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Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability

Published online by Cambridge University Press:  20 April 2006

D. I. Pullin
D. I. Pullin
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Parkville 3052, Victoria, Australia Present address: Department of Mechanical Engineering, University of Queensland, St Lucia, 4067, Queensland, Australia.
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Abstract

We consider the behaviour of an interface between two immiscible inviscid incompressible fluids of different density moving under the action of gravity, inertial and interfacial tension forces. A vortex-sheet model of the exact nonlinear two-dimensional motion of this interface is formulated which includes expressions for an appropriate set of integral invariants. A numerical method for solving the vortex-sheet initial-value equations is developed, and is used to study the nonlinear growth of finite-amplitude normal modes for both Kelvin-Helmholtz and Rayleigh-Taylor instability. In the absence of an interfacial or surface-tension term in the integral-differential equation that describes the evolution of the circulation distribution on the vortex sheet, it is found that chaotic motion of, or the appearance of curvature singularities in, the discretized interface profiles prevent the simulations from proceeding to the late-time highly nonlinear phase of the motion. This unphysical behaviour is interpreted as a numerical manifestation of possible ill-posedness in the initial-value equations equivalent to the infinite growth rate of infinitesimal-wavelength disturbances in the linearized stability theory. The inclusion of an interfacial tension term in the circulation equation (which stabilizes linearized short-wavelength perturbations) was found to smooth profile irregularities but only for finite times. While coherent interfacial motion could then be followed well into the nonlinear regime for both the Kelvin-Helmholtz and Rayleigh-Taylor modes, locally irregular behaviour eventually reappeared and resisted subsequent attempts at numerical smoothing or suppression. Although several numerical and/or physical mechanisms are discussed that might produce irregular behaviour of the discretized interface in the presence of an interfacial-tension term, the basic cause of this instability remains unknown. The final description of the nonlinear interface motion thus awaits further research.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 119 , June 1982 , pp. 507 - 532
Copyright
© 1982 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.
Ahlberg, J. H., Nilson, E. H. & Walsh, J. L. 1967 The Theory of Cubic Splines and their Application. Academic.
Baker, G. R., 1980 J. Fluid Mech. 100, 209–220.
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Phys. Fluids 23, 1485–1490.
Bellman, R. & Pennington, R. H. 1954 Quart. Appl. Math. 12, 151.
Birkhoff, G. 1962 Proc. Symp. Appl. Math. Am. Math. Soc. 13, 55–76.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Daly, B. J. 1967 Phys. Fluids 10, 297–307.
Daly, B. J., 1969 Phys. Fluids 12, 1340–1354.
Fink, P. T. & Soh, W. K. 1978 Proc. R. Soc. Lond. A 362, 195–209.
Gear, C. W. 1971 Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall.
Harlow, F. H. & Welch, J. E. 1966 Phys. Fluids 9, 842–851.
Hirt, C. W., Cook, J. L. & Butler, T. D. 1970 J. Comp. Phys. 5, 103–124.
Holyer, J. Y. 1979 J. Fluid Mech. 93, 433–448.
International Mathematical and Statistical Libraries (IMSL) 1978 vol. 2.
Kelvin, Lord, 1910 Mathematical and Physical Papers, vol. 4, pp. 76–100. Cambridge University Press.
Lewis, D. J. 1950 Proc. R. Soc. Lond. A 202, 81–96.
Longuet-Higgins, M. S. & Cokelet, E. E. 1976 Proc. R. Soc. Lond. A 350, 1–26.
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, p. 376. Macmillan.
Moore, D. W. 1974 J. Fluid Mech. 63, 225–235.
Moore, D. W. 1979 Proc. R. Soc. Lond. A 365, 105–119.
Moore, D. W. 1980 On the point vortex method. Preprint.
Pullin, D. I. 1981 J. Fluid Mech. 108, 401–421.
Rosenhead, L. 1931 Proc. R. Soc. Lond. A 134, 170–192.
Saffman, P. G. & Baker, G. R. 1979 Ann Rev. Fluid Mech. 11, 95–122.
Taylor, G. I. 1950 Proc. R. Soc. Lond. A 201, 192–196.
Thorpe, S. A. 1969 J. Fluid Mech. 39, 25–48.
Van De Vooren, A. I. 1980 Proc. R. Soc. Lond. A 373, 67–91.
Yih, C-S. 1969 Fluid Mechanics. McGraw-Hill.
Zalosh, R. G. 1976 A.I.A.A. J. 14, 1517–1523.
Zaroodny, S. J. & Greenberg, M. D. 1973 J. Comp. Phys. 11, 440–452.