Uncertainty in Elastic Turbulence

TL;DR

The paper extends an uncertainty framework from inertial to viscoelastic flows by analyzing two perturbed realizations of a two-dimensional elastic-turbulence system modeled with a conformation tensor. It derives evolution equations for the velocity and polymer-deformation differences, defining $E_{\Delta}$ and $\Gamma_{\Delta}$ and decomposing their dynamics into inertial, viscous, and polymeric contributions, with detailed analysis of the terms $I_{\Delta}$, $D_{\Delta}$, $P_{\Delta}$, $A_{\Delta}$, $UC1_{\Delta}$, $UC2_{\Delta}$, $UC3_{\Delta}$, and $R_{\Delta}$. Through extensive DNS across parameter scans, four regimes of uncertainty evolution are identified: an initial rapid large-scale transfer with $\tau^{6}$ growth, a short dissipative phase, a regime of exponential growth with a Lyapunov-rate-like exponent $\lambda\approx 0.68$, and a late-time saturation, with the growth rate scaling approximately as $Wi^{0.7}$ and depending on $\kappa$, $n$, and $Re$. The results reveal that uncertainty can propagate from polymer deformation to the flow and back, offering a framework for assessing predictability and informing uncertainty-aware closures in viscoelastic turbulence and related chaotic flows.

Abstract

Elastic turbulence can lead to to increased flow resistance, mixing and heat transfer. Its control -- either suppression or promotion -- has significant potential, and there is a concerted ongoing effort by the community to improve our understanding. Here we explore the dynamics of uncertainty in elastic turbulence, inspired by an approach recently applied to inertial turbulence in Ge et al. (2023) \textit{J. Fluid Mech.} 977:A17. We derive equations for the evolution of uncertainty measures, yielding insight on uncertainty growth mechanisms. Through numerical experiments, we identify four regimes of uncertainty evolution, characterised by I) rapid transfer to large scales, with large scale growth rates of $τ^{6}$ (where $τ$ represents time), II) a dissipative reduction of uncertainty, III) exponential growth at all scales, and IV) saturation. These regimes are governed by the interplay between advective and polymeric contributions (which tend to increase uncertainty), viscous, relaxation and dissipation effects (which reduce uncertainty), and inertial contributions. In elastic turbulence, reducing Reynolds number increases uncertainty at short times, but does not significantly influence the growth of uncertainty at later times. At late times, the growth of uncertainty increases with Weissenberg number, with decreasing polymeric diffusivity, and with the logarithm of the maximum length scale, as large flow features adjust the balance of advective and relaxation effects. These findings provide insight into the dynamics of elastic turbulence, offering a new approach for the analysis of viscoelastic flow instabilities.

Figures (23)

• Figure 1: Left panel: The kinetic energy spectra of the flow in the reference configuration, for three resolutions $\left(96n\right)^{2}$,$\left(128n\right)^{2}$ and $\left(192n\right)^{2}$ (solid lines), and for lower ($\kappa=5\times10^{-5}$, dash-dot line) and higher ($\kappa=10^{-5}$, dashed line) polymeric diffusivity. Right panel: a snapshot of the vorticity field for the reference configuration.
• Figure 2: Left panel: the evolution of $\left\langle{E}_{\Delta}\right\rangle/E_{avg}^{\left(tot\right)}$ and $\left\langle\Gamma_{\Delta}\right\rangle/Wi^{2}$ for the reference configuration ($Re=10^{-2}$, $\beta=1/2$, $\varepsilon=0$, $\kappa=2.5\times{10}^{-5}$, $n=4$, $Wi=2$). Red lines indicate the evolution for individual simulations, whilst blue lines indicate the average. Right panel: the evolution of $A_{0}^{-2}\left\langle{E}_{\Delta}\right\rangle/E_{avg}^{\left(tot\right)}$ and $A_{0}^{-2}\left\langle\Gamma_{\Delta}\right\rangle/Wi^{2}$ for a range of values of $A_{0}$. In both panels, the inset shows the evolution of uncertainty at small $\tau$.
• Figure 3: The evolution of the orientation of uncertainty for the reference configuration ($Re=10^{-2}$, $\beta=1/2$, $\varepsilon=0$, $\kappa=2.5\times{10}^{-5}$, $n=4$, $Wi=2$). Left panel: the evolution of $\left\langle\theta^{\prime}\right\rangle$ and $\left\langle\theta^{\prime\prime}\right\rangle$. Right panel: the evolution of components of $\left\langle\psi^{\prime}\right\rangle$ and $\left\langle\psi^{\prime\prime}\right\rangle$. Note the components of $\left\langle\psi^{\prime\prime}\right\rangle$ may be greater than unity, and where we have plotted $\left\langle1-\psi_{22}^{\prime\prime}\right\rangle$ and this quantity is negative, it is plotted with a dotted line. The black line shows the evolution of $-\left(\left\langle\Gamma_{\Delta}\right\rangle-\left\langle\Pi_{\Delta}\right\rangle\right)/2\left\langle\Gamma_{\Delta}\right\rangle$. The different regimes of uncertainty evolution are indicated separated by dotted red vertical lines.
• Figure 4: The evolution of $d\left\langle{E}_{\Delta}\right\rangle/dt$ and terms in \ref{['eq:akeu']} for the reference configuration ($Re=10^{-2}$, $\beta=1/2$, $\varepsilon=0$, $\kappa=2.5\times{10}^{-5}$, $n=4$, $Wi=2$). The left panel shows the sum of terms in \ref{['eq:akeu']} and the calculated values of $d\left\langle{E}_{\Delta}\right\rangle/dt$. Where $d\left\langle{E}_{\Delta}\right\rangle/dt>0$, a solid red line is used, and where $d\left\langle{E}_{\Delta}\right\rangle/dt<0$, a dotted red line. The right panel shows ratios of the terms in \ref{['eq:akeu']}. The lower inset highlights the early-time evolution. The upper inset shows the evolution of the individual terms $\left\langle{I}_{\Delta}\right\rangle$, $\left\langle{D}_{\Delta}\right\rangle$, and $\left\langle{P}_{\Delta}\right\rangle$.
• Figure 5: The evolution of $d\left\langle{\Gamma}_{\Delta}\right\rangle/dt$ and terms in \ref{['eq:dG']} for the reference configuration ($Re=10^{-2}$, $\beta=1/2$, $\varepsilon=0$, $\kappa=2.5\times{10}^{-5}$, $n=4$, $Wi=2$). The left panel shows the sum of terms in \ref{['eq:dG']} and the calculated values of $d\left\langle{\Gamma}_{\Delta}\right\rangle/dt$. Where $d\left\langle{\Gamma}_{\Delta}\right\rangle/dt>0$, a solid red line is used, and where $d\left\langle{\Gamma}_{\Delta}\right\rangle/dt<0$, a dotted red line. The right panel shows the individual terms in \ref{['eq:dG']}. The inset shows these same terms normalised by $\left\langle\Gamma_{\Delta}\right\rangle$. Where $\left\langle{A}_{\Delta}\right\rangle$ is negative, it is plotted with a dotted red line, and with a solid red line where positive.