English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This article describes two spectral methods for solving problems in interfacial fluid mechanics. These are illustrated here for the Rayleigh--Taylor instability, in which a layer of heavy fluid lies above a light fluid. Disturbances to the interface between them are unstable, and grow with time. The first spectral method solves time dependent, free-surface problems in inviscid flow theory. It is capable of following the development of the interface almost to the time at which the curvature at the interface becomes infinite, and the inviscid model then ceases to be valid. A second spectral method is presented, and solves the viscous Boussinesq equations. It shows that the curvature singularity in the inviscid model is associated with regions of large vorticity at precisely these same points on the interface. Consequently, the interface rolls over at these points, forming delicate overhanging structures, leading ultimately to mixing of the two fluid layers.   References G. Baker, R. E. Caflisch and M. Siegel, Singularity formation during Rayleigh--Taylor instability. J. Fluid Mech. , 252 (1993) 51--78. doi:10.1017/S0022112093003660 G. R. Baker and L. D. Pham, A comparison of blob-methods for vortex sheet roll-up. J. Fluid Mech. , 547 (2006) 297--316. doi:10.1017/S0022112005007305 G. K. Batchelor, An Introduction to Fluid Dynamics . Cambridge University Press, Cambridge (1977). D. E. Farrow and G. C. Hocking, A numerical model for withdrawal from a two-layer fluid. J. Fluid Mech. , 549 (2006) 141--157. doi:10.1017/S0022112005007561 L. K. Forbes, M. J. Chen and C. E. Trenham, Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow. J. Comput. Phys. , 221 (2007) 269--287. doi:10.1016/j.jcp.2006.06.010 L. K. Forbes and G. C. Hocking, Unsteady draining flows from a rectangular tank. Phys. Fluids , 19 , 082104 (2007) 14 pages. doi:10.1063/1.2759891 R. Krasny, Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. , 65 (1986) 292--313. doi:10.1016/0021-9991(86)90210-X D. W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. London A , 365 (1979) 105--119. doi:10.1098/rspa.1979.0009 P. Ramaprabhu, G. Dimonte, Y.-N. Young, A. C. Calder and B. Fryxell, Limits of the potential flow approach to the single-mode Rayleigh--Taylor problem. Phys. Rev. E , 74 , 066308 (2006) 10 pages. doi:10.1103/PhysRevE.74.066308 Lord Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. London Math. Soc. , 14 (1883) 170--177. G. I. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, I. Proc. Roy. Soc. London Ser. A , 201 (1950) 192--196. doi:10.1098/rspa.1950.0052 G. Tryggvason and S. O. Unverdi, Computations of three-dimensional Rayleigh--Taylor instability. Phys. Fluids A , 2 (1990) 656--659. doi:10.1063/1.857717

Spectral solution methods for free-surface flow: the Rayleigh--Taylor problem