A model for the formation of “optimal” vortex rings taking into account viscosity

Kaplanski, F.; Rudi, Ylo

doi:10.1063/1.1996928

Cited by 48 publications

(62 citation statements)

References 18 publications

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“…While in the present case vortex growth was extended until T*Ϸ8, recent theoretical studies suggest that vortex formation can, in principle, be extended by another 35%, approaching T*Ϸ11 (Kaplanski and Rudi, 2005). This room for improvement may already be occupied by jetters whose performance is known to surpass that of Nemopsis bachei, such as Aglantha digitale (Colin and Costello, 2002).…”

Section: Discussionmentioning

confidence: 53%

Fast-swimming hydromedusae exploit velar kinematics to form an optimal vortex wake

Dabiri

Colin

Costello

2006

Journal of Experimental Biology

114

123

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SUMMARY Fast-swimming hydromedusan jellyfish possess a characteristic funnel-shaped velum at the exit of their oral cavity that interacts with the pulsed jets of water ejected during swimming motions. It has been previously assumed that the velum primarily serves to augment swimming thrust by constricting the ejected flow in order to produce higher jet velocities. This paper presents high-speed video and dye-flow visualizations of free-swimming Nemopsis bachei hydromedusae, which instead indicate that the time-dependent velar kinematics observed during the swimming cycle primarily serve to optimize vortices formed by the ejected water rather than to affect the speed of the ejected flow. Optimal vortex formation is favorable in fast-swimming jellyfish because,unlike the jet funnelling mechanism, it allows for the minimization of energy costs while maximizing thrust forces. However, the vortex `formation number'corresponding to optimality in N. bachei is substantially greater than the value of 4 found in previous engineering studies of pulsed jets from rigid tubes. The increased optimal vortex formation number is attributable to the transient velar kinematics exhibited by the animals. A recently developed model for instantaneous forces generated during swimming motions is implemented to demonstrate that transient velar kinematics are required in order to achieve the measured swimming trajectories. The presence of velar structures in fast-swimming jellyfish and the occurrence of similar jet-regulating mechanisms in other jet-propelled swimmers (e.g. the funnel of squid) appear to be a primary factor contributing to success of fast-swimming jetters, despite their primitive body plans.

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Section: Discussionmentioning

confidence: 53%

Fast-swimming hydromedusae exploit velar kinematics to form an optimal vortex wake

Dabiri

Colin

Costello

2006

Journal of Experimental Biology

114

123

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“…Weigand & Gharib (1997) have shown that an appropriate choice of these constants leads to a close match to (1.3) with their original experimental data and the results of rigorous numerical analysis by Stanaway et al (1988). Both experimental data reported by Weigand & Gharib (1997) and the model by Kaplanski & Rudi (2005) predict the Gaussian distribution of the vorticity in the vortex ring. Also, it was shown that the formulae obtained in the limit of small vortex ring Reynolds numbers can be applicable for the description of vortex rings with realistic values of these numbers (see Fukumoto & Kaplanski 2008).…”

Section: Introductionmentioning

confidence: 85%

“…An approximate, linear first-order solution of the Navier-Stokes equation for the axisymmetric geometry and arbitrary time was reported by Kaltaev (1982), Berezovski & Kaplanski (1995) and Kaplanski & Rudi (1999). Based on this solution, Kaplanski & Rudi (2005) derived an expression for the translational velocity of the vortex ring for arbitrary times. In the limit of small and large times this expression reduces to those described by (1.1) and (1.2) respectively (Kaplanski & Rudi 1999;Fukumoto & Kaplanski 2008).…”

Section: Introductionmentioning

confidence: 98%

“…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%

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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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“…Therefore, there is room for improvement of our understanding of the relationship between the perturbation response of models for isolated vortex rings and dipoles, and the pinch-off phenomenon observed in laboratory Rows and in the field, by considering more realistic models for che vonices. ln two dimensions, more realistic vortex models with continuous distributions of vorticity have been previously studied by Boyd & Ma ( 1990), Kizner & Khvoles (200.t ), Khvoles,Berson & Kizner (200.5 ) and Albrecht, Elcrat & Miller (2011 ), and others; whereas Kaplanski & Rudi (2005) and Fukumoto & Kaplanski (2008) have considered similar models for vortex rings. Unlike the inviscid solutions of Norbury ( 1971) andPierrehumbert ( 1980), these models were viscous and studying their perturbation response required the use of viscous flow solvers.…”

Section: Introductionmentioning

confidence: 99%

Nested contour dynamics models for axisymmetric vortex rings and vortex wakes

O’Farrell

Dabiri

2014

J. Fluid Mech.

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Inviscid models for vortex rings and dipoles are constructed using nested patches of vorticity. These models constitute more rcalislic approximations to experimcncal vortex rings and dipoles than the single-contour models of Norbury and Pierrehumbert, and nested contour dynamics algorithms allow their simulation with low computational cost. In two dimensions, nested-contour models for the analytical Lamb dipole are constructed. In the axisymmetric case, a family of models for vortex rings generated by a piston-cylinder apparatus at different stroke ratios is constructed from experimental data. The perturbation response of this family is considered by the introduction of a small region of vorticity at the rear of the vortex, which mimics the addition of circulation to a growing vortex ring by a feeding shear layer. Model vortex rings are found to either accept the additional circu lation or shed vorticity into a tail, depending on the perturbation size. A change in the behaviour of the model vortex rings is identified at a stroke ratio of three, when it is found that the maximum relative perturbation size vortex rings can accept becomes approximately constant. We hypothesise that this change in response is related to p inch-off, and that pinch-off might be understood and predicted based on the perturbation responses of model vortex rings. In particu lar, we suggest that a perturbation response-based framework can be useful in understanding vortex formation in biological flows.

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A model for the formation of “optimal” vortex rings taking into account viscosity

Cited by 48 publications

References 18 publications

Fast-swimming hydromedusae exploit velar kinematics to form an optimal vortex wake

Fast-swimming hydromedusae exploit velar kinematics to form an optimal vortex wake

A generalized vortex ring model

Nested contour dynamics models for axisymmetric vortex rings and vortex wakes

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