Rayleigh Taylor Instability Challenge

OBJECTIVE: To perform the Rayleigh Taylor Instability Simulation.

INTRODUCTION: 

The Rayleigh-Taylor instability occurs when a light fluid is accelerated into a heavy fluid, and is a fundamental fluid-mixing mechanism. Any perturbation along the interface between the two fluids will grow. The width of the mixing layer at a given time is reduced by the development of turbulence. The growth rate of the instability and the rate of mixing between the two fluids depends on the effective viscosity of the two fluids.

This notion for a fluid in a gravitational field was first discovered by Lord Rayleigh in the 1880s and later applied to all accelerated fluids by Sir Geoffrey Taylor in 1950.

Understanding the rate of mixing caused by Rayleigh-Taylor instabilities is important to a wide variety of applications, including inertial confinement fusion, nuclear weapons explosions and stockpile management, and supernova explosions. Avoiding Rayleigh-Taylor instability in inertial confinement fusion applications requires both very high precision in the target manufacture, and very high uniformity in the heating of the outside of the capsule--that is, very high symmetry.

Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves are-

Richtmyer-Meshkov instability (RMI) -
The Richtmyer—Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shove wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This instability can be considered the impulsive-acceleration limit of the Rayleigh-Taylor Instability.

The Kelvin-Helmholtz instability -
The Kelvin—Helmholtz instability can occur when there is velocity shear in a single continuous fluid or where there is a velocity difference across the interface between two fluids. An example is wind blowing over water.

The Plateau-Rayleigh instability-
The Plateau—Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh-Taylor Instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup.

Case-1: Mesh size = 0.5mm 

Mesh:

Residuals:

Animation:

Case-2: Mesh size = 0.3mm 

Mesh:

Residuals:

Animation:

Conclusion:

With increase in the mesh size the results are more accurate at the interface as seen in the animation and also bubble formation increases.

Atwood Number :

The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as-

Atwood number for above cases-A=(1000-1.25)/(1000+1.25)=0.9975

For Atwood number close to 0, RT instability flows take the form of symmetric "fingers" of fluid;

For Atwood number close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.