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FULLY 3D RAYLEIGH–TAYLOR INSTABILITY IN A BOUSSINESQ FLUID

Part of: Hydrodynamic stability Basic methods in fluid mechanics

Published online by Cambridge University Press:  01 July 2019

S. J. WALTERS Open the ORCID record for S. J. WALTERS [Opens in a new window]  and
L. K. FORBES Open the ORCID record for L. K. FORBES [Opens in a new window]
S. J. WALTERS*
Affiliation:
School of Mathematics and Physics, University of Tasmania, P.O. Box 37, Hobart, 7001, Tasmania, Australia email stephen.walters@utas.edu.au, larry.forbes@utas.edu.au
L. K. FORBES
Affiliation:
School of Mathematics and Physics, University of Tasmania, P.O. Box 37, Hobart, 7001, Tasmania, Australia email stephen.walters@utas.edu.au, larry.forbes@utas.edu.au
*
✉stephen.walters@utas.edu.au
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Abstract

Rayleigh–Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present a linearized theory for arbitrary three-dimensional (3D) initial disturbances that grow in time, and calculate the evolution of the interface for early times. A new spectral method is introduced for the fully 3D nonlinear problem in a Boussinesq fluid, where the interface between the light and heavy fluids is approximated with a smooth but rapid density change in the fluid. The results of large-scale numerical calculation are presented in fully 3D geometry, and compared and contrasted with the early-time linearized theory.

Keywords

viscous flowRayleigh–Taylor instabilitylinearizationspectral method

MSC classification

Primary: 76E17: Interfacial stability and instability
Secondary: 76M22: Spectral methods
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 3 , July 2019 , pp. 286 - 304
Copyright
© 2019 Australian Mathematical Society 

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References

Abarzhi, S. I., “Review of theoretical modelling approaches of Rayleigh–Taylor instabilities and turbulent mixing”, Philos. Trans. R. Soc. Lond. Ser. A 368 (2010) 1809–1828; doi:10.1098/rsta.2010.0020.Google Scholar
Atkinson, K. E., An introduction to numerical analysis, 2nd edn (John Wiley, New York, 1978).Google Scholar
Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967).Google Scholar
Boyd, J. P., Chebyshev and Fourier spectral methods (Dover, New York, 2001).Google Scholar
Daly, B. J., “Numerical study of two fluid Rayleigh–Taylor instability”, Phys. Fluids 10 (1967) 297–307; doi:10.1063/1.1762109.Google Scholar
Farrow, D. E. and Hocking, G. C., “A numerical model for withdrawal from a two-layer fluid”, J. Fluid Mech. 549 (2006) 141–157; doi:10.1017/S0022112005007561.Google Scholar
Forbes, L. K., “The Rayleigh–Taylor instability for inviscid and viscous fluids”, J. Engrg. Math. 65 (2009) 273–290; doi:10.1007/s10665-009-9288-9.Google Scholar
Forbes, L. K. and Brideson, M. A., “On modelling the transition to turbulence in pipe flow”, ANZIAM J. 59 (2017) 1–34; doi:10.1017/S1446181117000256.Google Scholar
Lee, H. G. and Kim, J., “Numerical simulation of the three-dimensional Rayleigh–Taylor instability”, Comput. Math. Appl. 66 (2013) 1466–1474; doi:10.1016/j.camwa.2013.08.021.Google Scholar
Liang, H., Li, Q. X., Shi, B. C. and Chai, Z. H., “Lattice Boltzmann simulation of three-dimensional Rayleigh–Taylor instability”, Phys. Rev. E 93 (2016) 033113; doi:10.1103/PhysRevE.93.033113.Google Scholar
Lim, T. T. and Nickels, T. B., “Instability and reconnection in the head-on collision of two vortex rings”, Nature 357 (1992) 225–227; doi:10.1038/357225a0.Google Scholar
PGI Community Edition, https://www.pgroup.com/products/community.htm (accessed 28 March 2019).Google Scholar
Lord Rayleigh, “Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density”, Proc. Lond. Math. Soc. 14 (1883) 170–177; doi:10.1112/plms/s1-14.1.170.Google Scholar
Sandlin, D., “Two vortex rings colliding in SLOW MOTION” (online video clip: YouTube, uploaded 20 June 2018, accessed 19 March 2019);https://www.youtube.com/watch?v=EVbdbVhzcM4.Google Scholar
Shadloo, M. S., Zainali, A. and Yildiz, M., “Simulation of single mode Rayleigh–Taylor instability by SPH method”, Comput. Mech. 51 (2013) 699–715; doi:10.1007/s00466-012-0746-2.Google Scholar
Sharp, D. H., “An overview of Rayleigh–Taylor instability”, Phys. D: Nonlinear Phenomena 12 (1984) 3–18; doi:10.1016/0167-2789(84)90510-4.Google Scholar
Taylor, G. I., “The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I”, Proc. R. Soc. Lond. Ser. A 201 (1950) 192–196; doi:10.1098/rspa.1950.0052.Google Scholar
Wilbraham, H., “On a certain periodic function”, Cambridge Dublin Math. J. 3 (1848) 198–201; https://books.google.com.au/books?id=JrQ4AAAAMAAJ&pg=PA198&redir_esc=y#v=onepage&q&f=false.Google Scholar
Zhou, Y., “Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I”, Phys. Rep. 720–722 (2017) 1–136; doi:10.1016/j.physrep.2017.07.005.Google Scholar
Zhou, Y., “Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II”, Phys. Rep. 723–725 (2017) 1–160; doi:10.1016/j.physrep.2017.07.008.Google Scholar