Noncontinuous Froude Number Scaling for the Closure Depth of a Cylindrical Cavity

Gekle, Stephan; Bos, Arjan van der; Bergmann, Raymond; Meer, Devaraj van der; Lohse, Detlef

doi:10.1103/physrevlett.100.084502

Cited by 46 publications

(51 citation statements)

References 17 publications

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“…Continuing the simulation, these nodes eventually form the tip of the top and bottom jets. These numerical simulations have shown excellent agreement with experiments for different impact geometries [4,20] and we verified carefully that our results are independent of numerical parameters such as node density and time stepping. The influence of air is neglected.…”

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confidence: 69%

High-Speed Jet Formation after Solid Object Impact

Gekle

Gordillo

Meer

et al. 2009

Computational Fluid Dynamics 2008

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A circular disc hitting a water surface creates an impact crater which after collapse leads to a vigorous jet. Upon impact an axisymmetric air cavity forms and eventually pinches off in a single point halfway down the cavity. Two fast sharp-pointed jets are observed shooting up-and downwards from the closure location, which by then has turned into a stagnation point surrounded by a locally hyperbolic flow pattern. This flow, however, is not the mechanism feeding the jets. Using high-speed imaging and numerical simulations we show that jetting is fed by the local flow around the base of the jet, which is forced by the colliding cavity walls. We show how the well-known theory of a collapsing void (using a line of sinks on the symmetry axis) can be continued beyond pinch-off to obtain a new and quantitative model for jet formation which agrees well with numerical and experimental data. DOI: 10.1103/PhysRevLett.102.034502 PACS numbers: 47.55.NÀ, 47.11.Hj, 47.55.DÀ, 47.55.df The most prominent phenomenon when a solid object hits a water surface is the high-speed jet shooting upwards into the air. The basic sequence of events leading to this jet has been studied since Worthington over a century ago: After impact, the intruder creates an air-filled cavity in the liquid which due to hydrostatic pressure immediately starts to collapse, eventually leading to the pinch-off of a large bubble. Two very thin jets are ejected up-, respectively, downwards from the pinch-off point. This finite-time singularity has been intensively studied in recent time [1][2][3][4][5]. Such singularities have been shown to lead to a hyperbolic flow pattern after collapse and thus to the formation of liquid jets [6][7][8][9].As we show in the present work, however, the radial energy focusing towards the singular pinch-off point alone is not sufficient to explain the extreme thinness of jets observed after the impact of a solid object. Instead, this jet formation is shown to depend crucially on the kinetic energy contained in the entire collapsing wall of the cavity even far above the pinch-off singularity. This is in contrast to jets observed in many other situations where narrow confining cavity walls are not present, e.g., for bubbles bursting on a free surface or near a solid wall [9][10][11], wave focusing [12,13], or jets induced by pressure waves [14]. In addition, surface tension in our case turns out to be irrelevant in contrast to capillary-driven scenarios as suggested for Faraday waves [7,8]. In all these cases jetting seems thus to be accomplished by a mechanism different from the one in this Letter. In the case of drop impact [15,16] or gas injection through a needle [3,17], however, the formation of a cavity and its subsequent inertial collapse can sometimes be observed and the present mechanism might be of relevance.Our experimental setup consists of a circular disc with radius R 0 that is pulled through a water surface with velocity V 0 as described in [4]. The velocity V 0 is kept constant throughout the whole process. Globa...

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confidence: 69%

High-Speed Jet Formation after Solid Object Impact

Gekle

Gordillo

Meer

et al. 2009

Computational Fluid Dynamics 2008

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“…The numerical details, including the "surface surgery" needed to accurately capture the transition from the cavity collapse process to the jet ejection, are given elsewhere [see Gekle et al (2009a); Bergmann et al (2009)]. These simulations have shown excellent agreement with experimental high-speed recordings and particle image velocimetry measurements [Bergmann et al (2006); Gekle et al (2008Gekle et al ( , 2009a; Bergmann et al (2009)]. The simulation stops when the downward jet hits the disc surface.…”

Section: Disc Impact Simulationssupporting

confidence: 50%

“…The differences pointed out above set our system somewhat apart from similar studies [Duclaux et al (2007); Glasheen & McMahon (1996)]. The experimental realization of the setup to which the numerical simulations presented are referred, is described by Bergmann et al (2006Bergmann et al ( , 2009Gekle et al (2008Gekle et al ( , 2009a, who show that boundary-integral simulations are in excellent agreement with experiments. In addition, potential flow numerical simulations to study of the type of Worthington jets ejected after bubble pinch-off from an underwater nozzle sticking into a quiescent pool of water [Manasseh et al (1998) This paper is organized as follows: In section 2 we present the three different numerical methods used.…”

Section: Introductionmentioning

confidence: 99%

Generation and breakup of Worthington jets after cavity collapse. Part 1. Jet formation

2010

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At the beginning of the last century Worthington and Cole discovered that the high-speed jets ejected after the impact of an axisymmetric solid on a liquid surface are intimately related to the formation and collapse of an air cavity created in the wake of the impactor. In this paper, we combine detailed boundary-integral simulations with analytical modelling to describe the formation of such Worthington jets after the impact of a circular disk on water. We extend our earlier model in Gekle et al. (Phys. Rev. Lett., vol. 102, 2009a, 034502), valid for describing only the jet base dynamics, to describe the whole jet. We find that the flow structure inside the jet may be divided into three different regions: the axial acceleration region, where the radial momentum of the incoming liquid is converted to axial momentum; the ballistic region, where fluid particles experience no further acceleration and move constantly with the velocity obtained at the end of the acceleration region; and the jet tip region, where the jet eventually breaks into droplets. From our modelling of the ballistic region we conclude that, contrary to the case of other physical situations where high-speed jets are also ejected, the types of Worthington jets studied here cannot be described using the theory of hyperbolic jets of Longuet-Higgins (J. Fluid Mech., vol. 127, 1983, p. 103). Most importantly, we find that the velocity and the shape of the ejected jets can be well predicted at any instant in time with the only knowledge of quantities obtained before pinch-off occurs. This fact allows us to provide closed expressions for the jet velocity and the sizes of the ejected droplets as a function of the velocity and the size of the impactor. We show that our results are also applicable to Worthington jets emerging after the collapse of a bubble growing from an underwater nozzle, although this system creates thicker jets than the disk impact.

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“…Figure 10 shows a parametric plot of the solutions just before pinch-off (10a) and an experimental image at the same time (10b). We can improve on this result by using the same axisymmetric boundary integral code that was used in (Bergmann et al 2006;Gekle et al 2008Gekle et al , 2009a) to obtain the undisturbed cavity profile R(z, t) which has been found to be in very good agreement with the experimental results (Bergmann et al 2009b) and again use equation (3.11) to superimpose the effect of the disturbance in exactly the same manner as described above. This procedure gives the shape in figure (10c) which is very similar to the experimental picture, capturing even small details.…”

Section: The Structure Of the Cavitymentioning

confidence: 99%

“…The kinetic energy of the flow is focused into a vanishing small volume with a velocity whose magnitude diverges as the pinch-off moment is approached. Several experimental and theoretical scenarios have been recently considered in the study of this problem: a bubble rising from a capillary (Longuet-Higgins, Kerman & Lunde 1991;Oguz & Prosperetti 1993;Burton, Waldrep & Taborek 2005;Thoroddsen, Etoh & Takehara 2007), bubbles in a co-flowing liquid (Gordillo, Sevilla, Rodríguez-Rodríguez & Martínez-Bazán 2005;Bergmann, Andersen, van der Meer & Bohr 2009a), an initially necked bubble (Eggers, Fontelos, Leppinen & Snoeijer 2007), and cavities created through impact (Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti & Lohse 2006;Gekle, van der Bos, Bergmann, van der Meer & Lohse 2008;Bergmann, van der Meer, Gekle, van der Bos & Lohse 2009b). Depending on the case, the collapse might be initiated by surface tension, external flow, or hydrostatic pressure.…”

Section: Introductionmentioning

confidence: 99%

Collapse and pinch-off of a non-axisymmetric impact-created air cavity in water

et al. 2012

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The axisymmetric collapse of a cylindrical air cavity in water follows a universal power law with logarithmic corrections. Nonetheless, it has been suggested that the introduction of a small azimuthal disturbance induces a long term memory effect, reflecting in oscillations which are no longer universal but remember the initial condition. In this work, we create non-axisymmetric air cavities by driving a metal disc through an initially-quiescent water surface and observe their subsequent gravity-induced collapse. The cavities are characterized by azimuthal harmonic disturbances with a single mode number m and amplitude a m . For small initial distortion amplitude (1 or 2 % of the mean disc radius), the cavity walls oscillate linearly during collapse, with nearly constant amplitude and increasing frequency. As the amplitude is increased, higher harmonics are triggered in the oscillations and we observe more complex pinch-off modes. For small amplitude disturbances we compare our experimental results with the model for the amplitude of the oscillations by Schmidt et al. (2009) and the model for the collapse of an axisymmetric impact-created cavity previously proposed by Bergmann et al. (2009b). By combining these two models we can reconstruct the three-dimensional shape of the cavity at any time before pinch-off. IntroductionThe pinch-off of an axisymmetric air cavity in water is characterized by a finite-time singularity. The kinetic energy of the flow is focused into a vanishing small volume with a velocity whose magnitude diverges as the pinch-off moment is approached. Several experimental and theoretical scenarios have been recently considered in the study of this problem: a bubble rising from a capillary . Depending on the case, the collapse might be initiated by surface tension, external flow, or hydrostatic pressure. However, irrespective of the cause, towards the end it is the inertia of the fluid that takes over in every case, and the collapse is accelerated as the radius of the cavity shrinks.The time it takes each of these systems to reach the inertial collapse regime varies by orders of magnitude . Hence, it was not an easy task to determine whether there was indeed a universal behaviour underlying this phenomenon. The first proposed model was a power law where the radius decreased proportionally to the arXiv:1109.5823v2 [physics.flu-dyn]

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Noncontinuous Froude Number Scaling for the Closure Depth of a Cylindrical Cavity

Cited by 46 publications

References 17 publications

High-Speed Jet Formation after Solid Object Impact

High-Speed Jet Formation after Solid Object Impact

Generation and breakup of Worthington jets after cavity collapse. Part 1. Jet formation

Collapse and pinch-off of a non-axisymmetric impact-created air cavity in water

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