A long, smooth cylinder is dragged through a water surface to create a cavity
with an initially cylindrical shape. This surface void then collapses due to
the hydrostatic pressure, leading to a rapid and axisymmetric pinch-off in a
single point. Surprisingly, the depth at which this pinch-off takes place does
not follow the expected Froude$^{1/3}$ power-law. Instead, it displays two
distinct scaling regimes separated by discrete jumps, both in experiment and in
numerical simulations (employing a boundary integral code). We quantitatively
explain the above behavior as a capillary waves effect. These waves are created
when the top of the cylinder passes the water surface. Our work thus gives
further evidence for the non-universality of the void collapse

Arjan van der Bos

Detlef Lohse

Devaraj van der Meer

F. Cioffi

G. I. Barenblatt

Raymond Bergmann

Stephan Gekle

English

arXiv

Crossref

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

A long, smooth cylinder is dragged through a water surface to create a cavity with an initially cylindrical shape. This surface void then collapses due to the hydrostatic pressure, leading to a rapid and axisymmetric pinch-off in a single point. Surprisingly, the depth at which this pinch-off takes place does not follow the expected Froude1/3 power law. Instead, it displays two distinct scaling regimes separated by discrete jumps, both in experiment and in numerical simulations (employing a boundary integral code). We quantitatively explain the above behavior as a capillary wave effect. These waves are created when the top of the cylinder passes the water surface. Our work thus gives further evidence for the nonuniversality of the void collapse

Gekle, Stephan

van der Bos, Arjan

Bergmann, Raymond

van der Meer, Deveraj

Lohse, Detlef

NARCIS 

University of Twente Research Information

Noncontinuous Froude Number Scaling for the Closure Depth of a Cylindrical CavityStephan Gekle, Arjan van der Bos, Raymond Bergmann, Devaraj van der Meer, and Detlef LohsePhysics of Fluids Group and J. M. Burgers Centre for Fluid Dynamics, University of Twente,P.O. Box 217, 7500AE Enschede, The Netherlands(Received 18 December 2006; revised manuscript received 20 July 2007; published 27 February 2008)A long, smooth cylinder is dragged through a water surface to create a cavity with an initiallycylindrical shape. This surface void then collapses due to the hydrostatic pressure, leading to a rapid andaxisymmetric pinch-off in a single point. Surprisingly, the depth at which this pinch-off takes place doesnot follow the expected Froude1=3 power law. Instead, it displays two distinct scaling regimes separated bydiscrete jumps, both in experiment and in numerical simulations (employing a boundary integral code).We quantitatively explain the above behavior as a capillary wave effect. These waves are created when thetop of the cylinder passes the water surface. Our work thus gives further evidence for the nonuniversalityof the void collapse.DOI: 10.1103/PhysRevLett.100.084502 PACS numbers: 47.55.D, 47.11.Hj, 47.35.PqMany phenomena in fluid dynamics are known to beself-similar [1] and universal, allowing physicists to de-scribe their final outcome without precise knowledge of theinitial conditions. Prime examples for such universality arethe breakup of an elongated fluid filament inside anotherviscous fluid [2] and the pinch-off of a liquid droplet in air[3–5]. For the inverse problem [6–10], i.e., when an airbubble pinches off inside a liquid, the dynamics retains amemory of its creation until the very end, indicating non-universality. As an example for such a breakup, we exam-ine the air-filled cavity created when a solid object israpidly submerged through a water surface. The walls ofthe cavity subsequently collapse due to hydrostatic pres-sure from the liquid bulk. When the colliding walls meet, aviolent jet shoots up into the air. Regardless of the non-universality of the pinch-off [7], the location at which ittakes place has been reported (experimentally and theo-retically) to scale in a continuous fashion with the objectvelocity for such different systems as spheres on preflui-dized sand [11], solid disks [12], spheres and cylinders [13]on water, and even water columns on water [14]. Ourexperimental and numerical evidence shows the lowerlimit where this universal scaling is broken through theinterference of a second phenomenon unrelated to hydro-static pressure. Surface waves created as the object passesthe water surface significantly alter the pinch-off locationin a noncontinuous manner. Similar effects for the break-down of a universal behavior due to wave interaction havebeen observed in, e.g., magnetohydrodynamics [15] andturbulence [16].In our experiment we drag a cylinder with radius R0 20 mm and length l  147 mm through the surface of alarge water tank using a linear motor connected to thecylinder bottom by a rod. We prescribe a constant cylindervelocity V between 0.5 and 2:5 m=s. With the kinematicviscosity  the global Reynolds number Re  R0V= is ofthe order of 104, while the local Reynolds number Re R _R= defined with the cavity radius R for the point ofminimum radius lies between 102 and 105, demonstratingthat inertia dominates viscous effects. Further, with thesurface tension coefficient , gR0  =R0; i.e., gravitydominates over surface tension. The relevant dimension-less parameter is thus the Froude number Fr  V2=R0gwith g  9:81 m=s2, which in our experiment ranges be-tween 1.2 and 32.The shape of the axisymmetric cavity is imaged with ahigh-speed camera at up to 10; 000 frames= sec , with thevertical coordinate z pointing upwards along the cylinderaxis and r being the radial coordinate. Figure 1 shows atypical sequence of the cavity dynamics. We choose thestarting position of the lower edge of the cylinder slightlybelow the water surface to suppress the splash. From thereit is pulled downwards with high acceleration such that ithas reached its prescribed speed before the top passes thewater surface at t  0. The submerging cylinder creates anair-filled cavity with an initially cylindrical shape whoseside walls immediately start to collapse 1(a). Magnificationof the cavity walls as in 1(b) shows that they are not smoothsurfaces, but exhibit pronounced ripples. Note that theseripples are of different origin than those observed afterpinch-off [17].From the high-speed video images we extract the closuredepth of the cavity zc for cylinder velocities up to 2:5 m=s.Plotting the closure depth over the Froude number as inFig. 2, we find two asymptotic regimes. The regime for lowFroude numbers obeys a scaling behavior zc=R0  Frwith an exponent   1  0:1. This is in contrast toearlier experiments [13], which showed a single continu-ous scaling behavior with   13 , for which a theoreticalexplanation in terms of inertia and gravity could be given[11,13,14]. At the end of the low Froude regime, the dataabruptly depart to an intermediate regime without anyexperimentally clearly discernible structure. After a pro-nounced jump we find a third regime, which due to experi-mental limitations can be observed only betweenFr  22:4 and 35.5. While this is too short to ascribe aPRL 100, 084502 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending29 FEBRUARY 20080031-9007=08=100(8)=084502(4) 084502-1 © 2008 The American Physical Societydefinite power law behavior, we can nevertheless concludethat this last regime—in contrast to the preceding tworegimes—is not disagreeing with the theoretically pre-dicted value of a scaling exponent of 1=3.To understand the underlying mechanism leading to thisdiscontinuous behavior, we conducted boundary integralsimulations. Our numerical results match very well withthe high-speed videos without the use of any adjustableparameter; see Fig. 1. The agreement of the wave pattern inFig. 1(b) can be even further improved when at the expenseof introducing a free parameter we allow for a deviationfrom the precise 90 angle between the cylinder wall andthe free surface, accounting for the free surface beingdragged down and resulting in a slightly curved profileclose to the cylinder wall.The numerical simulations allow us to study the ripplesin Fig. 1(b) in great detail. As the cylinder top passes thewater surface, the rectangular corner between the cylinderwall and the water surface is no longer held in place by thesolid boundary of the cylinder. The newly created freesurface thus possesses a corner with very high (initiallyinfinite) curvature. Surface tension immediately tries toflatten this surface by pulling the corner diagonally in-wards into the fluid bulk. This results in a shock similarto throwing a stone onto a lake, which consequently leadsto the formation of capillary waves traveling out- anddownward on the free surface. The downward waves canbe observed in Fig. 1(b). The shock creates a wave packetcontaining waves of different frequencies. Each of thesewaves spreads with a velocity c  !=k given by the dis-persion relation !2  =k3, where we assume planecapillary waves (since kR 1 during all but the very rapidcollapse at the end of the cavity evolution) and ! desig-nates the angular frequency, k the wave vector, and  thedensity of the liquid.Using stationary phase approximation, we calculate thedominant wave vector k	 at a given distance x from thesource at time t: k	  23 x=t2=. Our simulations allowfor an accurate estimate of the dominant wavelength from the shape of the surface; cf. Fig. 3(a). Comparisonwith the above equation as in the inset of Fig. 3(a) confirmsthat the observed ripples on the surface are indeed capillarywaves originating from the corner point as the cylinder toppasses the water surface. The damping for capillary wavescan be estimated as tdamp  1=k2 obtaining, with awavelength of the order of 6 mm [see Fig. 1(b)], a damping0 0.5 1 1.500.050.10.150.20.250.30.350.4α1=0.1αtheory = 1/3log10(Fr)log 10(z c/R0)low intermediate highFIG. 2 (color online). The experimental closure depth as afunction of the Froude number. The asymptotic regime for lowFroude numbers scales with 1  0:1 (red solid line). Only inthe limit of high Froude numbers does the data not contradict anexponent of 13 (blue dashed line). For the intermediate regime theexperiments show no systematic behavior.1cmt = 36.9ms(a)1cm(b)1cmt = 53.3ms(c)1cmt = 51.5ms(d)1cmt = 51.9ms(e)1cmt = 51.7ms(f)1cmt = 51.8ms(g)FIG. 1 (color online). (a)–(c) Snapshots of the cavity for Fr  11:5. The area designated by the yellow rectangle in (a) is magnifiedin (b) illustrating the rippled surface of the cavity walls. Red lines show the numerical simulation. In (a) and (b) video frames areconnected to simulation time by matching the cylinder position; for (c) it was more convenient to match the cavity closure timesinstead. (d)–(g) The transition from one closing ripple to the other. At Fr  18:8, (d) and (e), the upper of the two marked ripplescloses. At Fr  20:6, (f) and (g), it is the lower of the two ripples that pinches.PRL 100, 084502 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending29 FEBRUARY 2008084502-2time of 900 ms, which is well beyond the closure time ofthe cavity.The closing of the cavity is driven by hydrostatic pres-sure acting on every point of the free surface accelerating itinward as soon as the cylinder has passed. This accelerat-ing force increases with the depth. Thus, points near the topsurface start moving early with a small acceleration, whiledeeper points start with increasing delay, but higher accel-eration [11,13]. On a rippled surface this process favors thewave crests over the other points. The resulting closuredepth will thus be determined by a combination of (i) hy-drostatic pressure induced by gravity and (ii) capillarywaves created by surface tension.The numerically obtained closure depth shown in Fig. 4reasonably reproduces the experimental results. The finalregime scales with 2  0:43. Considering that the theoryfrom [11,13,14] assumes a purely radial flow pattern and aninitially perfectly cylindrical cavity shape, the predictionof 13 is reasonably close to our observed exponent. Oursimulations even make the identification of an intermediateregime possible, which due to its small range in Froudenumbers cannot be clearly observed in experiments. Theshift between numerical and experimental data on theFroude axis can be attributed to the fragility of the mea-surement process in and around the intermediate regime aswell as to small contaminations that lower the surfacetension of the water.The insets in Fig. 4 elucidate precisely how the capillarywaves lead to the discontinuous jumps between the differ-ent regimes of the closure depth: For Froude numbers nearthe transition point three different local minima of theradius come very close to meeting their counterpart onthe opposite side. In the first regime the uppermost of thethree minima closes first and thus determines the closuredepth, while in the second regime the one located in themiddle is the fastest to reach the central axis. Finally, whenthe lowest minimum closes before the other two, the thirdregime is attained. Figures 1(d)–1(g) show experimentalphotographs of this effect at the transition point from theintermediate to the high Froude regime.Since the behavior of the capillary waves is determinedby surface tension, one expects that modifications of thesurface tension coefficient  should significantly alter theclosure depth in the first and any intermediate regimes. Forthe last regime capillary waves are irrelevant and the scal-ing behavior of the closure depth can be derived indepen-dent of surface tension [11,13,14]. We have performedsimulations with a tenfold increase and decrease of  ascompared to the natural value of 72:8 mN=m for water aswell as for a hypothetical liquid without any surface ten-sion (  0). As Fig. 5 demonstrates, the Froude numberranges for the three regimes are found to depend indeedstrongly on the value of the surface tension coefficient. Thelength of the first regime significantly enlarges for a highersurface tension coefficient. As expected, the last regime isalmost uninfluenced by changes in surface tension. For lowsurface tension merely the two asymptotic regimes existand the intermediate regime is not observed anymore. Thelimiting case completely without surface tension possesses0 0.5 1 1.5 2 2.5 300.10.20.30.40.50.60.70.80.91log10(Fr)log 10(z c/R0)α1 = 0.08α2 = 0.43−4 −2 0 2 4−40−35−30−25r [mm]z [mm]FIG. 4 (color online). Comparison of the experimental closuredepth (black crosses) with the numerical data (blue diamonds).The insets illustrate the shape of the cavity at pinch-off for onerepresentative of each regime (axes are the same for all insets).The regimes are determined by which local minimum first meetsits counterpart on the central axis.18 20 22 24 26 28−8−6−4−20r [mm]z [mm]t = 5.35ms(a)2 6 100123x [mm]λ [mm]FIG. 3 (color online). (a) Capillary waves for V  3 m=s(Fr  45:9) as obtained from the simulation. Wave crests aremarked by red circles and allow an estimate of the dominantwavelength   2=k at a given position. The inset comparesthese wavelengths for the downward (blue circles) and outward(black crosses) waves to the theoretically expected behavior forcapillary waves (red line). (b) The natural cavity dynamics (redsolid) as a superposition of two hypothetical settings: withoutsurface tension (black dashed) and without gravity (blue dotted)for V  1 m=s (Fr  5:1).PRL 100, 084502 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending29 FEBRUARY 2008084502-3only one regime in reasonable agreement with the theo-retical Fr1=3 scaling.With x  zc and k	  1=zc we estimate the relevant timescale for the capillary waves as tw =pz3=2c while thetime scale for cavity closure is tc  zc=V. The onset of thehigh Froude number regime is now readily found by equat-ing the ratio tw=tc to a constant of order 1. Introducing theBond number Bo  gR20= and making use of the theo-retically expected scaling zc=R0  Fr1=3, this yieldsFrtrans  Bo3=4, which is in good agreement with ournumerical observations as shown in the inset of Fig. 5.The cavity dynamics completely without surface tensionand thus deprived of all capillary waves is illustrated by theblack dashed lines in Fig. 3(b). The blue dotted lines depictthe evolution of a free surface starting with a (nearly)rectangular corner without gravity, allowing us to studythe formation of capillary waves in an isolated setting. Thecavity dynamics under realistic conditions (g  9:81 m=s2and   72:8 mN=m) is shown in red and can clearly beidentified as a superposition of the above-mentioned limit-ing cases. In the first instants, the real dynamics is almostidentical to the one without gravity with its main featurebeing the capillary waves. Later, hydrostatic pressure be-comes more important until finally the cavity approachesthe shape of the pure gravity simulation with the capillarywaves superposed on the cavity walls.In conclusion, we have shown that capillary wavescreated when a submerging object passes the water surfacehave a strong and lasting influence on the dynamics of thecavity. This influence remains observable until the very endof the cavity collapse. manifesting itself in clearly distinctregimes of the closure depth as a function of the submerg-ing velocity. We have thus illustrated the lower limit to thecontinuous inertial-gravitational scaling regime observedin [11–14]. Since capillary waves are an unavoidable con-sequence of disturbances on a water surface, we expect thatthe effects elaborated in this Letter will be of relevance to awide range of related phenomena.We thank L. van Wijngaarden and A. Prosperetti fordiscussions. This work is part of the program of theStichting FOM, which is financially supported by NWO.[1] G. I. Barenblatt, Scaling, Self-similarity, and IntermediateAsymptotics (Cambridge University Press, Cambridge,England, 1996).[2] M. Tjahjadi, H. A. Stone, and J. M. Ottino, J. Fluid Mech.243, 297 (1992).[3] M. P. Brenner, X. D. Shi, and S. R. Nagel, Phys. Rev. Lett.73, 3391 (1994).[4] J. Eggers, Rev. Mod. Phys. 69, 865 (1997).[5] P. Doshi et al., Science 302, 1185 (2003).[6] J. C. Burton, R. Waldrep, and P. Taborek, Phys. Rev. Lett.94, 184502 (2005).[7] R. Bergmann, D. van der Meer, M. Stijnman, M. Sandtke,A. Prosperetti, and D. Lohse, Phys. Rev. Lett. 96, 154505(2006).[8] J. M. Gordillo and M. Pérez-Saborid, J. Fluid Mech. 562,303 (2006).[9] N. C. Keim, P. Møller, W. W. Zhang, and S. R. Nagel,Phys. Rev. Lett. 97, 144503 (2006).[10] J. Eggers, M. A. Fontelos, D. Leppinen, and J. H. Snoeijer,Phys. Rev. Lett. 98, 094502 (2007).[11] D. Lohse, R. Bergmann, R. Mikkelsen, C. Zeilstra, D. vander Meer, M. Versluis, K. van der Weele, M. van der Hoef,and H. Kuipers, Phys. Rev. Lett. 93, 198003 (2004).[12] J. W. Glasheen and T. A. McMahon, Phys. Fluids 8, 2078(1996).[13] V. Duclaux, F. Caillé, C. Duez, C. Ybert, L. Bocquet, andC. Clanet, J. Fluid Mech. 591, 1 (2007).[14] H. N. Oguz, A. Prosperetti, and A. R. Kolaini, J. FluidMech. 294, 181 (1995).[15] A. Basu and J. K. Bhattacharjee, J. Stat. Mech. (2005)P07002.[16] F. Cioffi, F. Gallerano, and G. Troiani, J. Hydraul. Res. 44,155 (2006).[17] T. Grumstrup, J. B. Keller, and A. Belmonte, Phys. Rev.Lett. 99, 114502 (2007).0 0.5 1 1.5 2 2.5 300.10.20.30.40.50.60.70.80.91α1 ≈ 0.47α2 ≈ 0.43α3 ≈ 0.39α4 ≈ 0.40log10(Fr)log 10(z c/R0)0 1 2 30.511.522.5log10(Bo)log 10(Fr trans)FIG. 5 (color online). The numerical closure depth as a func-tion of the Froude number and corresponding scaling exponentsfor different surface tension coefficients: 1  728 mN=m(black circles), 2  72:8 mN=m (blue diamonds), 3 7:28 mN=m (red crosses), and 4  0 (magenta squares). For  0 no data at low Froude numbers are shown, since axialvelocity components during the collapse prohibit the applicationof the Fr1=3 theory. The inset shows the onset of the high Froudenumber regime as a function of the Bond number with the redline depicting the expected scaling law Frtrans  Bo3=4.PRL 100, 084502 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending29 FEBRUARY 2008084502-4

https://research.utwente.nl/en/publications/672a59bc-7567-4c1f-abd7-3a3e4c22efc2

Non-continuous Froude number scaling for the closure depth of a
  cylindrical cavity

Non-continuous Froude number scaling for the closure depth of a cylindrical cavity

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