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The Japan Society of Fluid Mechanics

Fluid Dynamics Research doi:10.1088/0169-5983/46/3/031415

Fluid Dyn. Res. 46 (2014) 031415 (16pp)

The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid
Alexey V Borisov1,2, Alexander A Kilin1,2, Ivan S Mamaev1,2 and Valentin A Tenenev
1 2

1,3

Udmurt State University, Universitetskaya 1, Izhevsk 426034, Russia Institute of Mathematics and Mechanics of the Ural Branch of RAS, S Kovalevskaja str. 16, Ekaterinburg 620990, Russia 3 Kalashnikov Izhevsk State Technical University, 30 let Pobedy 2, Izhevsk 426069, Russia E-mail: borisov@rcd.ru, aka@rcd.ru, mamaev@rcd.ru and tenenev@istu.ru Received 20 July 2013, revised 14 April 2014 Accepted for publication 17 April 2014 Published 27 May 2014 Communicated by Y Fukumoto Abstract

We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible and viscous fluid. Using the numerical simulation of the NavierStokes equations, we confirm the existence of leapfrogging of three equal vortex rings and suggest the possibility of detecting it experimentally. We also confirm the existence of leapfrogging of two vortex rings with opposite-signed vorticities in a viscous fluid. (Some figures may appear in colour only in the online journal) 1. Introduction In this paper we address a number of issues involving the classical problem of motion of symmetric vortex rings in an ideal fluid. The problem of motion of two vortex rings was first considered by Helmholtz (1858) in his famous paper, which laid the foundations for the whole vortex dynamics. In particular, Helmholtz qualitatively described a particularly interesting effect, namely, leapfrogging motion of vortex rings, i.e., the motion where the rings move along a common symmetry axis, alternately passing through each other. However, he did not present a theoretical analysis of this problem. Hicks (1883, 1885, 1899, 1922) and Dyson (1893a, 1893b) explored, in more or less detail, the dynamics of this system and pointed out a few particular conditions for the onset of leapfrogging. Later Chaplygin also tried to explain the phenomenon of leapfrogging of vortex rings in his remarks to the Russian
0169-5983/14/031415+16$33.00 © 2014 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK 1

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Figure 1. A vortex ring of radius R with a small circular cross-section of radius a.

translation of Helmholtzs paper and presented his own analysis of this problem (Chaplygin 1902). The general problem of description of the motion of two axisymmetric vortices and, in particular, their leapfrogging motion, was considered in several publications (see Boyarintsev et al 1985, Brutyan and Krapivskii 1984, Gurzhii et al 1988, Lamb 1932). A more detailed list of references and historical comments can be found in Meleshko (2010). It should be noted that the results of the above-mentioned papers concerning the analysis of this system are incomplete in spite of its integrability. In our previous paper (Borisov et al 2013) we made a complete qualitative analysis of the dynamics of a system of two rings, based on the topological approach and the concept of a bifurcation complex (Bolsinov et al 2010), and obtained a complete description of the motion for different initial conditions and intensities of vortex rings. In addition to the classical leapfrogging, we point out some more families of periodic solutions to the reduced system, one of which corresponds to quite exotic motions called by us pseudo-leapfrogging. In the same paper we showed that in the case of three vortex rings there also exists leapfrogging motion in a wide range of values of the first integrals of the system. In this work we present results of numerical simulation of the systems of two and three vortex rings using the NavierStokes equations for a viscous fluid. The calculations confirm the existence of leapfrogging of three vortex rings in a viscous fluid and suggest the possibility of detecting it experimentally. We note that the system of two vortex rings was studied in detail both experimentally (Maxworthy 1977, Oshima et al 1975, Satti and Peng 2013) and by simulating the NavierStokes equations (Riley and Stevens 1993, Riley and Weidman 1993). At the same time we have not found any papers devoted to experiments and simulations of three vortex rings.

2. Equations of motion for vortex rings
2.1. Equations of motion

To describe the interaction of vortex rings in an ideal fluid, it is common to use the model of thin toroidal vortex rings. In this model each ring is a regular torus with a circular crosssection of radius ai whose center is at distance Ri from the symmetry axis (see figure 1), where ai Ri . The vorticity is taken to be zero everywhere outside the torus, while inside the torus
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the vorticity distribution depends only on the distance s to the center of the cross-section of the torus = ( s ) (figure 1). There exist several models of thin vortex rings (e.g., a homogeneous ring, a hollow vortex ring etc), which differ from each other by the distribution of vorticity ( s ). All of them are characterized by an approximately constant velocity of self-propulsion along the axis Oz without noticeable changes in form (Lamb 1932). Without dwelling on each of them, we write down the general formula for the velocity of self-propulsion of thin toroidal vortex rings, which was obtained by Saffman (1970):

8R 1 ln - + , Vz( 0) = 4R a 2

(1)

where depends on the vorticity distribution ( s ) inside the ring. In the case of a hollow vortex ring, the vorticity inside the ring is absent, therefore, = 0 and (1) coincides with the Hicks formula (Hicks 1883, 1885). In the case of uniform vorticity distribution inside the ring, = 1 and (1) coincides with the formula for the velocity obtained by Lord Kelvin 4 (Thomson 1867). The most complete results have been obtained for the dynamics of homogeneous vortex rings. Therefore, throughout this paper we shall confine ourselves to this case by setting = 1 . 4 When several rings interact in the process of motion, their radii and thicknesses become variable, however (by the Helmholtz theorem of circulation and incompressibility of a fluid), the following relation is satisfied for each ring

Rai2 = Bi2 = const , i

(2)

where Bi are the constants characterizing the volumes of the rings. We shall assume that for the whole time of motion the vortices remain thin ( ai Ri ) and are far enough away from each other not to affect the toroidicity of the form and the vorticity distribution inside the rings. We shall not dwell here on a detailed derivation of the equations of motion, which can be found in Akhmetov (2009), Borisov et al (2013), Cheremnykh (2008), Lamb (1932), Saffman (1992), and Vasilev (1913). The equations of motion of vortex rings can be represented in Hamiltonian form

1 H Z i = { Z i , H} = , Ri Ri i

1 H Ri = { Ri , H} = - , Ri Zi i

(3)

where the Poisson bracket of coordinates Zi , Ri satisfies the relations

{

Zi, R

j

}

=

ij R i
i

,

{

Zi, Z

j

}={

Ri , R

j

}

= 0,

and the Hamiltonian can be written as

H=

1 2

N

i=1

8R i 3 i2Ri ln Bi

2

-

7 + 4

ij

jG Zi, Ri ; Zj, Rj , i

(

)

(4)

3

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where

rr 2 2 - k K ( k ) - E ( k ) , G ( z, r, z , r ) = 2 k k

1 2 4rr (5) k= , 2 2 (z - z) + (r + r)

and K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively. The equations of motion for vortex rings in Hamiltonian form were first presented in Vasilev (1913).
2.2. Integrals of motion and reduction in symmetry

In the general case, equation (3) possesses, in addition to the Hamiltonian, another first integral of motion
N

P=

i=1

R i2 , i

( 6)

which is the absolute value of the momentum of the vortex flow (Saffman 1992). This integral can be used to reduce the system by one degree of freedom. The reduction can be performed by various means. Here we describe the method similar to the Jacobi method, used in Borisov et al (2005b) for the reduction of the problems of N N point vortices on a plane and a sphere. We confine ourselves to the case i = 1i 0 . To do this, we determine the center of vorticity of the entire system of rings on the axis Oz
N

0 =

i=1

Zi i

N

i=1

-1 , i
-1 j ,
i

(7)

and instead of the coordinates Zi and Ri we introduce the coordinates
i

i = Z

i+1

-

j=1

jZj

i

j=1

i = 1...N - 1 ,
1

i = R

i 2 i+1

-

j=1

jR
2 j

j=1

- j

i

i +1

j=1

j

i+1

j=1

-1 j .

(8)

For brevity we denote Z = ( Z1, ... , ZN ), R = ( R1, ... , RN ), = ( 1, ... , N - 1), = ( 1, ... , N - 1). It can be shown by direct calculation that the transformation

(

Z, R) ( , 0, , P ) is reversible and the following proposition holds.

Proposition 1. The equations of motion of the reduced system have the canonical form

H , i = i

i = -

H , i

i = 1...N - 1 ,

(9)

where the Hamiltonian H = H ( , , P ) depends parametrically on the value of the first integral (6). For a solution of the reduced system ( t ), ( t ), the dynamics of a full system is restored by means of an additional quadrature
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H . 0 = P

(10)

The reduced system (9) actually describes the motion in a coordinate system related to the center of vorticity. This suggests an interesting geometric interpretation of absolute motion for periodic solutions to the reduced system (9). This interpretation is based on the concept of choreography, which was introduced by Moore (1993), Chenciner and Montgomery (2000), SimС (2002) for the N-body problem of celestial mechanics (for the case of point vortices on a plane, choreographies are pointed out in Borisov et al (2005a).
Proposition 2. Let ( t), ( t) be a periodic solution to the reduced system (9) with period

T. Then, in a moving coordinate system related to the vorticity center (7), the vortex rings move along closed (generally different) trajectories with period T. Thus, periodic solutions to the reduced system correspond to relative choreographies of vortex rings (Borisov et al 2005a, SimС 2002).

(

)

3. Two vortex rings
3.1. Leapfrogging and the problem of classifying motions

The problem of motion of two vortex rings was first investigated by Helmholtz (1858), where he pointed to the existence of the phenomenon of leapfrogging motion of vortex rings. Later, this problem was studied in more detail by Hicks (1922), GrЖbli (1877), Dyson (1893), Gurzhii et al (1988) and others (further references can be found in Meleshko (2010)). In our previous paper (Borisov et al 2013) we provided a sufficiently detailed classification for motions of two vortex rings depending on the values of the first integrals of the system and gave a detailed analysis of conditions for the onset of leapfrogging. Below we present the main results on the classification of possible types of motion of two rings, which we obtained when preparing this paper. (1) Repulsion of rings. At t = - the rings are infinitely far away from each other and, as time increases, move towards each other (in the system of the center of vorticity). After approaching each other up to a minimum distance, the rings repulse each other and fly apart again. A monotone decrease of the radius of one ring and increase of the radius of the other ring occur during the whole motion. The larger P, the less the radii of the rings change. This type of motion is possible only for the case of rings which have circulation of the same sign. (2) Single passage of one ring through the other. In this case, after approaching each other the rings do not repulse but pass through each other alternately and then fly apart to infinity. This scenario of motion occurs at all values of vortex intensities and the integral P. We note that before the interaction (t -) the dimensions of each of the rings are the same as after the interaction (t +); nevertheless, during the interaction they can change depending on the initial conditions. Thus, for example, motions are possible for which the larger ring first becomes a smaller one, passes through the second ring and then becomes a larger one again.
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Figure 2. Trajectories of vortex rings on the plane ( Z , R ) in a system of the center of vorticity (on the left) and in a moving space (on the right) for = 2 = -0.5 and for 1 various values of P: (a) P = 0.001, capture of a negative vortex ring by a positive one, (b) P = 5, capture of a positive vortex ring by a negative one, (c) P = 0.02 and E < Ea , leapfrogging, (d) P = 0.02 and Ea < E < Ec , leapfrogging. The critical values of energy Ea and Ec correspond to a change of the type of vortex motion. For their calculation and the description of an appropriate classification of motions, see Borisov et al (2013).

(3) Leapfrogging. In the problem of vortex rings, nonuniform motion in which the rings pass through each other alternately is called leapfrogging. This type of motion is of particular interest from the viewpoint of dynamics and is possible in the case of vortices with circulations of the same sign and in the case of circulations of opposite signs under the condition that the integral P is positive. For both cases the condition for the existence of leapfrogging depending on the values of the first integrals and the initial conditions is presented in Borisov et al (2013). (4) Oscillations with propulsion. This is another type of periodic constrained relative motion of two vortex rings. This motion is possible only for rings of opposite-signed vorticity and arises in a neighborhood of equilibrium positions corresponding to the ring configurations at which they are located in each other and move as a unit along the axis with constant velocity. This equilibrium position corresponds to the uniform motion of a vortex pair described in Riley and Weidman (1993). And the fact of existence of periodic motions close to it is pointed out in Gurzhii et al (1988). These solutions are motions in which the larger ring moves along the axis with some nonconstant velocity, and the smaller ring oscillates inside the larger one, alternately overtaking and lagging behind it. In this case the larger ring can be said to capture the smaller one. We also note that in some cases the closest approach of the rings occurs with the maximal velocity. As a result, during fast motion the impression arises that the rings pass through each other, while in fact this is not the case. We call such motions pseudo-leapfrogging.
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Figure 3. The PoincarИ maps of the problem of three vortex rings for = 2 = = 1, 1 3

B1 = B2 = B3 = 1, P = 10 and different values of the Hamiltonian H. The regions in which motion is impossible are shown in grey. The small Latin letters denote the fixed points of the map.

The above solutions are illustrated in figure 2 showing a typical view of the trajectories of vortex rings in a fixed coordinate system and in a coordinate system related to the center of vorticity. We note that the trajectories of rings in absolute space depend not only on the form of the curves along which the rings move in a system of the center of vorticity but also on the velocity of the center of velocity, which is given by (10). In particular, if one selects the corresponding initial conditions and intensities of the rings, the displacement of the center of vorticity within one period of the reduced system may become equal to zero. Thus, the vortex rings will execute leapfrogging motion while staying in the same place.

4. Three vortex rings We now consider the problem of motion of three coaxial vortex rings. By proposition 1, the equations of motion (3) for three vortex rings can be reduced to a two-degree-of-freedom system by changing

1 = Z 2 - Z1, 2 = Z 3 -

1 = R 22 - R12

(

)

2 1 , + 2 1

Z1 + 2Z 2 1 , + 2 1

R 2 + 2R 22 3 ( + 2 ) 1 2 = R 32 - 1 1 . + 2 + 2 + 3 1 1

Using the standard approach (see, e.g., Borisov and Mamaev (2005)) for a reduced system, one can construct the PoincarИ cross-section at the level of the Hamiltonian
H 1, 2, 1, 2 , P = const . As a secant we choose the plane 1 = 0 . The section of the level

(

)

of the Hamiltonian formed by the intersection of this level with the plane defines in the threedimensional space 3 = surface
7

{(

2, 2 , 1

)}

a rather complicated (noncompact) two-dimensional

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Figure 4. Trajectories of vortex rings in a system related to the center of vorticity (a), (b), and (c) and in a fixed coordinate system (d) and (e). Figures (a) and (d) correspond to the periodic solution a in figure 3(b), figures (b) and (c) correspond to the solutions b and c, respectively, and figure (e) corresponds to the chaotic leapfrogging motion of vortex rings. Equal numbers in figure (d) indicate the positions of vortices at equal instants in time.

Mh

, P0

= 2 , 2 , 1

(

)

H 1, 2 , 1, 2 , P

(

)

1= 0P = P0

= h .

It is obvious that this, rather arbitrary choice of the plane of section does not guarantee a global recurrence of trajectories back onto the surface of section. In addition, a part of the trajectories can go to infinity, which makes analysis of the mapping difficult. However, it is sufficient to search for periodic trajectories. Figure 3 shows a series of PoincarИ maps. For convenience, they are projected onto the plane 2, 2 in the case of three equal vortex rings. The bold points show the fixed points of the map, which correspond to periodic solutions of the reduced system. These periodic solutions correspond to motions of vortex rings where in the system of the center of vorticity they move along closed trajectories. Examples of such trajectories corresponding to different periodic solutions of the reduced system are depicted in figure 4. At large values of energy the dynamics of the system of three vortex rings breaks down --just as in the case of two rings--into constrained and unconstrained motion. The boundary between them is the loop of separatrices asymptotic to the unstable fixed point b (bold line in figure 3(a)). As can be seen from the figure, the constrained motion can be both regular and chaotic. As energy decreases, the invariant tori are destroyed and the chaotic layers grow both around the origin of coordinates and in the vicinity of the point b. After the destruction of the
8

(

)

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last invariant torus separating the chaotic layers, they merge (see figure 3(b)), and questions can be raised about the chaotic scattering, the probability of one ring being captured by a pair of others etc. We note that practically on the entire interval of the change of energy in the vicinity of the origin of coordinates there exist stable fixed points denoted on the phase portraits by the letters a and a. Next, the corresponding trajectories of vortex rings are shown in a system related to the center of vorticity (figure 4(a)) and in a fixed coordinate system (figure 4(b)). As can be seen from figures 4(a) and (b), the solutions a represent a leapfrogging motion of three vortex rings. The assumptions of its existence were made already by Maxwell (1990). As the analysis of the PoincarИ map shows, these periodic solutions corresponding to threefold leapfrogging motion exist and are stable in a wide range of values of the integrals P and H. This points, among other things, to the possibility of experimentally observing this phenomenon. Another property of the periodic solutions a and a is the following: in a system of the center of vorticity all three rings move along one and the same trajectory lagging behind by 1 3 of a period (figure 4(a)). We note that in addition to the solutions a, almost for all values of energy there exist six stable periodic solutions c, which are obtained from each other by combinatorial rearrangements of vortices and their trajectories in a system of the center of vorticity. These periodic trajectories correspond to the leapfrogging motion as well, but in this case the rings move already along different trajectories (see figure 4(c)). Thus, in the system integrals P and H th vorticity which corre system, i.e. motion two others.

of three equal vortex rings in a wide range of values of the ere exist stable periodic motions in the system of the center of spond to a regular leapfrogging motion of the rings in a fixed in which each of the rings alternately passes through the

As already mentioned, for large values of energy there exist bounded chaotic trajectories (see figure 3(a)). These trajectories correspond to motions of vortex rings where they chaotically pass through each other without scattering to infinity. Such motions can be called chaotic leapfrogging. An example of the trajectories of vortex rings, which correspond to chaotic leapfrogging, in a fixed coordinate system is shown in figure 4(e). Thus, in the problem of three equal vortex rings, for large values of energy there exists a region of initial conditions that corresponds to chaotic leapfrogging. We note that at small values of energy the chaotic leapfrogging can be observed as well. However, in this case there are no invariant tori separating the constrained and unconstrained motion (see figure 3(b)), therefore, after a certain time period the vortex rings can scatter and go to infinity.

5. Numerical simulation for a viscous fluid We now consider the possibility of experimentally observing the above-mentioned behavior of two and three vortex rings. Recent experimental work along these lines is confined, as a rule, to the investigation of two vortex rings in different settings (Maxworthy 1977, Oshima et al 1975, Satti and Peng 2013, Shariff 1992). A substantial body
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of published research is devoted to obtaining and exploring the phenomenon of leapfrogging, however, in spite of the large number of experiments, we have not found any mention of the detection of the phenomenon of pseudo-leapfrogging or capture of one ring by the other in the literature. This is possibly due to the fact that the solutions found by us lie in the region of `nonphysicality' of the model of thin vortex rings. For example, during motion the thickness of one of the rings becomes comparable or larger than its radius, or the distance between the rings becomes smaller than the sum of their radii (i.e., the rings intersect). In particular, the hypothesis of nonphysicality of the solutions follows from the analysis of phase portraits and bifurcation diagrams presented in our previous paper (Borisov et al 2013). However, these solutions may exist in the case of significantly different dimensions and vorticities of the rings ( = 2 1, 1, 1). To prove or
1

disprove this hypothesis, it is necessary to carry out additional investigations involving the projection of the region of `physicality' of the model into the space of first integrals and an analysis of the positions of branches of the bifurcation diagram relative to this region. Below w e present results of numerical simulation of t he leapfroggin g m otion of t wo and t hree vorte x r ing s for a viscous fluid u sing the N avierStokes equa tion s. Following Riley and Stevens ( 1993) , Ri l ey and W e id man ( 1993 ), we write the axisymmetric NavierStokes equations in te rms of t he vorticitystream function variab le s ( - ) in a cylindrical coordinate system:

( r ) ( rv ) ( rvr) r ( r ) z , + + = + z z t z r r r 2 1 = -r , +r 2 z r r r

(11)

where vz = 1 r r and vr = -1 r z are the components of the velocity vector, and is the coefficient of viscosity. We shall solve equations (11) in a cylindrical region with length L and diameter H. Choosing a scale corresponding to the region using coordinates (H) and time (T), we rewrite equations (11) in dimensionless form

( rvz ) ( rvr ) 1 ( r) r ( r) = + + + , z r r r Re z z 2 + r r z 2

1 = -r , r r

(12)

where the tilde stands for dimensionless values, which are related to the initial ones by

z r = r = H, z

t = = T,

vz vr H = v =U= T, vz r

= HU .

(13)

The Reynolds number takes the form Re = HU , and the values of circulation i will be scaled as i = HU . Below we shall consider only nondimensioned values, so we shall omit the i tildes for simplicity.

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We choose the boundary conditions to be

= 0, = 0; 2 L 2 z = l= : = 0, = 0; H z 2 z 2 r = 0: = 0, = 0; r r = 1: = 0, = 0.

z = 0:

The initial conditions corresponding to the given positions of N rings (Riley and Stevens 1993, Riley and Weidman 1993) are
N

= 0,

( 0, z , r ) =

i=1

( z - z )2 + ( r - r )2 i i i exp - , 2 2 2 2

(14)

where is the value specifying the dimensionless thickness of the ring, and zi and ri are the dimensionless cylindrical coordinates of the ith ring. We shall assume that this thickness is determined by the diffusion of infinitely thin vortex rings for the typical time 0 . Then, following Lamb (1932), 2 = 2T0 , and the initial conditions can be written as

( 0, z , r ) =

Re 40

N

i=1

( z - z i)2 + ( r - ri)2 exp - Re . i 40

(15)

The first equation of the system (12) was solved numerically using a monotone thirdorder k-scheme in spatial variables and a fourth-order RungeKutta time-stepping scheme (Wesseling 2000). And the second one was solved using a multigrid method. The following values of parameters were taken in the calculations: U = 1, H = 50, l = 2, Re = 2000, = 0.1. We consider the motion of two vortex rings with equal radii r1 = r2 = 0.234 and with equal intensity = 2 = 1. The distance between the rings is 0.156. Under these conditions 1 the leapfrogging motion regime is realized in which the rings alternately pass inside each other (figure 5). The difference from the motion in an ideal fluid is the presence of dissipation and diffusion of the rings. The rings with vorticities equal in sign merge with time. Figure 6 presents simulation results for the initial conditions

z1 = 0.178 , z 2 = 0.178 ,

r1 = 0.2144 , r2 = 0.1732 ,

= 0.02 , 1 2 = -0.004 ,

(16)

corresponding (in the model of an ideal fluid) to the leapfrogging of vortices with opposite signs of vorticities. Figure 7 presents a comparison of the trajectories of vortex rings for ideal and viscous fluids. As evident from the figures, leapfrogging is observed in the viscous fluid as well, but after the first turn the second ring (with negative vorticity) envelops the first ring, which then continues its motion as a single one. In conclusion, we present results of calculations (see figure 8) with initial conditions (15) for three rings of equal circulation = 2 = 3 = 0.02 with the initial positions 1

z1 = 0.139 , r1 = 0.2008 ,

z 2 = 0.139 , r2 = 0.3044 ,

z 3 = 0.256 , r3 = 0.2588 .

(17)

A comparison of the trajectories of motion of the vorticity centers of the rings obtained from the numerical solution of the NavierStokes equations and for an ideal fluid is presented in figure 9.
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Figure 5. Evolution of the lines of equal vorticity for the leapfrogging of two identical

vortex rings.

Figure 6. Evolution of the field of lines of equal vorticity for the initial conditions (16).

The solid lines correspond to the inviscid fluid, and the markers indicate the successive positions of the rings for the viscous model. The circles correspond to the positions of the first ring, the rhombuses correspond to the second ring and the triangles indicate the third ring. At first the trajectories for the viscous and inviscid motions coincide. Then, as the vortices diffuse (see figure 8), the rings begin to merge and the trajectories diverge. The viscous interaction of the rings can be decreased by increasing the distances between the rings.

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opposite-signed vorticities for the cases of viscous and ideal fluids. Lines 1 and 2 correspond to trajectories of vortices in the model of an ideal fluid, and the lines with markers 3 and 4 to those in the model of a viscous fluid.

Figure 7. Comparison of the trajectories of the leapfrogging motion of vortex rings with

Figure 8. Evolution of the lines of equal vorticity for the leapfrogging of three identical vortex rings.

The results of comparative calculations of the trajectories for the doubled spatial dimensions and coordinates of the rings are shown in figure 9(b). Conclusion In this work we have supplemented the results of the previous paper (Borisov et al 2013) with the numerical experiments using the NavierStokes equations. In particular, it is shown that for the initial conditions corresponding to the leapfrogging of two and three thin vortex rings
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Figure 9. Comparison of the trajectories of vortex rings for ideal and viscous models.

The markers indicate positions of the rings at equal time intervals for the viscous model. The lines denote trajectories of vortices in the model of an ideal fluid (a) for the initial conditions (17); (b) for the doubled dimensions of the rings (as compared with (17)).

in simulating the NavierStokes equations the leapfrogging motion is observed as well. Thus, despite the fact that the equations of motion for thin vortex rings are approximate and have been obtained under some unobvious assumptions, they are quite suitable to describe the dynamics of vortex rings at high Reynolds numbers. In the future it would be of interest to find motions like pseudo-leapfrogging or capture of one ring by the other in the model of a viscous fluid, or to prove the nonexistence of such motions. Another issue that requires further study is that of the motion of vortex rings and their influence on the motion of axisymmetric bodies in a fluid (see, e.g., Tallapragada and Kelly (2013)).

Acknowledgments The work of A V Borisov, I S Mamaev and A A Kilin was carried out within the framework of the state assignment to the Udmurt State University `Regular and chaotic dynamics'. The work of V A Tenenev was supported by the RFBR grant 14-01-00395-a.

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