Preferential Sampling of Elastic Chains in Turbulent Flows
Jason R. Picardo, Dario Vincenzi, Nairita Pal, Samriddhi Sankar Ray
PPreferential Sampling of Elastic Chains in Turbulent Flows
Jason R. Picardo, ∗ Dario Vincenzi, † Nairita Pal, ‡ and Samriddhi Sankar Ray § International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India Universit´e Cˆote d’Azur, CNRS, LJAD, Nice 06100, France Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.
A string of tracers, interacting elastically, in a turbulent flow is shown to have a dramaticallydifferent behaviour when compared to the non-interacting case. In particular, such an elastic chainshows strong preferential sampling of the turbulent flow unlike the usual tracer limit: an elasticchain is trapped in the vortical regions. The degree of preferential sampling and its dependence onthe elasticity of the chain is quantified via the Okubo-Weiss parameter. The effect of modifying thedeformability of the chain, via the number of links that form it, is also examined.
PACS numbers: 47.27.Gs, 05.20.Jj
The development of Lagrangian techniques, in exper-iments and theory, has lead to major advances in ourunderstanding of the complexity of turbulent flows, es-pecially at small scales [1–3]. What makes this possibleis the use of tracer particles which uniformly sample theflow and hence access the complete phase space in whichthe dynamics resides. This feature of tracers depends,crucially, on the assumption that the particles remain inertia-less and point-like . When some of these assump-tions are relaxed, it may lead to dissipative particle dy-namics and preferential sampling of the structures in aflow [4–10]. This is, for instance, the case for heavy, iner-tial particles, which show small-scale clustering and con-centrate away from vortical regions. Various phenomenacan influence the properties of inertial clustering in tur-bulence, such as gravity [11, 12], turbophoresis [13, 14], orthe non-Newtonian nature of the fluid [15]. Preferentialsampling in turbulent flows may also emerge as a result ofthe motility of particles, as in the case of gyrotactic [16],interacting [17], or jumping [18] micro-swimmers.We now propose a novel mechanism for preferentialsampling in turbulent flows which is induced by exten-sibility. A simple model of an extensible object whichretains enough internal structure is a chain of tracerswith an elastic coupling between the nearest neighbours.We show, remarkably, that turning on such elastic inter-actions amongst tracers leads to very different dynam-ics: unlike the case of non-interacting tracers, an elasticchain preferentially samples vortical regions of the flow.We perform a systematic study of this phenomenon andquantify, via the Okubo-Weiss parameter, the level ofpreferential sampling and its dependence on the elastic-ity and deformability of the chain.Harmonic chains have been at the heart of several im-portant problems in the areas of equilibrium and non-equilibrium statistical physics. These have ranged fromproblems in crystalline to amorphous transitions [19],electrical and thermal transport both in and out-of-equilibrium [20], as well as understanding structuralproperties of disordered and random systems [21]. Giventhe ubiquity and usefulness of the elastic chain, it is sur- prising that the effect of a turbulent medium on long chains has not been studied as extensively as in otherareas of non-equilibrium statistical physics.There is another reason why this study is important.The last decade or more has seen tremendous advancesin our understanding of heavy inertial particles and theirdynamics. These were helped primarily by the pioneeringresults on the issue of preferential concentration, whichplays a dominant role in every aspect of turbulent trans-port. In contrast, similar studies of extended objectssuch as fibers are recent [22–25], even though they arejust as important and common place in nature and in-dustry. However, the effect of elasticity, which is intrinsicto extensible objects (just as inertia is to finite-sized par-ticles), in determining their dynamics in a turbulent flowis an open question. In this Letter, we settle this questionthrough a model which is amenable to detailed numericalsimulations.We generalize a well-studied model for polymericchains, the Rouse model [26–28], which consists of a se-quence of N b identical beads connected through (phan-tom) elastic links with its nearest left and right neigh-bours; the two end beads are free. Starting from New-ton’s equation for a single bead and incorporating theeffect of the fluid Stokesian drag, elastic interactions,and thermal noise, the dynamics is most conveniently ex-pressed in terms of the center of mass of the chain X c =( x + · · · + x N b ) /N b ( x , . . . , x N b denote the positionsof the beads) and the separation vector r j = x j +1 − x j ( j = 1 , . . . , N b −
1) between the j -th and ( j + 1)-th bead.For arbitrary r j , the general form of the equations ofmotion of such a chain in a velocity field u ( x , t ), in theabsence of inertia, are:˙ X c = 1 N b (cid:34) N b (cid:88) i =1 u ( x i , t ) + (cid:114) r τ N b (cid:88) i =1 ξ i ( t ) (cid:35) (1a)˙ r j = u ( x j +1 , t ) − u ( x j , t ) + (cid:114) r τ [ ξ j +1 ( t ) − ξ j ( t )] − τ (2 f j r j − f j +1 r j +1 − f j − r j − ) (1b) a r X i v : . [ phy s i c s . f l u - dyn ] J a n ( a ) ( b ) ( c ) - - - FIG. 1. Representative snapshots showing the positions of a subset of chains ( N L = 9 and L m = 4) overlaid on the vorticityfield of the carrier turbulent flow (with (cid:96) f = 2 and t f = 2 . Wi = 0 .
04 and Wi = 0 . Wi = 0 .
9. Parameter values: ν = 10 − , µ = 10 − , and F = 0 . where f j = (1 − | r j | /r ) − are the standard FENE(Finitely-Extensible-Nonlinear-Elastic) interactions [26],which are linear for small separations and divergequadratically for larger r j to ensure that inter-bead sep-arations remain bounded by the maximum length r m .Thus, at any given instant in time, the length of thechain R ≡ (cid:80) N L j r j < L m , where L m = N L r m is themaximum or contour chain length ( N L = N b − τ ,which, in turn, determines the effective relaxation time τ chain = ( N b + 1) N b τ / ξ i ( t ) ( i = 1 , . . . , N b ), whose amplitude r sets the equilibrium length of the chain in the absence offlow. Although the effect of thermal fluctuations on themotion of the center of mass is negligible in a turbulentflow, their effect on the separation vectors is essential:thermal fluctuations prevent the chain from collapsinginto a point-like particle and hence a tracer. Finally,inter-bead hydrodynamic interactions are ignored.We note that such a generalised model ensures thatwhen the separations between the beads are smallenough for the velocity differences to be approximatedas u ( x j +1 , t ) − u ( x j , t ) ≈ A · r j , where A ik = ∂ k u i isthe velocity gradient tensor, then Eqs. (1) reduce to theRouse model with FENE links smaller than the viscousscale at all times. This model is commonly used in simu-lations of polymer chains in turbulent flows [29–33]. Byconsidering the full velocity difference between adjacentbeads, Eqs. (1) describe an elastic chain that may extendinto the inertial range of turbulence or even beyond theintegral scale (see Refs. [34–37] for an analogous general-ization of the dumbbell ( N b = 2) model).What is the effect of a turbulent velocity field u on themotion of such chains? To answer this, we solve, by usinga pseudo-spectral method with a 2/3 de-aliasing rule, the two-dimensional Navier-Stokes equation on a square gridwith 1024 collocation points and 2 π periodic boundaryconditions. We drive the system to a homogeneous andisotropic, turbulent, statistically steady state through anexternal, deterministic force f = − F k f cos( k f x ) ( F isthe amplitude and k f the energy-injection scale in Fourierspace, which sets the typical size of the vortices (cid:96) f =2 πk − f ). Forcing at small wavenumbers ensures that thevortices are fairly large: this allows us to clearly illustratethe issues of preferential sampling which are central tothis work. We use a small Ekmann-friction coefficient µ (in addition to a coefficient of kinematic viscosity ν ) toprevent pile-up of energy at the large scales due to inversecascade. Consequently the turbulent flow is in the direct-cascade regime. The definition of (cid:96) f also allows us tocharacterise the stretching ability of the flow in terms ofthe dimension-less Weissenberg number Wi = τ chain /t f ,where t f = (cid:96) f / √ E is the turnover time scale of the largevortices ( E is the mean kinetic energy of the flow).The temporal evolution of the chain (Eqs. (1)) is doneby a second-order Runge-Kutta scheme, augmented bya rejection algorithm [28] to avoid numerical instabilitiesdue to the divergence of the nonlinear force for | r j | ap-proaching r m . A bilinear scheme is used to interpolate,from the Eulerian grid, the fluid velocity at the typicallyoff-grid positions of the beads [38, 39]. In order to observethe regime of preferential sampling, we choose parame-ters for the chain which ensure that its equilibrium lengthin a quiescent flow is similar to the enstrophy dissipationscale, while (cid:96) f < L m < π .This model of an elastic chain in a turbulent flow is theideal setting, theoretically and numerically, to investigatethe natural interplay between the relative importance ofLagrangian (uniform) mixing and the elasticity of thelinks. It is this competing effect that leads to a surprisingpreferential sampling of the flow by the chain, hitherto - - - - - - FIG. 2. Lagrangian pdfs of (a) the Okubo-Weiss parameter Λ and (b) the scaled lengths of the chains
R/(cid:96) f for different valuesof Wi . Parameter values: (cid:96) f = 1 . , t f = 1 . , L m = 3 . , N L = 9, ν = 10 − , µ = 10 − , and F = 0 . not observed.It is important to stress here that we chose two-dimensional turbulent flows to take advantage of theirlong-lived vortical structures, which help to convincinglyillustrate this new phenomenon of preferential sampling.We have checked in several simulations that this phe-nomenon persists even in three-dimensional turbulence,becoming increasingly prominent as the Reynolds num-ber is raised and intense vortex filaments proliferate.However, as in the case of preferential concentration ofinertial particles, the effect is most convincingly broughtout in two-dimensional flows.To illustrate this phenomenon, we begin by randomlyseeding 5 × chains into the flow and study their evo-lution in time for different Wi . (We evolve a large num-ber of chains simultaneously for the purpose of visual-izing their sampling behaviour and for obtaining goodstatistics of the chain dynamics; we do not describe thecollective motion of an ensemble of chains, which wouldinteract with each other hydrodynamically or by directcontact.) In Fig. 1 we show the center-of-mass positionsat an instant of time overlaid on the vorticity field ofthe turbulent flow for Wi = 0 .
04 and Wi = 0 .
9. It isimmediately apparent that for the case of small elastic-ity the chains behave like tracers and distribute evenly(Fig. 1(a)). However, for larger Wi there is a preferentialsampling of the vortical regions (Fig. 1(b)).Figure 1(c) demonstrates the coupling between thetranslational and the extensional dynamics of the chainby showing a snapshot in which the entire chains, andnot just the centers of mass, are overlaid on the vortic-ity field. This figure emphasises the strong correlationbetween the positions of elongated chains with regionsof low vorticity, where the straining flow stretches outthe chains. In contrast, the chains that encounter vor-tices tend to curl up and contract to a much smaller size.These strikingly different phenomena are best seen in avideo of the time evolution of the chains [40]. All of this suggests the following picture: a stretched chain is morelikely to leave straining zones and coil up in vortical re-gions.The above observations can be quantified via a La-grangian approach by measuring the statistics of the ex-tension R and the Okubo-Weiss parameter Λ along thetrajectories of the centers of mass of the chains. We re-call that, for incompressible flows, Λ ≡ det A / (cid:104) ω (cid:105) =( ω − σ ) / (cid:104) ω (cid:105) (here rescaled by the mean enstrophy (cid:104) ω (cid:105) ), where ω = ∇ × u is the vorticity and σ is thestrain rate, given by σ = 2 S ij S ij where S = ( A + A T ) / R for different valuesof Wi . The Λ distribution for tracers ( Wi = 0) is alsoshown for comparison (its positive skewness is a conse-quence of the strongest velocity gradients occurring in in-tense vortical zones [42–44]). We find strong quantitativeevidence that increasing elasticity leads to a chain prefer-entially sampling vortical regions of the flow (Fig. 2(a)).This is accompanied by an increase in the probability ofhighly stretched configurations (Fig. 2(b)).The key to understanding the phenomenon of prefer-ential sampling lies in the correlation between the trans-lational and the extensional dynamics of a chain. Thisis quantified through the joint pdf P ( R, Λ), which showsthat when its center of mass is in vortical regions, a chainis in a contracted state (Fig. 3). The velocity terms inEq. (1a) can therefore be Taylor-expanded about X c ( t ),and Eq. (1a) reduces to the equation of motion of a tracer.Thence, the center of mass follows the flow and remainstrapped in the vortex. Contrastingly, away from vorticalregions, i.e., in straining or shearing regions, an extensi-ble chain is highly stretched (Fig. 3). As time proceeds, - - - - - FIG. 3. Joint pdf of Λ sampled by the centers of mass of thechains and
R/(cid:96) f for Wi = 1 .
22 and (inset) Wi = 0 . a large- Wi chain stretches to lengths so long that even-tually it is unable to follow the rapidly evolving strainingzones. On departing from these zones, such a chain islikely to encounter a vortex and begin to coil up (seenclearly in the video [40]). The links of the chain that enterthe vortex shrink and follow the rotational flow, eventu-ally leading to the entrapment of the entire chain withinthe vortex. A stiff chain (small- Wi ), which remains shortin straining zones, samples the negative values of Λ more,leading to annular contours (Fig. 3, inset).The stretching out of a chain in straining zones maythus be seen as a precursor to its entrapment inside vor-tices. This explains why strong preferential sampling ofvortices occurs only when Wi is so large that there is asignificant probability for chains to be stretched beyond (cid:96) f (see Fig. 2 for Wi = 0 . , . L m < (cid:96) f , then one would expectthis mechanism to fail and the chain to uniformly samplethe flow for all Wi . We have confirmed this hypothesis inour simulations but do not show the results for brevity.The results, so far, suggest a complete picture for thedynamics of an elastic chain in a turbulent flow: A chainwith a sufficient degree of elasticity—defined as the ra-tio of elastic and fluid time scales—preferentially sam-ples the flow. But is this effect truly independent of thecharacteristic length scales present in the system? Theshort answer is no, and the ability of a chain to pref-erentially sample the flow is determined by the relativemagnitudes of its typical inter-bead separation (which fora fixed value of L m depends only on N b ) and the charac-teristic fluid length scale (cid:96) f .For large values of Wi , as we have seen, the typicalinter-bead separation approaches L m /N L . For the resultsreported so far, N b was such that L m /N L (cid:28) (cid:96) f . As N b decreases, however, we will eventually obtain character-istic inter-bead separations of the order of L m /N L > (cid:96) f .In this setting, typically, no two neighboring beads willbe able to reside in a vortex simultaneously. Hence themechanism for preferential sampling will fail, and thechain will start sampling the flow uniformly once more.This suggests that, apart from the role of elasticity, thedynamics of a chain for large values of Wi ought to de-pend on a second dimensionless number Φ ≡ L m / ( N L (cid:96) f ),such that for Φ > < (cid:104) Λ (cid:105) , as a function of Wi , for different values of Φ. Forsmaller Φ, there is an increase in (cid:104) Λ (cid:105) that eventually satu-rates for Wi (cid:38)
5. However, for larger values of Φ, (cid:104) Λ (cid:105) in-creases initially [45]—indicating preferential sampling ofthe vortical regions of the flow—before decreasing againto reflect uniform sampling (see video [46]). The chain,however, continues to stretch as the elasticity increasesfor all values of Φ, and its mean square extension doesnot show any non-monotonic behavior (inset of Fig. 4).At large Wi a chain is indeed typically longer for largervalues of Φ owing to the reduced level of preferential sam-pling of vortices and correspondingly lower probability ofbeing in a contracted state.Figure 4 lends itself to an intuitively appealing pic-ture of the motion of a chain. For a given turbulent flow(characterised by t f and (cid:96) f ), whether or not a chain ofa given elasticity and maximum length L m may coil—and hence preferentially sample—depends on the num-ber of links which form the chain. In particular, for ahighly extended chain (large Wi ), decreasing the num-ber of links limits the ability of the chain to coil up intovortices. This result highlights the importance of de-formability of extensible chains. In our study, we haveconsidered a freely-jointed chain. However, the bendingstiffness of the chain may be modeled by introducing apotential force that is a function of the internal anglebetween neighboring links and restores them to an anti-parallel configuration [26]. Such stiffness would preventthe chain from coiling up and, therefore, would reducepreferential sampling in a manner analogous to that ofincreasing Φ in our freely-jointed model.The elastic chain is a simplified model for various phys-ical systems, which include, inter alia , fibers, micro-tubules, and algae in marine environments. Such sys-tems of course present additional properties that werenot taken into account here, such as the inertia andthe stiffness of the system, hydrodynamic and excluded-volume interactions between different portions of it, andthe modification of the flow generated by the motion ofthe system. These effects will certainly change quantita-tive details of the dynamics, but the mechanism at theorigin of preferential sampling identified here is of generalvalidity. It indeed relies on a few basic ingredients: the FIG. 4. (cid:104) Λ (cid:105) vs Wi for different values of Φ ≡ L m / ( N L (cid:96) f ); weuse N L = 9, 4, 2, and 1 (dumbbell), while keeping L m = 3 . (cid:96) f = 1 .
25 fixed. Inset: The variance of
R/(cid:96) f vs Wi . Theflow parameters are the same as those in Fig. 2. system must be extensible, its equilibrium size should besmaller than (cid:96) f and its contour length greater than (cid:96) f .Finally, we cannot avoid mentioning that the prefer-ential concentration of inertial particles through dissipa-tive dynamics is completely different from the mechanismthat we report here. Hence it is tempting to investigatethe interplay between the competing effects of inertia andelasticity in future studies of inertial extensible objectsin turbulent flows.We thank Abhishek Dhar, Anupam Kundu, and RahulPandit for useful suggestions and discussions. We ac-knowledge the support of the Indo-French Centre forApplied Mathematics (IFCAM). SSR acknowledges DST(India) project ECR/2015/000361 for financial support.JRP acknowledges travel support from the ICTS InfosysCollaboration grant. The simulations were performed onthe ICTS clusters Mowgli and
Mario , as well as the workstations from the project ECR/2015/000361:
Goopy and
Bagha . ∗ [email protected] † [email protected] ‡ [email protected] § [email protected][1] P. K. Yeung, Annu. Rev. Fluid Mech. , 115 (2002).[2] F. Toschi and E. Bodenschatz, Annu. Rev. Fluid Mech. , 375 (2009).[3] J. P. L. C. Salazar and L. R. Collins, Annu. Rev. FluidMech. , 405 (2009).[4] E. K. Longmire and J. K. Eaton, J. Fluid Mech. ,217 (1992)[5] J. Bec, Phys. Fluids , 81 (2003); J. Fluid Mech. ,255 (2005). [6] J. Bec. A. Celani, M. Cencini, and S. Musacchio, Phys.Fluids , 073301 (2005).[7] J. Chun, D. L. Koch, S. L. Rani, A. Ahluwalia, and L.R. Collins, J. Fluid Mech. , 219 (2005).[8] J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio,and F. Toschi, Phys. Rev. Lett. , 084502 (2007).[9] R. Monchaux, M. Bourgoin, and A. Cartellier, Int. J.Multiphase Flow , 1 (2012).[10] K. Gustavsson and B. Mehlig, Adv. Phys. , 1 (2016).[11] J. Bec, H. Homann, and S. S. Ray, Phys. Rev. Lett. ,184501 (2014).[12] G. H. Good, P. J Ireland, G. P. Bewley, E. Bodenschatz,L. R. Collins, and Z. Warhaft, J. Fluid Mech. , R3(2014).[13] S. Belan, I. Fouxon, and G. Falkovich, Phys. Rev. Lett. , 234502 (2014).[14] F. De Lillo, M. Cencini, S. Musacchio, and G. BoffettaPhys. Fluids , 035104 (2016).[15] F. De Lillo, G. Boffetta, S. Musacchio, Phys. Rev. E ,036308 (2012).[16] W. M. Durham, E. Climent, M. Barry, F. De Lillo, G.Boffetta, M. Cencini, and R. Stocker, Nature Communi-cations , 2148 (2013).[17] A. Choudhary, D. Venkataraman, and S. S. Ray, Euro-phys. Lett. , 24005 (2015).[18] H. Ardeshiri, I. Benkeddad, F. G. Schmitt, S. Souissi,F. Toschi and E. Calzavarini, Phys. Rev. E , 043117(2016).[19] Z. H. Stachurski, Materials , 1564 (2011); B. Illing, S.Fritschi, H. Kaiser, C. L. Klix, G. Maret, and P. KeimProc. Nat. Acad. Sci. , 1856 (2017).[20] A Dhar, Phys. Rev. Lett. , 5882 (2001); S. Lepri, R.Livi, A. Politi, Phys. Rep. , 1 (2003); A Dhar, Adv.Phys. , 457 (2008); A. Kundu, S. Sabhapandit, and A.Dhar, J. Stat. Mech., P03007 (2011).[21] A. M. Mayes, Macromolecules , 3114 (1994); K.Binder, J. Baschnagel, and W. Paul, Prog. Polymer Sc. , 115 (2003); P. Chaudhuri, S. Karmakar, C. Dasgupta,H. R. Krishnamurthy, and A. K. Sood, Phys. Rev. Lett., , 248301 (2005); T. Sridhar, D. A. Nguyen, R. Prab-hakar, and J. R. Prakash, Phys. Rev. Lett. , 167801(2007).[22] C. Brouzet, G. Verhille, and P. Le Gal, Phys. Rev. Lett. , 074501 (2014).[23] G. Verhille and A. Bartoli, Exp. Fluids , 117 (2016).[24] M. E. Rosti, A. A. Banaei, L. Brandt, and A. Mazzino,Phys. Rev. Lett. , 044501 (2018).[25] S. Allende, C. Henry, and J. Bec, arXiv:1805.05731[26] R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Has-sager, Dynamics of Polymeric Liquids (John Wiley andSons, New York, 1977), Vol. 2[27] M. Doi and S. F. Edwards,
The theory of polymer dy-namics (Oxford University Press, 1986).[28] H. C. ¨Ottinger,
Stochastic Processes in Polymeric Fluids (Springer, Berlin, Germany, 1996).[29] H. Massah, K. Kontomaris, W. R. Schowalter, and T. J.Hanratty, Phys. Fluids A , 881 (1993)[30] Q. Zhou and R. Akhavan, J. Non-Newtonian Fluid Mech. , 115 (2003).[31] V. K. Gupta, R. Sureshkumar, and B. Khomami, Phys.Fluids , 1546 (2004).[32] J. Jin and L. R. Collins, New J. Phys. , 360 (2007).[33] T. Watanabe and T. Gotoh, Phys. Rev. E , 066301 (2010).[34] M. De Lucia, A. Mazzino, and A. Vulpiani, Europhys.Lett. , 181 (2002).[35] M. F. Piva and S. Gabanelli, J. Phys. A: Math. Gen. ,4291 (2003).[36] J. Davoudi and J. Schumacher, Phys. Fluids , 025103(2006).[37] A. Ahmad and D. Vincenzi, Phys. Rev. E , 052605(2016).[38] P. Perlekar, S. S. Ray, D. Mitra, and R. Pandit, Phys.Rev. Lett. , 12231 (2017) [40] https://youtu.be/etLuK6ovAqk [41] A. Okubo, Deep-Sea Res. Oceanogr. Abstr. , 445(1970); J. Weiss, Physica (Amsterdam) , 273 (1991).[42] P. Perlekar and R. Pandit, New J. Phys. , 073003(2011)[43] A. Gupta, P. Perlekar, and R. Pandit, Phys. Rev. E ,033013 (2015)[44] D. Mitra and P. Perlekar, Phys. Rev. Fluids , 044303(2018)[45] See also Refs. [34, 35] for dumbbells in cellular flows.[46] Online movie showing uniform sampling of highlystretched dumbbells:, 044303(2018)[45] See also Refs. [34, 35] for dumbbells in cellular flows.[46] Online movie showing uniform sampling of highlystretched dumbbells: