Theor. Comput. Fluid Dyn. DOI 10.1007/s00162-017-0425-1

O R I G I NA L A RT I C L E

V. Sadri · P. S. Krueger

Formation and behavior of counter-rotating vortex rings

Received: 22 April 2016 / Accepted: 13 February 2017 © Springer-Verlag Berlin Heidelberg 2017

Abstract Concentric, counter-rotating vortex ring formation by transient jet ejection between concentric cylinders was studied numerically to determine the effects of cylinder gap ratio, R R , and jet stroke lengthL to-gap ratio, R , on the evolution of the vorticity and the trajectories of the resulting axisymmetric vortex L in pair. The flow was simulated at a jet Reynolds number of 1000 (based on R and the jet velocity), R R the range 1–20, and R in the range 0.05–0.25. Five characteristic flow evolution patterns were observed and L and R classified based on R R . The results showed that the relative position, relative strength, and radii of the vortex rings during and soon after formation played a prominent role in the evolution of the trajectories of their vorticity centroids at the later time. The conditions on relative strength of the vortices necessary for them to travel together as a pair following formation were studied, and factors affecting differences in vortex circulation following formation were investigated. In addition to the characteristics of the primary vortices, the stopping vortices had a strong influence on the initial vortex configuration and effected the long-time flow L L R evolution at low R and small R R . For long R and small R , shedding of vorticity was sometimes observed and this shedding was related to the Kelvin–Benjamin variational principle of maximal energy for steadily translating vortex rings. Keywords Vortex rings · Vortex dynamics · Vortex interactions 1 Introduction Vortex rings have proven useful for investigating vortical flows because they may be generated easily, and individual (isolated) vortex rings have an extensive theoretical background. Moreover, interactions of vortex rings with each other or with solid structures provide valuable insight into the development and evolution of more complex flows due to the highly nonlinear interaction between vortices. Likewise, various aspects of vortex ring evolution and interactions are related to coherent structures in some types of turbulent flows, particularly turbulent jets [1]. Although isolated vortex rings have been studied extensively [2,3], multi-vortex ring interactions have received less attention, including the problem of concentric, opposite-signed (counter-rotating) vortex rings. Communicated by Tim Colonius. P. S. Krueger (B) Southern Methodist University, Dallas, TX 75275, USA E-mail: [email protected] V. Sadri Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology and Emory University, Atlanta, GA 30332, USA

V. Sadri, P.S. Krueger

This flow can be readily generated by adding an inner cylinder to the standard piston-cylinder vortex ring generator and ejecting the flow between the two cylinders. When flow is suddenly initiated through the cylinder (e.g., by pushing a piston), the boundary layers inside the tubes separate from the lip of each cylinder at the nozzle exit and roll-up into a pair of counter-rotating, co-axial vortex rings with different radii. In this case, the mutual interaction between the rings is in the same direction as the outer ring self-induced velocity. However, the selfinduced velocity of the inner ring is in the opposite direction. The balance between the ring self-induced velocities and the mutual interaction between the rings dictates the vortex pair behavior and can lead to complex flow L evolution [4]. This balance mainly is function of cylinder gap ratio ( R R ) and jet stroke length-to-gap ratio ( R ). Previous work on concentric, opposite-signed vortex rings have illustrated interesting behavior. Weidman and Riley [4] conducted a series of experiments and numerical simulations on counter-rotating, co-planar vortex rings. In the experiments, pairs of vortex rings were formed by the impulsive motion of piston through an annular circular orifice and vortex rings of opposite circulation were formed at the inner and outer orifice lips. They observed that the axisymmetric vortex pairs followed common trajectories that seemed to be Reynolds number independent for the conditions investigated. They reported that the behavior of the rings depended on the impulse applied to the flow during the formation and also the gap width of the annulus. Based on these parameters, the rings may separate with the inner ring reversing its direction of propagation and the outer continuing as a single ring or, in rare cases, the two rings may propagate together as a vortex ring pair. In some cases, apparently influenced by vortex ring stability, portions of the inner ring were observed to wrap around the outer ring. In other work, Wakelin and Riley [5] reproduced the results of Weidman and Riley [4] numerically and conclude that the inner ring was always slightly weaker following formation compared to outer ring and in a few cases observed a rising vortex pair (corresponding to increased diameter of both rings). Kambe and Takao [6] also reported an experiment of counter-rotating vortex rings in air using an annular orifice. After formation of both rings by impulsive piston displacement, they reported that the inner ring reversed direction immediately and collided with the orifice wall. These three studies [4–6] focused mainly on the convection of the vortices (such as whether or not the vortices remained together as a pair) and gave little information about the flow structure beyond flow trajectories. Also, they focused on relatively large annulus gap width and considered parameter ranges corresponding to relatively short jet pulses in order to produce compact vortices. There are several other types of co-axial vortex ring interactions, in addition to those involving counterrotating rings, including head-on collision of two vortex rings and leapfrogging. These interactions have some similarities with the current study in terms of competition between self-induced and mutually induced vortex motion, but in the case of leapfrogging the two vortex rings have the same sign of vorticity and in the head-on collision case, the two vortex rings have the same radii. Details about head-on collision and leapfrogging of vortex rings can be found in dedicated studies of these phenomena, including [7–10], and are summarized in the review papers by Shariff and Leonard [2] and Lim and Nickels [3]. The case of counter-rotating axisymmetric vortex rings considered here becomes qualitatively similar to the Cartesian counterpart of a vortex pair system when the vortex rings become close together. A key difference, however, is that counter-rotating axisymmetric vortex rings can still experience vortex stretching so that the axisymmetric system is less constrained than the Cartesian counterpart [11]. Nevertheless, in planar two-dimensional flows, a coherent structure similar to the axisymmetric vortex ring is often observed: the symmetric vortex dipole (counter-rotating vortices). Dipolar vortices have been observed experimentally in flows in which three-dimensional motions have been suppressed by stratification [12], at a two-fluid interface [13], or by rotation of the ambient fluid [14]. Dipolar vortices of equal but opposite strength will travel together, but if the strength is unequal, the weaker vortex will orbit around the stronger. Additionally, the strain field of each vortex has a strongly nonlinear interaction with neighboring vortices [15,16] that distorts the vortex cores. Stability of vortex pairs has received considerable attention as indicated in the review by Leweke et al. [17]. A line vortex is susceptible to the growth of both an elliptic (short-wavelength or Widnall) instability and a long-wavelength (Crow) instability. As the focus of this investigation is on the interaction of the vortices with each other, the numerical approach used in this study enforces axisymmetry and consequently avoids some of the complications of the aforementioned instabilities. The main objective of the present investigation is to study the behavior of concentric counter-rotating vortex rings for a wider range of parameters than has been studied previously and to investigate the origins of different L R behaviors observed. The effects of the key flow generation parameters ( R , R ) on the flow evolution will be considered, and the underlying physics of the observed behaviors will be discussed.

Formation and behavior of counter-rotating vortex rings

Pressure Outlet

2D

Wall u(t)

R

Ri

∆R 2D

Axis

Velocity Inlet

5D Fig. 1 Numerical domain and boundary conditions for axisymmetric flow simulation

2 Numerical approach In this study, the vortex formation process was simulated numerically. For practical reasons, a commercial code, ANSYS FLUENT, was used. The flow evolution was determined by solution of the unsteady, incompressible, axisymmetric Navier-Stokes equations, without swirl, namely ∂ 1 ∂ (1) (vx ) + (r vr ) = 0 ∂x r ∂r      ∂vx ∂ 1 ∂ 1 ∂ 1 ∂p 1 ∂ ∂vr ∂vx 1 ∂ + + 2r ν + rν (2) (vx ) + (r vx vx ) + (r vr vx ) = − ∂t r ∂x r ∂r ρ ∂x r ∂x ∂x r ∂r ∂r ∂x      1 ∂ 1 ∂ 1 ∂p 1 ∂ ∂ ∂vx ∂vr 1 ∂ ∂vr vr + + rν + 2r ν − 2ν 2 (vr ) + (r vx vr ) + (r vr vr ) = − ∂t r ∂x r ∂r ρ ∂r r ∂x ∂x ∂r r ∂r ∂r r (3) The solution domain is illustrated in Fig. 1. The coupled algorithm (solving the momentum and continuity equations simultaneously) in FLUENT was used for pressure-velocity coupling, and the Green-Gauss node-based method [18] (second-order spatial accuracy) was used for gradient evaluation. Also, the second– third-order MUSCL [19] scheme was used for interpolation of the spatial convective terms at locations away from control volume centers, and a second-order scheme was used for the pressure term. Time integration used a second-order finite difference implicit scheme (three time level method [20]). The dimensions of the computational domain in the x and r directions were 5D × 2D where D = 2R is the diameter of the outer cylinder. The boundary conditions are also indicated in Fig. 1. To generate the jet flow, the inlet boundary condition between the two cylinders (see Fig. 1) was set at a specified constant velocity during the jet pulse and zero velocity afterward (impulsive jet velocity program). Specifically, ⎧

t ≤ uL0 = t p u (t) ⎨ 1

= (4) ⎩0 u0 t > uL0 = t p where the pulse duration, t p , was specified to provide the desired pulse length L, and u 0 is the specified piston/inlet velocity. The Reynolds number, ReR = ρu 0μR ,was set to 1000 for all results presented here, where R is the difference between outer cylinder and inner cylinder radii. Preliminary simulations conducted for Reynolds numbers of 500, 1000, and 2000 for two different R showed qualitatively similar behavior, so only ReR = 1000 was investigated in detail. At this Reynolds number, the boundary layer at the exit of the concentric tubes is thin and as a result, the length of the tubes (2D in this investigation) does not have a strong effect on the results. The parameters varied in this study were the jet length (L)-to-cylinder gap (R) ratio u0t p L = R R

(5)

V. Sadri, P.S. Krueger

Table 1 Parameters for computational cases R/R

L/R

ReR

0.05 0.1 0.15 0.2 0.25

2,3,5,8,10,15,20 2,3,5,8,10,15,20 2,3,5,8,10,20 2,3,5,10,20 2,3,5,8,10,15

1000 1000 1000 1000 1000

R and the ratio of the cylinder gap to outer radius, R R . For fixed ReR and R , the impulse ejected into the L L flow is determined by the jet stroke length-to-gap ratio ( R ). Here, R is preferred over the impulse used in L Weidman and Riley [4] for characterizing the flow because for sufficiently large R the vorticity can break into several pieces (as will be shown), so the total flow impulse is no longer a fundamental characteristic of the primary (initial) vortex pair. The conditions for all the cases considered are listed in Table 1. To demonstrate spatial convergence of the solutions, three grid resolutions (3.2, 2.2, and 1.1 million quadrilateral cells) were considered with computational domain error analysis determined based on the grid convergence index (GCI) method [21] using a dimensionless time step of 0.005 (time was non-dimensionalized by the convective time, uD0 ). The main interest of this study was the location and the circulation of the vortex rings, so the average vortex location and the vortex circulation were considered as quantities for the grid convergence study. Based on the grid convergence study of the vortex with positive vorticity, the GCI for the 3.2 million cell grid was 3.4% for circulation and 1.7% for vortex centroid location. Overall, the results indicate that the spatial grid convergence was achieved for two relevant quantities. Also, a temporal convergence study was performed for dimensionless time steps of 0.05, 0.01, and 0.005 using the 3.2 million cell mesh. In this case the GCI was 1.3% for circulation, so a time step of 0.005 was chosen for this study. Finally, the uncertainty due to the domain size was studied for domain sizes of 3D × 1D, 5D × 2D, and 8D × 4D using the same mesh density at the exit of the orifice for all cases. The error was 0.26% for circulation for the 5D × 2D domain compared to the 8D × 4D domain, so the domain size of 5D × 2D was chosen for this study. For identifying the vortices and tracking their location, the vorticity field, ωθ , was computed from the velocity field using a central finite difference scheme to approximate the derivatives. In order to distinguish between the primary positive and negative vortices and the rest of vorticity field, a threshold of 5% of the peak vorticity was applied to the vorticity field for both positive and negative vorticity. The vortex locations were identified as the centroids of the resulting positive and negative vorticity domains:   r ωθ dS S xωθ dS xc = , rc = S (6)   where the circulation  was determined from  = ωθ dxdr (7)

and the integrals were computed using a 2D version of the trapezoidal rule (second-order truncation error). Additional dynamical quantities of interest for the vortices were the hydrodynamic impulse (I ) and kinetic energy (E), determined for axisymmetric flows from [22] E I (8) = π ωθ ψdxdr, and = π ωθ r 2 dxdr, ρ ρ where ψ, the Stokes stream function, is determined from the governing equation,   1 ∂ 2ψ ∂ 1 ∂ψ + = −ωθ . r ∂x2 ∂r r ∂r

(9)

Equation (9) was solved using a second-order accurate finite deference method with ψ = 0 on the centerline of the domain. The integrals in Eq. (8) were determined using the same method as for Eqs. (6) and (7). The selected vorticity cutoff level introduces some variability in the computed values. The optimum cutoff level is the minimum contour level that distinguishes between the vortex rings and the rest of the vorticity field, which may vary from case to case and frame to frame. Selecting the cutoff level in this manner produced results for the above integrals within 5% of the values obtained using the fixed 5% level, so a fixed 5% cutoff was used throughout as an acceptable compromise threshold level.

Formation and behavior of counter-rotating vortex rings

r/R

1.5

Us +

1

U sUm-

0.5

0

Um+

0

L Fig. 2 Typical results for R = 10, R R = 0.05, at dashed lines indicate negative vorticity

0.5 tu 0 D

x/R

1

1.5

= 1.25. Vorticity contour levels (min = −40, max = 40, increment = 10),

3 Results L Typical simulation results are shown in Fig. 2 for R = 10 and R R = 0.05, where vorticity has been nonu0 dimensionalized by D here and in the following. The impulsive jet flow between the concentric cylinders separates at both cylinder lips generating a positive vortex (positive ωθ ) from the outer cylinder and a negative vortex (negative ωθ ) from the inner cylinder. To simplify the discussion of the vortices, the top positive vortex will be denoted V + and the lower negative vortex will be denoted V − . The flow evolution following vortex formation is determined primarily by the competing effects of the self-induced velocity, Us , and mutual interaction of the vortex pair (which involves a mutually induced velocity, Um , on each vortex core). The radii and strength (circulation) of each vortex are the primary factors determining the self-induced velocity, while the distance between the vortices and their strength are the primary factors determining the mutual induction. The approximate direction and relative strength of these two components are illustrated in Fig. 2.

3.1 Vortex ring pair behavior L L For fixed ReR and nozzle exit geometry, the flow evolution depends on R R and R . When R was less R than 8, the vortex pair would separate regardless of the value of R (Case 1, illustrated in Fig. 3). Due to the L , the vortex rings were weak (small circulation and impulse) and strongly influenced by the stopping small R vortices generated at jet termination, placing them in a configuration where V + was leading (downstream of) V − (see Fig. 3a) so that the mutually induced velocity Um would drive them toward the symmetry axis where their self-induced velocities would ultimately dominate for any R R . The stopping vortex interaction is discussed in more detail below. Similar results were observed by Weidman and Riley [4] and Wakelin and Riley [5] under corresponding conditions, as will be discussed below. L R For larger R , a variety of behaviors were observed over the range of R R studied. At R = 0.05 and 8 L ≤ R < 15, the vortex pair exhibited a rising trajectory (corresponding to increased vortex radii) as illustrated L in Fig. 4 (Case 5). For R R = 0.1 and 8 ≤ R < 15, the vortex pair followed a nearly straight trajectory initially, and then gradually moved toward the axis and ultimately the vortices separated (Case 3). The vortex trajectories for this case are shown in Fig. 9a (the vorticity evolution is similar to Case 2 and so is not presented) and represents transition behavior between Case 1 and Case 5. L + Interestingly, when R ≥ 15 and R R = 0.05, the vortex pair would rise at early time, but then V would L = 20 in Fig. 5 (Case 4) where the vortex shedding shed some vorticity and become weaker, as illustrated for R is shown in Fig. 5b. Subsequently, the vortices were no longer equally paired and the V + vortex was pulled forward by V − , resulting in a downward trajectory of the vortex pair toward the axis (the influence of the relative vortex circulation on this behavior is discussed below). Consequently, the vortex pair would approach the axis and when the V − radius was reduced enough it would begin to move rearward by its own self-induced L velocity and the vortex pair would separate. The same dynamic occurred for R R = 0.1 and R ≥ 20. The

V. Sadri, P.S. Krueger

(a)

(b)

L Fig. 3 Flow evolution for R = 5, R R = 0.05 (Case 1). Vorticity contour levels (min = −100, max = 100, increment = 10), dashed lines indicate negative vorticity, a tuD0 = 0.5; b tuD0 = 5.75

1.4

1.4

1.2

r/R

r/R

1.2

1

1

0.8

0.8

0.6

0.6 -0.2

0

0.2

x/R

0.4

(a)

0.6

0.8

0.4

0.6

0.8

x/R

1

1.2

1.4

(b) R R

Fig. 4 Flow evolution for = 10, = 0.05 (Case 5). Vorticity contour levels (min = −90, max = 90, increment = 10), dashed lines indicate negative vorticity, a tuD0 = 0.75; b tuD0 = 1.75 L R

L larger value for R at this R R indicates the increased separation between the vortices required more vorticity to observe the same behavior for this case. L Distinctly different flow evolution was observed for R R ≥ 0.15 and R ≥ 10 (although the boundL has not been fully explored). In this case, the smaller radius of the V − vortex increased the V − ary on R L self-induced velocity, so the V + V − mutual interaction was not dominant. However, with increasing R , V+ − became stronger and the V vortex became distorted (by the larger core size and smaller radius), allowing the V + vortex to pull away or “strip” some part of the V − vortex. This much weaker and smaller secondary V − , hereafter referred to as V2− , was strongly influenced by the induced velocity from V + , causing V2− to loop − L = 20, R around V + , as illustrated in Fig. 6 for R R = 0.15 (Case 2). The V2 vortex was too weak to support the strain field generated by V + , and thus it was stretched and lost most of its vorticity by an elongation process

Formation and behavior of counter-rotating vortex rings

1.4 1.4 1.2 r/R

r/R

1.3 1.2

1 1.1 0.8

1 0.9

0.6 0.2

0.4

0.6

0.8

0.8

1

0.8

1

x/R

x/R

(a)

(b)

1.2

1.4 1 1.3 r/R

r/R

0.8

1.2 0.6 1.1 0.4 1

1.3

1.4

1.5

1.6

1.7

1.6

1.8

2

x/R

x/R

(c)

(d)

2.2

2.4

L Fig. 5 Flow evolution for R = 20, R R = 0.05 (Case 4). Vorticity contour levels (min = −50, max = 50, increment = 10), dashed lines indicate negative vorticity, a tuD0 = 0.9; b tuD0 = 1.25; c tuD0 = 1.75; d tuD0 = 2.75. Note vorticity shedding from the positive vortex in (a) and (b)

similar to that described by Trieling et al. [23]. Finally, at later time the vortices moved in opposite directions under their own self-induced velocities. Vortex trajectories for this case are shown in Fig. 9b. The vortex pair behavior for the ranges of parameters considered in this study (Table 1) can be arranged in a map of vortex behavior shown schematically in Fig. 7. It is interesting to compare the current study result with previous studies Weidman and Riley [4] and Wakelin and Riley [5]. For comparison with the present results, L the results of these prior studies were re-categorized according to R R and R . In the experimental study [4], L R R was in the range 1–5.5 and R was in the range 0.128–0.425. Based on the provided vortex trajectories, all of their results, except one observation, correspond to Case 1, as expected from the large R R values. For the one exception, referred to as a “balanced ring pair” (vortex pair traveling together, Fig. 12 in their paper), the decrease in the inner vortex ring radius suggests it is possible that this case would also exhibit separation of the vortices if observed long enough. Also, the results of this study were for trajectories very close to the exit plane due to the ultimate, rapid decay of the vortices, so it is difficult to have a complete comparison with results of the current study, which focused on long-time evolution. Nevertheless, for the data that are presented, the results of Weidman and Riley [4] are in overall good agreement with the current study.

V. Sadri, P.S. Krueger

1.4 1.5

1.2 1

r/R

r/R

V2-

0.8

1 0.6 0.4

V-

0.5 0.5

0.2 1

0

1.5

1.5

2 x/R

(a)

(b)

1.2

1.2 r/R

1.4

r/R

1

x/R

1.4

1

0.8

0.6

0.6

0.4

0.4 2

2.5

2.5

1

0.8

1.5

V-

1.5

2

2.5

x/R

x/R

(c)

(d)

L Fig. 6 Flow evolution for R = 20, R R = 0.15 (Case 2). Vorticity contour levels (min = −70, max = 70, increment = 10), dashed lines indicate negative vorticity, a tuD0 = 1.75; b tuD0 = 2.25; c tuD0 = 2.5; d tuD0 = 3.5

L For the numerical study of Wakelin and Riley [5], on the other hand, R was in the range 6.4–10.6, and R R L was in the range 0.14–0.15. Their simulations showed Case 1 for R = 0.15 and = 10.6 (in agreement R R L with the current study), Case 3 for R R = 0.15 and R = 9.6 (close to but slightly displaced from the Case 3 domain in the current study), and rising and looping behaviors that not seen in current study. The reason for this discrepancy could be related to the differences between their approach and the approach presented here. Specifically, Wakelin and Riley [5] define a specific velocity profile a distance of 0.4R upstream of the jet exit plane, which is much shorter than the 2D = 4R upstream of the jet exit plane used in the present study. This may have impacted the ability of the flow to develop naturally (particularly the asymmetry in the top and bottom vortices, discussed below) before reaching the jet exit plane. Wakelin and Riley [5] also had vertical walls near at the jet exit plane, so the vortices would have interacted with the walls, especially when they remained close to the jet exit plane as in their simulations. As discussed below, the initial development of the vortices following formation strongly influences their relative locations and, hence, their ultimate trajectories, suggesting the behavior of this system is sensitive to the flow and boundary conditions at the jet exit plane. To provide a more direct comparison of some of the behaviors observed, Fig. 8 plots the vortex trajectoL ries for R R = 0.05 and three different R . Figure 8 shows an interesting progression from vortices initially

Formation and behavior of counter-rotating vortex rings

x

x

0.25

0.2

x

0.15

x

Case 1 Case 2 Case 3 Case 4 Case 5

x

∆R/R 0.1

0.05

5

10

15

20

L/∆R

Fig. 7 General characteristics of the vortex pair evolution with respect to the generator parameters ( R R , circles represent positive and negative vortices in the schematic insets 2.2

Positive Vorticies L/ Negative Vorticies L/ Positive Vorticies L/ Negative Vorticies L/ Positive Vorticies L/ Negative Vorticies L/

2 1.8

R=5 (Case 1) R=5 (Case 1) R=20 (Case 4) R=20 (Case 4) R=10 (Case 5) R=10 (Case 5)

1.6

r/R

1.4 1.2 1 0.8 0.6 0.4 0.2

Fig. 8 Vorticity trajectories for

0

R R

0.5

1

= 0.05 at different

1.5 L R

x/R

2

2.5

3

3.5

L R ).

Black and white

V. Sadri, P.S. Krueger

1.1

1.4

Positive Vortex Negative Vortex

Positive Vortex Main Negative Vortex Secondary Negative Vortex

1 1.2 0.9 0.8

1

r/R

r/R

0.7 0.8

0.6 0.5

0.6

0.4 0.4 0.3 0.2 -0.5

0

0.5

1

x/R

1.5

2

2.5

3

0.2

0

0.5

(a) Fig. 9 Vorticity trajectories for a

L R

1

1.5

x/R

2

2.5

3

(b) = 10,

R R

= 0.1 (Case 3) b

L R

= 20,

R R

= 0.15 (Case 2)

L L directed toward the axis at low R to vortices rising away from the axis indefinitely at intermediate R , to L R L rising followed by turning toward the axis for large R at this R . The effect of R on the behavior of the flow can be illustrated with a new parameter called penetration depth (introduced in [4]), x p . The penetration depth is defined as the axial location at which the translation velocity of V − changes from positive to negative. Physically, it indicates the point at which the mutual interaction between the positive vortex and negative vortex L is no longer dominant and V − moves primarily due to its self-induced velocity. For R ≤ 10, a generally R L smooth progression of behavior is observed as R and R are varied, with larger penetration observed for L L due to the higher momentum associated with these flows. Note that the data for R = 10 does not larger R R R go below R = 0.1 because for smaller R the vortices remain together as a pair indefinitely. Additionally, by L increasing R R the penetration depth was decreased for these R due to the stronger self-induced velocity of L R = 20 in that at R V − . The observed trend shifts dramatically for R R = 0.1 x p is more than for R = 0.05. + This shift is associated with the transition to vorticity shedding from V (Case 4). Results are not plotted for R L − R > 0.1 at R = 20 because V splits into two (Case 2), making determination of x p ambiguous. Experimental data for x p from Weidman and Riley [4] have been included in Fig. 10 for comparison. The current study is in fair agreement with the experimental results, although the experimental results tend to L produce larger x p than observed here for a given R . This is likely due to differences in the system geometry, as Weidman and Riley [4] used an orifice geometry with vertical walls and a large contraction before the jet exit plane. It is worth noting that there is some ambiguity in Weidman and Riley [4] as to the definition of their parameter W (compare the caption to their Table 1 with the text at the bottom of their p. 323). Based on the plots of vortex trajectories, it appears that W = R was used in generating the data in their Table 2, and this was utilized for the data plotted in Fig. 10.

4 Discussion In the following sections, the different behaviors of concentric vortex rings observed in these results are discussed. The first question concerns conditions necessary for the vortex rings to travel together as a pair. The results show a key requirement is a stronger V + compared to V − . This condition leads to the question of how the asymmetry in vortex ring strength is achieved, which is also discussed. Then, the effects of stopping vortices on the concentric vortex rings are investigated to explain the reason that concentric vortices do not remain together for short jet stroke ratios at smaller gap size. Finally, the Kelvin–Benjamin variational principle is used to explain the shedding of V + observed in long jet stroke ratios for smaller gap size.

Formation and behavior of counter-rotating vortex rings

2.5

L/ L/ L/ L/ L/ L/ L/

2

R=3 R=5 R=10 R=20 R~4.8-5.45 (Exp.) R~2.4-2.9 (Exp.) R~4.3( Exp.)

xp/R

1.5

1

0.5

0

0

0.05

0.1

0.15

0.2

0.25

0.3

R/R Fig. 10 Penetration depth versus are shown for comparison

R R .

These results include Cases 1, 3, and 4. Experimental data from Weidman and Riley [4]

4.1 Requirement for the vortices to remain paired The dynamics leading to Case 1 changes with gap size. For small gap size, the mutual interaction of the primary vortices plays a strong role, so in this subsection the mutual interaction between closely spaced vortex rings is analyzed analytically. Specifically, the question is what conditions are required for the mutually induced velocity of two co-axial vortex rings to allow them to remain together or separate. In order for the vortices to remain together as a pair (as observed in Case 5, for example), they must travel parallel to the flow symmetry axis or have a (rising) trajectory away from the axis. This behavior can be achieved (even if the initial configuration following formation has Us directed toward the axis) if V + is stronger than V − , in which case the induced velocity of V + on V − can move V − ahead of V + . This condition can lead to the vortices moving radially outward even if Us is initially directed toward the axis following formation, provided V + is strong enough to change the orientation of the vortices before they approach the axis too closely. To avoid the complications of changing relative axial locations of the vortices during the formation and initial evolution process so as to highlight the main factors involved in directing the vortex pair away from the axis, initially co-planar vortices will be considered as shown in Fig. 11. The additional simplification of inviscid vortex ring motion is also employed. Similar conditions were considered by Weidman and Riley [4], but their results are limited to large ring radius and small ring core size compared to the ring separation. The vortex rings in Fig. 11 have equal core radii, δ, and outer and inner ring radii of ro and ri , respectively. The equal core radii assumption is well approximated for small gap sizes. For V − to overtake V + and become the leading vortex, the axial velocity of V − should be larger than that of V + , namely Um− + Us− > Um+ + Us+ .

(10)

Note that Eq. (10) is a necessary but not a sufficient condition for the vortices to move away from the axis because if the velocity of V − is too large, it may loop around V + rather than the vortices gradually moving away from the axis as a pair. The self-induced velocity of a thin-core inviscid vortex ring is [22]  

  δ 8r  1 Us = ln − +O (11) 4πr δ 4 r Additionally, the mutually induced velocity of V + on V − is [24]

V. Sadri, P.S. Krueger

r V+ Γp

δ Γn

ro

x

ri

Vδ Fig. 11 Schematic diagram of concentric vortex rings

p ro 4π

   ri A I2 − I1 r o + ri B B

(12)

n = ri 4π

   A ro I2 − I1 ri + r o B B

(13)

Um− = and of V − on V + is Um+ where,

4ro ri 4 4 E (m) , m = 2 , a 2 = r o + ri , K (m) , I2 = 3 a a (1 − m) a 2 A = a + B, B = −2ro ri

I1 =

and K (m) and E (m) represent complete elliptic integrals of the first and second kind, respectively. Substituting Eqs. (11–13) into Eq. (10) gives   

 p > n

K (m)

2− r ro

−

E(m)

r ro

+

+ − K (m) r 2− ro

1

2 1− r ro E(m)

r ro

+

1 2

ln

8 1− r ro

  ln

−

δ ro

 8

δ ro

1 4

 −

 ≡G

1 4

r δ , ro ro

 (14)

where, r = ro − ri . In Fig. 12, the function G is plotted for different 

δ ro

and non-dimensional vortex separation ratio, r ro . In this 

figure, circulation ratios, np , above the curve produce rising vortex pairs ( np > 1 means V + is the stronger ring), p n

below the curve produces sinking vortex pairs (which will ultimately lead to the vortices separating). As seen in Fig. 12, for small vortex separation (r ) there is little dependence on the core size. However, vortex separation tends to increase with larger vortex core size. Also, viscosity increases the core size at the rate 1 of (νt) 2 so that the actual circulation ratio required for V − to become the lead vortex under the stated conditions may be somewhat greater than this analysis suggests. Nevertheless, the analysis provides general trends that are consistent with the results, illustrates the general magnitude of circulation asymmetry required for the vortex pairs to change direction away from the axis, and shows that the vortex core size is not a strong parameter.

and

Formation and behavior of counter-rotating vortex rings

2.8

δ/ro=0.01 δ/ro=0.02 δ/ro=0.05 δ/ro=0.1

2.6 2.4 2.2

G

2 1.8 1.6 1.4 1.2 1 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Δr /ro Fig. 12 Circulation ratio required for

V−

to lead

V+

for different

δ r ro , ro

1.25 G = 1.64

1.6

G = 1.23

1.55

1.2

1.5

/ n

1.15 p

1.4

p

/ n

1.45

1.35 1.1 1.3 1.25 1.05

1.2 1.15 1.1 0.2

0.22

0.24

r/r0

0.26

0.28

0.3

1 0.07

0.08

0.09

(a) Fig. 13 Circulation ratio versus vortex distance during the formation phase for (Case 5)

r/r0

0.1

0.11

0.12

(b) L R

= 10 a

R R

= 0.15 (Case 1), b

R R

= 0.05

For example, as illustrated previously, for R R larger than 0.15 the vortex pair is short-lived and the vortices quickly separate (Case 1), which is consistent with the model because a large circulation ratio would be required to prevent this. For small gap ratios, on the other hand, the required circulation ratio is relatively small. Consequently, tiny asymmetry in the velocity profile which can produce different circulation for each ring, which can lead to different behavior (e.g., transition from Case 1 to Case 5). As the vortex trajectories evolve, ro will typically change some since the initial vortex trajectories are not perfectly horizontal, and the vortices may move relative to each other (e.g., according to Eq. 10), so that the co-planar assumption of the model is no longer valid. As a result, Fig. 12 is not completely predictive of the overall behavior, but it does provide guiding trends. To illustrate this point, the evolution of the ratio of the vortex circulation magnitude during the formation process is illustrated in Fig. 13 for two cases. In both cases, the circulation ratio is less than the required G

V. Sadri, P.S. Krueger

150

150

d +/dt

d +/dt

-

d -/dt

d /dt 125

(d /dt)/(u R/t ) 0 p

(d /dt)/(u0R/tp )

125

100

100

75

50

50

25

25

0

75

0 0

0.2

0.4

t/tp

0.6

0.8

1

0

0.1

0.2

0.3

d dt ,

0.5

t/t

0.6

0.7

0.8

0.9

1

p

(b)

(a) Fig. 14 Circulation rate,

0.4

versus time during the formation phase for

R R

= 0.05 a

L R

= 10, b

L R

= 20

value given by Eq. (14). The required G value for each case was calculated for conditions promptly following flow initiation to have a nearly co-planer vortex pair and the core size used for calculating G was determined based on a uniform core with vorticity equal to half the peak vorticity observed in the simulations. For the results in Fig. 13b (Case 5), however, the circulation ratio value is initially only slightly less than the necessary value, consistent with the observed behavior that the vortices remain together for this case. The fact that the circulation ratio continues to decrease is not a serious concern as the relatively large initial circulation ratio tends to move V − forward initially, so that the vortices are no longer co-planar and a smaller circulation ratio would be required at later time to keep the vortices together, which is not accounted for in Eq. (14). Figure 13a (Case 1) on the other hand, shows that the circulation ratio is much lower than required for the vortices to move away from the axis and stay together as a pair (even at initiation where the vortices are nearly co-planar), which is consistent with the observation that the vortices move toward the axis and separate for this case. 4.2 Origin of asymmetry in the symmetric concentric vortex rings generator As explained in the previous section, for the vortex pair to travel together (Case 5) the outer ring should be  stronger than the inner ring ( np > 1, Fig. 12). Achieving this result implies that the formation conditions for the two rings must be different. Wakelin and Riley [5] also reported that the inner ring is always weaker after formation. Motivated by these observations and the desire to explain why the required circulation ratio for larger gap size cannot be produced in this configuration, the reasons for the asymmetric vortex ring strength are investigated. The difference in vortex circulation for the two vortices during formation is related to the difference in the rate of circulation supplied to each vortex. Following Didden [25], the rate of circulation supplied to the vortices can be determined from the flux of vorticity at the nozzle exit plane according to d + = dt d − = dt



ro

r∗ r∗ ri

ωu|x=0 dr

(15)

ωu|x=0 dr

(16)

where r ∗ determines the division between positive and negative vorticity. Using these results to determine the rate of circulation applied to each vortex produces results like those shown in Fig. 14. As illustrated in this figure, early in the formation process V + has larger circulation flux compared to V − and consequently has larger circulation when the jet stopped.

Formation and behavior of counter-rotating vortex rings

0.05

0.05

t/tp=0.2

0.045

t/tp=0.4

0.045

t/tp=0.8

0.04

t/tp=0.6

o

0.035

t/tp=1

o o

o

(r-Ri)/R

0.03

i

0.02

0.015

0.015

0.01

0.01

0.005

0.005 1.05

1.1

1.15

1.2

1.25

t/t p=0.8

o

t/t p=1

o o

0.025

0.02

1

t/t p=0.6

0.03

o

0.025

0

t/t p=0.4

0.035 (r-R )/R

0.04

t/t p=0.2

0

1.3

u(t)/u0

1

1.05

(a)

1.1

1.15 u(t)/u

1.2

∂v ∂x

−

∂u ∂r ,

o

1.25

1.3

0

(b)

Fig. 15 Axial velocity versus radial distance at the nozzle exit. Circles show the maximum velocity location a L = 20, R 0.05, b R R = 0.05

Using ωθ =

o

L R

= 10,

R R

=

Eqs. (15) and (16) reduce to  ∂v  u  dr ∂ x x=0 r∗  r∗ 1 2 ∗  ∂v  d − = − u r ,t + u  dr dt 2 ∂ x x=0 ri  d + 1  = u2 r ∗, t + dt 2



ro

(17) (18)

So, the u ∂u ∂r term feeds equal circulation to each vortex pair and any differences between the two arise from ∂v the u ∂ x term and asymmetry in the velocity profile must be investigated. In Fig. 15, the velocity profile time history for the same cases as Fig. 14 are shown (both cases have the same gap size to remove the effect of geometry differences on the velocity profile). In both cases, the maximum velocity (located at r ∗ ) was near the outer cylinder throughout the jet pulse and gradually moved toward the center of the annulus as time proceeded. The analytical solution for starting flow in an infinitely long pipe [26] exhibits similar behavior, indicating the flow has inherent asymmetry with respect to the mid-plane of the annulus. This is a key element of the flow generating system and influences the overall flow evolution. In Fig. 16, profiles of ∂∂vx are shown. Similar to the velocity profile, ∂∂vx also shows a slight asymmetry with ∂∂vx being larger near the outer cylinder, especially at early time. Near jet termination, ttp = 1, ∂∂vx became

smaller and its effect on total the vorticity contribution was less prominent. Hence, both components of u ∂∂vx tend to be larger near the outer cylinder, providing greater vorticity flux, at least initially. If these conditions persist throughout the formation, greater total circulation can be provided to V + as in Fig. 13, which can lead to a rising vortex pair (Case 5). Also, the conditions leading to high vorticity flux for the outer vortex ring have a very short duration (occurring primarily near jet initiation), making it difficult to achieve substantial asymmetry in vortex circulation. These dynamics prevent the circulation required for Case 5 behavior from being achieved for larger gap size.

4.3 Effect of stopping vortices L Although for small stroke ratio ( R ), all results for the present investigation fell in the same category (Case R 1), regardless of R , the physical reasons for these were different for large and small gap size. The R R ≥ 0.15 results were addressed previously and related to the asymmetry in vortex strength that develops during vortex

V. Sadri, P.S. Krueger

0.05 0.045 0.04

(r-Ri)/R

0.035

t/t p=0.1 t/t p=0.2 t/t p=0.3 t/t p=0.5 t/t p=1

0.03 0.025 0.02 0.015 0.01 0.05 0 -80

Fig. 16 Gradient

∂v ∂x

-60

-40

-20

0

20

( v/ x)/(u0/D)

versus radial distance at the nozzle exit for

L R

= 10,

R R

40

60

80

= 0.05

formation. To emphasize the contrast with small gap-size behavior, note that for R R = 0.05 in Fig. 13b (which shows results during the formation phase before jet termination), the circulation ratio decreases throughout L the jet pulse, indicating that stopping the jet short (at a smaller ttp ), corresponding to a smaller R , would give a larger circulation ratio more favorable for a rising vortex pair (Case 5). Therefore, any shift from Case L 5 to Case 1 as R is decreased for smaller gap size is due to what happens after the formation phase. This introduces the importance of stopping vortices for small gap sizes, which will be discussed in this section. In Figs. 17 and 18, the effect of the stopping vortices on the dynamics of the vortex rings is illustrated. When L the stroke ratio ( R ) was small (5 or less), the induced velocities of the stopping vortices had a strong impact on the initial vortex configuration following formation. In the cases where R R was less than 0.1, if the locations of the vortices after jet termination were outside the domain of influence of the stopping vortices, then the primary L vortices tended to go straight at early time. This occurred for large enough R (see Fig. 17). Conversely, when L was small, the primary and stopping vortices were both close to the nozzle exit plane at jet termination so R that the induced velocity of the stopping vortices changed the relative location and shape of the primary vortices, which altered the symmetry of vortex pair relative to the gap between the two cylinders (Fig. 18). This change influenced the evolution of the vortex trajectories, causing the vortex pair to move toward the symmetry axis. The nature of the interaction between the primary and stopping vortices is illustrated in Fig. 19. In the L + cases where R R was less than 0.1 and the stroke ratio ( R ) was small, V was stronger during formation (see + − Fig. 13b). So, the stopping vortex generated by V , denoted SV , was stronger compared to the stopping vortex from V − , denoted SV + . Thus, the induced velocity of SV − was stronger than that from SV + . The influence of the stopping vortices is to slow the primary vortices as shown in Fig. 19, but V − was slowed more because SV − was stronger than SV + and V − was also slowed by its own self-induced velocity. The net effect is that V + became the leading vortex and with the vortex pair directed downward toward the axis. The effect of the stopping vortices is also clearly visible in the primary vortex behavior following jet pulse termination. Figure 20 shows the centroid x-location difference for the primary vortices (xC p − xCn , xC p is for L the positive vortex and xCn is for the negative vortex) for R R = 0.05 and R = 5, 10. Note that x C p − x Cn > 0 indicates a vortex pair angled downward toward the axis and vice versa. As shown in Fig. 20a, there was a L rapid change of the centroid relative x-location of the vortices soon after jet termination ( ttp = 1) for R = 5, L = 10, there illustrating effect of the stopping vortices, which form quickly following jet termination. For R was no significant change in xC p − xCn for some time, so the stopping vortices had a diminished effect when L R increased (Fig. 20b).

Formation and behavior of counter-rotating vortex rings

1.2

1.15

1.15

1.1

1.1

1.05

1.05

r/R

r/R

1.2

1

1

0.95

0.95

0.9

0.9

0.85

0.85

0.8

0.8 0

0.1

0.2

0.3

0.4

0

0.1

x/R

0.2

0.3

0.4

0.3

0.4

x/R

(a)

(b)

1.15

1.15

1.1

1.1

1.05

1.05

r/R

1.2

r/R

1.2

1

1

0.95

0.95

0.9

0.9

0.85

0.85

0.8

0.8 0

0.1

0.2

0.3

0.4

0

0.1

x/R

0.2 x/R

(c)

(d) R R

Fig. 17 Flow evolution for = 10, = 0.05 (Case 5). Vorticity contour levels (min = −150, max = 150, increment = 10), dashed lines indicate negative vorticity: a ttp = 1.2; b ttp = 1.4; c ttp = 1.7; (d) ttp = 2 L R

4.4 Vortex Shedding and the Kelvin–Benjamin variational principle in Case 4 Sudden or spontaneous shedding of vorticity similar to that observed in Case 4 has been observed in other vortex ring systems and related to constraints imposed by the Kelvin–Benjamin variational principle [27]. According to this principle, a steady vortex ring (more generally, an axisymmetric vorticity distribution) possesses a local maximum in kinetic energy relative to equivalent rearrangements of its vorticity that maintain the same total impulse. [Although the Kelvin–Benjamin variational principle is formulated for inviscid flows, it has been successfully applied to axisymmetric flows with sufficiently high Reynolds number (e.g., Gharib et al. [28])]. Consistent with this principle, Gharib et al. [28] observed that vortex rings generated by starting jets could not be made arbitrarily large and that the vortex rings stopped forming when the rate at which energy was supplied by the generating jet dropped below a critical value of the dimensionless energy of the leading vortex ring (suggesting further supply of energy would not satisfy the maximal energy requirements of the ring). They further observed that rings which had entrained excess circulation would later reduce it by vortex shedding to increase their dimensionless energy. They suggested that vortex shedding would necessarily occur when the energy of a ring is too low to sustain a steady state.

V. Sadri, P.S. Krueger

1

1

r/R

1.05

r/R

1.05

0.95

0.95

0.9

0.9

-0.05

0

0.05

0.1

-0.05

0.15

(a)

(b)

1

r/R

1

r/R

1.05

0.95

0.95

0.9

0.9

0

0.05 x/R

1.05

-0.05

0

x/R

0.05

0.1

0.15

-0.05

0

0.05

x/R

x/R

(c)

(d)

0.1

0.15

0.1

0.15

L Fig. 18 Flow evolution for R = 5, R R = 0.05 (Case1). Vorticity contour levels (min = −150, max = 150, increment = 10), dashed lines indicate negative vorticity a ttp = 1.2; b ttp = 1.4; c ttp = 1.6; d ttp = 2

V+ SV

-

-

Self-induction

+ Mutual Induction

Effect of stopping vortices

SV+ + Stopping vortices

-

V-

Self-induction

Fig. 19 Illustration of the effect of the stopping vortices on the evolution of the primary vortex rings. Size of vortex cores indicates vortex strength/circulation

0.02

0.02

0.015

0.015

0.01

0.01

(xCp-xCn)/R

(xCp-xCn)/R

Formation and behavior of counter-rotating vortex rings

0.005

0.005

0

0

-0.005

-0.005

-0.01

0

0.5

1

1.5

t/tp

2

-0.01

0

(a) Fig. 20 Centroid x-location difference for

0.5

1

t/tp

1.5

2

(b) R R

= 0.05 a

L = 5, b R = 10

L R

Nitsche [29] investigated vortex ring shedding during the roll-up of an initially spherical vortex sheet using axisymmetric computations. As the initial vortex sheet evolved to form a vortex ring, it shed vorticity which formed another, smaller ring, which in turn shed another ring and so forth, producing an apparently self-similar cascade of vortex rings. Nitsche [29] reported that with each shedding of vortices, the non-dimensional energy of the vortex rings increased and approached a local maximum at which the rings translated steadily. As in Gharib et al. [28], this behavior was attributed to the Kelvin–Benjamin variational principle. These previous results suggest the Kelvin–Benjamin variational principle may govern the spontaneous vorticity shedding observed in Case 4. To investigate further, the total dimensionless energy of the V + V − vortex pair system is defined following Gharib et al. [28] as αt =

(It /ρ)

3/2

E t /ρ    p + |n |

(19)

where E t is total energy of the combined vortex pair, It is total impulse of the vortex pair, and  p and n are circulation of the positive and negative vortices, respectively. The quantities in Eq. (19) are computed using Eqs. (7) and (8). Note that  p and |n | are both positive quantities. Figure 21 shows αt vs. ttp for three difference cases together with vorticity plots to illustrate the character of the vorticity field for different points in the flow evolution. The trend for the cases in Fig. 21b, c is similar. Following shedding of vorticity, αt increases sharply and maintains an approximately constant value, similar to the behavior observed in Nitsche [29]. That is, the observed vorticity shedding appears to be related to the need for the vortices to rearrange their vorticity distribution to achieve maximal energy in order to approach steady behavior, in accordance with the Kelvin–Benjamin variational principle. From this perspective, if the vortex pair has maximal total energy it will travel together (Case 5), otherwise vorticity will be shed (Case 4) to maximize the energy. Following vorticity shedding, the vortices may remain together or separate according to the relative strengths and ultimate evolution of the vortices. Significantly, the “final” αt was reasonably consistent for the results in Fig. 21b, c, achieving a value between 0.35 and 0.45. If it was below 0.3, vortex shedding was observed to increase this value. After shedding, αt maintained a relatively consistent value, indicating the shedding was successful for energy maximization. L For R = 10 (Fig. 21a), on the other hand, the trend for αt was smooth. In this case, the vortex pair was at its maximum energy (above the initial value for the cases in Fig. 21b, c and closer to the maximum value observed in these cases) and at steady state. Therefore, there was no need to shed vorticity and the rings maintained their initial configuration.

V. Sadri, P.S. Krueger

0.5

0.45

0.45

0.4

0.4

t

t

0.5

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2 2

4

6

t/tp

8

10

12

1

1.5

2

2.5

(a)

3

t/tp

3.5

4

4.5

5

(b)

0.5 0.45 0.4

t

0.35 0.3 0.25 0.2 0.15

1

2

3

t/tp

4

5

6

(c) Fig. 21 Non-dimensional energy of the vortex pair for

R R

= 0.05 a

L R

= 10, b

L R

= 15, c

L R

= 20

5 Summary and conclusions This work presents numerical investigation of the interactions of concentric counter-rotating vortex rings. L Several different generating conditions ( R and R R ) were simulated for ReR = 1000. The results showed L and R that the trajectories of the vortex rings strongly depended on both R R through the resulting position of vortices relative to each other and their relative strength during and soon after formation. In all cases, V + was the stronger vortex compared to V − . This difference in formation was related to asymmetry of the vorticity flux relative to the mid-plane of the annulus. An inviscid model was presented to indicate the impact of the circulation ratio between the vortices on whether or not the vortex pair would remain together, if initially coplanar. The model showed that as R R increased, the required circulation ratio became large and the asymmetry in circulation was insufficient for the vortex pair to remain together. For late time, the flow evolution was more L L complex, and five characteristic flow patterns were observed based on both R and R R . When R was less than 8, the self-induced velocity of each vortex dominated and the main vortex pair was separated at later time L for all R R simulated. Also, the stopping vortices had a prominent effect at small R . The stopping vortex interactions disturbed the orientation of the vortex pair (directing it toward the axis) and affected the evolution

Formation and behavior of counter-rotating vortex rings L + − of the flow at the later time. For R R = 0.05 and 8 ≤ R ≤ 10, the mutual interaction between V V overcame the self-induced velocity of the vortices and the vortex pair moved together on an upward (increasing radii) trajectory. In this case, the vortex ring pair becomes qualitatively similar to the two-dimensional Cartesian counterpart, a vortex dipole, and provided a less constrained model for studying vortex dipole dynamics [11]. L By increasing R , initially the vortex pair was directed away from the axis, but due to shedding of vorticity + from V , the mutual interaction between V + V − reoriented the vortex pair trajectory toward the axis, so the self-induced velocity of each vortex came to dominate the flow evolution as the pair approached the axis L and the primary vortex pair separated. By increasing R R to 0.1, with 8 ≤ R ≤ 15, the vortex pair went L larger than 20 straight and remained together longer, but eventually the primary vortices separated. For R R + the same dynamic as the R = 0.05 case was observed and V vortex shedding occurred. The observed V + shedding was related to the need for the flow to rearrange its vorticity distribution to achieve maximal energy in accordance with the Kelvin–Benjamin variational principle for steadily translating ring vortices. Finally, when R R was larger than 0.15, the self-induced velocity of each vortex dominated and the main vortex pair L L simulated. However, for R ≥ 10, the mutual interaction between V + V − sheared was separated for all R − − and split V . The smaller, secondary V looped around the V + and ultimately this newly generated vortex pair was separated. The emphasis of this investigation has been on axisymmetric flow to investigate the long-time vortex interactions. However, it could be expected that at least some of these flows might be susceptible to three-dimensional, non-axisymmetric instability [11]. With respect to the Crow mechanism (long-wavelength instability), Klein et al. [30] and Fabre [31] demonstrated that all counter-rotating pairs are unstable to at least some disturbance L wavelengths. For the extreme case of R R = 0.05 and R = 10, the results presented in Leweke et al. [17] suggest wavelengths as short as five times the separation distance (or λ ∼ R/4) may be unstable, which could be activated by simple asymmetries in the cylinders of the generating mechanism. For larger R, much longer wavelengths are required for instability and the Crow instability may be largely suppressed for R R approaching 0.2 or larger. The elliptic (short wavelength) instability, on the other hand, has a critical Reynolds L number (/ν) near 500 for R R = 0.05 and R = 10 (based on Fig. 17c in Leweke et al. [17]), which exceeds the Reynolds number for this case (/ν ≈ 490). As R R increases, however, the critical Reynolds number decreases and the elliptic instability may become a factor for these cases, at least initially before  decays. Investigation of the effect of these instabilities on the behavior of counter-rotating vortex rings is beyond the scope of the present investigation, but an important topic for future study.

Acknowledgements This material is based on work supported by the National Science Foundation under Grant No. 1133876. This support is gratefully acknowledged.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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