Abstract
The disorganized fluctuations of turbulence are crucial in the transport of particles or chemicals1,2 and could play a decisive role in the formation of rain in clouds3, the accretion process in protoplanetary disks4, and how animals find their mates or prey5,6. These and other examples7 suggest a yet-to-be-determined unifying structure of turbulent flows8,9. Here, we unveil an important ingredient of turbulence by taking the perspective of an observer who perceives its world with respect to three distant neighbours all swept by the flow. The time evolution of the observer’s world can be decomposed into rotation and stretching. We show that, in this Lagrangian frame, the axis of rotation aligns with the initially strongest stretching direction, and that the dynamics can be understood by the conservation of angular momentum. This ‘pirouette effect’ thus appears as an important structural component of turbulence, and elucidates the mechanism for small-scale generation in turbulence.
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To an observer who perceives its world with respect to three distant fluid tracers, all carried by the flow, the seemingly random turbulent motion modifies the distances to and between them. Turbulent motion, on average, separates two tracers10,11,12. However, as shown in Fig. 1, given a set of four tracers initially located on a regular tetrahedron, that is, with all pairs equally separated by a distance R0, the growth of the distance between pairs is very uneven, resulting in strong shape deformation13,14,15,16. As first observed in ref. 17, the resulting ‘minimal’ four-point description provides important insights into the dynamics of turbulence.
Remarkably, our study of the relative motion between these neighbouring particles, as shown in Fig. 1, reveals the alignment of the rotation towards the direction of the initially strongest stretching, while the angular momentum remains constant (statistically, see Supplementary Information). This is a manifestation of the ‘pirouette effect’, well known from classical ballet or ice-skating.
We used a particle tracking technique to follow several hundreds of nearly neutrally buoyant, 30 μm size polystyrene particles as tracers in a turbulent water flow11,18,19 with high turbulence intensities as defined by the Taylor microscale Reynolds number10,20 350≤Rλ≤815 . We recorded tracer motion in volumes as large as (5 cm)3 with a spatial resolution of approximately 20 μm and a time resolution of 0.04 ms by stereoscopic observation using three high-speed cameras. We accurately determined the trajectories and velocities of millions of tracers in three dimensions, from which we extracted the dynamics of initially regular tetrahedra by conditioning statistics on four tracers with nearly equal mutual distances (as in ref. 15 and Supplementary Information). We complemented the experiments by direct numerical simulations (DNS) of the Navier–Stokes equations for 100≤Rλ≤170 (ref. 14).
The time evolution of the observer’s world can be decomposed into rotation and stretching, which is measured by the perceived velocity gradient M, defined as:
where ρia and uia represent the component in the direction i (i=1,2 or 3) of the position and velocity of the ath tracer (see Supplementary Information). The perceived velocity gradient, M, can then be uniquely decomposed as a sum of a straining motion, S, and of a rotation, : , with S≡(M+MT)/2 and . In addition, the straining motion can be understood as a superposition of simple stretching or compression along three orthogonal directions, denoted . The corresponding three stretching rates λi are arranged here in decreasing order: λ1≥λ2≥λ3 . A positive (respectively negative) value of λi corresponds to stretching (respectively compression) along the direction given by . The rotation matrix is ‘characterized’ by a local rotation vector with its direction denoted as (note that the conventional vorticity vector is ). The angular momentum of the system of four points, , defined by:
is related, to a good approximation (see Supplementary Information), to the rotation vector through the relation , where I is the moment of inertia tensor familiar in classical mechanics21. In qualitative terms, the angular momentum is proportional to the rotation vector , multiplied by the square of a distance, which characterizes the extent of the set of points in the direction transverse to the direction of rotation vector . For a Lagrangian tetrahedron, angular momentum is not strictly conserved (as studied in the Supplementary Information). However, as we demonstrate below, the conservation is a good approximation for the short times we examined.
The shape dynamics can be conveniently studied in the basis given by the eigenvectors of the strain at a chosen initial time at which the tetrads are equilateral. The strong stretching in the direction results in the shape becoming thinner in the direction 1, thus leading to a reduction of the component I11, compared to the components in the other directions I22 and I33. If angular momentum is conserved, then it implies that the direction of rotation at a time t+Δt should align preferentially with the initial stretching direction . The effect is clearly demonstrated in our experiments, as shown in Fig. 2.
The time evolution of the alignment between the initial stretching and rotation can be quantified by studying the square of the cosine of the angles between the unit vectors:
where the brackets 〈·〉 denote an average over many configurations, starting with an initially regular tetrahedron of fixed size. As a point of reference, we note that when the two vectors and are randomly oriented with respect to each other, the distribution of is uniform, and the value of is equal to 1/3. A value of greater or less than 1/3 indicates that the two vectors tend to be oriented preferentially parallel or perpendicular to each other.
Figure 3 quantifies the alignment effect between the direction of the rotation vector with the strongest initial stretching direction for different turbulence levels and regular tetrahedra with side length R0. The distances R0 were chosen in the inertial range, where neither the viscosity nor the large-scale details of the flow are expected to play a role20. As shown in Fig. 3a, the function C1,ω(t) starts at a value very close to 1/3 at time t=0, indicating little correlation between the two vectors at time t=0. Then the function C1,ω(t) increases significantly with time, to a value of approximately 0.45 at a time t≈0.2t0, where t0=(R02/ɛ)1/3 is the characteristic turbulent time at scale R0 for a flow with energy dissipation rate per unit mass ɛ in the classical Kolmogorov theory20. As shown in Fig. 3b, at t≈t0/4 and t0/2 the probability density functions (PDFs) of peaks at , whereas at t≈0 it is flat. This demonstrates beautifully the preferential alignment between the two vectors for t>0 . The origin of the alignment can be traced to the weakening of the component of the moment of inertia tensor matrix I11, due to the strong stretching along the direction compared with the two other components I22 and I33. Remarkably, the relative value of I11 reaches a minimum at tmax≈0.2−0.3t0 (Fig. 3c). This is the time needed to reach the best alignment between rotation and the strongest initial stretching direction . This is expected from angular momentum conservation. Figure 3d demonstrates that for short times, up to 0.1t0, angular momentum is conserved. The mean value of , divided by its initial value at t=0, is shown as a function of t/t0 for several initial sizes R0. A slow increase of I11(t), induced by the increase in inter-particle separation at later time (relative turbulent dispersion9,10,11,12), caused the observed drift in . Our results thus suggest that, in the inertial range, the dynamics of alignment is statistically self-similar, and depends simply on time and initial scales through the reduced variable t/t0. In addition, our numerical simulations show that the alignment of the vectors and is observed even at smaller scales, where viscosity dominates, albeit with a viscous timescale.
The alignment properties between the rotation direction and the stretching direction at the same time have been investigated for the true velocity gradient tensor22,23,24,25,26,27,28,29. On the basis of the intuitive notion that stretching is strongest in the direction , it was expected that there would be a strong alignment of with . In contrast, it was found that the direction is essentially independent of the equal-time strongest stretching , but rather aligns preferentially with the direction of the intermediate rate of strain, , which is associated on average with weaker stretching22,23. This misalignment between and weakens the nonlinear interaction between strain and rotation that is known to be necessary for the generation of smaller scales from larger ones, which is the hallmark of turbulence10. This puzzling observation has since been the subject of numerous studies24,25,26,27,29.
Our results shed new light on this puzzle. The strain does indeed align rotation with the strongest initial stretching direction , albeit with a spatial-scale dependent delay of ∼0.2−0.3t0. Our DNS results show this conclusion to extend down to the Kolmogorov scale, where M reduces to the true velocity gradient tensor. Thus, the alignment with the initially strongest stretching direction is a dynamical process: not surprisingly, the alignment between and builds up over time. Understanding the alignment properties between and the eigenvectors of the strain at equal time (Eulerian point of view) requires a proper description of the rotation of the eigenvectors , which is affected by nonlocal (pressure) effects28. The time of decorrelation between the direction of and is found to be on the order of 0.2t0, comparable to the time of alignment of and , thus explaining the lack of observed alignment between and . Our numerical data demonstrates that the much-reduced picture discussed extends to the true velocity gradient tensor and thus adequately captures the physics of turbulent small scale generation17,30.
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Acknowledgements
We are grateful to B. Shraiman and E. Siggia for discussions. A.P. thanks IDRIS for providing the computation resources and the ANR for financial support through the contract DSPET. E.B. and H.X. thank the Max Planck Society for support. This research was supported in part by the US National Science Foundation under Grant No. NSF PHY05-51164 and the European COST Action MP0806.
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H.X. and E.B. designed and performed the experiments. H.X. and E.B. analysed the experimental data. A.P. conducted the numerical simulations and analyzed the numerical data. All authors discussed the physics and wrote the manuscript.
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Xu, H., Pumir, A. & Bodenschatz, E. The pirouette effect in turbulent flows. Nature Phys 7, 709–712 (2011). https://doi.org/10.1038/nphys2010
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DOI: https://doi.org/10.1038/nphys2010
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