Viscous Rayleigh-Taylor instability with and without diffusion effect

doi:10.1007/s10483-017-2169-9

Abstract

Abstract:

The approximate but analytical solution of the viscous Rayleigh-Taylor instability(RTI) has been widely used recently in theoretical and numerical investigations due to its clarity. In this paper, a modified analytical solution of the growth rate for the viscous RTI of incompressible fluids is obtained based on an approximate method. Its accuracy is verified numerically to be significantly improved in comparison with the previous one in the whole wave number range for different viscosity ratios and Atwood numbers. Furthermore, this solution is expanded for viscous RTI including the concentration-diffusion effect.

Key words: dispersion relation, diffusion effect, viscous Rayleigh-Taylor instability(RTI)

2010 MSC Number:

O357.1

Chenyue XIE, Jianjun TAO, Ji LI. Viscous Rayleigh-Taylor instability with and without diffusion effect. Applied Mathematics and Mechanics (English Edition), 2017, 38(2): 263-270.

TrendMD

References

[1] Rayleigh, L. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proceedings of the London Mathematical Society, 14(1), 170-177(1883)
[2] Taylor, G. I. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proceedings of the Royal Society of London Series A, 201(1065), 192-196(1950)
[3] Conrad, C. P. and Molnar, P. The growth of Rayleigh-Taylor-type instabilities in the lithosphere for various rheological and density structures. Geophysical Journal International, 129, 95-112(1997)
[4] Houseman, G. A. and Molnar, P. Gravitational (Rayleigh-Taylor) instability of a layer with nonlnear viscosity and convective thinning of continental lithosphere. Geophysical Journal International, 128, 125-150(1997)
[5] Michioka, H. and Sumita, I. Rayleigh-Taylor instability of a particle packed viscous fluid:implications for a solidifying magma. Geophysical Research Letters, 32, L03309(2005)
[6] Ribeyre, X., Tikhonchuk, V. T., and Bouquet, S. Compressible Rayleigh-Taylor instabilities in supernova remnants. Physics of Fluids, 16(12), 4661-4670(2004)
[7] Lindl, J. D., McCrory, R. L., and Campbell, E. M. Progress toward ignitition and burn propagation in inertial confinement fusion. Physics Today, 45, 32-40(1992)
[8] Kilkenny, J. D., Glendinning, S. G., Haan, S. W., Hammel, B. A., Lindl, J. D., Munro, D., Remington, B. A., Weber, S. V., Knauer, J. P., and Verdon, C. P. A review of the ablative stabilization of the Rayleigh-Taylor instability in regimes relevant to inertial confinement fusion. Physics of Plasmas, 1(5), 1379-1389(1994)
[9] Regan, S. P., Epstein, R., Hammel, B. A., Suter, L. J., Scott, H. A., Barrios, M. A., Bradley, D. K., Callahan, D. A., Cerjan, C., Collins, G. W., Dixit, S. N., Döppner, T., Edwards, M. J., Farley, D. R., Fournier, K. B., Glenn, S., Glenzer, S. H., Golovkin, I. E., Haan, S. W., Hamza, A., Hicks, D. G., Izumi, N., Jones, O. S., Kilkenny, J. D., Kline, J. L., Kyrala, G. A., Landen, O. L., Ma, T., MacFarlane, J. J., Mackinnon, A. J., Mancini, R. C., McCrory, R. L., Meezan, N. B., Meyerhofer, D. D., Nikroo, A., Park, H. S., Ralph, J., Remington, B. A., Sangster, T. C., Smalyuk,V. A., Springer, P. T., and Town, R. P. Hot-spot mix in ignition-scale inertial confinement fusion targets. Physics Review Letters, 111(4), 045001(2013)
[10] Harrison, W. J. The influence of viscosity on the oscillations of superposed fluids. Proceedings of the London Mathematical Society, 2, 396-405(1908)
[11] Bellman, R. and Pennington, R. H. Effects of surface tension and viscosity on taylor instability. Quarterly Journal of Mechanics and Applied Mathematics, 12, 151-162(1954)
[12] Chandrasekhar, S. The character of the equilibrium of an incompressible heavy viscous fluid of variable density. Mathematical Proceedings of the Cambridge Philosophical Society, 51, 162-178(1955)
[13] Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Oxford University Press, London (1961)
[14] Mikaelian, K. O. Rayleigh-Taylor instabilities in stratified fluids. Physical Review A, 26(4), 2140-2158(1982)
[15] Mikaelian, K. O. Time evolution of density perturbations in accelerating stratified fluids. Physical Review A, 28(3), 1637-1646(1983)
[16] Mikaelian, K. O. Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shell. Physical Review A, 42(6), 3400-3420(1990)
[17] Goldston, R. J. and Rutherford, P. H. Introduction to Plasma Physics, Institute of Physics Publishing, Bristol (1997)
[18] Ramaprabhu, P., Karkhanis, V., and Lawrie, A. G. W. The Rayleigh-Taylor instability driven by an accel-decel-accel profile. Physics of Fluids, 25, 115104(2013)
[19] Liang, H., Shi, B. C., Guo, Z. L., and Chai, Z. H. Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows. Physical Review E, 89, 053320(2014)
[20] Sagert, I., Howell, J., Staber, A., Strother, T., Colbry, D., and Bauer, W. Knudsen-number dependence of two-dimensional single-mode Rayleigh-Taylor fluid instabilities. Physical Review E, 92, 013009(2015)
[21] Hide, R. The character of the equilibrium of an incompressible heavy viscous fluid of variable density:an approximate theory. Mathematical Proceedings of the Cambridge Philosophical Society, 51, 179-201(1955)
[22] Reid, W. H. The effects of surface tension and viscosity on the stability of two superposed fluids. Mathematical Proceedings of the Cambridge Philosophical Society, 57(2), 415-425(1961)
[23] Menikoff, R., Mjolsness, R. C., Sharp, D. H., and Zemach, C. Unstable normal mode for RayleighTaylor instability in viscous fluids. Physics of Fluids, 20(12), 2000-2004(1977)
[24] Mikaelian, K. O. Effect of viscosity on Rayleigh-Taylor and Richtmyer-Meshkov instabilities. Physical Review E, 47, 375-383(1993)
[25] Nie, X. B., Qian, Y. H., Doolen, G. D., and Chen, S. Y. Lattice Boltzmann simulation of the two-dimensional Rayleigh-Taylor instability. Physical Review E, 58, 6861-6864(1998)
[26] He, X. Y., Chen, S. Y., and Zhang, R. Y. A lattice boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability. Journal of Computational Physics, 152, 642-663(1999)
[27] Kadau, K., Germann, T. C., Hadjiconstantinou, N. G., Lomdahl, P. S., Dimonte, G., Holian, B. L., and Alder, B. J. Nanohydrodynamics simulations:an atomistic view of the Rayleigh-Taylor instability. Proceedings of the National Academy of Sciences, 101(16), 5851-5855(2004)
[28] Barber, J. L., Kadau, K., Germann, T. C., and Alder, B. J. Initial growth of the Rayleigh-Taylor instability via molecular dynamics. The European Physical Journal B, 64, 271-276(2008)
[29] Duff, R. E., Harlow, F. H., and Hirt, C. W. Effects of diffusion on interface instability between gases. Physics of Fluids, 5(4), 417-425(1962)
[30] Batchelor, G. K. and Nitsche, J. M. Instability of stationary unbounded stratified fluid. Journal of Fluid Mechanics, 227, 357-391(1991)
[31] Kurowski, P., Misbah, C., and Tchourkine, S. Gravitational instability of a fictitious front during mixing of miscible fluids. Europhysics Letters, 29(4), 309-314(1995)
[32] Brouillette, M. and Sturtevant, B. Experiments on the Richtmyer-Meshkov instability:single-scale perturbations on a continuous interface. Journal of Fluid Mechanics, 263, 271-292(1994)
[33] Fournier, E., Gauthier, S., and Renaud, F. 2D pseudo-spectral parallel Navier-Stokes simulations of compressible Rayleigh-Taylor instability. Computers and Fluids, 31, 569-587(2002)
[34] Amiroudine, S., Boutrouft, K., and Zappoli, B. The stability analysis of two layers in a supercritical pure fluid:Rayleigh-Taylor-like instabilities. Physics of Fluids, 17, 054102(2005)
[35] Boutrouft, K., Amiroudine, S., and Ambari, A. Stability diagram and effect of initial density stratification for a two-layer system in a supercritical fluid. Physics of Fluids, 18, 124106(2006)
[36] Tartakovsky, A. M. and Meakin, P. A smoothed particle hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh-Taylor instability. Journal of Computational Physics, 207, 610-624(2005)
[37] Schneider, N., Hammouch, Z., Labrosse, G., and Gauthier, S. A spectral anelastic Navier-Stokes solver for a stratified two-miscible-layer system in infinite horizontal channel. Journal of Computational Physics, 299, 374-403(2015)
[38] Wei, T. and Livescu, D. Late-time quadratic growth in single-mode Rayleigh-Taylor instability. Physical Review E, 86, 046405(2012)
[39] Zhao, Y. P. Moving contact line problem:advances and perspectives. Theoretical and Applied Mechanics Letters, 4, 034002(2014)
[40] Gueyffier, D., Li, J., Nadim, A., Scardovelli, R., and Zaleski, S. Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. Journal of Computational Physics, 152, 423-456(1999)
[41] Chevalier, M., Schlatter, P., Lundbladh, A., and Henningson, D. S. SIMSON:A Seudo-Spectral Solver for Incompressible Boundary Layer Flows, Technical Report TRITA-MEK 2007:07, KTH Mechanics, Stockholm (2007)