Rectification of chaotic fluid motion in two-dimensional turbulence

Turbulence is a mechanism leading to energy dissipation, however it also accumulates energy by spreading it over a range of scales. This valuable energy reservoir is known as the inertial interval. The broader this interval is, the more energy is stored and an interesting question is whether it is possible to efficiently use this energy. Recent advances in the understanding of turbulence rely on the trajectory-based or Lagrangian description of the flow. Here we show how to extract energy from the inertial interval of two-dimensional turbulence by taking advantage of its fine Lagrangian structure. A floating object in wave-driven turbulence can exploit the fluid erratic motion to fuel either directional propulsion or rotation. The shape of the object controls its ability to become a vehicle or a rotor that can tap the energy of correlated bundles of fluid trajectories. These findings offer methods of creating self-propelled devices or turbines utilizing the energy of turbulence.

Figure 1

Disk floater and self-propelled vehicle in turbulent flows. (a) Fluid particle streaks around a 40-mm-diam floating disk. The trajectory of the disk floater (red dots) in the turbulent flow (the parameters are $U=3.7\times {10}^{-2}\text{m}{\text{s}}^{-1}$ and $L_f=4.4\text{mm}$) is shown with a scale bar equal to 40 mm. The floater motion is tracked for 15 s; for clarity, particle streaks are only 0.7 s long. (b) Mean-square displacement of fluid particles and of the large disk floater. (c) Probability distribution function (PDF) of the components $U_x$ and $U_y$ of the flow velocity along the $x$ and $y$ axes. The orange and blue dashed lines are Gaussian fits. (d) Wave-number spectrum $E_k\left(k\right)$ of the horizontal kinetic energy of the wave driven flow for two different supercriticality factors $\epsilon$. The red arrow indicates the direction of the energy flux and the orange arrow indicates the forcing wave number $k_f=2\pi /L_f$. (e) Schematic of the large neutrally buoyant object at the water surface. (f) Trajectory of the winged vehicle (red dots) in the turbulent flow (the wing span of the vehicle is 40 mm and the red arrow indicates the direction of motion). The vehicle motion is tracked for 24 s; for clarity, particle streaks are only 0.5 s long (the scale bar is equal to 40 mm).

Figure 2

Rotors. (a) and (b) A single beam and a $\mathsf{Z}$ rotor in a 2D turbulent flow. The two structures can rotate freely about the $z$ axis but are fixed in the $x\text{−}y$ plane (the blue dot marks the axis of rotation and the scale bar is equal to 20 mm). (c) Cumulative angular displacement $S_{\theta }=\int \dot{\theta }\left(t\right)dt$ versus time. The beam rotates erratically with a mean angular position $\langle \theta \left(t\right)\rangle$ close to zero while the random rotation of the $\mathsf{Z}$ rotor is biased along a direction prescribed by its chirality. (d) Angular displacement $S_{\theta }$ versus time measured for increasing rms flow velocity $U$ and when the rotor handedness is reversed [rms velocity range $U={10}^{-3}\text{–}(5\times {10}^{-2})\text{m}{\text{s}}^{-1}]$. (e) Angular kinetic energy $J{\langle V_{\theta }\rangle }^2$ of the rotor versus mean kinetic energy $U^2$ of the flow (the dashed blue line is a linear fit; the parameters are $L_f=4.4\text{mm}$, $L_r=25\text{mm}$, and $J=3.1\text{g}{\text{cm}}^2$). The inset shows the PDF of the fluctuations $V_{\theta }-\langle V_{\theta }\rangle$ of the $\mathsf{Z}$ rotor measured in flows with different mean kinetic energies $U^2$ and forcing scale $L_f$ [the parameters are $(U_5=3.7\times {10}^{-2}\text{m}{\text{s}}^{-1},L_f=3\text{mm})$ and $(U_6=4.7\times {10}^{-2}\text{m}{\text{s}}^{-1},L_f=4.4\text{mm})]$. (f) Angular kinetic energy $J{\langle V_{\theta }\rangle }^2$ versus the ratio $L_r/L_f$ at fixed $U$ [the dashed blue line is a linear fit; the parameters are $L_f=3\text{–}8\text{mm}$, $L_r=5\text{–}25\text{mm}$, and $J=(1.9\times {10}^{-2})\text{–}3.1\text{g}{\text{cm}}^2]$.

Figure 3

Trajectories bundles: the fine fabric of turbulent flows. (a) Lagrangian autocorrelation ${\rho }_{11}$ (single particle) and cross-correlation ${\rho }_{12}$ (two particles) functions versus time. (b) Cross correlation ${\rho }_{12}$ versus time for increasing initial distance $d_{12}(\tau =0)$ between the two particles. Time is expressed in $T_L$ units ($T_L$ is the single-particle autocorrelation time) and $d_{12}$ in $L_f$ units. (c) Bundle correlation time $T_B$ versus the flow mean kinetic energy $U^2$; $T_B$ is computed by integration of ${\rho }_{12}$ over time for a distance $d_{12}=L_f/4$. The measurements give the relation $T_B\approx 40T_L$. (d) Fluid particle streaks in a turbulent flow and close-up of a trajectory bundle detected via the braid analysis (the parameters are 0.7-s-long streaks, $L_f=4.4\text{mm}$, and $U=3.7\times {10}^{-2}\text{m}{\text{s}}^{-1}$; the scale bar is equal to 20 mm). Details on the temporal evolution of the bundle shown in (d) are shown at different time intervals: (e) $4T_L$–$8T_L$, (f) $8T_L$–$12T_L$, (g) $12T_L$–$16T_L$ (scale bars show the forcing scale $L_f$).

Figure 4

Flow-rotor coupling. (a) Fluid particle streaks around a fixed $\mathsf{Z}$ rotor. The rotor rotation is prevented by two anchoring axes (blue dots) (the parameters are streaks recorded over 0.66 s, $L_f=4.4\text{mm}$, and $U=9.3\times {10}^{-3}\text{m}{\text{s}}^{-1}$; the scale bar is equal to 25 mm). (b) Velocity field measured using PIV in the corner of a fixed rotor ($T_L\approx 0.2$ s, the velocity field is averaged over 1 s, and the arrow length is proportional to the velocity magnitude). (c) Schematic of the modeling of the torque $P$ exerted by the fluid on the rotor. Here ${\mathbf{F}}_{\mathbf{p}}$ is the force resulting from the bending of a fluid bundle (green trajectories) in the inside part of the rotor. The trajectories in the external domain (in red) are not guided by the rotor corner. (d) A half rotor fixed in its corner rotates randomly with a mean angular velocity equal to zero; the scale bar is equal to 26 mm. (e) A half rotor fixed at one of its extremities shows biased random rotation about its anchoring point; the scale bar is equal to 33 mm. [In (d) and (e) the stroboscopic images of the half rotor, the motion is followed over 70 s at time intervals of 10 s; the parameters are $L_f=4.4\text{mm}$ and $U=2.8\times {10}^{-2}\text{m}{\text{s}}^{-1}$.]