Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-25T05:11:40.331Z Has data issue: false hasContentIssue false
  • English
  • Français

Jetting in finite-amplitude, free, capillary-gravity waves

Published online by Cambridge University Press:  17 December 2020

Saswata Basak ,
Palas Kumar Farsoiya  and
Ratul Dasgupta Open the ORCID record for Ratul Dasgupta [Opens in a new window]
Saswata Basak
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai400 076, India
Palas Kumar Farsoiya
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai400 076, India
Ratul Dasgupta*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai400 076, India
*
†Email address for correspondence: dasgupta.ratul@iitb.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a theoretical and computational study of the mechanics of a jet formed from a large amplitude, axisymmetric, capillary-gravity wave on the surface of a liquid pool in a cylindrical container. Jetting can cause a pronounced overshoot of the interface at the axis of symmetry. A linear theory presented earlier in Farsoiya et al. (J. Fluid Mech., vol. 826, 2017, pp. 797–818) was shown to be incapable of describing this jet. To understand its mechanics, we present here the inviscid, weakly nonlinear solution to the initial value problem where the initial surface perturbation is a single Fourier–Bessel mode on quiescent fluid. The theory predicts that energy injected into a primary (Bessel) mode initially is transferred nonlinearly to a spectrum of modes. The extent of the theoretically predicted energy transfer is found to be very accurate for modes up to the second harmonic. We show using numerical simulations that the jet originates as a small dimple formed at the trough of the wave, analogous to similar observations in bubble cavity collapse (Duchemin et al., Phys. Fluids, vol. 14, 2002, pp. 3000–3008; Lai et al., Phys. Rev. Lett. vol. 121, 2018, 144501). The theory is able to describe the jet overshoot and the velocity and pressure fields in the liquid qualitatively, but does not capture the temporal evolution of the dimple or the thinning of the jet neck leading to pinchoff. Modal analysis shows that the latter phenomenon requires higher-order approximations, beyond the second order presented here. The nonlinear theory yields explicit analytical expressions without any fitting parameters which are systematically tested against numerical simulations of the incompressible Euler equation. The theory contains cylindrical analogues of the singularities corresponding to second harmonic resonance (Wilton, Lond. Edinb. Dubl. Phil. Mag. J. Sci., vol. 29, 1915, pp. 688–700). The connection of these to triadic resonant interactions among capillary-gravity waves in a cylindrical confined geometry is discussed.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 909 , 25 February 2021 , A3
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Agbaglah, G., Delaux, S., Fuster, D., Hoepffner, J., Josserand, C., Popinet, S., Ray, P., Scardovelli, R. & Zaleski, S. 2011 Parallel simulation of multiphase flows using octree adaptivity and the volume-of-fluid method. CR Mecanique 339 (2–3), 194–207.CrossRefGoogle Scholar
Antkowiak, A., Bremond, N., Le Dizès, S. & Villermaux, E. 2007 Short-term dynamics of a density interface following an impact. J. Fluid. Mech. 577, 241–250.CrossRefGoogle Scholar
Bagrov, V. G, Belov, A. A., Zadorozhyni, V. N. & Trifonov, A. Yu. 2012 Methods of mathematical physics-special functions. http://portal.tpu.ru:7777/SHARED/a/ATRIFONOV/eng/academics/Tab3/FTI\_Bagrov\_Belov\_Zadorozhnyi\_Trifonov\_EMathPh-1e.pdf.Google Scholar
Bartolo, D., Josserand, C. & Bonn, D. 2006 Singular jets and bubbles in drop impact. Phys. Rev. Lett. 96 (12), 124501.CrossRefGoogle ScholarPubMed
Basak, S., Farsoiya, P. & Dasgupta, R. 2020 Nonlinear capillary-gravity wave. http://basilisk.fr/sandbox/farsoiya/capillary\_gravity\_jetting.c, [Online; accessed 12-June-2020].Google Scholar
Bell, J. B, Colella, P. & Glaz, H. M. 1989 A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85 (2), 257–283.CrossRefGoogle Scholar
Blake, J. R. & Gibson, D. C. 1981 Growth and collapse of a vapour cavity near a free surface. J. Fluid Mech. 111, 123–140.CrossRefGoogle Scholar
Blanchard, D. C. 1963 The electrification of the atmosphere by particles from bubbles in the sea. Prog. Oceanogr. 1, 73–202.CrossRefGoogle Scholar
Blanchard, D. C. 1989 The size and height to which jet drops are ejected from bursting bubbles in seawater. J. Geophys. Res.: Oceans 94 (C8), 10999–11002.CrossRefGoogle Scholar
Blanchard, D. C. 2004 From Raindrops to Volcanoes: Adventures with Sea Surface Meteorology. Courier Corporation.Google Scholar
Blanchard, D. C. & Syzdek, L. D. 1972 Concentration of bacteria in jet drops from bursting bubbles. J. Geophys. Res. 77 (27), 5087–5099.CrossRefGoogle Scholar
Blanco-Rodríguez, F. J. & Gordillo, J. M. 2020 On the sea spray aerosol originated from bubble bursting jets. J. Fluid Mech. 886, R2.CrossRefGoogle Scholar
Boulton-Stone, J. M. & Blake, J. R. 1993 Gas bubbles bursting at a free surface. J. Fluid Mech. 254, 437–466.CrossRefGoogle Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.CrossRefGoogle Scholar
Castillo-Orozco, E., Davanlou, A., Choudhury, P. K. & Kumar, R. 2015 Droplet impact on deep liquid pools: Rayleigh jet to formation of secondary droplets. Phys. Rev. E 92 (5), 053022.CrossRefGoogle ScholarPubMed
Cheny, J. M. & Walters, K. 1996 Extravagant viscoelastic effects in the worthington jet experiment. J. Non-Newtonian Fluid Mech. 67, 125–135.CrossRefGoogle Scholar
Christodoulides, P. & Dias, F. 1994 Resonant capillary–gravity interfacial waves. J. Fluid Mech. 265, 303–343.CrossRefGoogle Scholar
Das, S. P. & Hopfinger, E. J. 2008 Parametrically forced gravity waves in a circular cylinder and finite-time singularity. J. Fluid Mech. 599, 205–228.CrossRefGoogle Scholar
Dasgupta, R., Tomar, G. & Govindarajan, R. 2015 Numerical study of laminar, standing hydraulic jumps in a planar geometry. Eur. Phys. J. E 38 (5), 45.CrossRefGoogle Scholar
Debnath, L. 1994 Nonlinear Water Waves. Academic Press.Google Scholar
Deike, L., Ghabache, E., Liger-Belair, G., Das, A. K, Zaleski, S., Popinet, S. & Séon, T. 2018 Dynamics of jets produced by bursting bubbles. Phys. Rev. Fluids 3 (1), 013603.CrossRefGoogle Scholar
Dhar, M., Das, G. & Das, P. K. 2020 Planar hydraulic jumps in thin film flow. J. Fluid Mech. 884, A11.CrossRefGoogle Scholar
Duchemin, L., Popinet, S., Josserand, C. & Zaleski, S. 2002 Jet formation in bubbles bursting at a free surface. Phys. Fluids 14 (9), 3000–3008.CrossRefGoogle Scholar
Farsoiya, P. K., Mayya, Y. S. & Dasgupta, R. 2017 Axisymmetric viscous interfacial oscillations–theory and simulations. J. Fluid Mech. 826, 797–818.CrossRefGoogle Scholar
Fultz, D. & Murty, T. S. 1963 Experiments on the frequency of finite-amplitude axisymmetric gravity waves in a circular cylinder. J. Geophys. Res. 68 (5), 1457–1462.CrossRefGoogle Scholar
Fuster, D. & Popinet, S. 2018 An all-mach method for the simulation of bubble dynamics problems in the presence of surface tension. J. Comput. Phys. 374, 752–768.CrossRefGoogle Scholar
Gañán-Calvo, A. M. 2017 Revision of bubble bursting: universal scaling laws of top jet drop size and speed. Phys. Rev. Lett. 119 (20), 204502.CrossRefGoogle ScholarPubMed
Gañán-Calvo, A. M. 2018 a Gañán-calvo replies. Phys. Rev. Lett. 121 (26), 269402.CrossRefGoogle ScholarPubMed
Gañán-Calvo, A. M. 2018 b Scaling laws of top jet drop size and speed from bubble bursting including gravity and inviscid limit. Phys. Rev. Fluids 3 (9), 091601.CrossRefGoogle Scholar
Ganán-Calvo, A. M. & Lopez-Herrera, J. M. 2019 Capillary soft singularities and ejection: application to the physics of bubble bursting. arXiv:1911.08844.Google Scholar
Gekle, S. & Gordillo, J. M. 2010 Generation and breakup of worthington jets after cavity collapse. Part 1. Jet formation. J. Fluid Mech. 663, 293–330.CrossRefGoogle Scholar
Gekle, S., Gordillo, J. M., van der Meer, D. & Lohse, D. 2009 High-speed jet formation after solid object impact. Phys. Rev. Lett. 102 (3), 034502.CrossRefGoogle ScholarPubMed
Ghabache, E., Antkowiak, A., Josserand, C. & Séon, T. 2014 a On the physics of fizziness: How bubble bursting controls droplets ejection. Phys. Fluids 26 (12), 121701.CrossRefGoogle Scholar
Ghabache, E. & Séon, T. 2016 Size of the top jet drop produced by bubble bursting. Phys. Rev. Fluids 1 (5), 051901.CrossRefGoogle Scholar
Ghabache, É., Séon, T. & Antkowiak, A. 2014 b Liquid jet eruption from hollow relaxation. J. Fluid Mech. 761, 206–219.Google Scholar
Goodridge, C. L., Hentschel, H. G. E. & Lathrop, D. P. 1999 Breaking faraday waves: critical slowing of droplet ejection rates. Phys. Rev. Lett. 82 (15), 3062.CrossRefGoogle Scholar
Goodridge, C. L., Shi, W. T. & Lathrop, D. P. 1996 Threshold dynamics of singular gravity-capillary waves. Phys. Rev. Lett. 76 (11), 1824.CrossRefGoogle ScholarPubMed
Gordillo, J. M. 2008 Axisymmetric bubble collapse in a quiescent liquid pool. I. Theory and numerical simulations. Phys. Fluids 20 (11), 112103.CrossRefGoogle Scholar
Gordillo, J. M. & Rodríguez-Rodríguez, J. 2018 Comment on ‘Revision of bubble bursting: universal scaling laws of top jet drop size and speed’. Phys. Rev. Lett. 121 (26), 269401.CrossRefGoogle Scholar
Gordillo, J. M. & Rodríguez-Rodríguez, J. 2019 Capillary waves control the ejection of bubble bursting jets. J. Fluid Mech. 867, 556–571.CrossRefGoogle Scholar
Gordillo, J. M. & Gekle, S. 2010 Generation and breakup of worthington jets after cavity collapse. Part 2. Tip breakup of stretched jets. J. Fluid Mech. 663, 331–346.CrossRefGoogle Scholar
Govindarajan, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Intl J. Multiphase Flow 30 (10), 1177–1192.CrossRefGoogle Scholar
Govindarajan, R. & Narasimha, R. 1999 Low-order parabolic theory for 2D boundary-layer stability. Phys. Fluids 11 (6), 1449–1458.CrossRefGoogle Scholar
Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25 (1), 55–97.CrossRefGoogle Scholar
Hogrefe, J. E., Peffley, N. L., Goodridge, C. L., Shi, W. T., Hentschel, H. G. E. & Lathrop, D. P. 1998 Power-law singularities in gravity-capillary waves. Physica D 123 (1-4), 183–205.CrossRefGoogle Scholar
Ismail, A. S., Gañán-Calvo, A. M., Castrejón-Pita, J. R., Herrada, M. A. & Castrejón-Pita, A. A. 2018 Controlled cavity collapse: scaling laws of drop formation. Soft Matt. 14 (37), 7671–7679.CrossRefGoogle ScholarPubMed
Jacobs, J. W. & Catton, I. 1988 Three-dimensional Rayleigh–Taylor instability. Part 1. Weakly nonlinear theory. J. Fluid Mech. 187, 329–352.CrossRefGoogle Scholar
James, A. J., Vukasinovic, B., Smith, M. K. & Glezer, A. 2003 Vibration-induced drop atomization and bursting. J. Fluid Mech. 476, 1–28.CrossRefGoogle Scholar
Kang, Y. J. & Cho, Y. 2019 Gravity–capillary jet-like surface waves generated by an underwater bubble. J. Fluid Mech. 866, 841–864.CrossRefGoogle Scholar
Kientzler, C. F., Arons, A. B., Blanchard, D. C. & Woodcock, A. H. 1954 Photographic investigation of the projection of droplets by bubbles bursting at a water surface. Tellus 6 (1), 1–7.CrossRefGoogle Scholar
Kim, N. & Park, H. 2019 Water entry of rounded cylindrical bodies with different aspect ratios and surface conditions. J. Fluid Mech. 863, 757–788.CrossRefGoogle Scholar
Krishnan, S., Hopfinger, E. J. & Puthenveettil, B. A. 2017 On the scaling of jetting from bubble collapse at a liquid surface. J. Fluid Mech. 822, 791–812.CrossRefGoogle Scholar
Kroon, L. A. M. 2012 On Faraday waves and jets. Master's thesis, University of Twente.Google Scholar
Lai, C.-Y., Eggers, J. & Deike, L. 2018 Bubble bursting: universal cavity and jet profiles. Phys. Rev. Lett. 121 (14), 144501.CrossRefGoogle ScholarPubMed
Lake, B. M. & Yuen, H. C. 1977 A note on some nonlinear water-wave experiments and the comparison of data with theory. J. Fluid Mech. 83 (1), 75–81.CrossRefGoogle Scholar
Liger-Belair, G., Cilindre, C., Gougeon, R. D., Lucio, M., Gebefügi, I., Jeandet, P. & Schmitt-Kopplin, P. 2009 Unraveling different chemical fingerprints between a champagne wine and its aerosols. Proc. Natl Acad. Sci. USA 106 (39), 16545–16549.CrossRefGoogle ScholarPubMed
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics. Proc. R. Soc. Lond. A 360 (1703), 471–488.Google Scholar
Longuet-Higgins, M. S. 1983 Bubbles, breaking waves and hyperbolic jets at a free surface. J. Fluid Mech. 127, 103–121.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1994 Inertial shocks in surface waves and collapsing bubbles. In Bubble Dynamics and Interface Phenomena, pp. 383–396. Springer.CrossRefGoogle Scholar
MacIntyre, F. 1972 Flow patterns in breaking bubbles. J. Geophys. Res. 77 (27), 5211–5228.CrossRefGoogle Scholar
Mack, L. R. 1962 Periodic, finite-amplitude, axisymmetric gravity waves. J. Geophys. Res. 67 (2), 829–843.CrossRefGoogle Scholar
mat 2018 MATLAB Version 9.4.0.813654 (R2018a). The Mathworks, Inc.Google Scholar
McGoldrick, L. F. 1970 On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42 (1), 193–200.CrossRefGoogle Scholar
Milgram, J. H. 1969 The motion of a fluid in a cylindrical container with a free surface following vertical impact. J. Fluid Mech. 37 (3), 435–448.CrossRefGoogle Scholar
Morton, D., Rudman, M. & Jong-Leng, L. 2000 An investigation of the flow regimes resulting from splashing drops. Phys. Fluids 12 (4), 747–763.CrossRefGoogle Scholar
Natarajan, R. & Brown, R. A. 1986 Quadratic resonance in the three-dimensional oscillations of inviscid drops with surface tension. Phys. Fluids 29 (9), 2788–2797.CrossRefGoogle Scholar
Penney, W. G., Price, A. T., Martin, J. C., Moyce, W. J., Penney, W. G., Price, A. T. & Thornhill, C. K. 1952 Part II. Finite periodic stationary gravity waves in a perfect liquid. Phil. Trans. R. Soc. Lond. A 244 (882), 254–284.Google Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190 (2), 572–600.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228 (16), 5838–5866.CrossRefGoogle Scholar
Popinet, S. 2018 Numerical models of surface tension. Annu. Rev. Fluid Mech. 50, 49–75.CrossRefGoogle Scholar
Popinet, S. 2020 Basilisk flow solver and PDE library. http://basilisk.fr, [Online; accessed 20-January-2020].Google Scholar
Price, W. G. & Penneyand, A. T. 1952 Finite periodic stationary gravity waves in a perfect fluid. Part 2. Phil. Trans. R. Soc. Lond. A 244, 254.Google Scholar
Prosperetti, A. & Oguz, H. N. 1993 The impact of drops on liquid surfaces and the underwater noise of rain. Annu. Rev. Fluid Mech. 25 (1), 577–602.CrossRefGoogle Scholar
Raja, D. K., Das, S. P. & Hopfinger, E. J. 2019 On standing gravity wave-depression cavity collapse and jetting. J. Fluid Mech. 866, 112–131.CrossRefGoogle Scholar
Ray, B., Biswas, G. & Sharma, A. 2015 Regimes during liquid drop impact on a liquid pool. J. Fluid Mech. 768, 492–523.CrossRefGoogle Scholar
van Rijn, C. J. M. 2018 Emanating jets as shaped by surface tension forces. Langmuir 34 (46), 13837–13844.CrossRefGoogle ScholarPubMed
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31 (1), 567–603.CrossRefGoogle Scholar
Schwartz, L. W. & Whitney, A. K. 1981 A semi-analytic solution for nonlinear standing waves in deep water. J. Fluid Mech. 107, 147–171.CrossRefGoogle Scholar
Segur, H. & Stewart, A. 2020 Lecture 4: Zhakharov formulation of water waves. https://gfd.whoi.edu/wp-content/uploads/sites/18/2018/03/lecture4-harvey\_136525.pdf, online; accessed 29 February 2020.Google Scholar
Séon, T. & Liger-Belair, G. 2017 Effervescence in champagne and sparkling wines: from bubble bursting to droplet evaporation. Eur. Phys. J.: Spec. Top. 226 (1), 117–156.Google Scholar
Shi, W. T., Goodridge, C. L. & Lathrop, D. P. 1997 Breaking waves: bifurcations leading to a singular wave state. Phys. Rev. E 56 (4), 4157.Google Scholar
Shin, J. & McMahon, T. A. 1990 The tuning of a splash. Phys. Fluids A 2 (8), 1312–1317.CrossRefGoogle Scholar
Singh, M., Farsoiya, P. K. & Dasgupta, R. 2019 Test cases for comparison of two interfacial solvers. Intl J. Multiphase Flow 115, 75–92.CrossRefGoogle Scholar
Strutt, J. W. 1915 Deep water waves, progressive or stationary, to the third order of approximation. Proc. R. Soc. Lond. A 91 (629), 345–353.Google Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. Lond. A 218 (1132), 44–59.Google Scholar
Thoroddsen, S. T., Etoh, T. G. & Takehara, K. 2007 Microjetting from wave focusing on oscillating drops. Phys. Fluids 19 (5), 052101.CrossRefGoogle Scholar
Thoroddsen, S. T., Takehara, K., Nguyen, H. D. & Etoh, T. G. 2018 Singular jets during the collapse of drop-impact craters. J. Fluid Mech. 848, R3.CrossRefGoogle Scholar
Tjan, K. K. & Phillips, W. R. C. 2007 On impulsively generated inviscid axisymmetric surface jets, waves and drops. J. Fluid Mech. 576, 377–403.CrossRefGoogle Scholar
Truscott, T. T., Epps, B. P. & Belden, J. 2014 Water entry of projectiles. Annu. Rev. Fluid Mech. 46, 355–378.CrossRefGoogle Scholar
Tsai, C., Mao, R., Tsai, S., Shahverdi, K., Zhu, Y., Lin, S., Hsu, Yu.-H., Boss, G., Brenner, M., Mahon, S., et al. . 2017 Faraday waves-based integrated ultrasonic micro-droplet generator and applications. Micromachines 8 (2), 56.CrossRefGoogle ScholarPubMed
Veron, F. 2015 Ocean spray. Annu. Rev. Fluid Mech. 47, 507–538.CrossRefGoogle Scholar
Viola, F., Brun, P.-T. & Gallaire, F. 2018 Capillary hysteresis in sloshing dynamics: a weakly nonlinear analysis. J. Fluid Mech. 837, 788–818.CrossRefGoogle Scholar
Wilton, J. R. 1915 LXXII. On ripples. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 29 (173), 688–700.CrossRefGoogle Scholar
Woodcock, A. H., Kientzler, C. F., Arons, A. B. & Blanchard, D. C. 1953 Giant condensation nuclei from bursting bubbles. Nature 172 (4390), 1144–1145.CrossRefGoogle Scholar
Worthington, A. M. & Cole, R. S. 1897 V. Impact with a liquid surface, studied by the aid of instantaneous photography. Phil. Trans. R. Soc. Lond. A 189, 137–148.Google Scholar
Worthington, A. M. & Cole, R. S. 1900 IV. Impact with a liquid surface studied by the aid of instantaneous photography. Paper II. Phil. Trans. R. Soc. Lond. A 194 (252–261), 175–199.Google Scholar
Yamamoto, K., Motosuke, M. & Ogata, S. 2018 Initiation of the worthington jet on the droplet impact. Appl. Phys. Lett. 112 (9), 093701.CrossRefGoogle Scholar
Yukisada, R., Kiyama, A., Zhang, X. & Tagawa, Y. 2018 Enhancement of focused liquid jets by surface bubbles. Langmuir 34 (14), 4234–4240.CrossRefGoogle ScholarPubMed
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.CrossRefGoogle Scholar
Zeff, B. W., Kleber, B., Fineberg, J. & Lathrop, D. P. 2000 Singularity dynamics in curvature collapse and jet eruption on a fluid surface. Nature 403 (6768), 401.CrossRefGoogle ScholarPubMed

Basak et al. supplementary movie

Jet in a cylindrical pool: three dimensional visualisation of an axisymmetric simulation for large initial wave steepness (ϵ).

Download Basak et al. supplementary movie(Video)
Video 30.1 MB
Supplementary material: PDF

Basak et al. supplementary material

Supplementary Material - Jetting in finite-amplitude, free, capillary-gravity waves

Download Basak et al. supplementary material(PDF)
PDF 2.5 MB