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Singularities in water waves and the Rayleigh–Taylor problem

Published online by Cambridge University Press:  30 April 2010

M. A. FONTELOS  and
F. DE LA HOZ
M. A. FONTELOS*
Affiliation:
Instituto de Ciencias Matemáticas (ICMAT, CSIC-UAM-UCM-UC3M), C/ Serrano 123, 28006 Madrid, Spain
F. DE LA HOZ
Affiliation:
Departamento de Matemática Aplicada, Escuela Universitaria de Ingeniería Técnica Industrial, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Plaza de la Casilla 3, 48012 Bilbao, Spain
*
†Email address for correspondence: [email protected]
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Abstract

We describe, by means of asymptotic methods and direct numerical simulation, the structure of singularities developing at the interface between two perfect, inviscid and irrotational fluids of different densities ρ1 and ρ2 and under the action of gravity. When the lighter fluid is on top of the heavier fluid, one encounters the water-wave problem for fluids of different densities. In the limit when the density of the lighter fluid is zero, one encounters the classical water-wave problem. Analogously, when the heavier fluid is on top of the lighter fluid, one encounters the Rayleigh–Taylor problem for fluids of different densities, with this being the case when one of the densities is zero for the classical Rayleigh–Taylor problem. We will show that both water-wave and Rayleigh–Taylor problems develop singularities of the Moore-type (singularities in the curvature) when both fluid densities are non-zero. For the classical water-wave problem, we propose and provide evidence of the development of a singularity in the form of a logarithmic spiral, and for the classical Rayleigh–Taylor problem no singularities were found. The regularizing effects of surface tension are also discussed, and estimates of the size and wavelength of the capillary waves, bubbles or blobs that are produced are provided.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 651 , 25 May 2010 , pp. 211 - 239
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Ambrose, D. M. & Masmoudi, N. 2005 The zero surface tension limit of two-dimensional water waves. Comm. Pure Appl. Math. 58, 1287–1315.Google Scholar
Baker, G., Caflisch, R. E. & Siegel, M. 1993 Singularity formation during Rayleigh–Taylor instability. J. Fluid Mech. 252, 51–78.CrossRefGoogle Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.Google Scholar
Banner, M. L. & Peregrine, D. H. 1993 Wave breaking in deep water. Annu. Rev. Fluid Mech. 25, 373.CrossRefGoogle Scholar
Beale, J. T., Hou, T. Y. & Lowengrub, J. 1996 Convergence of a boundary-integral method for water waves. SIAM J. Numer. Anal. 33 (5), 1797–1843.Google Scholar
Caflisch, R. E. & Orellana, O. F. 1989 Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20 (2), 293–307.CrossRefGoogle Scholar
Ceniceros, H. D. & Hou, T. Y. 1999 Dynamic generation of capillary waves. Phys. Fluids 11 (5), 1042–1050.CrossRefGoogle Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Clavin, P., Duchemin, L. & Josserand, C. 2005 Asymptotic behaviour of the Rayleigh–Taylor instability. Phys. Rev. Lett. 94 (22), 224501.Google Scholar
Cowley, S. J., Baker, G. R. & Tanveer, S. 1999 On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech. 378, 233–267.Google Scholar
Craig, W. & Wayne, C. E. 2007 Mathematical aspects of surface water waves. Russ. Math. Surv. 62, 453–473.Google Scholar
Crapper, G. D. 1970 Non-linear capillary waves generated by steep gravity waves. J. Fluid Mech. 40, 149.CrossRefGoogle Scholar
Darrigol, O. 2006 Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press.Google Scholar
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 73–79.Google Scholar
Hou, T. Y., Lowengrub, J. S. & Shelley, M. J. 1994 Removing the stiffness from interfacial flow with surface-tension. J. Comput. Phys. 114 (2), 312–338.Google Scholar
Hou, T. Y., Lowengrub, J. S. & Shelley, M. J. 1997 The long-time motion of vortex sheets with surface tension. Phys. Fluids 9 (7), 1933–1954.CrossRefGoogle Scholar
de la Hoz, F., Fontelos, M. A. & Vega, L. 2008 The effect of surface tension on the Moore singularity of vortex sheet dynamics. J. Nonlinear Sci. 18, 463–484.Google Scholar
Joo, S. W., Messiter, A. F. & Schultz, W. W. 1991 Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant. J. Fluid Mech 229, 135–158.Google Scholar
Kambe, T. 1989 Spiral vortex solution of Birkhoff–Rott equation. Physica D 37, 463–473.CrossRefGoogle Scholar
Koga, M. 1982 Bubble entrainment in breaking wind waves. Tellus 34, 481.Google Scholar
Moore, D. W. 1979 Spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A. 365, 105–119.Google Scholar
Sirviente, A. I. & Song, C. 2004 A numerical study of breaking waves. Phys. Fluids 16 (7), 2649–2667.Google Scholar
Tanveer, S. 1991 Singularities in water waves and Rayleigh–Taylor instability. Proc. R. Soc. Lond. A 435, 137–158.Google Scholar
Tanveer, S. 1993 Singularities in the classical Rayleigh–Taylor flow: formation and subsequent motion. Proc. R. Soc. Lond. A 441, 501–525.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201, 192–196.Google Scholar
Tsoukala, V. K. & Moutzouris, C. I. 2008 Gas transfer under breaking waves: experiments and an improved vorticity-based model. Ann. Geophys. 26, 2131–2142.CrossRefGoogle Scholar
Wang, J. & Baker, G. 2009 A numerical algorithm for viscous incompressible interfacial flows. J. Comput. Phys. 228 (15), 5470–5489.CrossRefGoogle Scholar
Wu, S. 1997 Well-posedness in Sobolev spaces of the full water wave problem in two-dimensional. Invent. Math. 130 (1), 39–72.CrossRefGoogle Scholar
Wu, S. 2009 Almost global wellposedness of the two-dimensional full water wave problem. Invent. Math. 177 (1), 45–135.CrossRefGoogle Scholar