Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-21T18:26:01.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Phase-field model for the Rayleigh–Taylor instability of immiscible fluids

Published online by Cambridge University Press:  10 March 2009

ANTONIO CELANI ,
ANDREA MAZZINO ,
PAOLO MURATORE-GINANNESCHI  and
LARA VOZELLA
ANTONIO CELANI
Affiliation:
Institut Pasteur, CNRS, URA 2171, 25 Rue du docteur Roux, 75015 Paris, France
ANDREA MAZZINO
Affiliation:
Department of Physics – University of Genova, and CNISM & INFN – Genova Section, via Dodecaneso 33, 16146 Genova, Italy
PAOLO MURATORE-GINANNESCHI
Affiliation:
Department of Mathematics and Statistics – University of Helsinki, PO Box 4, 00014 Helsinki, Finland
LARA VOZELLA*
Affiliation:
Department of Physics – University of Genova, and CNISM & INFN – Genova Section, via Dodecaneso 33, 16146 Genova, Italy Department of Mathematics and Statistics – University of Helsinki, PO Box 4, 00014 Helsinki, Finland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Rayleigh–Taylor instability of two immiscible fluids in the limit of small Atwood numbers is studied by means of a phase-field description. In this method, the sharp fluid interface is replaced by a thin, yet finite, transition layer where the interfacial forces vary smoothly. This is achieved by introducing an order parameter (the phase-field) continuously varying across the interfacial layers and uniform in the bulk region. The phase-field model obeys a Cahn–Hilliard equation and is two-way coupled to the standard Navier–Stokes equations. Starting from this system of equations we have first performed a linear analysis from which we have analytically rederived the known gravity–capillary dispersion relation in the limit of vanishing mixing energy density and capillary width. We have performed numerical simulations and identified a region of parameters in which the known properties of the linear phase (both stable and unstable) are reproduced in a very accurate way. This has been done both in the case of negligible viscosity and in the case of non-zero viscosity. In the latter situation, only upper and lower bounds for the perturbation growth rate are known. Finally, we have also investigated the weakly nonlinear stage of the perturbation evolution and identified a regime characterized by a constant terminal velocity of bubbles/spikes. The measured value of the terminal velocity is in agreement with available theoretical prediction. The phase-field approach thus appears to be a valuable technique for the dynamical description of the stages where hydrodynamic turbulence and wave-turbulence come into play.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 622 , 10 March 2009 , pp. 115 - 134
Copyright
Copyright © Cambridge University Press 2009

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

Alon, U., Hecht, J., Ofer, D. & Shvarts, D. 1995 Power laws and similarity of Rayleigh–Taylor and Richtmyer–Meshkov mixing fronts at all density ratio. Phys. Rev. Lett. 74 (4), 534–537.CrossRefGoogle Scholar
Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.Google Scholar
Badalassi, V. E., Ceniceros, H. D. & Banerjee, S. 2003 Computation of multiphase systems with phase field models. J. Comput. Phys. 190, 371–397.CrossRefGoogle Scholar
Berti, S., Boffetta, G., Cencini, M. & Vulpiani, A. 2005 Turbulence and coarsening in active and passive binary mixtures. Phys. Rev. Lett. 95, 224501-1–224501-4.CrossRefGoogle ScholarPubMed
Bray, A. J. 2002 Theory of phase-ordering kinetics. Adv. Phys. 51, 481–587.CrossRefGoogle Scholar
Cabot, W. H. & Cook, A. W. 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type-Ia supernovae. Nature phys. 2, 562–568.Google Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a non uniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Series in Computational Physics. Springer.CrossRefGoogle Scholar
Celani, A., Mazzino, A. & Vozella, L. 2006 Rayleigh–Taylor turbulence in two-dimensions. Phys. Rev. Lett. 96, 134504-1–134504-4.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chertkov, M. 2003 Phenomenology of Rayleigh–Taylor turbulence. Phys. Rev. Lett. 91, 115001-1–115001-4.CrossRefGoogle ScholarPubMed
Chertkov, M., Kolokolov, I. & Lebedev, V. 2005 Effects of surface tension on immiscible Rayleigh–Taylor turbulence. Phys. Rev. E 71, 055301-1–055301-4.Google ScholarPubMed
Cook, A. W. & Zhou, Y. 2002 Energy transfer in Rayleigh–Taylor instability. Phys. Rev. E 66, 026312-1–026312-12.Google ScholarPubMed
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12, 304–321.CrossRefGoogle Scholar
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 2078–2095.CrossRefGoogle Scholar
Di Prima, R. C. & Swinney, H. L. 1981 Hydrodynamic Instabilities and the Transition to Turbulence. (ed. Swinney, H. L. & Gollup, J. P.) Springer.Google Scholar
Ducea, M. & Saleeby, J 1998 A case for delamination of the deep batholithic crust beneath the Sierra Nevada, California. Intl Geol. Rev. 40, 78–93.CrossRefGoogle Scholar
Goncharov, V. N. 2003 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88, 134502-2–134502-4.Google Scholar
Kull, H. J. 1991 Theory of the Rayleigh–Taylor instability. Phys. Rep. 206 (5), 197–325.CrossRefGoogle Scholar
Kundu, P. K. & Cohen, I. M. 2001 Fluids Mechanics, 2nd edn.Academic.Google Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155, 96–127.Google Scholar
Landau, L. D. & Lifshitz, E. M. 2000 Fluid Mechanics. Course of Theoretical Physics. 2nd edn, Revised. Butterworth–Heinemann.Google Scholar
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 1–12.Google Scholar
Lee, C.-T., Rudnick, R. L. & Brimhall, G. H. Jr, 2001 Deep lithospheric dynamics beneath the Sierra Nevada during the Mesozoic and Cenozoic as inferred from xenolith petrology. Geochem. Geophys. Geosys. 2, 2001GC000152.CrossRefGoogle Scholar
Liu, C. & Shen, J. 2003 A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179, 211–228.Google Scholar
Menikoff, R., Mjolsness, R. C., Sharp, D. H. & Zemach, C. 1977 Unstable normal mode for Rayleigh–Taylor instability in viscous fluids. Phys. Fluids 20 (12), 2000–2004.Google Scholar
Ramabrabhu, P. & Andrews, M. J. 2004 Experimental investigation of Rayleigh–Taylor mixing at small Atwood numbers. J. Fluid Mech. 502, 233–271.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177.Google Scholar
Schultz, D. M., Kanak, K. M., Straka, J. M., Trapp, R. J., Gordon, B. A., Zrnić, D. S., Bryan, G. H., Durant, A. J., Garrett, T. J., Klein, P. K. & Lilly, D. K. 2006 The mysteries of Mammatus clouds: observations and formation mechanisms. J. Atmos. Sci. 10, 2409–2435.Google Scholar
Sethian, A. J. 1999 Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge University Press.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 3–18.Google Scholar
Smolianski, A., Haario, H. & Luukka, P. 2005 Vortex shedding behind a rising bubble and two-bubble coalescence: a numerical approach. Appl. Math. Model 29, 615–632.CrossRefGoogle 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, 192–197.Google Scholar
Waddell, J. T., Niederhaus, C. E. & Jacobs, J. W. 2001 Experimental study of Rayleigh–Taylor instability: low Atwood number systems with single-mode initial perturbations. Phys. Fluids 13 (5), 1263–1273.CrossRefGoogle Scholar
Young, Y. N. & Ham, F. E. 2006 Surface tension in incompressible Rayleigh–Taylor mixing flow. J. Turbulence. 71 (7), 1–23.Google Scholar
Young, Y. N., Tufo, H., Dubey, A. & Rosner, R. 2001 On the miscible Rayleigh–Taylor instability: two and three dimensions. J. Fluid Mech. 447, 377–408.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317.Google Scholar