The Rayleigh–Taylor instability for inviscid and viscous fluids
The classical Rayleigh–Taylor instability occurs when two inviscid fluids, with a sharp interface separating them, lie in two horizontal layers with the heavier fluid above the lighter one. A small sinusoidal disturbance on the interface grows rapidly in time in this unstable situation, as the heavier upper fluid begins to move downwards through the lighter lower fluid. This paper presents a novel numerical method for computing the growth of the interface. The technique is based on a spectral representation of the solution. The results are accurate right up to the time when a curvature singularity forms at the interface and the inviscid model loses its validity. A spectral method is then presented to study the same instability in a viscous Boussinesq fluid. The results are shown to agree closely with the inviscid calculations for small to moderate times. However, the high interface curvatures that develop in the inviscid model are prevented from occurring in viscous fluid by the growth of regions of high vorticity at precisely these singular points. This leads to over-turning of the interface, to form mushroom-shaped profiles. It is shown that different initial interface configurations can lead to very different geometrical outcomes, as a result of the flow instability. These can include situations when detached bubbles form in the fluid.
KeywordsBoussinesq approximation Curvature singularity Spectral methods Surface roll-up Vorticity
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- 1.Lord Rayleigh (1883) Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc Lond Math Soc 14: 170–177Google Scholar
- 4.Kull HJ (1991) Theory of the Rayleigh-Taylor instability. Phys Lett 206: 197–325Google Scholar
- 5.Inogamov NA (1999) The role of Rayleigh-Taylor and Richtmyer-Meshkov instabilities in astrophysics: an introduction. Astrophys Space Phys 10: 1–335Google Scholar
- 21.Gradshteyn IS, Ryzhik IM (2000) Tables of integrals, series and products, 6th edn. Academic Press, San DiegoGoogle Scholar
- 23.Kreyszig E (2006) Advanced engineering mathematics, 9th edn. Wiley, New YorkGoogle Scholar
- 24.von Winckel G, lgwt.m, at: MATLAB file exchange website, written (2004) http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=4540&objectType=file