Leapfrogging of multiple coaxial viscous vortex rings

Abstract

A recent theoretical study [Borisov, Kilin, and Mamaev, "The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem," Regular Chaotic Dyn. 18, 33 (2013); Borisov et al., "The dynamics of vortex rings: Leapfrogging in an ideal and viscous fluid," Fluid Dyn. Res. 46, 031415 (2014)] shows that when three coaxial vortex rings travel in the same direction in an incompressible ideal fluid, each of the vortex rings alternately slips through (or leapfrogs) the other two ahead. Here, we use a lattice Boltzmann method to simulate viscous vortex rings with an identical initial circulation, radius, and separation distance with the aim of studying how viscous effect influences the outcomes of the leapfrogging process. For the case of two identical vortex rings, our computation shows that leapfrogging can be achieved only under certain favorable conditions, which depend on Reynolds number, vortex core size, and initial separation distance between the two rings. For the case of three coaxial vortex rings, the result differs from the inviscid model and shows that the second vortex ring always slips through the leading ring first, followed by the third ring slipping through the other two ahead. A simple physical model is proposed to explain the observed behavior. C 2015 AIP Publishing LLC.Vortex rings have been a subject of interest in vortex dynamics because of their frequent appearance in nature and technology. 1-4 Moreover, they can be easily produced experimentally. 5 The study of the interaction of multiple vortex rings has also attracted considerable attention 4-9 because of its relevance to the fundamental understanding of vortex-vortex interaction that occurs in many unsteady flow phenomena such as jets and plumes.Past studies 1-15 have shown that when two identical coaxial vortex rings travel in the same direction, it could lead to a process often referred to as leapfrogging of vortex rings. During this process, the induced velocity of the leading ring causes the rear ring to contract radially and accelerate, and concurrently, the induced velocity of the rear ring causes the leading ring to expand and slow down. The rear ring eventually catches up with the leading ring and slips through it, and the process repeats until they merge into a single ring. Although the leapfrogging of two vortex rings has been extensively investigated experimentally, numerically, and theoretically for many years (see an excellent review by Meleshko 6 ), the case involving more than two vortex rings has received little attention. In a recent theoretical study based on topological approach and concept of a bifurcation, complex leapfrogging of three inviscid vortex rings was reported for the first time. 16 The result reveals that each of the vortex rings alternately slips through the other two rings ahead. They also showed the similar leapfrogging of three viscous vortex rings with the same initial circulation, different radii, and different separation distances by solving the axisymmetric Navier-Stokes equations. 17 The...

On the existence of leapfrogging pair of circular vortex filaments

“…We prove that (iii) implies (iv) and vice versa. From (12), we see that ( , 0) takes its minimum value at = * in the interval ( , ∞). We also see from (12) and (13) that there is a unique ∈ (0, ) such that…”

“…Gurzhii et al, Boyarintsev et al, and Shariff et al gave theoretical observations as well as numerical calculations of the interaction of coaxial vortex rings. Riley and Stevens, Nakanishi et al, Lim, and Cheng et al gave further numerical studies of the leapfrogging rings, and Maxworthy, Oshima, Oshima et al, and Yamada and Matsui conducted laboratory experiments to study the leapfrogging phenomenon of interacting vortex rings. Notably, although the leapfrogging phenomenon was theoretically observed for a long time, the first experiment that successfully provided photographic proofs of the leapfrogging phenomenon in a laboratory setting was the one conducted by Yamada and Matsui in 1978.…”

A fast lattice Green's function method for solving viscous incompressible flows on unbounded domains

“…The time rate of change of K is related to E by the relationship d dt K = −2νE. Differences in the time history of d dt K between different numerical simulations of the same flow are commonly used to characterize the accuracy of solutions of unsteady flows [44][45][46]. In the absence of viscosity, the helicity, J, is an invariant of the flow and provides a measure for the degree of linkage of the vortex lines of the flow [47].…”

Collinear interaction of vortex pairs with different strengths—Criteria for leapfrogging

“…The rear ring then passes through the leading ring, with this process of 'leapfrogging' then repeating again and again [23]. We note that interactions of such vortex rings -primarily with filament cores -have been considered in Hicks [20] and more recently in Borisov, Kilin, and Mamaev [13], in Caplan, Talley, Carretero-González, and Kevrekidis [14], in Cheng, Lou, and Lim [15] and in Aiki [4].…”