Interaction of Two Vortex Rings along Parallel Axes in Air

Oshima, Yuko; Asaka, Saburo

doi:10.1143/jpsj.42.708

Cited by 72 publications

(31 citation statements)

References 6 publications

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“…To demonstrate that the algorithm is robust enough to handle more complex problems, we also computed an adaptive solution for a viscous vortex ring merger problem, similar to the ones studied computationally in [2,4,17] and experimentally in [5,18]. The initial conditions are two vortex rings which are angled toward each other by an inclination angle φ from horizontal.…”

Section: Vortex Merger Examplementioning

confidence: 99%

A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions

Martín

Colella

Graves

2008

Journal of Computational Physics

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We present a method for computing incompressible viscous flows in three dimensions using block-structured local refinement in both space and time. This method uses a projection formulation based on a cell-centered approximate projection, combined with the systematic use of multilevel elliptic solvers to compute increments in the solution generated at boundaries between refinement levels due to refinement in time. We use an L 0 -stable second-order semi-implicit scheme to evaluate the viscous terms. Results are presented to demonstrate the accuracy and effectiveness of this approach.

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Section: Vortex Merger Examplementioning

confidence: 99%

A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions

Martín

Colella

Graves

2008

Journal of Computational Physics

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“…9(b)). Since then, the reconnection of vortex rings were observed by many scientists, such as Kambe and Takao (1971), Fohl and Turner (1975) and Oshima and Asaka (1977) (see Fig. 9(c)).…”

Section: Reconnection Of Two Vortex Filamentsmentioning

confidence: 92%

Vortex Dynamics

Takaki¹

2015

Forma

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Dynamics of vortices of filament type is introduced for the following three cases: deformation of a vortex filament by self-induction, entanglement of two vortex filaments by self-and mutual inductions and reconnection of two vortex filaments. The first two cases are treated by the use of vorticity equation without the viscous term, while the third one requires a treatment with the effect of viscosity. Key words: Vorticity, Self-Induction, Mutual Induction, Entanglement, Reconnection What is Vortex?The vortex is a swirling motion of fluid, where the angular velocity of the motion is largest in its central region and decreases with the distance from the center. The vortex has been familiar to us since many years, and has been often expressed as artworks such as "Study of water falling into still water" by Leonardo da Vinci and "Panel with red and white apricot flowers" by Korin Ogata (Japanese artist in 18-19 c.). Recently, we often see satellite photos of typhoon, which is a large scale vortex with large angular velocity in the central part, called "eye of a typhoon".In most vortices including the typhoon the fluid velocity u(r ) vanishes at the center, increases linearly with the radius r and then decreases, as is shown in Fig. 1. The angular velocity u(r )/r is nearly uniform in the central region, which corresponds to the eye of a typhoon. On the other hand, the vorticity, which is defined below, vanishes out of this central region. Of course, vortices appearing in the nature receive various disturbances from the environment, and the distributions shown in Fig. 1 are much distorted.The vorticity ω(x) is obtained from a velocity vector u(x) = (u, v, w) by the following differential operation:where ∇ = (∂/∂ x, ∂/∂y, ∂/∂z). The physical meaning of the vorticity is a strength of rotation of a small fluid element around its center. The equation governing the vorticity can be derived from the Navier-Stokes equation (a basic equation governing the dynamics of viscous fluid), and the result is given below (refer some textbooks for its derivation, such as Batchelor (1967) and Imai (1973)).Copyright c Society for Science on Form, Japan.doi:10.5047/forma.2015.s005The second and the third terms in the left-hand side indicate, respectively, convection of the vorticity by the flow field u and increase of vorticity by stretching of vortex by u. The right-hand side indicates the diffusion of the vorticity due to the viscosity. If the fluid is inviscid (non-viscous), there are some conserved quantities in the vortex dynamics. In order to understand the conserved quantities, the Stokes' theorem given below is necessary. Consider a closed smooth surface in the 3D space with its edge denoted by C, then the following equation is satisfied by a vector field u defined in this space:where n is a normal vector on the surface and d is a line element of the edge C. If rot u is replaced by ω, it is rewritten aswhere is called a circulation and indicates a strength of the part of vorticity distribution within the closed curve C, as sho...

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“…A number of laboratory experiments have been performed to study the interaction of two vortex rings moving side by side in the same direction [58][59][60][61][62][63]. As time proceeds, the rings are drawn together and they merge into a single ring that later splits apart again into two rings.…”

Section: Vortex Ring Mergermentioning

confidence: 99%

A Particle Method and Adaptive Treecode for Vortex Sheet Motion in Three-Dimensional Flow

Lindsay¹,

Krasny²

2001

Journal of Computational Physics

142

171

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A particle method is presented for computing vortex sheet motion in three-dimensional flow. The particles representing the sheet are advected by a regularized BiotSavart integral in which the exact singular kernel is replaced by the RosenheadMoore kernel. New particles are inserted to maintain resolution as the sheet rolls up. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor approximation in Cartesian coordinates, and the necessary Taylor coefficients are computed by a recurrence relation. The adaptive features include a divide-andconquer evaluation strategy, nonuniform rectangular clusters, variable-order approximation, and a run-time choice between Taylor approximation and direct summation. Tests are performed to document the treecode's accuracy and efficiency. The method is applied to simulate the roll-up of a circular-disk vortex sheet into a vortex ring. Two examples are presented, azimuthal waves on a vortex ring and the merger of two vortex rings.

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Interaction of Two Vortex Rings along Parallel Axes in Air

Cited by 72 publications

References 6 publications

A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions

A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions

Vortex Dynamics

A Particle Method and Adaptive Treecode for Vortex Sheet Motion in Three-Dimensional Flow

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