunov exponent are different, we note that the Lyapunov dimension obtained by Plan et al. (Reference Plan, Gupta, Vincenzi and Gibbon2017) was given in the form $C\textit{Wi}^{\alpha }$ , with $\alpha \approx 0.7$ . At large $\textit{Wi}$ (figure 15 b) this increase in $\{\lambda \}_{(\textit{III})}$ ceases, and for $\kappa =10^{-4}$ and $\textit{Wi}\geqslant 8$ , the growth rate of uncertainty is roughly constant with further increases in $\textit{Wi}$ . We postulate that this limit is a consequence of the larger polymeric diffusivity, which acts to reduce uncertainty growth. For the simulations with larger $\kappa$ , we observe a much more significant decrease of uncertainty at low $\textit{Wi}$ during regime (II); in figure 15(b), $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ decreases significantly with decreasing $\textit{Wi}$ during the early stages when polymeric diffusion dominates the uncertainty evolution. We note here that the rapid transfer of uncertainty across scales during regime (I) occurs for all $\textit{Wi}$ investigated (even those at low $\textit{Wi}$ in the laminar regime), and the growth rate or large-scale uncertainty with $\tau ^{6}$ in this regime holds across the range of $\textit{Wi}$ .

3.5. The effect of increasing domain size $n$

Finally, we explore the influence of the domain size. We have observed in regime (I) rapid transfer of uncertainty across scales, and with the configuration used in this work, it is interesting to explore how uncertainty evolves differently as the domain size is changed. Note that our non-dimensionalisation is based on the forcing, and not on any large-scale flow features which may develop. An alternative non-dimensionalisation is possible, based on a fixed (unit) domain size, as was used in Plan et al. (Reference Plan, Gupta, Vincenzi and Gibbon2017). In that work they estimated the Lyapunov dimension for two different forcing wavelengths, and obtained a scaling with $\textit{Wi}$ which was independent of the forcing scale. Denoting the dimensionless quantities based on the domain size with a subscript $n$ , we can express the relationship ${\textit{Re}}_{n}=n{\textit{Re}}$ and $\textit{Wi}_{n}=\textit{Wi}/n$ . If we non-dimensionalise based on the domain size, for fixed fluid transport properties (i.e. viscosity, relaxation times) and forcing magnitude, the Reynolds number would increase with domain size and the Weissenberg number would scale with the inverse of domain size. Defining the elasticity number as $El=\textit{Wi}/{\textit{Re}}$ , we see that the domain-size-based elasticity number $El_{n}$ scales with $1/n^{2}$ . We conduct simulations all with the same (unit) forcing wavelength, for a range of domain sizes $n\in [2,16]$ , and for three configurations: (i) with all other parameters matching the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ); (ii) using the sPTT model ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ); and (iii) with the sPTT model and increased polymeric diffusivity ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =10^{-4}$ , $\textit{Wi}=2$ ). Results highlighting the effect of sPTT nonlinearity parameter $\varepsilon$ on uncertainty evolution are provided in Appendix B. Note that whilst the mean kinetic energy is independent of $n$ , high-energy intermittent events increase with increasing $n$ , and consequently for $n\gt 10$ , we use a smaller time step of $\delta {t}=2\times {10}^{-4}$ .

Figure 16. (a,b) Time evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ with increasing domain size $n$ . Other parameters are ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ). In (b), $\tau$ has been rescaled by $\ln (n^{0.5})$ .

Figure 17. (a) The variation of the growth rate $\{\lambda \}_{(\textit{III})}$ with increasing domain size $n$ , for three different configurations. The dashed line corresponds to $0.5\ln (n)$ . The inset shows $\{\textit{Wi}^{2}\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}\langle \varGamma _{\varDelta }\rangle \}_{(\textit{III})}$ . (b) The energy spectra of the reference field for different values of $n$ . All other parameters match the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ).

Figure 16(a) shows the evolution of $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ with increasing domain size $n$ . All other parameters match configuration (ii) described above ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ). In figure 16(b), the results are plotted against $\tau \ln (n^{0.5})$ . In figure 16(a), it is clear that increasing the maximum length scales of the flow results in a faster growth of uncertainty during regime (III). The collapse of the curves in figure 16(b) shows this increased growth rate follows a trend of approximately $\ln (n^{0.5})$ . Our numerical experiments in which we increase $n$ for a fixed forcing strength may be re-framed as a set in which we increase $El_{n}^{-1/2}$ whilst co-varying ${\textit{Re}}_{n}$ and $\textit{Wi}_{n}$ ; we are effectively moving from (at small $n$ ) a high- $El_{n}$ regime where inertial effects are negligible to (at large $n$ ) a low- $El_{n}$ regime in which inertial effects play an increasing role.

Figure 18. Snapshots of the normalised conformation tensor trace ( $c_{\textit{ii}}^{(1)}/\textit{Wi}$ ) for increasing $n$ . All other parameters match the reference configuration ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ).

We denote the average growth rate during regime (III) as $\{\lambda \}_{(\textit{III})}$ , and this is plotted against $n$ in figure 17, for several parameter configurations. For the cases with $\varepsilon =10^{-2}$ (red and black lines), we see that $\{\lambda \}_{(\textit{III})}$ scales with $0.5\ln (n)$ over the range $n\in [4,16]$ . For small $n$ , the growth rate drops below this. For the reference configuration (blue lines/symbols) the growth rate follows this trend up to $n=6$ , and for larger $n$ drops below this trend. For the configuration with $\varepsilon =0.01$ and an increased $\kappa =10^{-4}$ , a growth rate is lower (due to the effect of polymeric diffusivity to reduce uncertainty growth), but the variation with $n$ still follows the logarithmic trend. For the configuration with increased polymeric diffusivity ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =0.01$ , $\kappa ={10}^{-4}$ , $\textit{Wi}=2$ ), we can obtain converged results with a resolution of $(64n)^{2}$ modes, allowing us to reach larger domain sizes at reasonable computational costs. For $n=32$ , the results deviate from the $0.5\ln (n)$ scaling. The inset of figure 17 shows the variation of $\{\textit{Wi}^{2}\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}\langle \varGamma _{\varDelta }\rangle \}_{(\textit{III})}$ with $n$ for the three different configurations. In the Oldroyd B limit, this quantity represents the ratio of uncertainty in kinetic energy to that in elastic energy, and we see for all three configurations that this quantity increases with increasing domain size. This is despite the average energies of the reference field being independent of domain size.

It is known that in two-dimensional inertial turbulence, large-scale condensates develop (see e.g. Chertkov et al. Reference Chertkov, Connaughton, Kolokolov and Lebedev2007, Svirsky, Herbert & Frishman Reference Svirsky, Herbert and Frishman2023) in finite domains, due to the inverse energy cascade and accumulation of energy at the largest scales. In two-dimensional elasto-inertial turbulence, the inverse cascade is present (Gupta, Perlekar & Pandit Reference Gupta, Perlekar and Pandit2015), and such condensates may form in settings with out-of-equilibrium forcing. The inverse cascade in two-dimensional (inertial, Newtonian) turbulence arises as a consequence of the additional constraint on the conservation of (squared) vorticity (Svirsky & Frishman Reference Svirsky and Frishman2025). The addition of polymers provides a mechanism which can draw energy from large to small scales, weakening or reversing the cascade depending on the relative balance of inertial and elastic effects (Gillissen Reference Gillissen2019). In the present investigation, inertial effects are small, so the inverse cascade which leads to energy condensation is absent: the dynamics is a consequence of elastic effects drawing energy from large to small scales, with a scale-by-scale balance between this transfer and viscous dissipation. Figure 17(b) shows the energy spectra of the reference field for different domain sizes $n$ . We see that there is energy below the forcing wavenumber, as in the simulations of Plan et al. (Reference Plan, Gupta, Vincenzi and Gibbon2017). In our simulations the energy spectra below the forcing wavenumber scale with $k^{-\alpha }$ for an $\alpha \gt 1$ ; there must be some transfer of energy from smaller to large scales. Furthermore, given the spectra above the forcing wavenumber closely match for all $n\geqslant 3$ , we postulate that it is the large-scale flow structures which drive the increase in the rate of uncertainty growth with increasing domain size. Figure 18 shows snapshots of the conformation tensor trace for increasing domain sizes. We see that for $n=2$ , the flow is laminar. With increasing domain size, there are increasing regions exhibiting the arrowhead-like flow structures (Page et al. Reference Page, Dubief and Kerswell2020) and increasingly large-scale patterns in the polymer deformation. A consequence of the rapid transfer of uncertainty across scales is that if one region of the flow has characteristics which result in a faster uncertainty growth, this will be transferred throughout the domain increasing the uncertainty growth rate everywhere. In figure 18 for $n=16$ , we see some form of large-scale structure consisting of alternating regions of arrowhead structures and cellular flow. Whilst the mechanisms driving the formation of large-scale flow features are uncertain, understanding their formation and structure may provide further insight into the dynamics of elastic turbulence. Such investigations via numerical simulations become prohibitively expensive, and are left for future work. Note that when inspecting the conformation tensor trace fields for the case with sPTT nonlinearity $\varepsilon =10^{-2}$ , we observe large-scale patterns spanning the domain, but we do not observe the formation of arrowhead-like structures.

Figure 19. (a) The time evolution of terms in (2.8) contributing to the evolution of ${\rm d}\langle {E}_{\varDelta }\rangle /{\rm d}t$ for a range of domain sizes $n$ . Time is scaled by $0.5\ln {n}$ . The inset shows the variation of ratios of these terms with $n$ . (b) The time evolution of the ratio $\langle {R}_{\varDelta }\rangle /\langle {\textit{PD}}_{\varDelta }\rangle$ , with time scaled by $0.5\ln {n}$ . The inset shows the short-time evolution of this ratio, which collapses with time scaled by $n^{0.2}$ . Other parameters are ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ).

We next consider the terms in (2.8) and (2.15) for varying $n$ . Figure 19(a) shows the individual terms in (2.8) for increasing $n$ , plotted against $\tau \ln ({n}^{0.5})$ . The same collapse is observed as in $\langle {E}_{\varDelta }\rangle /E_{\textit{a}v\textit{g}}^{(\textit{tot})}$ and $\langle \varGamma _{\varDelta }\rangle /\textit{Wi}^{2}$ in figure 16. The inset shows the averages during regime (III) of ratios of these terms. Both quantities $\langle {I}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle$ , which represents the ratio of inertial production to viscous dissipation of uncertainty, and $\langle {P}_{\varDelta }\rangle /\langle {D}_{\varDelta }\rangle -1$ , which indicates whether polymeric propagation of uncertainty outweighs viscous dissipation, increase with increasing $n$ , and this increase is roughly linear. However, the ratio $\langle {I}_{\varDelta }\rangle /(\langle {P}_{\varDelta }\rangle -\langle {D}_{\varDelta }\rangle )$ decreases with increasing $n$ , implying that the relative influence of inertial effects on uncertainty growth decreases with increasing $n$ .

Figure 20. The time evolution of terms in (2.15) contributing to the evolution of $\langle \varGamma _{\varDelta }\rangle$ . (a) The evolution of the ratio $\langle {A}_{\varDelta }\rangle /\langle {R}_{\varDelta }\rangle$ . (b) The evolution of the ratio $\langle {\textit{UC}1}_{\varDelta }\rangle /\langle {R}_{\varDelta }\rangle$ . In both cases, time is scaled by $0.5\ln {n}$ . The insets show the average values of these ratios over the period of exponential growth. Other parameters are ( ${\textit{Re}}=10^{-2}$ , $\beta =1/2$ , $\varepsilon =10^{-2}$ , $\kappa =2.5\times {10}^{-5}$ , $\textit{Wi}=2$ ).

Figure 19(b) shows the evolution of the ratio $\langle {R}_{\varDelta }\rangle /\langle {\textit{PD}}_{\varDelta }\rangle$ plotted against $\tau \ln (n^{0.5})$ . At short times, polymeric dissipation of uncertainty dominates in (2.15), and we see this in the inset with small values of $\langle {R}_{\varDelta }\rangle /\langle {\textit{PD}}_{\varDelta }\rangle$ at early times. Note that we have plotted the data in the inset against $\tau {n}^{0.2}$ and the early-time evolution of $\langle {R}_{\varDelta }\rangle /\langle {\textit{PD}}_{\varDelta }\rangle$ collapses under this scaling. At the start of regime (III), there is an increase in $\langle {R}_{\varDelta }\rangle /\langle {\textit{PD}}_{\varDelta }\rangle$ to approximately $6.5$ , a value that is roughly independent of $n$ , and persists through the exponential regime (III), before increasing further as a saturation of uncertainty is reached. In figure 20(a) we plot the evolution of $\langle {A}_{\varDelta }\rangle /\langle {\varGamma }_{\varDelta }\rangle$ . The collapse with $\ln (n^{0.5})$ still holds during regime (III), whilst (not shown) the collapse with $n^{0.2}$ observed for $\langle {R}_{\varDelta }\rangle /\langle {\textit{PD}}_{\varDelta }\rangle$ holds. At late times, into regime (IV), for all $n$ , $\langle {A}_{\varDelta }\rangle /\langle {\varGamma }_{\varDelta }\rangle$ tends towards the same value just below unity. During regime (III), the average value of $\langle {A}_{\varDelta }\rangle /\langle {\varGamma }_{\varDelta }\rangle$ (plotted in the inset) increases with increasing domain size. We see the same trend for the reference configuration with $\varepsilon =0$ (blue lines in the inset), but with lower magnitude. Figure 20(b) shows the evolution of $\langle {\textit{UC}1}_{\varDelta }\rangle /\langle {\varGamma }_{\varDelta }\rangle$ . This ratio appears to be roughly independent of $n$ during regime (III). We know from (2.17) that $\langle {\textit{UC}1}_{\varDelta }\rangle$ depends on the relative orientation of the conformation tensor difference with the reference flow field, and the independence of $\langle {\textit{UC}1}_{\varDelta }\rangle /\langle {\varGamma }_{\varDelta }\rangle$ with $n$ suggests changes in the domain size are not changing this orientation. We also note that for all three configurations for which we have conducted the investigation on increasing $n$ , the magnitudes of both $\langle {R}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle$ and $\langle {\textit{PD}}_{\varDelta }\rangle /\langle \varGamma _{\varDelta }\rangle$ are independent of $n$ during regime (III). The changes we see in growth rate with increasing $n$ appear to be driven by increases in the production of uncertainty via polymeric advection, which couples uncertainty across scales, and corresponding increases in inertial and polymeric propagation terms in the equation governing the evolution of $\langle {E}_{\varDelta }\rangle$ .