Three-dimensional Rayleigh-Taylor instability Part 1. Weakly nonlinear theory

Three-dimensional weakly nonlinear Rayleigh-Taylor instability is analysed. The stability of a confined inviscid liquid and an overlying gas with density much less than that of the liquid is considered. An asymptotic solution for containers of arbitrary cross-sectional geometry, valid up to order ε3 (where ε is the root-mean-squared initial surface slope) is obtained. The solution is evaluated for the rectangular and circular geometries and for various initial modes (square, hexagonal, axisymmetric, etc.). It is found that the hexagonal and axisymmetric instabilities grow faster than any other shapes in their respective geometries. In addition it is found that, sufficiently below the cutoff wavenumber, instabilities that are equally proportioned in the lateral directions grow faster than those with longer, thinner shape. However, near the cutoff wavenumber this trend reverses with instabilities having zero aspect ratio growing faster than those with aspect ratio near 1.

[1]  Francis H. Harlow,et al.  Numerical Study of Large‐Amplitude Free‐Surface Motions , 1966 .

[2]  J. Keller,et al.  Three‐Dimensional Standing Surface Waves of Finite Amplitude , 1962 .

[3]  S. Chandrasekhar.  Hydrodynamic and Hydromagnetic Stability , 1961 .

[4]  Ralph Menikoff,et al.  Rayleigh--Taylor instability and the use of conformal maps for ideal fluid flow , 1983 .

[5]  G. Taylor.  The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  J. Dienes.  Method of generalized coordinates and an application to Rayleigh-Taylor instability , 1978 .

[7]  J. Jacobs.  Three-dimensional Rayleigh-Taylor instability: Experiment and theory , 1986 .

[8]  H. Davis.  Introduction to Nonlinear Differential and Integral Equations , 1964 .

[9]  D. J. Lewis.  The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. II , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[10]  H. W. Emmons,et al.  Taylor instability of finite surface waves , 1960, Journal of Fluid Mechanics.

[11]  Bart J. Daly,et al.  Numerical Study of the Effect of Surface Tension on Interface Instability , 1969 .

[12]  B. J. Daly.  Numerical Study of Two Fluid Rayleigh‐Taylor Instability , 1967 .

[13]  D. Layzer,et al.  On the Instability of Superposed Fluids in a Gravitational Field. , 1955 .

[14]  Steven A. Orszag,et al.  Vortex simulations of the Rayleigh–Taylor instability , 1980 .

[15]  J. Freeman,et al.  Heuristic model of the nonlinear Rayleigh‐Taylor instability , 1981 .

[16]  A. Nayfeh.  On the non-linear Lamb-Taylor instability , 1969, Journal of Fluid Mechanics.

[17]  Richard Bellman,et al.  Effects of Surface Tension and Viscosity on Taylor Instability , 1954 .

[18]  D. I. Pullin,et al.  Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability , 1982, Journal of Fluid Mechanics.

[19]  John W. Miles,et al.  Nonlinear surface waves in closed basins , 1976, Journal of Fluid Mechanics.