Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits
Balachandra Suri, Logan Kageorge, Roman O. Grigoriev, Michael F. Schatz
CCapturing Turbulent Dynamics and Statistics in Experimentswith Unstable Periodic Orbits
Balachandra Suri, Logan Kageorge, Roman O. Grigoriev, and Michael F. Schatz IST-Austria, 3400 Klosterneuburg, Austria School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA (Dated: August 7, 2020)In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamicalrelevance of such solutions by demonstrating that turbulent flows in both experiment and numericstransiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). Weshow that UPOs we computed are also statistically significant, with turbulent flows spending asizable fraction of the total time near these solutions. As a result, the average rates of energy inputand dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.
Characteristic flow patterns (coherent structures) em-bedded in turbulence play critical roles in both moder-ately [1] and highly turbulent flows [2, 3], including cas-cade processes in two and three dimensions [4–6]. How-ever, inherently statistical descriptions of turbulence,which are currently widely accepted, fail to describe co-herent structures effectively. Consequently, they are un-able to quantitatively predict statistical averages of tur-bulent flows (e.g., energy dissipation rates).Recent studies suggest that coherent structures inturbulence can be described by recurrent (e.g., time-periodic) solutions to the deterministic equations govern-ing fluid flow [1, 7–11]. The existence of such solutionsembedded within a chaotic set suggests the possibilityof a fundamentally dynamical theory, inspired by Hopf’svision of turbulence as a walk between neighborhoods ofrecurrent solutions [8, 12]. For certain (e.g., uniformlyhyperbolic) low-dimensional dynamical systems exhibit-ing chaos, this viewpoint has been fleshed-out; chaotictrajectories in state space shadow (follow) a dense set ofrecurrent solutions in the form of unstable time-periodicorbits (UPOs). This property enables short-time fore-casting and the computation [via periodic orbit theory(POT)] of statistical averages from properly weightedsums evaluated over UPOs, with higher weights assignedto more frequently visited UPOs [13–15].Although the equations governing turbulence are for-mally infinite-dimensional, turbulent flows (due to dis-sipation ) can be represented as state space trajectoriesconfined to finite-dimensional chaotic sets [12]. This di-mension can be estimated, e.g., based on the number ofunstable directions of UPOs and can be relatively low[ O (10)] for transitional flows in domains of moderatesize [16–19]. While this qualitative similarity with low-dimensional chaos is encouraging, variability in the num-ber of unstable directions for UPOs suggests turbulentflows are nonhyperbolic [20]. The stable and unstablemanifolds of dynamically-invariant sets become tangentat some locations inside the chaotic set, destroying theshadowing property there and raising questions regarding the utility of UPOs for both forecasting and computingstatistical averages.To date, research devoted to developing and testing adynamical description of turbulence based on UPOs hasrelied exclusively on direct numerical simulations (DNS)[5, 7, 10, 18, 21–27]. Despite the likely presence of non-hyperbolicity, studies focusing on transitional flows (withdynamics and statistics dominated by coherent struc-tures) have generated valuable new insight. In canon-ical three-dimensional shear flows (e.g., plane-Couette)it was shown that UPOs capture salient dynamical as-pects (e.g., self-sustaining processes [28]) and statisti-cal averages (e.g., mean flow profile) of turbulent flows[5, 7, 10, 21, 22]. However, definitive evidence in supportof POT has not emerged even from studies that identifiedlarge sets of ( ≈
50) UPOs [17, 29].Previous numerical studies have imposed numerousflow restrictions, including spatially-periodic boundaryconditions, minimal-flow-unit domains and symmetry-invariance, that are not representative of experi-ment. Consequently, direct experimental evidence forshadowing–turbulent flows approaching UPOs and mim-icking their spatiotemporal evolution–has not been re-ported previously. Also, some amount of noise is alwayspresent in experiments and how it affects the dynamicalrelevance of UPOs is not currently understood. Lastly,the statistical significance of UPOs in laboratory flows isalso an outstanding question.In this Letter, we report clear evidence of UPOs inan experimental quasi-two-dimensional (Q2D) flow, in adomain whose size is much larger than a minimal flowunit. DNS of this moderately turbulent (transitional)flow is performed with no-slip boundary conditions andwithout imposing any symmetry constraints. In particu-lar, to test the shadowing hypothesis, we study the spa-tiotemporal evolution of turbulent flows that approachUPOs closely. We investigate the relation between sta-tistical “weights” predicted by POT and how frequentlyturbulent flow approaches UPOs. Finally, we comparetime-averaged properties of turbulent flows with those a r X i v : . [ phy s i c s . f l u - dyn ] A ug computed from UPOs.A Q2D Kolmogorov-like flow in the experiment is gen-erated in a shallow (6-mm thick) electrolyte-dielectricbilayer. The fluids lie in a rectangular container withlateral ( x and y ) dimensions 17.8 cm × B ∼ e − πz/w sin( πy/w ) ˆz , where w = 1 .
27 cm is thewidth of each magnet. Passing a direct current ( J ˆy )through the electrolyte layer generates a Lorentz force F = J ˆy × B ∼ e − πz/w sin( πy/w ) ˆx that drives a hori-zontal flow. The electrolyte-dielectric interface is seededwith glass microspheres and spatiotemporally resolved2D velocity fields u ( x, y, t ) that quantify the horizontalflow are measured using particle image velocimetry [30].Details of the experiment and DNS are provided in thesupplemental material (SM) [31].The Q2D flow in experiment is theoretically modeledusing the nondimensional 2D equation [32], ∂ t u + β u · ∇ u = −∇ p + 1 Re ( ∇ u − γ u ) + f , (1)which is derived by averaging 3D Navier-Stokes equationin the z direction. Here, u ( x, y, t ) is assumed to be in-compressible ( ∇ · u = 0) and corresponds to the velocityfield at the free surface in experiment. p is analogous tokinematic pressure. The spatial forcing profile f is ob-tained by depth-averaging and normalizing the Lorentzforce F . Prefactor β = 0 . − γ u ( γ = 3 .
86) capture the effects due to the solid boundaryat the bottom of the fluid layers. Reynolds number Re is related to the strength of electromagnetic forcing andis the parameter used to control the complexity of flow(cf. SM).DNS of the flow governed by Eq. (1) was performedusing a second-order (in space and time) finite differencecode previously employed in Refs. [11, 19, 33]. The di-mensions of the computational domain (14 w × w ), no- Magnet Array x J B F z –+ y Magnet Array x J B F z –+ Magnet Array x J B F z –+ agnetArray x BF z (a) JFM M
MagnetsDielectricElectrolyteAir Camera x z y (b) FIG. 1. Experimental setup to generate Q2D Kolmogorov-like flow (a) Top view indicating magnet array (dashed lines)and directions of magnetic field B , current density J = J ˆy ,and electromagnetic forcing F . (b) Side view showing stablystratified immiscible two-layer configuration. slip velocity boundary conditions, and electromagneticforcing in the DNS correspond to those in the experiment,facilitating direct quantitative comparison between thetwo. The 2D forcing profile f in the DNS is antisymmetricunder the inversion transformation R ( x, y ) → ( − x, − y ),i.e., R f = − f . Hence, Eq. (1) is equivariant under R .This two-fold symmetry ( R = ) is, however, weaklybroken in experiment due to imperfections.The Kolmogorov-like flow becomes weakly turbulentabove Re ≈
18. Results presented in this study corre-spond to Re = 23 . ± . Re ∈ [22 . , . Re = 0 .
5. The flow is chaotic for these Re ,which was validated in DNS by computing the Lyapunovexponents using continuous Gram-Schmidt orthogonal-ization (cf. SM) [34, 35]. The corresponding Kaplan-Yorke dimension is D KY ≈
12 and the Lyapunov time is τ l ≈
50 seconds. We analyzed a 36000 τ l -long turbulenttime series in the DNS and experiment to detect signa-tures of UPOs.Time-periodic flows are solutions to Eq. (1) that satisfythe condition u po ( t (cid:48) + T ) = u po ( t (cid:48) ). Here, t (cid:48) parametrizestime along the orbit with period T >
0. Due to equiv-ariance under R , Eq. (1) can also possess “pre-periodic”solutions such that u po ( t (cid:48) + T ) = R u po ( t (cid:48) ) [26]. However,it is not known a priori whether UPOs of either type ex-ist for our choice of parameters ( Re, β, γ ) and, whetherturbulent flow transiently approaches such solutions.To identify signatures of UPOs, we performed recur-rence analysis on the turbulent time series from DNS bycomputing [17, 26]: r ( t, τ ) = D − c min g (cid:107) g u ( t ) − u ( t + τ ) (cid:107) , g = {R , } . (2)Here, τ > (cid:107)·(cid:107) represents the L norm. The normal-ization constant D c = max t,τ (cid:107) u ( t ) − u ( t + τ ) (cid:107) is the em-pirically estimated diameter of the chaotic set which en-sures r ( t, τ ) ≤
1. Low recurrence values r ( t, τ ) (cid:28) t and t + τ are similar. Therefore, dur-ing the interval [ t, t + τ ], the turbulent trajectory in statespace is possibly shadowing an unstable periodic or pre-periodic orbit with period T ≈ τ . Initializing a Newton-Krylov solver [22] with 50 initial conditions u ( t ) that cor-responding to deep minima in recurrence ( r ≤ . ,UPO , UPO , UPO , UPO , UPO and UPO .Among these, UPO and UPO are R− invariant andhave been reported previously [19]. The rest lie in fullstate space; UPO are pre-periodic orbits that lie onthe same solution branch and UPO is the symmetry-related copy of UPO . Several properties of the UPOsare tabulated in the SM.To test the dynamical relevance of a UPO in exper-iment, i.e., whether turbulent flows u ( t ) approach the (a) (b) Exp (c) UPO FIG. 2. (a) Low-dimensional projection of state space showingturbulent trajectory from experiment (black curve) shadowingUPO (red loop). Each point on these curves representsa flow field. The segment in black (gray) lies in (outside)the neighborhood of UPO . The sphere and square indicateinstantaneously closest points on the turbulent trajectory andUPO . The corresponding flow snapshots are shown in (b)and (c), where color represents vorticity ω = ( ∇ × u ) z . Theprojection method is detailed in the SM. UPO, we computed the normalized distance [19, 36] D ( t ) = D − c min t (cid:48) (cid:107) u ( t ) − u po ( t (cid:48) ) (cid:107) . (3) D is the instantaneous separation between u ( t ) and theclosest point on the orbit u po ( t (cid:48) ), as shown in Fig. 2(a). D (cid:28) D ≈
1) implies the turbulent flow is veryclose to (far away from) the UPO in state space. Wepreviously identified that flow fields in physical spaceare visually similar when D ≤ .
45 [36]. Using thismetric, we found many instances when turbulent flowapproaches one of the computed UPOs. For example,Fig. 2 compares snapshots from experiment and UPO at an instant the turbulent trajectory is near UPO ( D = 0 . ; the period of this orbit is T = 113 . . τ l ).Using our closeness criterion, we estimated that the tur-bulent trajectory remains in the neighborhood of UPO for a duration equal to about four periods of UPO ( − < t/T < onto a low-dimensional subspace in Fig. 2. In-deed, the turbulent trajectory approaches UPO , shad-ows its evolution by tracing four loops, and subsequentlydeparts from the neighborhood of UPO . Video 1 inthe SM shows side-by-side comparison of turbulent flowand UPO in both physical space and state space.Since the shapes of the turbulent trajectory andUPO are similar, one may ask if the correspondingflows evolve at similar rates. To explore this, for eachpoint on the turbulent trajectory u ( t ), we identified the FIG. 3. (a) Instantaneous normalized separation D be-tween a turbulent trajectory in experiment and periodic orbitUPO . The dashed black line ( D = 0 .
45) indicates thelimit for closeness in state space. (b) t (cid:48) and t parametrizetime along UPO and the turbulent trajectory, respectively. closest point u po ( t (cid:48) ) on UPO (cf. Fig. 2). We thentested whether the time t (cid:48) increases at the same rate as t ; dt (cid:48) /dt = 1 implies identical rates of evolution for theturbulent flow and the UPO it is shadowing. Fig. 3(b)shows the relation between t and t (cid:48) during the intervalof shadowing. We defined t (cid:48) on the interval 0 < t (cid:48) < T due to periodicity of the UPO. For each of the four pe-riods, t (cid:48) (solid black line) follows the “diagonal” t mod T (dashed gray line). This shows the turbulent trajectoryand UPO evolve at comparable rates, on average. No-ticeable difference in the instantaneous rates of evolu-tion is related to turbulent trajectories not approachingUPO infinitesimally closely [36]. We also found thatturbulent trajectories in experiment shadow UPO andUPO for a duration that is nearly one and three timestheir respective periods (see Fig. S2 and S3 in the SM).Statistical significance of UPOs has received little at-tention in previous numerical studies [29, 37], and nonein experiments. To address this, we computed the frac-tion P ( (cid:15) ) of the total time turbulent trajectories visit the (cid:15) − neighborhood ( D ≤ (cid:15) ) of any UPO. Fig. 4(a) revealsthat particularly close passes ( (cid:15) ≤ .
2) to UPOs are rare(
P < (cid:15) = 0 .
45, we find that turbulent trajectoriesspend a sizable fraction of time near UPOs; about 30%in experiment and 23% in the DNS. The sensitivity of P to the choice of (cid:15) is comparable to that observed byKerswell et al. afor the statistical significance of travelingwave solutions in turbulent pipe flow at Re = 2400 [38].Since very close passes to UPOs are rare, quantifyingthe relative importance of various UPOs required coarsepartitioning of the state space. A turbulent trajectorycan be simultaneously close to several UPOs which areadjacent to each other in state space. To distinguish theirstatistical significance, we grouped the UPOs into threeclusters which are sufficiently far apart in state space:UPO , UPO , and UPO . These clusters wereidentified using pairwise separation between UPOs (cf.SM). For each cluster, we then computed the conditional ExpDNS
UPO
UPO
UPO
ExpDNSPOT
FIG. 4. Statistical significance of UPOs. (a) Probability (in%) to find a turbulent trajectory at a normalized distance D ≤ (cid:15) from the UPOs we computed. Dashed line indicatesthe upper limit (cid:15) = 0 .
45 for closeness in state space. (b) Con-ditional probabilities for turbulent trajectories visiting neigh-borhoods ( (cid:15) = 0 .
45) of UPO clusters. Error bars indicatechanges to probabilities when (cid:15) is varied between [0 . , . probability P c ( (cid:15) ) /P ( (cid:15) ) that a turbulent trajectory is nearthe UPOs in that cluster ( D ≤ (cid:15) ), given it is near oneof the seven UPOs.The probabilities for turbulent trajectories in experi-ment and DNS visiting the three UPO clusters are shownin Fig. 4(b) for (cid:15) = 0 .
45. Clearly, the R -invariant solu-tions UPO are rarely visited. In contrast, UPO clus-ters that do not lie in the symmetry subspace are visitedfrequently and hence are statistically significant. Chang-ing the neighborhood size between (cid:15) = 0 . (cid:15) = 0 . π i ∝ | Λ i | · | Λ i | · · · | Λ ik | , (4)where | Λ i | , · · · , | Λ ik | are the magnitudes of the unsta-ble Floquet multipliers of UPO i . The POT weight as-sociated with each cluster is then P c /P = (cid:80) i π i , wherethe summation is over the UPOs in that cluster. Theweights π i in Eq. (4) are defined to within a normaliza-tion constant, which we chose such that the cumulativeprobability for the three clusters is the same for POT andDNS. Fig. 4(b) shows that the statistical significance ofvarious UPO clusters predicted using POT is fairly con- FIG. 5. Energy input rate I versus the difference betweeninput and dissipation rates ( I − D ) for turbulent time seriesin experiment (scatter plot) and UPOs (closed loops). (Inset)Probability density function of I ( t ) for turbulent flow in ex-periment (solid gray) and DNS (dashed gray). Colored sym-bols show the mean values of I for each of the seven UPOs andthe dashed black lines represent the range of I for UPO and UPO . sistent with measurements in DNS. This is quite remark-able, given that turbulent trajectories do not visit theseUPOs infinitesimally closely. Lastly, alternative weight-ing formulas discussed in Refs. [17, 37, 39, 40] also yieldsimilar estimates for the statistical significance of UPOclusters (cf. SM).The motivation behind identifying UPOs and quanti-fying their statistical significance is to compare statisticalaverages of turbulent flows with those of UPOs. Follow-ing standard practice [5, 7, 22], we computed the instan-taneous energy input ( I ) and dissipation ( D ) rates I ( t ) = (cid:104) f · u (cid:105) Ω , D ( t ) = − Re (cid:104) u · ∇ u − γ u · u (cid:105) Ω , (5)for the turbulent flow and all the UPOs. Here, a · b isthe scalar product between vector fields a , b and (cid:104)·(cid:105) Ω represents the integral (cid:82) Ω ( · · · ) dx dy evaluated over theentire flow domain Ω. In Fig. 5, we plotted the differencebetween instantaneous input and dissipation rates ( I−D )versus the energy input rate I for the turbulent flow inexperiment. I and D are normalized by the temporalmean (cid:104)I(cid:105) t = (cid:104)D(cid:105) t . The corresponding quantities for eachUPO are overlaid. Additionally, the probability densityfunction for I from experiment (as well as DNS) is shownin the inset.For the statistically significant UPO andUPO , both energy input and dissipation ratescluster around the turbulent mean values, located at I / (cid:104)I(cid:105) t = 1 and I − D = 0 in Fig. 5. The I (and D ) val-ues for these UPOs vary over a narrow range (0.95,1.07)that is approximately ± σ I of the turbulent mean, where σ I = 0 .
055 is the standard deviation of I for turbulentflow. Consequently, the mean energy input (and dissipa-tion) rate for each of these five UPOs is within ± . σ I of the turbulent average (unity), as shown in the inset.In contrast, UPO , which are statistically insignificant,have mean values of I and D that deviate by over 2 σ I from the turbulent mean value.In this article, we provided unambiguous experimentalevidence for the dynamical relevance and statistical sig-nificance of UPOs in a moderately turbulent flow. Weshowed that turbulent trajectories in state space tran-siently approach UPOs closely and shadow their spa-tiotemporal evolution. We also quantified the statisti-cal significance of various UPOs by computing the frac-tion of time turbulent trajectories visit their neighbor-hoods. The estimates from DNS are consistent with the“weights” predicted by periodic orbit theory. Lastly, weshowed that statistically significant UPOs capture time-averaged properties of the turbulent flows in both exper-iment and DNS accurately.Our study identified that turbulent flows spend about30% of the time near the UPOs we computed. Thissuggests that UPOs with longer periods as well asother types of nonchaotic solutions–such as unstableequilibria, quasi-periodic orbits, and hetero/homoclinicconnections–may also play an important dynamical andstatistical role [10, 24, 41, 42]. Their existence and dy-namical relevance, at least in symmetry-invariant sub-spaces, was recently demonstrated for both 2D and three-dimensional shear flows [10, 19, 41, 42]. Hence, a dynam-ical framework based on UPOs, as well as other types ofrecurrent solutions, should ultimately enable forecasting[11, 36] and control (e.g., L¨uthje et al. a[43]) of turbu-lent dynamics, besides accurately predicting its statisti-cal properties.MS and RG acknowledge funding from the NationalScience Foundation (CMMI-1234436, DMS-1125302,CMMI-1725587) and Defense Advanced ResearchProjects Agency (HR0011-16-2-0033). B.S. acknowl-edges funding from the European Union’s Horizon2020 research and innovation program under the MarieSk(cid:32)lodowska-Curie grant agreement No 754411. [1] B. Hof, C. W. H. van Doorne, J. Westerweel, F. T. M.Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R. R.Kerswell, and F. Waleffe, Science , 1594 (2004).[2] S. K. Robinson, Annual Review of Fluid Mechanics ,601 (1991).[3] D. J. C. Dennis and F. M. Sogaro, Phys. Rev. Lett. ,234501 (2014).[4] G. Boffetta and R. E. Ecke, Annu. Rev. Fluid Mech. ,427 (2012). [5] L. van Veen, A. Vela-Martin, and G. Kawahara, Phys.Rev. Lett. (2019).[6] S. Goto, Progress of Theoretical Physics Supplement , 139 (2012).[7] G. Kawahara and S. Kida, J. Fluid Mech. , 291(2001).[8] J. F. Gibson, J. Halcrow, and P. Cvitanovi´c, J. FluidMech. , 107 (2008).[9] A. de Lozar, F. Mellibovsky, M. Avila, and B. Hof, Phys.Rev. Lett. , 214502 (2012).[10] M. Avila, F. Mellibovsky, N. Roland, and B. Hof, Phys.Rev. Lett. , 224502 (2013).[11] B. Suri, J. Tithof, R. O. Grigoriev, and M. F. Schatz,Phys. Rev. Lett. , 114501 (2017).[12] E. Hopf, Commun. Pur. Appl. Math. , 303 (1948).[13] D. Auerbach, P. Cvitanovi´c, J. P. Eckmann, G. Gu-naratne, and I. Procaccia, Phys. Rev. Lett. , 23 (1987).[14] P. Cvitanovi´c, Phys. Rev. Lett. , 2729 (1988).[15] Y. Lan, Commun. Nonlinear Sci. , 502 (2010).[16] J. F. Gibson, Channelflow: A spectral Navier-Stokes sim-ulator in C++ , Tech. Rep. (U. New Hampshire, 2014)
Channelflow.org .[17] G. J. Chandler and R. R. Kerswell, J. Fluid Mech. ,554 (2013).[18] N. B. Budanur, K. Y. Short, M. Farazmand, A. P. Willis,and P. Cvitanovi´c, J. Fluid Mech. , 274–301 (2017).[19] B. Suri, R. K. Pallantla, M. F. Schatz, and R. O. Grig-oriev, Phys. Rev. E , 013112 (2019).[20] E. J. Kostelich, I. Kan, C. Grebogi, E. Ott, and J. A.Yorke, Physica D: Nonlinear Phenomena , 81 (1997).[21] S. Toh and T. Itano, J. Fluid Mech. , 67–76 (2003).[22] D. Viswanath, J. Fluid Mech. , 339 (2007).[23] Y. Duguet, C. C. T. Pringle, and R. R. Kerswell, Phys.Fluids , 114102 (2008).[24] L. van Veen and G. Kawahara, Phys. Rev. Lett. ,114501 (2011).[25] T. Kreilos and B. Eckhardt, Chaos: An InterdisciplinaryJournal of Nonlinear Science , 047505 (2012).[26] A. P. Willis, P. Cvitanovi´c, and M. Avila, J. Fluid Mech. , 514 (2013).[27] J. Page and R. R. Kerswell, Journal of Fluid Mechanics , A28 (2020).[28] F. Waleffe, Phys. Fluids , 883 (1997).[29] D. Lucas and R. R. Kerswell, Phys. Fluids , 045106(2015).[30] B. Drew, J. Charonko, and P. P. Vlachos, “QI – Quan-titative Imaging (PIV and more),” (2013), available athttps://sourceforge.net/projects/qi-tools/.[31] See supplemental material for details regarding (i) ex-perimental setup, (ii) DNS, (iii) D KY computation, (iv)properties of UPOs, (v) state space projection procedure,(vi) turbulent trajectories shadowing UPO and UPO ,(vii) pairwise separation between UPOs, and (viii) com-parison of UPO weighting protocols. Videos 1, 2, and3 show side-by-side comparison of turbulent flows in ex-periment shadowing UPO , UPO , and UPO , respec-tively.[32] B. Suri, J. Tithof, R. Mitchell, R. O. Grigoriev, andM. F. Schatz, Phys. Fluids , 053601 (2014).[33] J. Tithof, B. Suri, R. K. Pallantla, R. O. Grigoriev, andM. F. Schatz, J. Fluid Mech. , 837 (2017).[34] D. A. Egolf, I. V. Melnikov, W. Pesch, and R. E. Ecke,Nature , 733 (2000).[35] A. Karimi and M. R. Paul, Phys. Rev. E , 046201 (2012).[36] B. Suri, J. Tithof, R. O. Grigoriev, and M. F. Schatz,Phys. Rev. E , 023105 (2018).[37] E. Kazantsev, Nonlinear Proc. Geoph. , 193 (1998).[38] R. R. Kerswell and O. R. Tutty, J. Fluid Mech. ,69–102 (2007).[39] S. M. Zoldi and H. S. Greenside, Phys. Rev. E , R2511(1998). [40] S. M. Zoldi, Phys. Rev. Lett. , 3375 (1998).[41] T. M. Schneider, B. Eckhardt, and J. Vollmer, Phys.Rev. E , 066313 (2007).[42] M. Farano, S. Cherubini, J.-C. Robinet, P. De Palma,and T. M. Schneider, Journal of Fluid Mechanics ,R3 (2018).[43] O. L¨uthje, S. Wolff, and G. Pfister, Phys. Rev. Lett.86