Cascades and Reconnection in Interacting Vortex Filaments
Rodolfo Ostilla-Mónico, Ryan McKeown, Michael P. Brenner, Shmuel M. Rubinstein, Alain Pumir
CCascades and Reconnection in Interacting Vortex Filaments
Rodolfo Ostilla-M´onico, Ryan McKeown, Michael P.Brenner, Shmuel M. Rubinstein, and Alain Pumir
3, 4 Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA School of Engineering and Applied Sciences,Harvard University, Cambridge, MA 02138, USA Universit´e de Lyon, ENS de Lyon, Universit´e Claude Bernard,CNRS, Laboratoire de Physique, 69342 Lyon, France Max Planck Institute for Dynamics and Self-Organization, 37077 G¨ottingen, Germany (Dated: February 23, 2021)At high Reynolds number, the interaction between two vortex tubes leads to intensevelocity gradients, which are at the heart of fluid turbulence. This vorticity amplificationcomes about through two different instability mechanisms of the initial vortex tubes, assumedanti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, thetubes destabilize via a Crow instability, with the nonlinear development leading to strongflattening of the cores into thin sheets. These sheets then break down into filaments whichcan repeat the process. At higher Reynolds number, the instability proceeds via the ellipticalinstability, producing vortex tubes that are perpendicular to the original tube directions. Inthis work, we demonstrate that these same transition between Crow and Elliptical instabilityoccurs at moderate Reynolds number when we vary the initial angle β between two straightvortex tubes. We demonstrate that when the angle between the two tubes is close to π/ β betweenthe two tubes, the breakdown mechanism changes to an elliptic cascade-like mechanism.Whereas the interaction of two vortices depends on the initial condition, the rapid formationof fine-scales vortex structures appears to be a robust feature, possibly universal at very highReynolds numbers. I. INTRODUCTION
Many experiments have demonstrated that the interaction of two vortex tubes coming closetogether eventually leads to a change of topology of the vortex lines through a process known asvortex reconnection [1–3]. Vortex reconnection is a fundamental process in fluid mechanics, andit has been postulated to play a significant role in fluid phenomena such as the turbulent energycascade [4], noise generation [5] and the transfer of helicity across topologically distinct vortices[6]. Reconnection is also interesting from a theoretical point of view, as the change of vortex linetopology appears to violate well-known conservation theorems in inviscid flows [7], which impliesthat viscosity must play a decisive role even at extreme Reynolds numbers.The process through which vortex pairs undergo reconnection has been well studied and char-acterized both numerically and theoretically. The early phase of the interaction, before viscosityplays the dominant role, is captured by the Biot-Savart equation, which keeps track of the lo-cation of the vortex tubes by assuming a circular structure of the cores, with a fixed vorticityprofile [8]. The numerical work of [9] showed that the resulting dynamics of a wide range of initialconditions spontaneously leads to the local pairing of antiparallel parts of nearby filaments. As aresult, the cores get close together, which generally leads to very rich dynamics. The Biot-Savartdescription, however, fails when the vortex tubes are so close that they deform each others’ cores,thereby making the initial assumption questionable [10, 11]. Therefore, to adequately capturethe initial dynamics of the interaction, it is necessary to undertake a full simulation of the Euler a r X i v : . [ phy s i c s . f l u - dyn ] F e b equations. The Biot-Savart equation (and the Euler equations) also fail to capture the viscousprocesses, essential to reconnection, which occur at scales much smaller than any other inviscidprocess scale [11].It has been known for a long time that several instability mechanisms may lead to the disruptionof two antiparallel vortex lines. Early studies based on the Biot-Savart equation suggested that thelong-wavelength Crow-instability [12] plays the dominant role in bringing together counter-rotatingparts of the tubes so reconnection can occur [13, 14]. This prompted a number of investigations,see e.g.[5, 15–20], based on numerical solutions of the Euler and Navier-Stokes equations whichimposed the symmetry of the most unstable mode corresponding to the long wave-number Crowinstability, namely that of two tubes with two mirror symmetry in the plane that separates them(denoted as the plane P ). With this symmetry, the components of velocity perpendicular to theplane P is uniformly zero, so if the vorticity component perpendicular to the plane is zero, then,reconnection is impossible in the absence of viscosity. The observation that the time necessary toachieve reconnection, in the limit of very small viscosity, ν , seems in practice to be independent ofviscosity, suggests that the limit ν → Re Γ → ∞ , where Re Γ ,the Reynolds number is defined as Re Γ = Γ /ν , Γ being the circulation of the vortex tubes and ν the kinematic viscosity) is singular. This observation has been interpreted as a signature pointingto the existence of singular solutions of the Euler equations. The study of simplified models, basedon the Biot-Savart equations, has suggested a large amplification of vorticity during the late stagesof reconnection, although the approximations necessary to derive the Biot-Savart model ultimatelybreak down [13, 21, 22]. Furthermore, decades of careful numerical work have not conclusivelyresolved the singularity issue for the corresponding initial value problem. The problem has beenparticularly studied in the inviscid limit, looking for signs of a diverging vorticity [15, 17, 18]. Still,DNS clearly show that very thin vortex sheets are formed on either side of the symmetry plane P ,both in the inviscid and in the viscous problem at large enough Re Γ . The formation of extremelythin vortex sheets, with a relatively slow growth of vorticity makes the problem very difficult tostudy numerically.While the quest to reach increasingly large values of Re Γ continues [23], numerical and exper-imental studies have recently identified another main mechanism in the interaction between twoantiparallel tubes, which leads to the breaking of the vortex tubes instead of to a reconnection [24].This mechanism can be observed in the head-on collision between two vortex rings, which leadsto a very rapid destruction of the vortices at large Reynolds numbers [25]. In this problem, thelong-wavelength Crow instability initially brings parts of the filaments together [26, 27]. However,the further interaction between the counter-rotating tubes, reveals the dominant role of an insta-bility whose wavenumber is comparable with the core size [28, 29]. This instability, known as theelliptic instability [26, 30, 31], involves a symmetry that completely differs from that of the long-wavenumber Crow instability. At sufficiently high Reynolds numbers, elliptic instabilities developon top of each other, leading to a cascade, and eventually, to a transient turbulent flow [24].The marked difference between these two cases leads us to investigate the onset of the mecha-nisms leading to reconnection. In this vein, we take inspiration from the studies of reconnectingmagnetic tubes at an angle in an astrophysical context [32], and from the recent study of vortexreconnection in superfluids, which has revealed the presence of non-universal features by compar-ing different classes of initial conditions [33]. However, we have to highlight the notable differencebetween these two cases, and ours, which arises from much larger degrees of freedom of the vortexcores in hydrodynamics and leads to a much richer phenomenology, not taken into account in thesimplified model of reconnection of skewed vortices in [34]. While it is clear that reconnections arestill present in hydrodynamical fluids with Re Γ (cid:29) b = cot( β/ β is the angle between the filaments. With this choice, b = 1 corresponds to two initiallyperpendicular filaments: β = π/
2. In this study, we consider only values of β ≤ π/ b ≥ β = π/ b = 1) has been studied, originally atmuch smaller Reynolds numbers than the ones considered here and with an additional hyperviscousdissipation term [36], and more recently at a much higher resolution [37].In all cases, we find an energy cascade during the interaction, reaching ever smaller scales as theReynolds number increases. This generation of small scales arises from deformations of the coreswhere the tubes intersect. We find that the interaction starts with the formation of characteristicvortex sheets for 67 . ◦ ≤ β ≤ ◦ (1 ≤ b ≤ / β ≤ . ◦ ( b ≥ β → b → ∞ ) due to the presence of the elliptical instability [24]. II. NUMERICAL PROCEDURES AND DATABASE
We simulate the incompressible Navier-Stokes equations: ∂ t u + ( u · ∇ ) u = − ρ − ∇ p + ν ∇ u (1) ∇ · u = 0 (2)in a triply periodic box, using pseudo-spectral methods. The details of the code have been describedin [38]. We vary the aspect ratio of the domain, which we take to be of size 2 π in the x and y direction, and of size 2 bπ in the z direction, where b is a control parameter that we take as b = 1, 5 /
4, 3 /
2, 2, 5 /
2, 3, and 4. No forcing is added to the Navier-Stokes equations, andthe flow is allowed to evolve from the initial conditions. These consist of two Gaussian vortices,where the initial position of the vortex cores are in two diagonal lines γ ± z = ± bx located at theplane y = ± d/ ω ± ( x , t = 0) = ± Ω exp( − ρ ± / σ )( e x ± b e z ) / √ b , where ρ ± is the distancebetween the point x to the two lines γ ± , and σ is the core radius. The resulting circulations, Γ ± ,are equal to Ω σ . A schematic of the initial condition can be seen in Fig. 1, which shows thatthe iso-contours of vorticity approximately concentrate in two tubes, at an angle of inclination β = 2 arctan(1 /b ).Our calculations are organized in two series of runs. In the first series, we fix the angle by setting b = 1, which results in β = 90 ◦ to each other. This configuration minimizes the strain directionthat triggers the elliptical instability. We then vary the Reynolds number, from Re Γ = 2200to Re Γ = 5400, which is around the Reynolds number range for which the elliptical instabilitysupersedes the Crow instability for antiparallel tubes, to study the genericity of the ellipticalinstability in the most disadvantageous configuration. In the second series of runs, we fix theReynolds number at Re Γ = 4000, and vary b from 1 to 4, which reduces the angle, β , and brings Runs 1 2 3 4 5 6 7 8 9 10 11 12
A/B/Cb / / / ∞ β ◦ ◦ ◦ ◦ ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ ◦ Re Γ N l
256 256 384 384 384 192 240 192 192 192 192 192 ∗ N h
384 512 512 512 512 384 400 320 320 320 320 320 ∗ TABLE I. Simulation parameters for the runs used in this work. The numerical domain is taken as − π ≤ x ≤ π , − π ≤ y ≤ π and − bπ ≤ z ≤ bπ . The Reynolds number is defined as the ratio of the initial circulation,Γ, divided by the kinematic viscosity, ν . Each run was started at low resolution, with N l × N l × bN l Fouriermodes. The runs were also conducted at a higher resolution, with N h × N h × bN h Fourier modes duringthe generation of small-scale flow structures. ∗ For these cases with β = 0 ◦ , the limits in the z direction aretaken as − π ≤ z ≤ π , and the z resolution is accordingly 4 N l or 4 N h . the tubes closer to being antiparallel, progressively amplifying the strain in the direction thatinduces the elliptical instability.In addition, we considered 3 runs with initially antiparallel vortex tubes, in the configurationstudied in [24, 27]. These runs were carried out in a box of aspect ratio 4, although they correspondformally to β = 0, hence b → ∞ . In these runs, we kept the Reynolds number to Re Γ = 4000, andwe slightly modulated the constant x and y -locations of the tubes by a sum of a few Fourier modes.We varied the overall coefficient of the perturbation by multiplying by 2 and 4. The runs werecarried out at low resolution (192 × N l × N l × ( b N l ) (or equivalently, with as many Fourier modes). When the vortex tubes come together, thevelocity field develops very fine scales, or equivalently, the Fourier spectrum extends to much largervalues of the wavenumbers, k . To simulate this phase of the dynamics, we extend the number ofFourier modes to N h × N h × ( N h b ). The parameters of the various simulations are shown in Table I.In the following, all quantities will be expressed in units defined with the box-size and a unitarycirculation. Note that the time scale associated with the inviscid evolution (Biot-Savart model) is ∼ d / Γ, which is of order 1.
Symmetries of the problem:
Although the planes P ( z = 0) and P ( y = 0), as indicated inFig. 1, play a particularly important role in the problem studied here, the velocity and vorticityfields in our simulations do not have any simple symmetry with respect to P or P . In config-urations with a symmetry with respect to P , as it is the case e.g. in [11, 15–20, 39, 40], thecomponent of the velocity field perpendicular to P is equal to 0 in P . As a consequence of thissymmetry, if the component of vorticity perpendicular to P is initially 0 in the symmetry plane P , then, a component of vorticity perpendicular to P cannot be generated without viscosity.Conversely, there is no particular symmetry plane between two vortex tubes undergoing theelliptic instability [24, 27], or in the configuration of two vortex tubes initially at a finite angle.Nonetheless, the fields in the present study are invariant after composing the two mirror symmetrieswith respect to P and to P , or equivalently, by a rotation with respect their intersection, i.e. thestraight line ∆ ( z = y = 0) shown in Fig. 1(a). This corresponds to the following symmetry:( x, y, z ) → ( x, − y, − z ) , ( u x , u y , u z ) → ( u x , − u y , − u z ) and ( ω x , ω y , ω z ) → ( ω x , − ω y , − ω z ) (3) ΓΓ x π − πz − bπ bπy − ππ ∆ P P Γ Γ z bπ − bπy − ππ d Γ Γ z bπ − bπxπ − π β FIG. 1. Schematic of DNS initial configuration. The thick red and blue lines represent the initial positionsof the two vortex filaments, with the arrow indicating the circulation direction. (a) 3D schematic. (b)side view of the yz -plane, which shows the definition of the spacing between the filaments, d . (c) top viewof the xz -plane, which shows the definition of β and b . The vorticity distribution is invariant under thesymmetry resulting from two mirror symmetries with respect to the planes P and P : ( x, y, z ) → ( x, − y, − z );( ω x , ω y , ω z ) → ( ω x , − ω y , − ω z ) and ( u x , u y , u z ) → ( u x , − u y , − u z ). The distance between the two tubes inthe y direction is d = 0 .
9; the circulation is chosen here to be Γ = 1, and the core radius σ = 0 . / √ ≈ . III. RESULTSA. Overview of reconnection
We begin by illustrating the phenomenon of reconnection for initially perpendicular vortextubes by analyzing run 3, with β = 90 ◦ ( b = 1), Re Γ = 4000. The vortices evolve in time fromthe initial horizontal conditions, as they approach one another and begin to deform. Fig. 2(a)shows the vorticity isosurface at t = 15 .
2, before the reconnection event. As the flow evolvesfurther, the vortices reconnect. This results in a changed topology, which can be inferred from thestructure of the vorticity field shown in panel (b) at t = 26 .
4. Vertical vortex structures, however,appear simultaneously with smaller scale, horizontal filaments perpendicular to the main tubes.To better characterize the large scale flow structures, present before and after reconnection, we usethe methods applied by Goto et al. [42], which consists of band-pass filtering the vorticity field.For the purpose of the present work, we found it convenient to isolate the wavenumbers in theband defined by √ k f ≤ k ≤ √ k f , where k f = 2 .
3. This filters out the small-scale featuresclearly seen in panel (a,b), but leaves apparent the change of topology due to the evolution: theoriginally horizontal tubes, parallel to the z axis in Fig.2(c), become parallel to the y axis at later (a) (b)(c) (d) FIG. 2. Vortex topology changes during a reconnection between initially perpendicular vortex tubes ( b = 1and Re Γ = 4000, run 3 in Table I). Panels (a) and (b) show iso-vorticity contours corresponding to ω th = 2 . t = 15 .
2, when the tubes begin to interact, and at t = 26 .
4, after the interaction. These dynamics reveala change of topology of the vortex lines and the generation of small-scale vortices during the reconnection.Panels (c) and (d) show the same vorticity field, band-pass filtered between k < = √ k f and k > = 2 √ k f ,with k f = 2 .
3, which removes the small-scale features of the flow and showcases only the change of topologyof the vortex tubes. The value of the isosurface is 1 in panels (c) and (d). Full videos of the process areavailable as supplementary material [41]. times (see Fig. 2(d)), signaling a topological transition in the vorticity field. Between the two timesshown in panels (a) and (b), we can observe not only changes in the large-scale vortex topology,but also the progressive appearance of small-scale vortical flow structures. At early times, we canobserve the appearance of slender vortex filaments which are perpendicular to the primary vortextubes and contain little circulation. These filaments are a well-known feature of reconnection whichhas already been discussed in previous studies, usually under the name of bridges [43–45]. In oursimulation, they are clearly visible at t = 15 .
2, Fig. 2(a). While the vorticity in the bridges isvery much amplified at early times through vortex stretching, the role that these slender filamentshave in the subsequent interaction of the main tubes during reconnection appears to be limitedbecause they contain little circulation. Conversely, the small scale features clearly visible at latertimes ( t = 26 .
4, Fig. 2(b) appear to be a reproducible feature of the interaction at high Reynoldsnumbers; recent DNS of a pair of vortex tubes, with imposed symmetry with respect to planes P and P (in our terminology), also led to the formation of a similar small scale structures, whichwas interpreted to be the result of a cascade [20].The reconnection process between two vortex tubes initially at an angle β = 90 ◦ , illustrated inFig. 2, exhibits similarities with reconnection of two tubes with symmetric initial conditions [11, 20],as noted e.g. in [36], and explained in more detail below. The dynamics leading to reconnection,however, are not universal. In fact, Fig. 3 shows an overview of the interaction between two tubesinitially at a much shallower angle, β ≈ . ◦ ( b = 4, run 11). Fig. 3(a) shows that the interactionoccurs over two closely paired sections of the vortex tubes. This is clearly illustrated by Figs. 3(c)and (d), which show iso-vorticity contours of the band-pass filtered solution for √ k F ≤ k ≤ √ k F ,with k F = 2 .
3, as shown in Figs. 2(c) and (d). The configuration with β = 28 . ◦ ( b = 4) exhibits adifferent type of dynamics, resulting in a larger portion of the vortices coming closer to each otherthan at β = 90 ◦ ( b = 1). The collision between these vortex tubes leaves behind a tangle of smallervortices, reminiscent of the breakdown that results from the collision of two vortex rings [24, 27].This points to a dependence on the initial orientation angle, which is examined further in thefollowing sections. B. Evolution of two nearly perpendicular tubes: sheet formation.
Early stage and sheet formation
The reconnection process starts with the pairing of the tubes,which locally aligns the vortices in an antiparallel manner, a feature clearly observed directly fromthe Biot-Savart equation [9], and consistent with all previous numerical observations. The localpairing of antiparallel filaments is accompanied by a significant deformation of the vortex tubes.Fig. 4(a-c) shows iso-surfaces of the vorticity magnitude for β = 90 ◦ and Re Γ = 4000 to illustratethis interaction. The three views from a perspective similar to that shown in Fig. 1(a), at t = 17(Fig. 4(a)), t = 19 (Fig. 4(b)) and t = 20 (Fig. 4(c)) indicate that the nearest regions of the tubescome together and flatten into thin vortex sheets. It is important to notice that the spatial extentof the vortex sheets, in the direction of the vortex tubes, is in fact rather limited. The vortexsheets are confined in the z -direction to a size smaller than that of the initial vortex cores. We alsostress that the sheets do not appear to perfectly align with the P ( y = 0) plane, as it happens inthe canonical problem of two initially weakly perturbed antiparallel vortex tubes, symmetric withrespect to the midplane. In fact, the tilt of the sheets increases from t = 17 to t = 20.The pairing shown in Fig. 4(a-c) with the formation of vortex sheets in the regions wherethe vortices interact, is qualitatively consistent with the simulations of [36]. As already stated, theformation of sheets is a robust feature in many simulations of interacting vortex tubes, starting withan initial configuration of almost parallel counter-rotating tubes with a slight perturbation [40].Further insight on these vortex sheets is provided by Fig. 4(d), which shows a magnified view at t = 20, from a slightly different perspective, showcasing the pronounced flattening of the vortexcores. As shown in Fig. 4(e-f), the isocontours of the z component of vorticity at the central plane, z = 0 (the plane P , as introduced in Fig. 1(a)) and at an adjacent plane parallel to P which isslightly off the symmetry plane, further indicate the formation of intense, thin vortex sheets. Upto the time shown in Fig. 4(d), the flattening of the sheets is not greatly affected by increasing theReynolds number. We view this as evidence that the formation of the narrow vortex sheets is onlya precursor of reconnection.We notice that Fig. 4(a-c) also demonstrates that the vortex filament “bridges”, clearly visible (a) (b)(c) (d) FIG. 3. Vortex topology changes during a reconnection between tubes at an acute angle ( β ≈ . ◦ , b = 4 and Re Γ = 4000, run 11 in Table I). Panels (a) and (b) show iso-vorticity contours correspondingto ω th = 2 .
35 at t = 16, when the tubes are paired, and at t = 25 .
5, after the interaction, respectively.These panels show the generation of fine-scale vortices, similar to the collision of two antiparallel vortices[24]. Panels (c) and (d) show the band-passed vorticity field, between the wavenumbers k < = √ k f and k > = 2 √ k f ( where k f = 2 . t = 25 .
5, and the tubes reconnect at the edges ofthe interaction zone. The value of the isosurface in panels (c) and (d) is 0 .
85. Full videos of the process areavailable as supplementary material [41]. at the earlier stages of the interaction when the tubes are drawn closer together, (Fig. 2(a)), are stillvisible at t = 17 (Fig. 4(a)). As the vortex tubes begin to flatten into sheets at t = 19 (Fig. 4(b)),these bridges become less pronounced and are no longer present at t = 20 (Fig. 4(c)). Part of thereason why the bridges are less visible at later times is the increase in the vorticity threshold, ω th used at the three different times. The vorticity magnitude increases locally at the reconnection sitewhere the cores become locally flattened into sheets. In fact, the bridges are concentrated in verynarrow regions of space; this implies that large velocity gradients are generated, but that viscosity (a) (b) (c)(d) β =90 o ; Re Γ =4000t=20; z=0 x -0.6-0.4-0.200.20.4 y (e) β =90 o ; Re Γ =4000T=20; z=0.235 x -0.6-0.4-0.200.20.4 y -15-10-5051015 (f) FIG. 4. Close-up view of sheet formation near the reconnection event of initially perpendicular vortex tubes(run 3, β = 90 ◦ , b = 1, Re Γ = 4000). Iso-vorticity contours of the flow, shown at t = 17 (panel (a), ω th = 3 . t = 19 (panel (b), ω th = 3 .
8) and t = 20 (panel (c), ω th = 4 . t = 20, ω th = 4 . P at z = 0 (panel e) and at z ≈ .
15 (panel f). acts very strongly to dissipate them. For these reasons, as already stated, the bridges do not playany appreciable role in the reconnection dynamics and are immaterial for the present discussion.
Late stage and reconnection
The flattening of the cores into sheets is the precursor of thereconnection process. Up until the latest time, shown in Fig. 4(c), the vortex lines are not broken;they are brought together and compressed into a narrow region. The change of topology of thevortex lines, clearly illustrated in Fig. 2, occurs at a later stage through the destruction of thevortex sheets. We stress that this process is strongly constrained by the symmetry imposed,as in the previously studied case of the two initially weakly perturbed, antiparallel vortex tubes(c.f. [20]). With our initial conditions, as previously noted, the sheets do not particularly align withany plane. In fact, as shown in Fig. 5, the vortex sheets strongly deform in a fully 3-dimensionalmanner at later times. The vortex sheets begin to twist around each other at t = 21 . P ( y = 0) plane, become almost verticalalong the P plane ( z = 0), as shown in Fig. 5(b). The continued twisting of the vortex sheetscauses them to become locally folded along both sides of the P plane, leading to the formationof transverse vortex filaments. Shortly afterward, at t = 23 .
2, shown in Fig. 5(c), the main sheetsin the P plane are anihilated, leaving behind a complicated tangle of small-scale vortices. The0 (a) (b)(c) (d) FIG. 5. Twisting of the vortex sheets and reconnection. Vorticity magnitude isosurface for initially per-pendicular vortex tubes in run 3 ( β = 90 ◦ , Re Γ = 4000) at three consecutive times, following the vortexsheet formation. (a) After the vortex cores flatten into sheets, they become twisted as they wrap aroundeach other. (b) The vortex sheets become folded and reorient along the z = 0 ( P ) plane, forming an arrayof transverse vortex filaments. (c) The two strong vortex sheets annihilate, leaving a tangle of small-scalevortex filaments. The last time shown, t = 23 . t = 21 . ω th = 5 . t = 22 . ω th = 5 . t = 23 . ω th = 6 . P ( z = 0) separates the box shown in the middle. results of Fig. 5 therefore show that once the vortices pair off and begin to interact, the evolutionof the reconnection dynamics differs significantly from those obtained with a much more symmetricinitial condition, such as [20] which cannot account for the twisting of the sheets. This differencemay affect the formation of small-scale vortices, which form at later times, as shown in Fig. 2(b). Evolution of global quantities
A quantitative measure of the production of small scales duringthe reconnection process is shown in Fig. 6 for the interaction of initially perpendicular vortextubes at several Reynolds numbers. Over the entire period of the simulation, the mean kineticenergy rate, shown in Fig. 6(a), decays by less than 20%, despite the rapid increase in the mean1 t h u i / × -3 Re Γ =2220Re Γ =3330Re Γ =4000Re Γ =4550Re Γ =5000 (a) t ν h ω i × -5 (b) t h ω i / (c) FIG. 6. Global dynamics for initially perpendicular vortex tubes at various Reynolds numbers. The evolutionof (a) the mean kinetic energy rate, (cid:104) u (cid:105) /
2, (b) the mean dissipation rate, ν (cid:104) ω (cid:105) , and (c) the mean 6 th moment of the vorticity, (cid:104) ω (cid:105) / , for runs 1-5, at fixed β = 90 ◦ and increasing values of Re Γ . The verticaldashed line in (b) and (c) corresponds to t = 23 .
2, which is the last time shown in Fig. 5(c), and is also veryclose to the peak dissipation rate. The dashed (respectively full) lines correspond to runs at low (respectivelyhigh) resolutions. dissipation rate, shown in Fig. 6(b). For each run, the initial mean dissipation rate decreases witha scaling of ∼ /Re Γ as the Reynolds number increases. It reaches a peak value at a time that isessentially independent of Re Γ . We note that the peak dissipation time t peak will approximatelycoincide with the time the small-scale vortices are most energetic, so we can use this time to studythe resulting small-scale structure.Notably, the height of the peak in the dissipation rate does not vary significantly as a function ofthe Reynolds number. Further information on the generation of motion at small scales is providedby higher moments of the vorticity distribution. Specifically, Fig. 6(c) shows the 6 th moment ofthe vorticity, taken to power 1 /
3. The 6 th moment is defined as: (cid:104) ω n (cid:105) = 1 V (cid:90) V d x ( ω ) n where n = 3 (4)We chose to show a moment of finite order of the vorticity distribution, rather than the maximumof the vorticity, which corresponds to the limit n → ∞ , as the latter is far more sensitive to finiteresolution effects.Since all runs share the same initial condition, the plots of (cid:104) ω (cid:105) / all start at the same initialvalue at 0. The peaks of the curves reach increasingly higher values with increasing Re Γ ; for the Re Γ = 5000 case the maximum is approximately 35 times greater than the initial value. Thistrend reflects the strong amplification of vorticity that occurs during reconnection as the corescontact and break down to fine scales. Similar results are obtained with values of other momentsof (cid:104) ω n (cid:105) /n , with n = 2 and 4. As expected, the peak amplification for these moments grows withthe order n . Note that on all plots in Fig. 6, the results of the runs at low resolution (with N l Fourier modes) are shown as a dashed lines, whereas the runs at higher resolution (with N h modes, N l and N h both given in Table I) are represented by solid lines. The deviations between the tworesolutions are small in the peak regions, even at the highest Reynolds number considered here.This gives us confidence in our numerical results. However, at Re Γ = 5000, the values of (cid:104) ω (cid:105) / at different resolutions diverge at later times (for t (cid:38) t h u i / Re Γ =4000 β =90 o β =77.3 o β =67.4 o (a) t ν h ω i × -4 Re Γ =4000 β =90 o β =77.3 o β =67.4 o (b) t h ω i / Re Γ =4000 β =90 o β =77.3 o β =67.4 o (c) FIG. 7. Global dynamics for vortex tubes at varying initial orientation angles. The evolution of (a) themean kinetic energy rate, (cid:104) u (cid:105) /
2, (b) the mean dissipation rate, ν (cid:104) ω (cid:105) , and (c) the mean 6 th moment of thevorticity, (cid:104) ω (cid:105) / , for runs 3 ( β = 90 ◦ , b = 1), 6 ( β ≈ . ◦ , b = 5 / β ≈ . ◦ , b = 3 / Re Γ = 4000. The local pairing of the vortex tubes leads, in all of these cases, to the formation of vortexsheets. Evolution at β (cid:38) . ◦ ( b ≤ / ) The evolution of the global quantities for runs 3, 6 and 7, allcorresponding to Re Γ = 4000, and 67 . ◦ ≤ β ≤ ◦ (1 ≤ b ≤ /
2) is shown in Fig. 7. The initialvalue of the mean kinetic energy slightly increases with b , as shown in Fig. 7(a), and only decaysby about 20% throughout the whole run, as it was the case at β = 90 ◦ (compare with Fig. 6). Themain differences between the runs is indicated by the evolution of the mean dissipation rate and of (cid:104) ω (cid:105) / , as shown in Fig. 7(b-c). Namely, as β decreases, the time required for these plots to reachtheir respective maxima also decreases. Visualization studies, comparable to what has been donein the case β = 90 ◦ , show that the peaks in Fig. 7(b-c) correspond to the time at which the vorticesreconnect and the tubes change topology. Notably, the reconnection dynamics look comparablewhen β = 90 ◦ and β ≈ . ◦ ( b = 5 / β decreases, the vortex tubes are initially closer to being antiparallel, and it therefore takesless time for them to locally align in an antiparallel manner and initiate the reconnection process.As explained earlier, this early phase can be captured with the Biot-Savart dynamics.Contrary to the peaks of (cid:104) ω (cid:105) / shown in Fig. 7(c), which are approximately constant, thepeak energy dissipation rate is approximately twice as large for the configuration where β ≈ . ◦ ( b = 3 /
2) than for β = 90 ◦ ( b = 1) or β ≈ . ◦ ( b = 5 / β , which will be discussed in the following subsection. C. Evolution at β ≤ . ◦ ( b ≥ ): short wavelength instability As the initial condition is varied to increase the alignment of the tubes (i.e. β → β ≈ . ◦ ( b = 4) and Re Γ = 4000. Becausethe angle between the two tubes is much smaller than the initially perpendicular case ( b = 1),the two tubes align, overlap, and interact over a significantly larger extent which is much largerthan the initial vortex core size. This is clearly visible in the left column of Fig. 8, as the extent3 (a) (b) (c)(d) (e) (f) FIG. 8. Interaction and breakdown of nearly antiparallel vortex tubes. The evolution of the vorticitymagnitude isosurface for run 11 ( β ≈ . ◦ , b = 4, Re Γ = 4000). Only a cubic subdomain, [ − . , . ,surrounding the regions where the vortices interact is shown. The isosurfaces of the vorticity field are shownat t = 15 . t = 20 . t = 22 . ω thr = 2 . t = 15 . . t = 20 .
4) and 3 . t = 22 . of the vortices in the z -direction is much larger than in the x - and y - directions. In fact, as theflow evolves from t = 15 . t = 20 . t (cid:38) . t = 20 . x, z )-plane and separated by a distance d in the y -direction, see Fig. 1. Forsmall enough values of β , as the flow evolves, the tubes move primarily in the y - and x -directionsas the vortex axis is almost parallel to the z direction. At each value of z along the axis of thetubes, we separate the y -domain into two subdomains, D ± , corresponding to the the two tubes, asclearly visible from the front view at T = 6 .
13 in Fig. 8. In practice, this is done by computing theintegral of ω over x : ζ ( y, z ) = (cid:82) dx (cid:48) ω ( x (cid:48) , y, z ) and by identifying, at each position z , the valueof y that separates the upper and lower part of the tube. We then determined the x -location ofthe centroids at each value of z by computing the moments x ± = (cid:0)(cid:82) D ± dxdy ω x (cid:1) / (cid:0)(cid:82) D ± dxdy ω (cid:1) ,with a similar definition for y ± . We note that this way of defining the location of the centerlines4 -0.6-0.4-0.200.20.4 z -0.200.20.40.60.8 x s h i f t β =28.1 o t=15t=17t=18t=19 (a) -0.6-0.4-0.200.20.4 z -1-0.500.51 y s h i f t β =28.1 o t=15t=17t=18t=19 (b) FIG. 9. Vortex centerline trajectories for nearly antiparallel vortex tubes for run 11 ( β ≈ . ◦ , b = 4, Re Γ = 4000). (a) Top view and (b) Side view. The solid lines correspond to the upper vortex and thedashed line correspond to the lower vortex. The centerlines were extracted by computing the moments of ω x , y and z . in a portion of the domain, including the central region where the vortices interact. fails as the two tubes begin to interpenetrate, as shown at t = 20 . β = 0), as found in [24]. This can be easily understood, given the relatively small size of thecores in interaction and the constraints on either side of the region of interaction. Nonetheless, thesmall distance between the cores, and the latest stage of the development shown in Fig. 9, whereperpendicular filaments are formed, suggests the prevalence as time progresses of a symmetry thatcorresponds more to the elliptic instability, than to the Crow instability, reminiscent of what wasobserved in [24, 27].The results shown in this subsection contrast sharply with those shown in Section III B for β (cid:38) . ◦ ( b < / β (cid:38) . ◦ , ( b < / β (cid:46) . ◦ , ( b ≥ β ≈ . ◦ , b = 3 /
2) does lead to thedevelopment of sheets, but the interaction mechanism ultimately differs from those shown in Fig. 4and in Fig. 8 because even if small-scale perpendicular filaments arise they come in small numbersand do not interact with each other significantly.Fig.10 shows the time-dependence of the kinetic energy of the runs (panel a), the dissipationrate (b), and the 6 th moment of vorticity, (cid:104) ω (cid:105) / for runs 8-11, where β ≤ . ◦ ( b ≥
2) and Re Γ = 4000. We have indicated by a cross in Fig. 10(b-c) the latest time corresponding to thevisualization in Fig. 8(c), which approximately coincides with the peak dissipation rate. As was thecase for the runs at β (cid:38) . ◦ ( b ≤ / t peak , varieswith β . Fig. 10b shows that t peak increases when β decreses. It should be kept in mind that thetime at which the interaction occurs is a consequence of the pairing process, which depends on the5 t h u i / Re Γ =4000 β =53.1 o β =43.6 o β =36.8 o β =28.1 o (a) t ν h ω i × -4 Re Γ =4000 (b) t h ω i / Re Γ =4000 (c) FIG. 10. Global dynamics for vortex tubes initially oriented at shallow angles. The evolution of (a) themean kinetic energy rate, (cid:104) u (cid:105) /
2, (b) the mean dissipation rate, ν (cid:104) ω (cid:105) , and (c) the mean 6 th moment ofthe vorticity, (cid:104) ω (cid:105) / , for runs 8 ( β ≈ . ◦ , b = 2), 9 ( β ≈ . ◦ , b = 5 / β ≈ . ◦ , b = 3), and 11( β ≈ . ◦ , b = 4), at fixed Re Γ = 4000. precise geometry of the problem, and more specifically, on the angle β between the vortex tubes.As the tubes become more parallel, it takes a longer time for the interaction between the tubesto initiate. In the limit of perfectly antiparallel filaments, this time becomes the time necessaryfor instabilities to grow, as observed in [24]. It depends on the amount of noise initially, and itcan be much longer than the time t ≈
25 for run 11 ( β ≈ . ◦ , b = 4). In fact, we checked thatthe triggering of the elliptic instability, leading to the strong interaction between two antiparallelvortex tubes, is delayed when decreasing the amplitude of the noise added to the solution. Thisis consistent with the intuitive notion that the interaction leading to turbulence starts with anexponential growth of a small perturbation of the two initially antiparallel vortex tubes.Interestingly, we also notice that the value of the peak dissipation rate tends to decrease when β decreases, for β (cid:46) . ◦ ( b (cid:38) / th moment, seeFig. 10(c) as well as for the fourth and 8 th moments (not shown). Contrary to runs at highervalues of β , we observe a stronger difference between the runs at low resolution (with N l Fouriermodes, shown as dashed lines), and the runs at a higher resolution (with N h Fourier modes, shownas full lines). This indicates stronger resolution requirements for these runs.
D. Discussion
Whereas the interaction between vortex tubes always leads to reconnection i.e. to a changeof topology of the vortex lines, the mechanisms involved when the initial conditions are close toanti-parallel ( β (cid:46) . ◦ or b > /
2, see subsection III C), appear to qualitatively differ from whatis observed when the vortices are closer to being perpendicular ( β (cid:38) . ◦ or b < /
2) as discussedin subsection III B. The dynamics observed in the former case are very reminiscent of what wasobserved during the interaction of two initially antiparallel vortex tubes [24, 27]. The qualitativesimilarity between reconnection when b = 4, occurring through the annihilation of a large fractionof the two locally antiparallel tubes, clearly shown in Fig. 3 and 8 and the dynamics resulting fromthe collision between two vortex rings [24] is an important aspect of our work.This configuration of two antiparallel tubes corresponds formally to β → ◦ ( b → ∞ ). In fact,the behaviors of the mean kinetic energy, dissipation rate, and 6 th moment of vorticity in runswith initially weakly perturbed antiparallel vortex tubes, see Fig. 11, are very comparable to thatshown for β = 28 . ◦ . The main difference is that the time at which the violent interaction leadsto the breakdown of the vortex tubes and generation of fine-scale flow structures begins at later6 t h u i / × -3 noise= φ noise= φ /2noise= φ /4 (a) t ν h ω i × -5 noise= φ noise= φ /2noise= φ /4 (b) t h ω i / noise= φ noise= φ /2noise= φ /4 (c) FIG. 11. Global dynamics for initially antiparallel vortex tubes with various noise levels. The evolutionof (a) the mean kinetic energy rate, (cid:104) u (cid:105) /
2, (b) the mean dissipation rate, ν (cid:104) ω (cid:105) , and (c) the mean 6 th moment of the vorticity, (cid:104) ω (cid:105) / , for run 12 with varying noise amplitudes where β = 0 ◦ and Re Γ = 4000. times, compared to what is seen in Fig. 10(b-c). Furthermore, the time at which the interactionoccurs depends on the level of noise. This can be clearly seen in Fig. 11(b-c), which compares 3simulations with three different noise levels, obtained from a weak level (upward triangle symbols),half its value (left pointing triangles) and a quarter of its value (downward triangles). The timewhere the dissipation rate peaks clearly depends on the destabilization of the initial noise amplitude:we observe a logarithmic dependence of this time, consistent with the intuitive notion that the firststage of the interaction comes from the exponential growth of an unstable perturbation, throughthe elliptic instability.This signals a clear qualitative difference between this configuration, where the interactionbetween the tubes leads to the annihilation of increasingly larger parts of overlapping tubes, andthat obtained for β > . ◦ ( b < / β , reconnection is overtaken by the mutual annihilation of the two tubes throughthe elliptical instability. This is because the elliptical instability requires the strain to be alignedalong the vortex core to begin to act (hence it barely acts for β = 90 ◦ ), but its growth rate ismuch larger than that of the Crow instability that leads to reconnection [24]. Even if the overalldynamics preceding reconnection appear to depend on the initial condition, the late stage of theinteraction leads at high enough Reynolds numbers in all cases studied, to an intense generationof small-scales, plausibly through a cascade as demonstrated in the case of parallel tubes in [24].It is tempting to postulate that this cascade through the generation of perpendicular, small-scalevortices, may in fact be universal [19, 42], independently of the initial conditions. IV. SUMMARY AND CONCLUSION
In this work, we have investigated the interaction between two initially straight, counterrotatingvortex tubes oriented at an angle β . We systematically varied β , hence the geometry of the initialflow configuration, and let the flow evolve. In all cases, we observe a change in topology of thevortex lines and the production of perpendicular small-scale vortices. The main result of our studyis that the dynamics which result in this outcome depend strongly on the initial orientation of theinteracting tubes.When the tubes are initially almost perpendicular to each other ( β ≈ π/ β ≈ π/
2, thesmall scales form after the sheets have annihilated, as observed in [20]. When the tubes are betteraligned in an antiparallel manner, the formation of small-scale vortex structures via a cascadeoccurs as soon as the vortices come together [24]. The idea that iterative mechanisms may lead toformation of a cascade has been suggested for theoretical reasons [19, 47]. Strong evidence for acascade driven by a hydrodynamic instability, namely the elliptic instability, has been presented inthe interaction of two vortex tubes. An interesting question for future work will be to understandwhether the mechanism leading to the proliferation of small-scale vortices is universal, based onthe physical mechanisms discussed in [24]. The possibility that such a cascade scenario may leadto a singularity, as already postulated [19, 47, 48] also deserves further attention.
Acknowledgments:
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