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Complex singularities near the intersection of a free surface and wall. Part 1. Vertical jets and rising bubbles

  • Thomas G. J. Chandler (a1) and Philippe H. Trinh (a2)

Abstract

It is known that in steady-state potential flows, the separation of a gravity-driven free surface from a solid exhibits a number of peculiar characteristics. For example, it can be shown that the fluid must separate from the body so as to form one of three possible in-fluid angles: (i)  $180^{\circ }$ , (ii)  $120^{\circ }$ or (iii) an angle such that the surface is locally perpendicular to the direction of gravity. These necessary separation conditions were notably remarked upon by Dagan & Tulin (J. Fluid Mech., vol. 51 (3), 1972, pp. 529–543) in the context of ship hydrodynamics, but they are of crucial importance in many potential-flow applications. It is not particularly well understood why there is such a drastic change in the local separation behaviours when the global flow is altered. The question that motivates this work is the following: outside of a formal balance-of-terms argument, why must cases (i)–(iii) occur and furthermore, what are the connections between them? In this work, we seek to explain the transitions between the three cases in terms of the singularity structure of the associated solutions once they are extended into the complex plane. A numerical scheme is presented for the analytic continuation of a vertical jet (or alternatively a rising bubble). It will be shown that the transition between the three cases can be predicted by observing the coalescence of singularities as the speed of the jet is modified. A scaling law is derived for the coalescence rate of singularities.

Copyright

Corresponding author

Email address for correspondence: p.trinh@bath.ac.uk

References

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Baker, G. R. 1990 Singularities in the complex physical plane. In Hyperbolic Problems (ed. Engquist, B. & Gustafson, B.). Studentlitteratur.
Baker, G. R. & Xie, C. 2011 Singularities in the complex physical plane for deep water waves. J. Fluid Mech. 685, 83116.
Birkhoff, G. & Carter, D. 1957 Rising plane bubbles. J. Ration. Mech. Anal. 6, 769779.
Chandler, T. G. J. & Trinh, P. H.2018 Complex singularities near the intersection of a free surface and wall. Part 2. Angled nozzles. (In preparation.)
Chapman, S. J. 1999 On the role of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension. Eur. J. Appl. Maths 10 (6), 513534.
Combescot, R., Hakim, V., Dombre, T., Pomeau, Y. & Pumir, A. 1986 Shape selection of Saffman–Taylor fingers. Phys. Rev. A 56 (19), 20362039.
Couët, B. & Strumolo, G. S. 1987 The effects of surface tension and tube inclination on a two-dimensional rising bubble. J. Fluid Mech. 184, 114.
Cowley, S. J., Baker, G. R. & Tanveer, S. 1999 On the formation of moore curvature singularities in vortex sheets. J. Fluid Mech. 378, 233267.
Crew, S. C. & Trinh, P. H. 2016 New singularities for Stokes waves. J. Fluid Mech. 798, 256283.
Dagan, G. & Tulin, M. P. 1972 Two-dimensional free-surface gravity flow past blunt bodies. J. Fluid Mech. 51 (3), 529543.
Daripa, P. 2000 A computational study of rising plane Taylor bubbles. J. Comput. Phys. 157 (1), 120142.
Farrow, D. E. & Tuck, E. O. 1995 Further studies of stern wavemaking. J. Austral. Math. Soc. B 36, 424437.
Garabedian, P. R. 1957 On steady-state bubbles generated by Taylor instability. Proc. R. Soc. Lond. A 241, 423431.
Garabedian, P. R. 1985 A remark about pointed bubbles. Commun. Pure Appl. Maths 38 (5), 609612.
Goh, K. H. M.1986 Numerical solution of quadratically non-linear boundary value problems using integral equation techniques with application to nozzle and wall flows. PhD thesis, University of Adelaide.
Goh, K. H. M. & Tuck, E. O. 1985 Thick waterfalls from horizontal slots. J. Engng Maths 19 (4), 341349.
Golubeva, N.2003 Singularities in the spatial complex plane for vortex sheets and thin vortex layers. PhD thesis, The Ohio State University.
Grant, M. A. 1973 The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59, 257262.
Lee, J. & Vanden-Broeck, J.-M. 1993 Two-dimensional jets falling from funnels and nozzles. Phys. Fluids 5 (10), 24542460.
Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769786.
McCue, S. W. & Forbes, L. K. 1999 Bow and stern flows with constant vorticity. J. Fluid Mech. 399, 277300.
McCue, S. W. & Forbes, L. K. 2002 Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity. J. Fluid Mech. 461, 387407.
Meiron, D. I., Baker, G. R. & Orszag, S. A. 1982 Analytic structure of vortex sheet dynamics. Part 1. Kelvin–Helmholtz instability. J. Fluid Mech. 114, 283298.
Milewski, P., Vanden-Broeck, J.-M. & Keller, J. B. 1998 Singularities on free surfaces of fluid flows. Stud. Appl. Maths 100 (3), 245267.
Moore, D. W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A 365 (1720), 105119.
Norman, A. C. 1974 Expansions for the shape of maximum amplitude stokes waves. J. Fluid Mech. 66 (02), 261265.
Tanveer, S. 1991 Singularities in water waves and Rayleigh–Taylor instablity. Proc. R. Soc. Lond. A 435, 137158.
Trinh, P. H. 2016 A topological study of gravity waves generated by moving bodies using the method of steepest descents. Proc. R. Soc. Lond A 472, 20150833.
Trinh, P. H. & Chapman, S. J. 2014 The wake of a two-dimensional ship in the low-speed limit: results for multi-cornered hulls. J. Fluid Mech. 741, 492513.
Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. 2011 Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413439.
Tuck, E. O. 1987 Efflux from a slit in a vertical wall. J. Fluid Mech. 176, 253264.
Tuck, E. O. & Roberts, A. J. 1997 Bow-like free surfaces under gravity. Phil. Trans. R. Soc. Lond. A 355, 655677.
Vanden-Broeck, J.-M. 1984a Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids 27 (5), 10901093.
Vanden-Broeck, J.-M. 1984b Rising bubbles in a two-dimensional tube with surface tension. Phys. Fluids 27 (11), 26042607.
Vanden-Broeck, J.-M. 1991 Axisymmetric jet falling from a vertical nozzle and bubble rising in a tube of circular cross section. Phys. Fluids A 3 (2), 258262.
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.
Vanden-Broeck, J.-M. & Tuck, E. O. 1994 Flow near the intersection of a free surface with a vertical wall. SIAM J. Appl. Maths 54 (1), 113.
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