Generation and Decay of Viscous Vortex Rings

Kambe, Tsutomu; Oshima, Yuko

doi:10.1143/jpsj.38.271

Cited by 33 publications

(12 citation statements)

References 8 publications

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“…The corrected formula exhibits an excellent agreement with the numerical result at R ⌫ 0 = ⌫ 0 / = 0.01, where ⌫ 0 is being the initial circulation carried by the ring, 19 and is therefore regarded as the lower bound on the translation velocity in the time range ͱ t Ӷ R. Limited by the viscous damping, a vortex ring cannot make an excursion to an indefinitely far place. 15 Our closed-form solution admits the deduction of the maximum traveling distance in a neat form.…”

Section: Introductionmentioning

confidence: 97%

“…13 Phillips' self-similar solution of the Stokes equations for a dipole flow 14 was exploited for a description of the decaying stage ͑ ͱ t ӷ R 0 ͒ and the temporal power law of slowing down of the motion was derived. 15 Rott and Cantwell 16,17 completed this program, by an elaboration of the Helmholtz-Lamb formula, 18 and manipulated the asymptotic drift velocity for a fat vortex ring.…”

Section: Introductionmentioning

confidence: 99%

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Global time evolution of an axisymmetric vortex ring at low Reynolds numbers

Fukumoto

Kaplanski

2008

Physics of Fluids

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An initial-value problem of the Navier-Stokes equation is solved, at small Reynolds numbers, for evolution of an axisymmetric vortex ring. The traveling speed is written down in closed form over the whole time range, in terms of the generalized hypergeometric functions, for a vortex ring starting with infinitely thin core. We make a thorough asymptotic analysis of this solution. Three stages are identified, namely, initial, matured, and decaying stages. At the initial stage when the core is very thin, correction terms are found to Saffman's early-time formula ͓Stud. Appl. Math. 449, 371 ͑1970͔͒. The augmented formula establishes a lower bound on traveling speed of vortex rings starting from delta-function cores and exhibits an excellent agreement with the numerical simulation, at a small Reynolds number, conducted by Stanaway et al. ͑NASA Technical Memorandum No. 101041, 1988͒. At the matured and decaying stages, the traveling speed is found to be closely fitted by Saffman's matured-stage formula, over a very wide time range, by an adjustment of disposable parameters in his formula. The traveling distance as a function of time is also deduced in closed form, and a simple relation of the maximum distance traversed during the whole life, being finite, is found with the viscosity, the initial circulation, and the initial ring radius. The formation number for an optimal vortex ring, estimated based on our solution, compares well with the experiments and numerical simulations.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Global time evolution of an axisymmetric vortex ring at low Reynolds numbers

Fukumoto

Kaplanski

2008

Physics of Fluids

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“…The importance of vortex rings was emphasized by Saffman (1992) who wrote, 'This commonly known phenomenon exemplifies the whole range of problems of vortex motion'. The properties of the vortex rings have been studied for over a century both theoretically and experimentally (Helmholtz 1858;Lamb 1932;Phillips 1956;Norbury 1973;Kambe & Oshima 1975;Saffman 1992;Shariff & Leonard 1992;Lim & Nickels 1995). Recent developments on the modelling side include Stanaway, Cantwell & Spalart (1988), Rott & Cantwell (1993a, b), Mohseni & Gharib (1998), Kaplanski & Rudi (1999, 2005, Fukumoto & Moffatt (2000), Shusser & Gharib (2000), Mohseni (2001Mohseni ( , 2006, Linden & Turner (2001) and Fukumoto & Kaplanski (2008).…”

Section: Introductionmentioning

confidence: 99%

A generalized vortex ring model

et al. 2009

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A conventional laminar vortex ring model is generalized by assuming that the time dependence of the vortex ring thickness is given by the relation = a t b , where a is a positive number and 1/4 6 b 6 1/2. In the case in which a = √ 2ν, where ν is the laminar kinematic viscosity, and b = 1/2, the predictions of the generalized model are identical with the predictions of the conventional laminar model. In the case of b = 1/4 some of its predictions are similar to the turbulent vortex ring models, assuming that the time-dependent effective turbulent viscosity ν * is equal to . This generalization is performed both in the case of a fixed vortex ring radius R 0 and increasing vortex ring radius. In the latter case, the so-called second Saffman's formula is modified. In the case of fixed R 0 , the predicted vorticity distribution for short times shows a close agreement with a Gaussian form for all b and compares favourably with available experimental data. The time evolution of the location of the region of maximal vorticity and the region in which the velocity of the fluid in the frame of reference moving with the vortex ring centroid is equal to zero is analysed. It is noted that the locations of both regions depend upon b, the latter region being always further away from the vortex axis than the first one. It is shown that the axial velocities of the fluid in the first region are always greater than the axial velocities in the second region. Both velocities depend strongly upon b. Although the radial component of velocity in both of these regions is equal to zero, the location of both of these regions changes with time. This leads to the introduction of an effective radial velocity component; the latter case depends upon b. The predictions of the model are compared with the results of experimental measurements of vortex ring parameters reported in the literature.

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“…For √ νt R 0 , (38) approaches Rott-Cantwell's decaying law (8) for which the velocity field is given by Phillips' spherical dipole [45], an exact solution of the Stokes equations. Given an initial delta function core (35), the early-time behavior (5) of the translation speed is common to O(ε), independently of the Reynolds number Γ/ν. At low Reynolds numbers, there is a solution that is valid over the whole time range (t ≥ 0), illustrating how the early time behavior (5) of a thin core crosses over to (8) [33,34].…”

Section: Low-reynolds-number Vortex Ringmentioning

confidence: 99%

Global time evolution of viscous vortex rings

Fukumoto¹,

福本²

2009

Iutam Bookseries

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This article gives an overview of growing knowledge of translation speed of an axisymmetric vortex ring, with focus on the influence of viscosity. Helmholtz-Lamb's method provides a shortcut to manipulate the translation speed at both small and large Reynolds numbers, for a vortex ring starting from an infinitely thin core. The resulting asymptotics significantly improve Saffman's formula (1970) and give closer lower and upper bounds on translation speed in an early stage. At large Reynolds numbers, Kelvin-Benjamin's kinematic variational principle achieves a further simplification. At small Reynolds numbers, the whole life of a vortex ring is available from the vorticity obeying the Stokes equations, which is closely fitted, over a long time, by Saffman's second formula.

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Generation and Decay of Viscous Vortex Rings

Cited by 33 publications

References 8 publications

Global time evolution of an axisymmetric vortex ring at low Reynolds numbers

Global time evolution of an axisymmetric vortex ring at low Reynolds numbers

A generalized vortex ring model

Global time evolution of viscous vortex rings

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