Dynamics of a long chain in turbulent flows: Impact of vortices
Abstract
We show and explain how a long beadspring chain, immersed in a homogeneous, isotropic turbulent flow, preferentially samples vortical flow structures. We begin with an elastic, extensible chain which is stretched out by the flow, up to inertialrange scales. This filamentary object, which is known to preferentially sample the circular coherent vortices of twodimensional (2D) turbulence, is shown here to also preferentially sample the intense, tubular, vortex filaments of 3D turbulence. In the 2D case, the chain collapses into a tracer inside vortices. In 3D, on the contrary, the chain is extended even in vortical regions, which suggests that it follows axiallystretched tubular vortices by aligning with their axes. This physical picture is confirmed by examining the relative sampling behaviour of the individual beads, and by additional studies on an inextensible chain with adjustable bendingstiffness. A highlyflexible, inextensible chain also shows preferential sampling in 3D, provided it is longer than the dissipation scale, but not much longer than the vortex tubes. This is true also for 2D turbulence, where a long inextensible chain can occupy vortices by coiling into them. When the chain is made inflexible, however, coiling is prevented and the extent of preferential sampling in 2D is considerably reduced. In 3D, on the contrary, bending stiffness has no effect, because the chain does not need to coil in order to thread a vortex tube and align with its axis.
I Introduction
A threedimensional (3D) incompressible turbulent flow has a peculiar geometrical structure which distinguishes it from a purely random field. The visualisation of the isosurfaces of enstrophy and energy dissipation indeed indicates that vorticity concentrates into tubular structures, while regions of intense strain are sheetlike skh12 (see Fig. 1 for a visualization based on the criterion Dubief ). Several Lagrangian studies have shown that the dynamics of objects smaller than the viscousdissipation scale can depend sensitively on the nature of the local velocity gradient. Heavy particles, for instance, are ejected from vortical regions because of centrifugal forces and thus concentrate in straindominated ones maxey ; se91 . In turbulent channel flows, highly stretched polymers are mainly found in the regions of strong biaxial extension that surround the nearwall streamwise vortices sg03 ; terrapon04 ; bagheri12 . Gyrotactic swimmers are trapped into the highshear zones of a turbulent flow boffetta16 , while ellipsoidal swimmers preferentially sample lowvorticity zones Pujara2018 . Much less studied, however, is the case of an object whose size lies in the inertial range of turbulence; its translation is coupled to its internal dynamics, which in turn is directly affected by coherent structures of the flow. This situation has only just begun to receive attention, especially in the context of flexible fibersverhille1 ; verhille2 ; gfv18 ; mazzino1 ; dc19 ; mazzino2 .
A beadspring chain is a simple physical system that allows us to investigate the interaction between an extended filamentary object and the geometrical structure of a turbulent flow. Such a system has been widely employed in polymer physics de86 and is here generalised in order to describe a chain longer than the dissipation scale. It consists of a sequence of beads connected by phantom springs. The extensibility of the chain can be tuned by varying the strength and the maximum length of the springs, and its stiffness is controlled by a bending potential that depends on the angle between each neighbouring pair of springs. Even though a beadspring chain can only be regarded as a very rudimentary model of an elastic filament, it has the basic properties needed to investigate the physical mechanisms that determine the motion of an extensible and flexible object in a turbulent flow. Moreover, its dynamics can be studied in detail with moderate numerical effort.
In picardo , an extensible beadspring chain was studied in a twodimensional (2D), homogeneous, and isotropic, turbulent flow. It was shown that the centre of mass of the chain preferentially samples the vortical regions. This is because the chain is strongly stretched in highstrain regions and thus becomes unable to follow their evolution. It eventually gets trapped into one of the largescale vortices that dominate a 2D turbulent flow. Inside the vortex, where straining is absent, it contracts and stays therein. Hence, in 2D turbulence, an extensible chain departs from straining regions but behaves like a tracer inside vortices, whence the preferential sampling of the latter.
Here we study the dynamics of a beadspring chain in 3D turbulence where, unlike in 2D, vorticitystretching leads to the formation of intense vortex tubesMoffat94 ; Davidson , with significant straining along the tube axis. We address the question of whether preferential sampling of vortical regions persists in three dimensions and, if so, how it depends on the extensibility and flexibility of the chain.
Ii Extensible chain
The system considered in the present study is a chain consisting of identical, inertialess beads connected to their nearest neighbours by nonlinear springs with equilibrium length , maximum length , and spring coefficient . The characteristic relaxation time of each spring is thus , which in turn sets the relaxation time of the chain collins . If , , denote the position vectors of the beads, it is convenient to describe the configuration of the chain in terms of the position of its centre of mass, , and the interbead separations , .
We first consider a freelyjointed (the links do not oppose resistance to bending), extensible chain, as in picardo . The equations of motion for such a chain are:
(1a)  
(1b) 
The elastic force on the th bead, , takes the form:
(2) 
with , and
(3) 
The divergence of the force at ensures that the length of the chain, , stays smaller than the maximum value . The vectors , , are independent multidimensional white noises; they serve the purpose of modeling the collisions between the beads and the molecules of the fluid, thereby setting the equilibrium endtoend length of the chain to . Equations (1) generalise the wellknown Rouse model of polymer physics de86 in such a way as to account for the nonlinearity of the velocity field. Indeed, in the original Rouse model the velocity differences between the beads are replaced by their firstorder Taylor expansion in the separation vectors, because polymers are assumed to be much shorter than the smallest scale of the velocity field. Here the full velocity differences are retained, since the chain is allowed to extend into the inertial range of turbulence. An analogous generalisation was applied in mazzino3 ; pg03 ; ds06 ; av16 to the elastic dumbbell model ().
Finally, the velocity field describes the motion of the fluid and is the solution of the incompressible Navier–Stokes equations,
(4) 
where is pressure, the kinematic viscosity of the fluid, and is a largescale body forcing which maintains a homogeneous, isotropic and statistically stationary turbulent flow. We perform direct numerical simulations (DNS) on a periodic domain, using a standard dealiased pseudospectral method. For 3D simulations, we use a spatial grid and a second order AdamsBashforth timeintegration scheme. The bodyforcing injects a constant amount of energy—equalling the mean dissipation rate —into the first two wavenumber shells. The Kolmogorov dissipation time and length scales are given by and (where is the maximum resolved wavenumber). The results presented below correspond to a flow with TaylorReynolds number (where is the mean kinetic energy), which is sufficiently large for a clear inertial range (albeit less than a decade) to emerge in the energy spectrum, and for the formation of distinct vortex tubes (cf. Fig. 1). We have checked that the sampling behaviour we describe persists even for smaller of and as well, though it does intensify with over this limited range.
While our primary focus is on chains in a 3D turbulent flow, it is helpful to compare their 3D dynamics with that in a 2D turbulent flow, especially when addressing the effects of extensibility and flexibility. For 2D flow simulations, a spatial grid and a second order RungeKutta timeintegration scheme is used, as in Ref. picardo . A constantintime forcing is applied, where is the forcing amplitude and is the forcing wavenumber, which sets the scale of the large coherent vortices, . An Ekman friction term Prasad2D ; Boffetta2D with coefficient is included in (4) to damp out the energy at the large scales (due to an inverse cascade).
After the flow (3D or 2D) attains a stationary state, we introduce a large number of chains, of order , each with beads. We evolve many chains simultaneously only to obtain good statistics; the dynamics we study are of a single chain in the flow. The chains are given a small initial length, close to the noflow equilibrium value. They are then allowed to be stretched out by the flow and attain a steadystate distribution before we begin recording statistics. The equations governing the dynamics of the centreofmass and separation vectors of the chain are integrated using a secondorder RungeKutta scheme, augmented by a rejection algorithm Ottinger that prevents the nonlinear spring force from diverging as approaches . The chains are evolved with a time step , where is the timestep of the fluid flow solver and is the number of substeps taken by the chain solver for every step of the fluid solver. The flow is assumed to be unchanging over the duration of the substeps. While is set to unity for the freelyjointed chain, used in this section, we use up to for accurately resolving the numericallystiff dynamics of inextensible and inflexible chains, to be introduced later in § III and IV.
The influence of elasticity on the dynamics of the chain is described in terms of the elastic Weissenberg number , which is the ratio of the chain relaxation time to the viscousdissipation timescale of the flow. For 3D turbulence, , whereas for 2D flows we choose the small timescale associated with the dissipation of enstrophy, , where is the mean enstrophy. For small , the chain is in a contracted configuration and acts like a tracer; for large , it is stretched out by the flow.
As mentioned in § I, the dynamics of an extensible chain was studied in picardo for a 2D turbulent flow forced at large spatial scales. The equilibrium size of the chain was assumed to be of the order of the dissipation scale, while was much longer and comparable to the scale of the coherent vortices, , or even greater than it. When was sufficiently large for the typical length of the chain to approach the size of the vortices, then the chain was shown to exhibit a marked preferential sampling of vortical regions. The mechanism that was proposed in picardo to explain this phenomenon, already sketched briefly in § I, can be summarised as follows:

Inside a vortex, where stretching is absent, the chain shrinks down to its equilibrium size. Since the equilibrium size of the chain is much smaller than the size of the vortex, the chain essentially behaves as a tracer and follows the vortex during its lifetime.

In a region of intense strain, the chain is stretched up to the extent that it cannot continue to follow the straining region. The chain eventually encounters a vortex that coils it up and ‘entraps’ it according to the dynamics described above.
The combination of these effects leads an extensible chain to stay inside large vortices and leave highstrain regions, which results in a strong preferential sampling of vortices. This phenomenon was quantified in picardo by studying the OkuboWeiss parameter okubo ; weiss evaluated at the position of the centre of mass of the chain,
(5) 
where and are the vorticity and the strain rate at the position of the centre of mass, respectively. We recall that positive values of correspond to vorticitydominated regions, whereas negative values indicate straindominated regions. For large enough values of , the probability of positive values of was found to be much higher than for a tracer transported by the same flow picardo . In addition, the joint probability density function (PDF) of and confirmed that the chain is contracted in vortical regions and extended in straining ones.
The first question we address here is whether or not an extensible chain exhibits preferential sampling of vortical regions also in a 3D turbulent flow. A substantial difference indeed exists between 2D and 3D flows. A 2D turbulent flow is characterised by large, longlived vortices, inside which stretching is weak. In a 3D turbulent flow, vorticity concentrates into intense tubular structures with significant stretching along their axes. The dynamics of the chain in vorticitydominated regions is therefore expected to be substantially different in the two cases.
In three dimensions, the local nature of the flow can be classified by using the  representation of the velocity gradient cantwell96 . If denotes the rescaled velocity gradient at the position of the centre of mass, let and be its second and third invariants. Since can be rewritten as
(6) 
the sign of discriminates between the regions dominated by vorticity () and those dominated by strain (). Thus, has a role analogous to that played by in a 2D flow. Figure 1 presents a snapshot of the isosurfaces of from our 3D simulation, which clearly reveal sheetlike straining zones (large negative in blue) in close proximity to intense tubular vortices (large postive in red).
Figure 2 compares the steadystate joint PDFs of and for a tracer particle (panel a) and for the extensible chain with a large (panel b). The chain has an equilibrium size of about and a maximum extension , so that when the chain is stretched lies in the inertial range and is comparable to the typical size of vortex tubes in our simulations (cf. the vortex tubes and scale bar in Figure 1). The shape of the joint PDF of the chain is similar to that of the tracer, with a majority of positive values occurring along with negative , which indicates vortex stretching rather than compression cantwell96 . However, the probability of sampling positive is significantly larger for the chain, than for the tracer. Therefore, we deduce that preferential sampling of vorticitydominated regions persists in three dimensions.
As mentioned above, however, a major difference exists between the chain dynamics in 2D and 3D turbulence.
This is reflected in the conditional PDF of the chain length given the value of , presented in figure 3b, which shows that contrary to the 2D case the chain is considerably stretched not only in straindominated but also in vorticitydominated regions. This difference is clear from the comparison of the joint PDF of and , shown in figure 3a, with its 2D analog (see figure 3 in picardo ) and has important consequences for preferential sampling. In two dimensions, indeed, the contraction of the chain inside vortices was identified as an essential element of the sampling dynamics picardo (though we shall revisit this idea in section IV). In three dimensions, the chain does not collapse into a tracer even when the flow has a strong vortical nature (figure 3b): Hence, the mechanism leading to preferential sampling ought to be different from that operating in 2D turbulence.
Now, the only way an elongated chain can remain inside a vortex tube of comparable length, is if it aligns itself along the vortex axis. Given that the majority of vortices are axially stretched (figure 2), this scenario is consistent with our observation that the chain is typically stretched out inside a vortex (figure 3b). In contrast, it is unlikely that the entire chain can be encapsulated into straining regions, given their less coherent, sheetlike topology. Moreover, the evolution of straining regions is much more nonlocal in nature than that of the vortex tubes Tsinober , which tend to move with the fluid, except for the effects of viscous diffusion (in the inviscid limit the tubes would be ’frozen’ into the fluid as a consequence of Kelvin’s circulation theorem Davidson ). Thus, we expect a chain to be able to enter and follow a vortex tube more easily than a straining region.
This explanation implies that there is an ideal chain elasticity for preferential sampling: If is too small then the chain acts like a tracer, whereas, if is too large then the chain may stretch out beyond the length of the vortex tubes and be unable to reside inside them. This intuition is corroborated by figure 4, which shows the effect of on preferential sampling (panel a), as well as on the mean length of the chain (panel b). We find that the third moment of the PDF of , whose positivity implies a higher probability of positive , varies nonmonotonically with , peaking at a value which corresponds to a mean chain length . For larger , all the chains approach the maximum chain length and cannot reside within the vortex tubes as effectively (cf. figure 1).
We shall present more direct evidence for our explanation of 3D preferential sampling in section IV. But first, let us address the question that naturally arises from figure 3a: Is elasticity essential for preferential sampling, given that the chains are stretched out even inside vortices? The physical picture of chains aligning along vortex tubes would hold even for long chains of a fixed length, and so we would expect such inextensible chains to show preferential sampling as well. We test this idea in the next section.
Iii Inextensible chain
In order to describe a ‘macroscopic’ inextensible chain, we disregard Brownian fluctuations in (1) and replace with
(7) 
where still has the meaning of equilibrium length of the springs. is set to a very small value to ensure that the chain is in effect inextensible. Its length is then (in the simulations, we set , which ensures that differs from by at most).
We now consider an inextensible chain having the same length as the maximum length of the extensible chain (with ) considered in § II, i.e. we take . The PDFs of for the two cases are compared in figure 5a. The absence of extensibility is seen to have only a small effect.
The influence of the length of the inextensible chain on the level of preferential sampling is depicted in figure 5b, in terms of the third moment of . Here, we see the same nonmonotonic variation of with , that we saw with for the extensible chain (figure 4a). Moreover, the value of for is very close to that of the extensible chain (horizontal line in panel b), which has a mean length (case of in figure 4b). These results clearly demonstrate that preferential sampling persists even when the chain is not collapsible. Indeed the only role of extensibility is to allow a chain with a short equilibrium length to be stretched by the flow and reach the appropriate length for staying in a vortex; it does not influence the preferentialsampling dynamics otherwise. This finding supports our intuition regarding the mechanism of preferential sampling whereby the chain resides in intense vortical regions by aligning with the axis of vortex tubes. Such a mechanism, indeed, does not rely on the extensibility of the chain, and is equally applicable to inextensible, but long, chains.
Iv Inflexible chain
So far we have considered a chain that does not oppose bending. However, taking the bending stiffness of the chain into account allows us to further understand preferential sampling in 3D turbulence. Specifically, by increasing the stiffness of the chain, we can check whether it is necessary for chains to be able to coil in order to enter vortex tubes. Coiling certainly plays an important role in preferential sampling in 2D turbulence, as demonstrated indirectly in picardo , where the deformability of the chain was controlled by changing while keeping fixed. Here, we directly incorporate the forces arising from bending stiffness into the equations of motion, by assuming that the chain has a bending energy given by gs06
(8) 
where determines the bending stiffness. (Note that our definition of the separation vectors differs from that used in gs06 .) The bending energy generates a force that depends on the angle between two neighbouring links and whose effect is to restore the chain into a rodlike configuration. The form of the force on the th bead is gs06
(9) 
where was defined in (3). The characteristic time associated with this force is , and a dimensionless measure of it is the ‘bending’ Weissenberg number . The chain is inflexible and rodlike for small , while the freelyjointed limit is recovered for large .
With the addition of the bending stiffness, the evolution equations for an inertialess, inextensible chain become:
(10a)  
(10b) 
We begin by assessing the effect of bending stiffness on preferential sampling in two dimensions. As we already appreciate the importance of coiling in 2D picardo , studying the impact of bending stiffness in this case will help us better understand the results in 3D. Moreover, this also provides us with the opportunity to check whether extensibility is crucial for preferential sampling in 2D as suggested in picardo , i.e., whether it is essential for a chain inside a vortex to collapse to a tracer, or if it is sufficient for a long chain to simply coil into the vortex.
Figure 6 presents snapshots of the inextensible chains overlaid on the vorticity field, for a highly flexible (panel a), a moderately flexible (panel b), and an inflexible (panel c) case (see also the movies in Movflex2D ). We see that the highly flexible chain is coiled up by the vortices and stays almost entirely within them. The stiff rodlike chain, however, is unable to coil, and thus while some portions of the chain linger within vortices, the entire chain can never be entrapped by a vortex and preferential sampling weakens considerably. This effect of bending stiffness, which is exactly as anticipated in picardo , is shown quantitatively by the PDFs of in figure 6d (see (5) for the definition of ).
Figure 6d also allows us to address the role of extensibility in 2D, by comparing the PDF of for an extensible, freelyjointed, chain (, studied in picardo ) with the PDF for the inextensible, highly flexible chain (). The extensible chain has a small equilibrium length of and a maximum length , which equals the fixed length of the inextensible chain. We see that while there is a small reduction of the degree of preferential sampling when the chain is unable to collapse to a tracerlike object, the long inextensible chains are still able to occupy vortices effectively (Figure 6a), provided of course that they are flexible. Therefore, extensibility is not a crucial ingredient for preferential sampling, either in 2D or in 3D (shown in section III). However, it does have a more significant effect in 2D because extensible chains can collapse to tracers inside 2D vortices, whereas this is not possible inside axiallystretched 3D vortex tubes.
Returning to the effect of the bending stiffness, we have shown above that in 2D it strongly affects the preferential sampling of vortical regions because chains must coil in order to enter vortices. We now examine the effect of the bending stiffness in three dimensions. Figure 7a shows the PDF of for a long inextensible chain () with different values of , corresponding to a highly flexible (), a moderately flexible (), and a rodlike inflexible () chain. The PDF clearly does not vary with , which is an unequivocal indication that, in three dimensions, the bending stiffness has no effect on preferential sampling. This fact provides further confirmation of our explanation of preferential sampling of vortical regions in 3D turbulence. If the main mechanism is the alignment of the chain with the axis of the tubular structures on which vorticity concentrates, the chain does not need to coil in order to reach and keep that configuration. It is then natural that the bending stiffness does not affect preferential sampling in three dimensions.
All our results, thus far, point to the ability of chains to thread vortex tubes and follow them by aligning with the vortex axis. In contrast, no such mechanism exists for chains to reside inside straining zones which have a complex geometry and evolve nonlocally, as pointed out earlier in section II. Indeed this difference between vortical and straining regions forms the basis for our explanation of preferential sampling. Therefore, it is important to support this claim with some direct evidence. To this end, we denote by the value of at the position of the th bead, . We then consider the mean and standard deviation of over the beads of the chain:
(11) 
with denoting the average over . Figure 7b compares the conditional PDFs of for chains in rotational () and straining () regions, for both flexible and inflexible (inextensible) chains. In both cases, we see that is typically greater when the centre of mass of the chain is in a highstrain region. This finding confirms that it is generally easier for a chain, even if it is rodlike, to reside within vortex tubes than within straining regions.
V Conclusion
We have shown that a long beadspring chain—a simple model for filamentary objects—preferentially samples vortical regions of both 2D and 3D turbulent flows. This behaviour is exhibited by a long inextensible chain, as well as an extensible one that has a small equilibrium length, but which is stretched out to inertialrange scales by the flow. The key difference between the underlying mechanisms in 2D and 3D is revealed by the contrasting behaviour of inflexible chains. In 2D, where the chain must coil in order to be trapped within vortices, bendingstiffness inhibits preferential sampling. In 3D, however, an inflexible chain exhibits the same sampling behaviour as a fullyflexible one, because the chain follows tubular vortices by aligning with their axes, and neither bending nor extensibility is essential for this to occur.
The extent of preferential sampling is much stronger in 2D that in 3D turbulence (compare the difference between the PDFs for a tracer and the chain in figure 6d and figure 7a). This is because 2D coherent vortices have much longer lifetimes than 3D vortex tubes, allowing the chain to spend a much larger fraction of time inside the former. On the other hand, the preferential sampling phenomenon is more robust in 3D, as it persists regardless of the extensibility or flexibility of the chain.
Filamentary objects in turbulent flows are encountered in several physical situations, from fibres in the paper industry, to microplastics and algae in oceanic environments. The understanding of the sampling behaviour of individual chains developed here sets the stage for future studies of the transport, dispersion, and settling of filaments in turbulent suspensions, for which, however, additional physical effects such as intra and interchain hydrodynamic interactions must be considered.
Acknowledgements.
We thank J. Bec for several useful discussions on this topic. DV acknowledges his Associateship with the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India. JRP, SSR, and DV acknowledge the support of the IndoFrench Centre for Applied Mathematics (IFCAM). JRP acknowledges funding from the IITBIRCC seed grant. SSR and RS acknowledge support of the DAE, Govt. of India, under project no. 12R&DTFR5.101100.References
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