The broken link between space and time in elastic turbulence

TL;DR

Elastic turbulence in viscoelastic flows at low inertia exhibits $E(k) \sim k^{-α}$ and $E(ω) \sim ω^{-β}$, with $α$ clustering near 4 while $β$ varies widely, challenging a universal space–time relation. Using direct numerical simulations in a triperiodic box, a channel, and a planar jet with Oldroyd–B dynamics, the authors demonstrate that $α \approx 4$ is robust across configurations but $β$ is not universal due to breakdown of Taylor’s hypothesis. A phenomenological analysis reveals multiple competing timescales and shows that, despite $u(r) \sim r^1$, a single dynamic exponent linking space and time does not exist in ET. The work reframes the problem by emphasizing the spatial scaling and suggests broader relevance to smooth chaotic flows and active turbulence.

Abstract

Elastic turbulence (ET), observed in flows of sufficiently elastic polymer solution at small inertia, is characterized by chaotic motions and power-law scaling of energy spectrum ($E$) in both wavenumber ($k$) and frequency ($ω$): $E(k) \sim k^{-α}$ and $E(ω) \sim ω^{-β}$. Experiments of ET have obtained a vast range of values for the exponent $β$. In inertial turbulence, Taylor's frozen-flow hypothesis implies $α= β$, i.e., spatial and temporal scales are linearly related to each other. In contrast, from high-resolution simulation in three different setups, a tri-periodic box, a channel, and a planar jet, we show that in ET $α\approx 4$ while $β$ varies significantly. Our analysis shows that in general Taylor's hypothesis does not hold in ET as there is no universal relation, linear or otherwise, between space and time. Thus, we clear the confusion of the different scaling exponents found in ET, and focus the attention of future research on understanding $α$. We introduce a fundamentally new way of looking not only at polymeric turbulence, but also at smooth chaotic flows in general, e.g., active turbulence.

Figures (6)

• Figure 1: Power-law spectrum of the turbulent kinetic energy measured in space and in time for different ET setups over the years. Orange symbols refer to spatial decay exponents $\alpha$ in the wavenumber domain ($k^{-\alpha}$) and blue symbols to temporal decay exponents $\beta$ in the frequency domain ($\omega^{-\beta}$). The only theoretical prediction is an inequality: $\alpha > 3$, denoted with an arrow. The ranges of variation of the exponents are shown as shaded regions, and highlight the sparsity of the measurements of $\beta$ compared to those of $\alpha$, which all lie in a narrow band centred around $4$. Notably, no trend for $\beta$ emerges even reordering the data in terms of the elasticity number (defined as the ratio of Deborah and Reynolds numbers).
• Figure 2: Turbulent kinetic energy spectra of elastic turbulence, for box (left), channel (middle) and jet (right), from left to right. Top row: visualizations of the vorticity magnitude, with darker colors denoting higher values. Middle row: spatial energy spectra, normalized with the mean-square of the velocity fluctuations. Bottom row: temporal energy spectra, normalized with the mean square of the velocity fluctuations. Our data is marked with black dots, while data from literature is highlighted with the same symbols of figure \ref{['fig:exponents']} and colors consistent with their spatial/temporal nature. A slight shift downward is introduced for ease of visualization. Power-law scalings are reported with dashed-dotted lines.
• Figure 3: Space-time relationship in the IT (top row) and ET (bottom row) channel flows. Contour plots (a,e) of the streamwise autocorrelation of the streamwise velocity component, $G_{xx}(r,t)$, see equation \ref{['eq:Cxx']}, and (b,f) streamwise autocorrelations of the spanwise velocity component, $G_{xy}(r,t)$, as a function of $r/\mathcal{L}$ and $t/\mathcal{T}$. The autocorrelations are computed at the center-plane. The contour lines are evenly spaced between $0.1$ (blue) and $0.9$ (yellow) times the maximum. We also plot (c,g) $G_{xy}$ as a function of $t$ for several different values of $r$. Black dots demarcate the time of the maximum of $G_{xy}$, $t_\bullet$. Blue triangles mark the time, $t_{\blacktriangledown}$, at which $G_{xy}$ falls below a threshold value of $0.75$. The insets show the attempted collapse following the ansatz in equation \ref{['eq:Gcol']}, assuming a linear relationship between space and time. We indeed obtain data collapse in IT, but not in ET. Finally, we plot (d,h) the two time-scales extracted from $G_{xy}$ for different length scales. For IT they exhibit the same linear behavior, but for ET different time scales follow different behaviors.
• Figure 4: Dynamic scaling in the box. (left) A representative plot of $\mathcal{F}(r,t)/\mathcal{F}(r,0)$ for two values of $r$ for ET in the box. The dashed horizontal line is at the value of ordinate equal to $0.25$. The shaded area is the integral scale $T_{\rm I}$ defined in equation \ref{['eq:TI']} for the case where the threshold is $0.25$. (right) The characteristic timescale $T_{\rm I}(r)$ as a function of $r$ for IT (squares) and ET (circles); blue symbols are for a threshold of $0.25$ and black symbols are for the zero--crossing. Note that, the black squares of IT overlap the blue ones. A dashed line with unitary slope is shown to guide the eye.
• Figure 5: Turbulent kinetic energy spectra with Pope's forcing in our box setup, normalised with the mean-square of the velocity fluctuations.