Scaling Regimes of Active Turbulence with External Dissipation

Active fluids exhibit complex turbulentlike flows at low Reynolds number. Recent work predicted that 2D active nematic turbulence follows scaling laws with universal exponents. However, experimentally testing these predictions is conditioned by the coupling to the 3D environment. Here, we measure the spectrum of the kinetic energy $E(q)$ in an active nematic film in contact with a passive oil layer. At small and intermediate scales, we find the scaling regimes $E(q)\sim q^{-4}$ and $E(q)\sim q^{-1}$, respectively, in agreement with the theoretical prediction for 2D active nematics. At large scales, however, we find a new scaling $E(q)\sim q$, which emerges when the dissipation is dominated by the 3D oil layer. In addition, we derive an explicit expression for the spectrum that spans all length scales, thus explaining and connecting the different scaling regimes. This allows us to fit the data and extract the length scale that controls the crossover to the new large-scale regime, which we tune by varying the oil viscosity. Overall, our work experimentally demonstrates the emergence of scaling laws with universal exponents in active turbulence, and it establishes how the spectrum is affected by external dissipation.

Popular Summary

Active fluids flow on their own, driven internally by their constituents. Examples of active fluids include bacterial suspensions, cell layers, and mixtures of cytoskeletal filaments and molecular motors. These fluids exhibit chaotic flows, which, given their visual similarity with turbulence, have been called active turbulence. However, the extent of this analogy remains debated. As first predicted by Kolmogorov, classical turbulence is characterized by scaling laws with universal exponents. Here, we report that active turbulence also follows universal scaling laws.

We measure the flow field in an active liquid-crystal film made of microtubules and kinesin motors. We experimentally verify that the energy spectrum exhibits two theoretically predicted scaling regimes characterized by universal exponents, and we find a new scaling regime that arises from the coupling of the active film with the surrounding passive fluids, which provide a source of external dissipation. By fitting the experiments to a new theoretical framework that explains all the scaling regimes, we also extract elusive properties of the active fluid, such as its viscosity.

Overall, our work experimentally demonstrates scaling laws with universal exponents in active turbulence, and it shows how these flows are affected by the coupling to the surrounding fluids.

Figure 1

Oil viscosity modifies the kinetic energy spectrum of active nematic turbulence. (a)–(c) Kinetic energy spectra of turbulent flows in an active nematic film in contact with an oil layer of low (a), intermediate (b), and high (c) viscosity. Averages are over 500 frames. Error bars are standard deviations. In each panel, the top inset shows a representative microtubule fluorescence micrograph. Scale bar is $100\mu \mathrm{m}$. The bottom inset in (a) shows a schematic of the experimental system. See also Movie 2 in Supplemental Material [52].

Figure 2

Scaling regimes of turbulent flows in an active nematic film in contact with an external fluid. The different regimes are predicted at length scales ($\sim 1/q$) either larger or smaller than the mean vortex radius $R_{*}$, the viscous length ${\ell }_v={\eta }_n/{\eta }_{\mathrm{ext}}$, and the thickness $H$ of the external fluid layer. This figure summarizes the scalings in the thick-layer limit $qH\gg 1$; see Fig. 6 for the predictions in the thin-layer limit $qH\ll 1$.

Figure 3

Oil viscosity tunes the scaling regimes of active nematic turbulence. (a) Kinetic energy spectra of turbulent flows in an active nematic film in contact with a layer of oil, for 20 different oil viscosities. The data are averaged over 500 frames. (b) Rescaling each spectrum by its maximum and its corresponding wave number clearly showcases the large-scale scaling regime. (c) Fit of Eq. (1) to a representative spectrum at intermediate oil viscosity (see fits for all oil viscosities in Fig. S1 in Supplemental Material [52]). As predicted by our theory (see Fig. 2), the spectrum features signatures of three scaling regimes, separated by two crossover lengths: the mean vortex size $R_{*}^s$ and the viscous length ${\ell }_{\mathrm{oil}}={\eta }_n/{\eta }_{\mathrm{oil}}$ (vertical dashed lines). Averages are over 500 frames. Error bars are standard deviations. (d),(e) Mean vortex radius (d) and oil viscous length (e) obtained from the spectral fits in the range of intermediate oil viscosities in which the theory fits the data well. Error bars are standard errors of the mean. The mean nematic viscosity obtained from these fits is indicated in (e) as an inset (see also Fig. S2 [52]).

Figure 4

Vortex size and correlation functions in active nematic turbulence. (a) Vortex area distributions in the active turbulence regime for 20 different oil viscosities. The distributions are obtained by measuring vortices in 500 frames. (b) Mean vortex radius obtained from the exponential tails of the vortex area distributions in (a) (see Appendix pp1). Error bars are standard errors of the mean. (c),(d) Spatial autocorrelation functions of the velocity (c) and vorticity (d) fields, for all 20 oil viscosities. The data are averaged over 500 frames. The insets show the corresponding correlation lengths, defined by the conditions $C_{vv}({\ell }_{vv})=0.5$ and $C_{\omega \omega }({\ell }_{\omega \omega })=0.5$, respectively. Error bars are standard errors of the mean.

Figure 5

Schematic of the experimental setup and flow fields (side view). The thicknesses of the fluid layers are not to scale. The actual thickness of the active nematic film is $h\approx 2\mu \mathrm{m}$, whereas the thicknesses of the passive fluid layers are $H_{\text{water}}\approx 40\mu \mathrm{m}$ and $H_{\mathrm{oil}}\approx 3\mathrm{mm}$. We treat the active nematic as a two-dimensional film. White and black arrows represent the flow fields in the active nematic film and in the passive fluid layers, respectively. The flow is planar and it penetrates into the oil and water layers. Here, we represent flow penetration according to Eq. (b9) for a planar wave number $q/(2\pi )=5\times {10}^{-3}\mu {\mathrm{m}}^{-1}$, which lies in the range of our experimental measurements.

Figure 6

Scaling regimes of turbulent flows in an active nematic film in contact with an external fluid. The different regimes are predicted at length scales ($\sim 1/q$) either larger or smaller than the mean vortex radius $R_{*}$, the viscous length ${\ell }_v={\eta }_n/{\eta }_{\mathrm{ext}}$, and the thickness $H$ of the external fluid layer. This figure summarizes the scalings in the thin-layer limit $qH\ll 1$; see the main text and Fig. 2 for the predictions in the thick-layer limit $qH\gg 1$.

Figure 7

Vortex size selection. (a) Growth rate [Eq. (c13)] along the most unstable direction for the spontaneous-flow instability in an active nematic film coupled to an external fluid layer. The viscosity ratio is set to $r=\gamma /{\eta }_n=1$, and flow alignment is ignored. The different curves correspond to different values of the external fluid viscosity ${\eta }_{\mathrm{ext}}$, expressed in terms of the viscous length ${\ell }_v={\eta }_n/{\eta }_{\mathrm{ext}}$. The blue line corresponds to an isolated active nematic film, without external fluid. Lengths and time are rescaled by the active length and time, ${\ell }_a$ and ${\tau }_a$ defined in Eq. (c12). (b) Wavelength at which the growth rate is maximum, as obtained from Eq. (c14), as a function of the viscous length. This wavelength is selected by the linear dynamics upon the spontaneous-flow instability. (c) Critical wavelength of the spontaneous-flow instability [Eq. (c15)] as a function of the viscous length.