Analysis of weakly nonlinear three‐dimensional Rayleigh–Taylor instability growth

1995 ◽  
Vol 2 (5) ◽  
pp. 1669-1681 ◽  
Author(s):  
M. J. Dunning ◽  
Steven W. Haan
Keyword(s):  
Three Dimensional ◽  
Taylor Instability ◽  
Weakly Nonlinear ◽  
Instability Growth ◽  
Rayleigh Taylor Instability

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

1988 ◽  
Vol 187 ◽  
pp. 329-352 ◽  
Author(s):  
J. W. Jacobs ◽  
I. Catton
Keyword(s):  
Aspect Ratio ◽  
Three Dimensional ◽  
Taylor Instability ◽  
Initial Surface ◽  
Surface Slope ◽  
Cross Sectional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Rayleigh Taylor Instability ◽  
The Stability

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.

Three-dimensional Rayleigh-Taylor instability Part 2. Experiment

1988 ◽  
Vol 187 ◽  
pp. 353-371 ◽  
Author(s):  
J. W. Jacobs ◽  
I. Catton
Keyword(s):  
High Speed ◽  
Small Volume ◽  
Circular Tube ◽  
Three Dimensional ◽  
Vertical Tube ◽  
Taylor Instability ◽  
Gravitational Acceleration ◽  
Lateral Direction ◽  
Weakly Nonlinear ◽  
Rayleigh Taylor Instability

Three-dimensional Rayleigh-Taylor instability, induced by accelerating a small volume of water down a vertical tube using air pressure, is investigated. Two geometries are studied: a 15.875 cm circular tube and a 12.7 cm square tube. Runs were made with initial disturbances in the form of standing waves forced by shaking the test section in a lateral direction. Accelerations ranging from 5 to 10 times gravitational acceleration and wavenumbers from 1 cm−1 to 8 cm−1 are studied. The resulting instability was recorded and later analysed using high-speed motion picture photography. Measurements of the growth rate are found to agree well with linear theory. In addition, good qualitative agreement between photographs and three-dimensional surface plots of the weakly nonlinear solution of Part 1 of this series (Jacobs & Catton 1988) is obtained.

scholarly journals Phase-field model for the Rayleigh–Taylor instability of immiscible fluids

2009 ◽  
Vol 622 ◽  
pp. 115-134 ◽  
Author(s):  
ANTONIO CELANI ◽  
ANDREA MAZZINO ◽  
PAOLO MURATORE-GINANNESCHI ◽  
LARA VOZELLA
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Stokes Equations ◽  
Phase Field Model ◽  
Terminal Velocity ◽  
Immiscible Fluids ◽  
Taylor Instability ◽  
Upper And Lower Bounds ◽  
Weakly Nonlinear ◽  
Rayleigh Taylor Instability

The Rayleigh–Taylor instability of two immiscible fluids in the limit of small Atwood numbers is studied by means of a phase-field description. In this method, the sharp fluid interface is replaced by a thin, yet finite, transition layer where the interfacial forces vary smoothly. This is achieved by introducing an order parameter (the phase-field) continuously varying across the interfacial layers and uniform in the bulk region. The phase-field model obeys a Cahn–Hilliard equation and is two-way coupled to the standard Navier–Stokes equations. Starting from this system of equations we have first performed a linear analysis from which we have analytically rederived the known gravity–capillary dispersion relation in the limit of vanishing mixing energy density and capillary width. We have performed numerical simulations and identified a region of parameters in which the known properties of the linear phase (both stable and unstable) are reproduced in a very accurate way. This has been done both in the case of negligible viscosity and in the case of non-zero viscosity. In the latter situation, only upper and lower bounds for the perturbation growth rate are known. Finally, we have also investigated the weakly nonlinear stage of the perturbation evolution and identified a regime characterized by a constant terminal velocity of bubbles/spikes. The measured value of the terminal velocity is in agreement with available theoretical prediction. The phase-field approach thus appears to be a valuable technique for the dynamical description of the stages where hydrodynamic turbulence and wave-turbulence come into play.

Simulation of the Weakly Nonlinear Rayleigh-Taylor Instability in Spherical Geometry

2020 ◽  
Vol 37 (5) ◽  
pp. 055201
Author(s):  
Yun-Peng Yang ◽  
Jing Zhang ◽  
Zhi-Yuan Li ◽  
Li-Feng Wang ◽  
Jun-Feng Wu ◽  
...  
Keyword(s):  
Spherical Geometry ◽  
Taylor Instability ◽  
Weakly Nonlinear ◽  
Rayleigh Taylor Instability

scholarly journals Rarefaction-driven Rayleigh–Taylor instability. Part 1. Diffuse-interface linear stability measurements and theory

2016 ◽  
Vol 791 ◽  
pp. 34-60 ◽  
Author(s):  
R. V. Morgan ◽  
O. A. Likhachev ◽  
J. W. Jacobs
Keyword(s):  
Linear Stability ◽  
Rarefaction Wave ◽  
Diffusion Model ◽  
Growth Rates ◽  
Three Dimensional ◽  
Mie Scattering ◽  
Taylor Instability ◽  
Diffuse Interface ◽  
Rayleigh Taylor Instability ◽  
Dynamic Diffusion

Theory and experiments are reported that explore the behaviour of the Rayleigh–Taylor instability initiated with a diffuse interface. Experiments are performed in which an interface between two gases of differing density is made unstable by acceleration generated by a rarefaction wave. Well-controlled, diffuse, two-dimensional and three-dimensional, single-mode perturbations are generated by oscillating the gases either side to side, or vertically for the three-dimensional perturbations. The puncturing of a diaphragm separating a vacuum tank beneath the test section generates a rarefaction wave that travels upwards and accelerates the interface downwards. This rarefaction wave generates a large, but non-constant, acceleration of the order of $1000g_{0}$, where $g_{0}$ is the acceleration due to gravity. Initial interface thicknesses are measured using a Rayleigh scattering diagnostic and the instability is visualized using planar laser-induced Mie scattering. Growth rates agree well with theoretical values, and with the inviscid, dynamic diffusion model of Duff et al. (Phys. Fluids, vol. 5, 1962, pp. 417–425) when diffusion thickness is accounted for, and the acceleration is weighted using inviscid Rayleigh–Taylor theory. The linear stability formulation of Chandrasekhar (Proc. Camb. Phil. Soc., vol. 51, 1955, pp. 162–178) is solved numerically with an error function diffusion profile using the Riccati method. This technique exhibits good agreement with the dynamic diffusion model of Duff et al. for small wavenumbers, but produces larger growth rates for large-wavenumber perturbations. Asymptotic analysis shows a $1/k^{2}$ decay in growth rates as $k\rightarrow \infty$ for large-wavenumber perturbations.

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